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PHILOSOPHICAL AND MATHEMATICAL
DICTIONARY.

VOL. I.

PRINTED BY S. HAMILTON,

Weybridge, Surrey.

PHILOSOPHICAL AND MATHEMATICAL

DICTIONARY:

CONTAINING

AN EXPLANATION OF THE TERMS, AND AN ACCOUNT OF THE SEVERAL SUBJECTS,

COMPRISED UNDER THE HEADS

MATHEMATICS, ASTRONOMY, AND PHILOSOPHY

BOTH NATURAL AND EXPERIMENTAL;

HISTORICAL ACCOUNT OF THE RISE, PROGRESS, AND PRESENT STATE OF THESE SCIENCES ;

fibSTON COLLEGE LIBRARY
also CHESTNUT HILL, MASS.

MEMOIRS OF THE LIVES AND WRITINGS OF THE MOST EMINENT AUTHORS,

BOTH ANCIENT AND MODERN,

WHO BY THEIR DISCOVERIES OR IMPROVEMENTS HAVE CONTRIBUTED TO THE ADVANCEMENT OF THEM.

L*£.r' r*

BY CHARLES J.UTTON, LL.D.

FELLOW OF THE ROYAL SOCIETIES OF LONDON AND EDINBURGH, AND OF THE PHILOSOPHICAL SOCIETIES OF HAARLEM

AND AMERICA ; AND EMERITUS PROFESSOR OF MATHEMATICS IN THE ROYAL

MILITARY ACADEMY, WOOLWICH.

IN TWO VOLUMES:

WITH MANY CUTS AND COPPER-PLATES.

A NEW EDITION,
WITH NUMEROUS ADDITIONS AND IMPROVEMENTS.

VOL. I.

LONDON:

PRINTED FOR THE AUTHOR;
F. C. AND J. RIVINGTON ; J. CUTHELL ; LAW AND WHITTAKER ; LONGMAN, HURST, REES, ORME, AND BROWN ;
CADELL AND DAVIES ; J. MAVVMAN ; BLACK, PARRY, AND CO ; BALDWIN, CRADOCK AND JOY ; J. BOOKER ;
G. AND S. ROBINSON; T. HAMILTON; WALKER AND EDWARDS J JOHN ROBINSON; B. REYNOLDS; AND
SIMPKIN AND MARSHALL,

1815.

DEC 2 3

q

<.*\Z*

BOSTON COURflf UBBAflY

cmsmm ml, am ami

PREFACE.

Among the Dictionaries of Arts and Sciences which have been published, of late years,
in various parts of Europe, it is matter of surprise that Philosophy and Mathematics
should have been so far overlooked as not to be thought worthy of a separate Treatise, in
this form. These sciences constitute a large portion of the present stock of human know-
ledge, and have been usually considered as possessing a degree of importance to which few
others are entitled ; and yet we have hitherto had no distinct Lexicon, in which their consti-
tuent parts and technical terms have been explained, with that amplitude and precision,
which the great improvements of the Moderns, as well as the rising dignity of the Subject,
seem to demand.

The only works of this kind in the English language, deserving of notice, are Harris's
Lexicon Technicum, and Stone's Mathematical Dictionary ; the former of which, though a
valuable performance at the time it was written, is now become too dry and obsolete to be
referred to with pleasure or satisfaction : and the latter, consisting only of one volume in 8vo,
must be regarded merely as an unfinished sketch, or brief compendium, extremely limited
in its plan, and necessarily deficient in useful information.

It became, therefore, the only resource of the Reader, in many cases where explanation
was wanted, to have recourse to Chambers's Dictionary, in four large volumes folio, or to
the Encyclopaedia Britannica, now in eighteen large volumes 4to, or the still more stupendous
performance of the French Encyclopedists ; and even here his expectations might be fre-
quently disappointed. These great and useful works, aiming at a general comprehension of
the whole circle of the Sciences, are sometimes very deficient in their descriptions of parti-
cular branches ; it being almost impossible, in such extensive undertakings, to appreciate,
with exactness, the due value of every article. They are, besides, so voluminous and hetero-

vi PREFACE.

geneous in their nature, as to render a frequent reference to them extremely inconvenient ;
and even if this were not the case, their high price puts them out of the reach of the gene-
rality of readers.

With a view to obviate these defects, the Public are here presented with a Dictionary
of a moderate size and price, which is devoted solely to Philosophical and Mathematical
subjects. It is a work for which materials have been collecting through a course of many
years ; and is the result of great labour and reading. Not only most of the Encyclopedias
already extant, and the various publications of the Learned Societies throughout Europe,
have been carefully consulted, but also all the original works, of any reputation, which have
hitherto appeared on these subjects, from the earliest writers down to the present times.

From the latter of these sources, in particular, a considerable portion of information has
been obtained, which the curious reader will find, in many cases, to be highly interesting and
important. The History of Algebra, for instance, which is detailed at considerable length
in the first volume, under the head of that article, will afford sufficient evidence to show
in what a superficial and partial way the inquiry has been hitherto investigated, even by
professed writers on the subject; the principal of whom are M. Montucla, our countryman
the celebrated Dr. Wallis, and the Abb6 De Gua, a late French author, who has pretended
to correct the Doctor's errors and misrepresentations.

Regular historical details are in like manner given of the origin and progress of each
of these Sciences, as well as of the inventions and improvements by which they have been
gradually brought from their first rude beginnings to their present advanced state.

It is also to be observed, that besides the articles common to the generality of Dictiona-
ries of this kind, an interesting Biographical Account is here introduced of the most cele-
brated Philosophers and Mathematicians, both ancient and modern; among which will be
found the Lives of eminent characters, who have hitherto been either wholly overlooked, or
very imperfectly recorded. Complete lists of their works are also subjoined to each article,
when they could be procured ; which cannot but prove highly acceptable to that class of
readers, who are desirous of obtaining the most satisfactory information on the subjects
of their particular inquiries and pursuits. On the head of Biography, however, the Author
has still to lament the want of many other respectable names which he was desirous to have

PREFACE. vii

added to his list of authors, not having been able to procure any circumstantial accounts of
their lives. He could have wished to have comprised in his list, the lives of all such public
literary characters as the University Professors of Astronomy, Philosophy, and Mathematics,
as well as those of the other more respectable class of authors on those sciences. He will
therefore thankfully receive the communication of any such memoirs from the hands of gen-
tlemen possessed of them; as well as hints and information on such useful improvements in
the sciences as may have been overlooked in this Dictionary, or any articles that may here
have been imperfectly or incorrectly treated ; that he may at some future time, by adding
them to this work, render it still more complete and deserving the public notice.

As this work is an attempt to separate the words in the sciences of Astronomy, Mathe-
matics, and Philosophy, from those of other arts or sciences, in several of which there are al-
ready separate Dictionaries ; as in Chemistry, Geography, Music, Marine and Naval affairs,
&c ; words sometimes occurred which it was rather doubtful whether they could be consi-
dered as properly belonging to the present work or not; in which case many of such words
have been here inserted. But such as appeared clearly and peculiarly to belong to any of
those other subjects, have been either wholly omitted, or else have had a very short account
only given of them. The readers of this work therefore, recollecting that it is not a General
Dictionary of all the Arts and Sciences, will not expect to find all sorts of words and subjects
here treated of; but such only as peculiarly appertain to the proper matter of the work.
And therefore, although some few words may inadvertently have been omitted ; yet when the
Reader does not immediately find every word which he wishes to consult, he will not always
consider them as omissions of the Author, but for the most part as relating to some other
science foreign to this Dictionary.

In all cases where it could be conveniently done, the necessary figures and diagrams are
inserted in the same page with the subjects which they are designed to elucidate; a method
which will be found much more commodious than that of putting them in separate plates at
the end of each volume, but, which has added very considerably to the expence of the un-
dertaking. Where the subjects are of such a nature that they could not be otherwise well re-
presented, they arc engraved on copperplates.

As the whole of this work was written before it was put to the press, the Reader will
find it of an equal and uniform nature and construction throughout; in which respect many

viii PREFACE.

publications of this kind are very defective, from the subjects being diffusely treated under
the first letters of the alphabet, while articles of equal importance in the latter part are
so much abridged as to be rendered almost useless, in order that the whole might be
comprised in a limited number of sheets, according to proposals made before the works were
composed. The present Dictionary having been completed without any of these unfavour-
able circumstances, will be found in most cases equally instructive and useful, and may be
consulted with no less advantage by the Man of Science than the Student.

1st addition to the foregoing original account of this work, we have to observe, that the
whole has been several times carefully perused, in order to .correct, enlarge, and amend the
several articles ; that all the new discoveries and improvements in the sciences, since the
first composing of the work, have been attentively collected and abridged, for introducing in
this new edition ; that many new articles have been inserted in their places through the
volumes, and even several of the chief articles of some of the other sciences &c,
have now been included, — as Electricity, Galvanism, Voltaic pile, Acid, Alkali, Freezing,
New Planets and Comets, with numerous other similar articles; that a large collection of
new and interesting lives of eminent authors have been inserted, as far as materials could
be obtained, and the merits of the authors were deemed worthy of notice in this work ; that
several new plates and figures have been added ; that the pages have been considerably en-
larged, and the printing so arranged and contrived as to include more matter in the
sheets. In short, the alterations, additions, and other improvements, are so numerous,
and so large, as entitle the present edition to be considered almost as an entire new work.

PHILOSOPHICAL AND MATHEMATICAL

DICTIONARY.

A.

x\_BACIST, an Arithmetician. In this sense we find
the word used by William of Malmesbury, in his History
de Gestis Anglorum, written about the year ] 150 ; where
he shows that one Gerbert, a learned monk of France,
who afterwards became pope of Rome in the year 998 or
999, by the name of Silvester the 2d, was the first who
got from the Saracens the abacus, and that taught such
rules concerning it, as the Abacists themselves could
hardly understand.

ABACUS, in Arithmetic, an ancient instrument used
by most nations for casting up accounts, or performing
arithmetical calculations. It is by some derived from the
Greek a?a£, which signifies a cupboard or beaufet, per-
haps from the similarity of the form of this instrument:
and by others it is derived from the Phoenician abak,
which signifies dust or powder, because it was said that
this instrument was sometimes a square board or tablet,
which was powdered over with fine sand or dust, in which
were traced the figures or characters used in making cal-
culations, which could thence be easily defaced, and the
abacus refitted for use. But Lucas Paciolus, in the first
part of his second distinction, thinks it is a corruption of
Arabicus, by which he meant their Algorithm, or the
method of numeral computation received from them.

We find this instrument in use for computation, under
some variations, with most nations; as the Greeks, Ro-
mans, Germans, French, Chinese, &c.

The Grecian Abacus was an oblong frame, over which
were stretched several brass wires, strung with little ivory
balls, like the beads of a necklace; by the various ar-
rangements of which, all kinds of computations were
easily made. Mahudel, in Hist. Acad. R. Inscr. torn. 3.
p. 390.

The Roman Abacus was a little varied from the Gre-
cian, having pins sliding in grooves, instead of strings or
wires and beads. Philos. Trans. No. 180.

The Chinese Abacus, or Shwan-pan, like the Grecian,
Consists of several series of beads strung on brass wires,
stretched from the top to the bottom of the instrument,
and divided in the middle by a cross piece from side to
side. In the upper space every string has two beads,
which an' each counted for 5; and in the lower space
every string has five beads, of different values, the first
fcing counted as 1, the second as 10, the third as 100,
and so on, as with us. See Shwan-pan.

Vol. I.

The Abacus chiefly used in European countries, is
nearly on the same principles, though the use of it is here
more limited, because of the arbitrary and unequal divi-
sions of money, weights, and measures, which, in China,
are all divided in a tenfold proportion, like our scale of
common numbers. This is made by drawing any number
of parallel lines, like paper ruled for music, at such a dis-
tance as may be at least equal to twice the diameter of
the calculus, or counter. Then the value of these lines,
and of the space's between them, increases, from the lowest
to the highest, in a tenfold proportion. Thus, counters
placed upon the first line, signify so many units or ones ;
on the second line 10's, on the third line 100's, on the
fourth line 1000's, and so on. In like manner a counter
placed in the first space, between the first and second line,
denotes 5, in the second space 50, in the third space 500,
in the fourth 5000, and so on. So that there are never
more than four counters placed on any line, nor more than
one placed in any space, this being of the same value as
five counters on the next line below. Thus, the counters
on the Abacus, in the following figure, express the num-
ber or sum 47382.

10000

-G—o-e-

000

•e — e-

1000
100
10
1

Besides the above instruments of computation, several
others have been invented by different persons: as, Na-
pier's Rods or Bones, described in his Rhabdologia, which
see, under the word Napier ; also the Abacus Rhabdo-
logicus, a variation of Napier's, which is described in the
first vol. of Machines et Inventions approuvees par l'Aca-
demie Hoyale des Sciences. An ingenious and general one
was also invented by Mr. Gamaliel Smethurst, and is de-
scribed in the Philosophical Transactions, vol. 40'; where
the inventor remarks that computations by it are much
quicker and easier performed than by the pen, are also
less burthensome to the memory, and can be performed
by blind persons, or in the dark as well as in the light.
A very comprehensive instrument of this kind was also
contrived by the late learned Dr. Nicholas Saunderson,
the blind professor of mathematics, by which he performed

B

ABA

[ 2 ]

ABA

very intricate calculations : an account of it is prefixed to
the first volume of his Algebra, where it is called by the
editor Palpable Arithmetic: which see.

In the Edinb. Rev. vol. 18, p. 205, is the following neat
account of this instrument : — It is curious to observe,
that the term Calculation itself claims no higher descent
than from calculus, a pebble. A table strewed with fine
sand, the pulvis eruditus of the poets, served both for
tracing geometrical diagrams, and teaching the elements
of writing; a very primitive contrivance, but universally
used throughout the East, from which it has lately been
imported into this country. The board on which arith-
metical operations were performed, at first by means of
pebbles, was named Abacus, from the Greek agccg ; pro-
bably because originally it was only a writing-table of a
compendious form, and designed for instructing children
in their alphabet.

Nee qui Abaco numeros, et secto in pulvere metas
Scit risisse vafer. Pers. Sat- i. 138.

The Abacus was divided, from the right to the left
hand, by vertical columns, on which the pebbles were
placed, to denote, units, tens, hundreds, thousands, &c.
The labour of counting and arranging those pebbles was
afterwards sensibly abridged, by drawing across the board
a horizontal line, above which, each single pebble had the
power of 5. In the progress of luxury, tali, or dies
made of ivory, were used instead of pebbles, and small
silver coins came to supply the place of counters. But
the operations with the abacus were rendered still more
commodious, by substituting for such tali or counters,
small bead-3 strung on parallel threads, and sometimes
pegs stuck along grooves. — Accountants by profession,
among the Romans, were hence styled calculatores, or ra-
tionarii. Various expedients seem to have been employed
for shortening the arithmetical operations. The different
positions of the fingers were, for that purpose, used to a
certain extent. Boethius treated largely of the subject ;
and even the venerable Bede has given very diffuse rules
for what was called digital arithmetic.

But the application of the abacus itself was not entirely
forgotten at a much later period. Witness the small vo-
lume of arithmetic, quaintly composed in the form of dia-
logue by Robert Recorde, teacher of mathematics and
practitioner in physic at Cambridge, during the reign of
Edward vi ; in which round dots, placed on perpendicu-
laV lines, and employed to express the succession of units,
tens, hundreds, &c, are made to perform some of the sim-
pler numerical operations. — A small instrument, entirely
resembling the abacus, has likewise at different times been
recommended, for teaching the elements of ciphering,
under the name of palpable arithmetic.

Abacus, Pythagorean, so denominated from its in-
ventor, Pythagoras, is a table of numbers, contrived for
readily learning the principles of arithmetic ; being pro-
bably what is now called the multiplication-table.

Abacus, or Abaciscus, in Architecture, the upper
part or member of the capital of a column; serving as
;i crowning both to the capital and to the whole column.
Vitruvius informs us tluit the Abacus was originally in-
tended to represent a square flat tile laid over an urn, or
a basket; and the invention is ascribed to Calimachus, an
ingenious statuary of Athens, who, it is said, adopted it on
observing a small basket, covered with a tile, over the
root ol an Acanthus plant, which grew on the grave of a

young lady ; the plant shooting up, encompassed the bas-
ket, till meeting with the tile, it curled back in a kind of
scroll: Calimachus passing by, took the hint, and imme-
diately executed a capital on this plan ; representing the
tile by the Abacus, the leaves of the acanthus by the vo-
lutes, and the basket by the vase or body of the capital.
See Acanthus.

( Abacus is also used by Scamozzi for a concave mould-
ing in the capital of the Tuscan pedestal. And the word
is used by Palladio for other members, which he describes.
In ancient architecture, the same term is used to denote
certain compartments in the incrustation or lining of the
walls of state-rooms, mosaic-pavements, and the like.
There were Abaci of marble, porphyry, jasper, alabaster,
and even glass; variously shaped, as" square, triangular,
and such-like.

Abacus Logisticus is a right-angled triangle, whose
sides, about the right-angle, contain all the numbers from

I to (SO ; and its area, the products of each two of the op-
posite numbers. This is also called a canon of sexagesi-
mals, and is no other than a multiplication- table carried
to 60 both ways.

Abacus et PalmultT, in the Ancient Music, denote the
machinery by which the strings of the polyplectra, or in-
struments of many strings, were struck, with a plectrum
made of quills.

Abacus Harmonicus is used by Kiicher for the struc-
ture and disposition, of the keys of a musical instrument,
either to be touched with the hands or feet.

Abacus, in Geometry, a table or slate upon which
schemes or diagrams are drawn.

ABATIS, or Abattis, from the French abattre, to
throw down, or beat down, in the Military Art, denotes
a kind of retrenchment made by a quantity of whole trees
cut down, and laid lengthways beside each other, the
closer the better, having all their branches pointed to-
wards the enemy, which prevents his approach, at the
same time that the. trunks serve as a breast-work before
the men. The Abattis is a very useful work on most oc-
casions, especially on sudden emergencies, when trees are
near at hand ; and has always been practised with consi-
derable success, by the ablest commanders in all ages and
nations.

ABAUZIT, Firmin, a learned and ingenious mathema-
tician and philosopher, born atUsez, in Languedoc, on the

I I th of November lu'79 ; his parents being protestants, he
was obliged to leave France by the revocation of the edict
of Nantes, and after suffering many hardships, at length
found an asylum in Geneva', where he prosecuted his stu-
dies with the greatest success ; and though his chief atten-
tion was directed to the mathematics, he nevertheless
made considerable progress in every department of litera-
ture ; he was acquainted with most of the learned men of
his time, and among others with our illustrious countryman
Sir Isa'ac Newton, who in an epistolary correspondence
does him the honour to say, " you area very fit person to
judge between Leibnitz and me." Me seems indeed to have
perfectly understood the Principia, having undertaken
the defence of it against father Castell, wherein he dis-
played great talents and ability: he discovered an error
in 1 hat work when it first appeared, which was corrected
in the next edition. After having signalized himself as
much for his virtue as his learning, he died on the '20th
of March 17()'?, at the advanced age of 8? years, lamented
and regretted by all whom he honoured with his friendship.

ABE

[ 3 ]

ABE

ABBREVIATE; to abbreviate fractions, in Arithmetic
and Algebra, is to lessen proportionally their terms, or the
numerator and denominator; which is performed by di-
viding those terms by any number or quantity, which will
divide them without leaving a remainder. And when the
terms cannot be any further so divided, the fraction is
said to be in its least terms.

c 16_8_4_2
24 _ 12 _ 6 _ 3 '

by dividing the terms continually by 2.

by dividing by 2, 3, and 7.

3x8x5 8x5

Also — - — = — = 4
by dividing by 3 and by 2.

5 = 20.

And - — = — -, by dividing by 4ax.

And

iacx

= — , by dividing by a ■+■ x.

ABBREVIATION, of fractions, in Arithmetic and Al-
gebra, is the reducing them to lower terms, as in the ex-
amples above.

ABERRATION, in Astronomy, an apparent motion of
the celestial bodies, occasioned by the progressive motion
of light, and the earth's annual motion in its orbit.

This effect may be explained and familiarized by the
motion of a line parallel to itself, much after the manner
of the composition and resolution of forces. For this pur-
pose, let the ratio of the velocity of light to that of the

*

Id B

earth in its orbit, be as the line b c
to a c ; then, by the composition
of these two motions, the particle
of light will seem to describe the
line ba or dc, instead of its real
course b c ; and will appear in the
direction ab or cd, instead of its
true direction cb. So that, if ab
represent a tube, carried with a
parallel motion by an observer
along the line ac, in the time that
a particle of light would move over
the space Bc.the different places of A. a c C E
the tube being ab, ab, cd, cd; and when the eye, or end
of the tube, is at a, let a particle of light enter the other
end at b; then, when the tube is at ab, the particle of
light will be at c, exactly in the axis of the tube ; and
when the tube is at cd, the particle of light will arrive at
/, still in the axis of , the tube; and lastly, when the tube
arrives at cd, the particle of light will arrive at the eye or
point c, and consequently will appear to come in the di-
rection dc of the tube, instead of the true direction bc;
and so on ; one particle succeeding another, and forming
a continued stream or ray of light in the apparent direc-
tion d c. So that the apparent angle made by the ray of
light with the line ae, is the angle dce, instead of the
true angle bce; and the difference, bcd or abc, is the
quantity of the aberration.

M. de Maupertuis, in his Elements of Geography, gives
also a familiar and ingenious idea of the aberration, in
this manner: " It is thus," says he, "concerning the
direction in which a gun must be pointed to strike a bird
in its flight; instead of pointing it straight to the bird, the
fowler will point a little before it, in the path of its

flight, and that so much the more as the flight of the bird
is more rapid, with respect to the flight of the shot." In
this way of considering the matter, the flight of the bird
represents the motion of the earth, or the line ac, in our
scheme above, and the flight of the shot represents the
motion of the ray of light, or the line b c.

M. Clairaut too, in the Memoirs of the Academy of
Sciences for the year 1746, illustrates this effect in a fa-
miliar way, by supposing drops of rain to fall rapidly and
quickly after each other from a cloud, under which a per-
son moves with a very narrow tube; in which case it is
evident that the tube must have a certain inclination, in
order that a drop which enters at the top, may fall freely
through the axis of the tube, without touching its sides ;
which inclination must be more or less according to the
velocity of the drops, in respect to that of the tube : then
the angle made by the direction of the tube and of the
falling drops, is the aberration arising from the combina-
tion of those two motions.

Another ingenious and familiar illustration of this effect
is given in the Encyclopedic Methodique ; which is nearly
as follows.

Let c f a c» represent a boat,
or other vessel, movingparallel ....fl..
to itself from c to D,andsuppose
a person at c to throw a stone
towards a, with such a veloci-
ty that it exactly arrives there
when the point a arrives at b •
by the motion of the boat: now
a person situated at a will na-
turally refer the direction of the stone to the diagonal c a or
d b of the boat, over which it has actually passed, not-
withstanding its real direction is that of cb; for the
same appearances would have taken place, had the stone
been cast from c to b, provided its velocity had been
such as to cause it to arrive at b, in the same time as
before.

This therefore explains the nature of the aberration of
the celestial bodies ; for if we now consider the stone
above mentioned as a particle of light issuing from a body
at c, in the direction cb, and that it arrives there in the
same time that a person moves from A to B ; then it is evi-
dent, from what is said above, that he will refer its direc-
tion to the line d b ; but all bodies appear in the direction
of the visual rays emitted by them, and consequently the
point d will be the apparent place of the body c. And
thus the effect of aberration is always to make bodies ap-
pear more advanced in the direction we are moving, than
they really are.

This discovery, which is one of the brightest that have
been made in the last age, we owe to the accuracy and in-
genuity of the late Dr. Bradley, afterwards Astronomer
Royal; to which he was occasionally led by the result of
some accurate observations which he had made with an-
other view, namely, to determine the annual parallax of
the fixed stars, or that which arises from the motion of the
earth in its annual orbit about the sun.

This motion of the earth, about the sun, had been
often doubted, and warmly contested. The defenders
of that doctrine, among other proofs of its reality, con-
ceived the idea of adducing an incontestable one from
the annual parallax of the fixed stars, if they should
be within such a distance, or if instruments and observa-
tions could be made with such accuracy, as to render that
B 2

ABE

[ * ]

ABE

parallax sensible; and with this view various attempts
have been made. Before the observations of M. Picard,
made in 1672, it was the general opinion, that the stars
did not change their position during the course of a year.
Tycho Brahe and Riccioli fancied that they had assured
themselves of it from their observations ; from which they
concluded that the earth did not move round the sun, and
that there was no annual parallax in the fixed stars. M.
Picard, in the account of his Voyage d'Uranibourg, made
in 16/2, says that the pole star, at different times of the
year, has certain variations which he had observed for
about 10 years, and which amounted to near 40" a year:
whence some, who favoured the annual motion of the
earth, were led to conclude that these variations were the
effect of the parallax of the earth's orbit. But it was im-
possible to explain it by that parallax ; because this mo-
tion was in a manner contrary to what ought to follow
only from the motion of the earth in its orbit.

In 1674 Dr. Hook published an account of observa-
tions, made by him in 1669, by which he had found
that the star y Draconis was 23" more northerly in July
than in October: observations which, at that time, seem-
ed to favour the opinion of the earth's motion, though
it be now known that there could not be any accuracy
in them.

Flamsteed having observed the pole star with his mural
quadrant, in 1689 and the following years, found that its
declination was 40" less in July than in December; which
observations, though very just, were yet, however, improper
for proving the annual parallax : and he recommended
the. making of an instrument of 15 or 20 feet radius, to
be firmly fixed on a strong foundation, for deciding a
doubt which was otherwise, not soon likely to be brought
to a conclusion.

In this state of uncertainty and doubt, Dr. Bradley, in
conjunction with Mr. Samuel Molineux, in the year 1725,
formed the project of verifying, by a series of new obser-
vations, those which Dr. Hook had communicated to the
public almost 50 years before. And as it was his attempt
that chiefly gave rise to this undertaking, so it was his me-
thod in making the observations, in some measure, that
they followed ; for they made choice of the same star,
and their instrument was constructed on nearly the same
principles: but had it not greatly exceeded the former in
exactness, they might still have continued in uncertainty
and doubt, as to the parallax of the fixed stars. And this
was chiefly owing to the accuracy of the ingenious Mr,
George Graham, to whom the lovers of astronomy are
also indebted for several other exact and convenient in-
struments.

The success then of the intended experiment, evidently
depending very much on the accuracy of the instrument,
that leading object was first to be well secured. Mr.
Molineux's apparatus having been completed, and fitted
for observing, about the end of November 1725, on the
third day of December following, the bright star in the
head of Draco, marked y by Bayer, was for the first time
observed, as it passed near (he zenith, and its situation
carefully taken with the instrument. The like observa-
tions were made frequently, on many days, both in that
and the. succeeding months, when they found a gradual
variation in the star's position towards the south. They
took care to examine very nicely, at the time of each ob-
servation, how much the variation was; till about the
beginning of March 1726', when the star was found to be

20" more southerly than at the time of the first jobserva-
tiou-: it now indeed seemed to have arrived at its utmost
limit southward, as in several trials, made about this
time, no sensible difference was observed in its situa-
tion. By the middle of April, it appeared to be return-
ing back again towards the north; and about the be-
ginning of June, it passed at the same distance from the
zenith, as it had done in December, when it was first
observed.

From the quick alteration in the declination of the star
about this time, increasing about one second 111 3 days, it
was conjectured that it would now proceed northward
of its present situation, as it had before gone southward ;
which happened accordingly ; for the star continued to
move northward till September following, when it became
stationary again ; being then near 20" more northefly
than in June, and upwards of 39" more northerly than it
had been in March. From Septeml er the star again re-
turned towards the south, till, in December, it arrived at
the same situation in which it had been observed twelve-
months before, allowing for the difference ot declination
on account of the precession of the equinox.

This was a sufficient proof that the instrument had not
been the cause of this apparent motion of the star ; aud
yet it seemed difficult to devise one that should be ade-
quate to such an unusual effect. A nutation of the
earth's axis was one of the first tilings that offered itself
on this occasion ; but it was soon found to be insuffi-
cient.

The great regularity of the observations left no room
to doubt, but that there was some uniform cause by
which this unexpected motion was produced, and which
did not depend on the uncertainty or variety of the seasons
of the year. But not being able to frame any hypothesis,
sufficient to account for all the phenomena, and being
very desirous to search a little further into this matter,
Dr. Bradley began to think of erecting an instrument for
himself at Wanstead ; that, having it always at hand, he
might with the more ease and certainty inquire into the
laws of this new motion. The consideration likewise of
being able, by another instrument, to confirm the truth of
the observations hitherto made with that of Mr. Moli-
neux's, was no small inducement to the undertaking ; but
the principal one was, the opportunity he should thereby
have of trying in what manner other stars should be af-
fected by the same cause, whatever it might be. For Mr.
Molineux's instrument being originally designed for ob-
serving y Draconis, to try whether it had any sensible pa-
rallax, it was so contrived, as to be capable of but little
alteration in its direction; not above 7 or 8 minutes of a
degree : and there being but few stars, within half that
distance from the zenith of Kew, bright enough to be
well observed, he could not, with his instrument, thorough-
ly examine how this cause affected stars that were diffe-
rently situated, with respect to the equinoctial and solsti-
cial points of the ecliptic.

These considerations determined him ; and by the
contrivance and direction of the same ingenious person,
Mr. Graham, his instrument was fixed upon the 19th of
August 1727- As he had no convenient place where he
could make use of so long a telescope as Mr. Molineux's,
lie contented himself with one of but little more than half
the length, namely, of 12 feet and a hall, the other being
24 feet and a half long, judging, from the experience he
had already had, that this radius would be long enough to

ABE

[ 5 ]

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adjust the instrument to a sufficient degree of exactness :
and he had no reason afterwards to change his opinion;
for by all his trials he was very well satisfied, that when
it was carefully rectified, its situation might be securely
depended on to half a second.

His instrument being fixed, he immediately began to
observe such stars as he judged most proper to give him
any light into the cause of the motion already mentioned.
There was a sufficient variety of small ones, and not less
than twelve (hat he could observe through all seasons of
the year, as they were bright enough to be seen in the day-
time, when nearest the sun.

When the year was completed, he began to examine
and compare his observations; and having pretty well sa-
tisfied himself as to the general laws of the phenomena,
he then endeavoured to find out the cause of them.
He was already convinced that the apparent motion of
the stars was not owing to a nutation of the earth's axis.
The next idea/that occurred to him, was an alteration "in
the direction of the plumb-line, by which the instrument
was constantly adjusted ; but this, upon trial, proved in-
sufficient. He then considered what refraction might do;
but here also he met with no satisfaction; At length,
through an. amazing sagacity, he conjectured that all the
phenomena hitherto mentioned, proceeded from the pro-
gressive motion of light, and the earth's annual motion in
its orbit : for he conceived, that if light were propagated in
time, the apparent place of a fixed object would not be
the same when the eye is at rest, as when it is moving in
any other direction except that of the line passing through
the object and the eye ; and that when the eye is moving
in different directions, the apparent place of the object
would be different.

He considered this matter in the following manner.
He imagined c a to be a ray of light, falling
perpendicularly on the line b d : then, if
the eye be at rest at a, the object must ap-
pear in the direction ac, whether light be
propagated in time, or instantaneously. But
if the eye be moving from b towards a, and
light be propagated in time, with a velocity
that is to the velocity of the eye, as a c to
ab; then, light moving from c to a, while
the eye moves from b to a, that particle of
it by which the object will be discerned,
when the eye in its motion comes to a, is
at c when the eye is at b. Joining the
points b,c, he supposed the line bc to be a
tube, inclined to the line bd in the angle L) A x>
c b d, and of such a diameter as to admit of but one
particle of light : then it was easy to conceive, that
the particle of light at c, by which the object must be
si-en when the eye arrives at a, would pass through the
tube bc, so inclined to the line bd, and accompanying
the eye iu its motion from b to a ; and that it would not
come to the eye, placed behind such a tube, if it had any
other inclination to the line bd. But if, instead of sup-
posing bc so small a lube, we conceive it to be the axis of
a larger; then, for the same reason, the particle of light
at c cannot pass through that axis, unless it be inclined
to b n in the same angle cbd.

In like manner, if the eye move the contrary way, from
i) towards a, with the same velocity ; then the tube
must be inclined in the angle line. Although therefore
I lie true or real place of an object, bc perpendicular to.

the line in which the eye is moving, yet the visible place
will not be so ; since that must doubtless be in the direc-
tion of the tube. But the difference between the true
and apparent place, will be, ceteris paribus, greater or
less, according to the different proportions between the
velocity of light and that of the eye : so that, if we could
suppose light to be propagated in an instant, then there
would be no difference between the real and visible place
of an object, though the eye were in motion; for in that
case, ac being infinite with respect to ab, the angle
ajcb, which is the difference between the true and visible
place, vanishes^ But if light be propagated in time,
which was then allowed by most philosophers, then it is
evident, from' the foregoing considerations, that there will
always be a difference between the true and visible place
of an object, except when the eye is moving either direct-
ly towards, or from the object. And in all cases, the
sine of the difference between the true and visible place of
the object, will be to the sine of the visible inclination of
the object to the line in which the eye is moving, as the
velocity of the eye, is to the velocity of light.

If light moved only 1000 times faster than the eye,
and an object, supposed to be at an infinite distance,
were really placed perpendicularly over the plane in
which the eye is moving ; it follows, from what has been
said, that the apparent place of such object will always
be inclined to that plane, in an angle of 89° 56' { ; so that
it will constantly appear 3 \ from its true place, and will
seem so much less inclined to the plane, that way towards
which the eye tends. That is, if a c be to a b or a d, as
1000 to 1, the angle abc will be S.9° 56' i, and the angle
ACS 3'i, and bcd or 2acb will be 7', if the direction
of the motion of the eye be contrary at one time to what
it is at another.

If the earth revolve about the sun annually, and the
velocity of light were to the velocity of the earth's mo-
tion in its orbit, as 1000 is to 1 ; then it is easy to con-
ceive, that a star really placed in the pole of the ecliptic,
would to an eye carried along with the earth, seem to
change its place continually ; and, neglecting the small
difference on account of the earth's diurnal revolution on
its axis, it would seem to describe a circle about that pole,
every where distant from it by 3 i. So that its longitude
would be varied through all the points of the ecliptic
every year, but its latitude would always remain the
same. Its right ascension would also change, and its de-
clination, according to the different situations of the sun
in respect of the equinoctial points ; and its apparent
distance from the north pole of the equator, would be 7'
less at the autumnal, than at the vernal equinox.

The greatest alteration of the place of a star, in the
pole of the ecliptic, or, which in effect amounts to the
same thing, the proportion between the velocity oflightand
the earth's motion in its orbit, being known, it will not be
difficult to find what would be the difference, on this ac-
count, between the true and apparent place of any other
star at any time: and, on the contrary, the difference be-
tween the true and apparent place being given, the pro-
portion between the velocity of light, and the earth's mo-
tion in its orbit, may be found.

After the history of this, curious discovery, related by
the author (in the Philos. Trans, of thai year), he gives
the results of a multitude of accurate observations, made
on a great number of stars, at all seasons of the year.
From all which observations, and the foregoing theory, he

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found that every star, in consequence of the earth's mo-
tion in its orbit and the progressive motion of light, ap-
pears to describe a small ellipse in the heavens, the trans-^
verse axis of which is equal to the same quantity for
every star, namely 40" nearly; and that the conjugate
axis of the ellipse, for different stars, varies as the sine of
the star's latitude; that is, radius is to the sine of the
star's latitude, as the transverse axis to the conjugate
axis : and consequently a star in the pole of the ecliptic,
, its latitude being there 90°, whose sine is equal to the ra-
dius, will appear to describe a small circle about that
pole as a centre, whose radius is equal to 20 . He also
gives the following law of the variation of the star's decli-
nation : if a denote the angle of position, or the angle at
the star made by two great circles drawn from it through
the poles of the ecliptic and equator, and b another angle,
whose tangent is to the tangent of a, as radius is to the
sine of the star's latitude ; then b will be equal to the dif-
ference of longitude between the sun and the star, when
the true and apparent declination of the star are the same.
And if the sun's longitude in the ecliptic be reckoned
from that point in which it is when this happens ; then
the difference between the true and apparent declination
of the star, will be always as the sine of the sun's longitude
from that point. It will also be found that the greatest
difference of declination that can be between the true and
apparent place of the star, will be to 20", the semitrans-
verse axis of the ellipse, as the sine of a to the sine of b.

The author then shows, by the comparison of a num-
ber of observations made on different stars, that they ex-
actly agree with the theory deduced from the progressive
motion of light, and that consequently it is highly proba-
ble that such motion is the cause of those variations in
the situation of the stars. From which he infers, that
the parallax of the fixed stars is much smaller than hither-
to supposed, by those who pretended to deduce it from ob-
servations. He thinks he may venture to say that, in the
stars he had observed, the parallax does not amount to 2'' ;
nay, that if it had amounted to 1", he should certainly
have perceived it, in the great number of observations
that he made, especially of y Draconis : which agreeing
with the hypothesis, without allowing any thing for paral-
lax, nearly as well when the sun was in conjunction with,
as in opposition to, this star, it seems very probable that
its parallax is not so much as one second ; and conse-
quently that it is above 400000 times farther from us
than the sun.

From the greatest variation in the place of the stars,
namely 40", Dr. Bradley deduces the ratio of the velocity
of light, in comparison with that of the earth in its orbit.
In the preceding figure, a c is to ab, as the velocity of
light to that of the earth in its orbit, the angle acb being
iqual to 20" ; so that the ratio of those velocities is that of
radius to the tangent of 20", or of radius to the arc of 20",
since in very small angles the tangent has no sensible dif-
ference from the arc itself: but the radius of a circle is
equal to the arc of 57°-^ nearly, or equal to 206260" ;
therefore the velocity of light is to the velocity of the
earth, as 206"260 to 20, or as 10313 to 1.

And hence also the time, in which light passes over the
space from the sun to the earth, is easily deduced; for
this time will be to one year, as a ii or 20" to 36'0°< or the
whole circle ; that is, 36'0° : 20" : : 365^ days : 8m 7s, that
is, light will pass from the sun to the earth in 8 minutes,
7 seconds; and this will be the same, whatever may be.

the distance of the sun. See Abridg. Philos. Trans, vol. ~ ,
p. 30S, and vol. 9, p. 417.

Dr. Bradiey having annexed to his theory the rules or
formulae for computing the aberration of the fixed stars in
declination and right ascension ; these rules have been va-
riously demonstrated, and reduced to other practical
forms, by Clairaut, in the Memoirs of the Academy of
Sciences for 1737; by Simpson, in his Essays in 1740; by
Fontaine, in 1744; and by several other persons. The re-
sults of these rules are as follow : Every star appears to
describe in the course of a year, by means of the aberra-
tion, a small ellipse, whose greater axis is 40", and the
less axis, perpendicular to the ecliptic, is equal to 40"
multiplied by the sine of the star's latitude, the radius
being 1. The eastern extremity of the longer axis, marks
the apparent place of the star, on the day pf the opposi-
tion ; and the extremity of the less axe, which is farthest
from the ecliptic, marks its situation three months after.

The greatest aberration in longitude, is equal to 20" di-^
vided by the cosine of its latitude, or multiplied by its
secant. And the aberration for any time, is equal to 20"
multiplied by the cosine of the elongation of the star
found for the same time, and divided by the cosine of its
latitude, or multiplied by its secant. This aberration is
subtractive in the first and last quadrants of the argument,
or of the difference between the longitudes of the sun and
star; and additive in the second and third quadrants.
The greatest aberration in latitude, is equal to 20" multi-
plied by the sine of the star's latitude. And the aberra-
tion in latitude for any time, is equal to 20" multiplied
by the sine of the star's latitude, and also by the sine of
the elongation. The aberration is subtractive before the
opposition, and additive after it.

The greatest aberration in declination, is equal to 20"
multiplied by the sine of the angle of position a, and di-
vided by the sine of b the difference of longitude between
the sun and star when the aberration in declination is no-
thing. And the aberration in declination at any other
time, will be equal to the greatest aberration multiplied
by the sine of the difference between the sun's place at the
given time and his place when the aberration is nothing.
Also the sine of the latitude of the star, is to radius, as the
tangent of a the angle of position at the star, is to the
tangent of b, the difference of longitude between the sun
and star when the aberration in declination is nothing.
The greatest aberration in right-ascension, is equal to 20"
multiplied by the cosine of a the angle of position, and
divided by the sine of c the difference in longitude, be-
tween the sun and star when the aberration in right-as-
cension is nothing. And the aberration in right-ascension
at any other time, is equal to the greatest aberration mul-
tiplied by the sine of the difference between the sun's
place at the given time, and his place when the aberration
is nothing. Also the sine of the latitude of the star is to
radius, as the cotangent of a the angle of position, to the
tangent of c.

In the first article of Mr. Simpson's Essays (an. 1740),
is given a geometrical demonstration of a star's apparent
path, resulting from the motion of light combined with
the earth's motion in its annual orbit ; accompanied with
rules thence deduced for practical calculations. Another
curious demonstration of the same was given by Dr.
Brook Taylor in the year 1729, as soon as Dr. Bradley's
discovery was known. It is preserved in Robins's Tracts,
vol. 2, p. 27t>, and is as follows. — Let abc be the orbit of

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L

the earth, in which it moves from Bloc and
a, round the. sun in d. Then, the earth
being in c, if the,, tangent cf, be drawn,
and on the transverse axis a b as a diame-
ter the circle aebf be described, ed being
drawn and continued to g, de is perpen-
dicular to the tangent ec; and therefore
the velocity of the earth in e, is to the ve-
locity in c, reciprocally as db to d e; that
is, directly as ad to dg, since the rectan-
gles under adb and edg are equal. Now
c ii being drawn from the earth in c, toward
the true place of any star, take ci to ad
as the velocity of light to the velocity of
the earth in b. Through I draw k i l in a
situation perpendicular to a b, and making
ki equal to- a d, and il equal to db,
on the diameter kl describe the circle
k m l n parallel to the plane acbf. Lastly,
make the angle under K I o equal to that
under a dg, which is nearly the arithme-
tical mean between the mean and true dis-
tance of the earth from the aphelion, and
draw co. — Now k i being in a situation perpendicular
to ab, and the angles under kio, adg equal, 01 will
be equal to dg, and be situated perpendicular to ge,
and therefore is parallel to ce, or to the direction of the
earth's motion in c ; and is to ki, as the velocity of the
earth, in c, to its velocity in b. Consequently, since ki
or a d is to c I as the velocity of the earth in b is to the
velocity of light, by equality 01 is to i c, as the velocity
'of the earth's motion in c to the velocity of light. There-
fore the star will be seen from the earth at c in the direc-
tion co. Hence it follows, that every fixed star is viewed
from the earth as moving in a circle parallel to the plane
of the ecliptic, the diameter of that circle which is parallel
to the lesser axis of the earth's orbit being divided by the
true place of the star, in the same proportion, as the
greater axis of the earth's orbit is divided by the sun.

Aberration of the Planets, is equal to their geocen-
tric motion, or the space they appear to move through
as seen from the earth, during the time that light employs
in passing from the planet to the earth. Thus, in the sun, the
aberration in longitude is constantly 20", that being the
space moved by the sun, or, which is the same thing, by
the earth, in Sm 7s, that being the time in which light
passes from the sun to the earth, as we have seen in the
foregoing article. In like manner, knowing the. distance
of any planet from the earth, by proportion it will be, as
the distance of the sun is to the distance of the planet, so
is 8m 73 to the time of light passing from the planet to the
earth : then computing the planet's geocentric motion in
this time, that will be the aberration of the planet, whether
it be in longitude, latitude, right-ascension, or declination.

It is evident that the aberration will be greatest in the
longitude, and very small in latitude, because the planets
deviate very little from the plane of the ecliptic or path
of the earth ; so that the aberration in the latitudes of the
planets, is commonly neglected, as insensible ; the greatest
in Mercury being otdy 4"-y, and much less in the other
planets. As to the aberrations in declination and right-
ascension, they must depend on the situation of the planet
in the zodiac. The aberration in longitude, being equal
to the geocentric motion, will be more or less according as
that motion is; it will therefore be least, or nothing at all,

Saturn -

27'

•0

Jupiter -

,29

•8

Mars

37

•8

Venus

43

•2

Mercury

59

0

The Moon

§

when the planet is stationary ; and greatest in the supe-
rior planets, Mars, Jupiter, Saturn, &c, when they are
in opposition to the sun ; but in
the inferior planets, Venus and
Mercury, the aberration is great-
est at the time of their superior
conjunction. These maxima of
aberration for the several planets,
when their distance from the sun
is least, are as annexed, viz, for

And between these numbers and nothing, the aberrations
of the planets, in longitude, vary according to their si-
tuations. But that of the sun varies not, being constantly
20", as has been before observed. And this may alter his
declination by a quantity, which varies from 0 to near 8";
being greatest or 8" about the equinoxes, and vanishing in
the solstices.

The methods of computing these, and the formulas for
all cases, are given by M. Clairaut, in the Memoirs of the
Academy of Sciences for the year 174-6, and by M. Euler
in the Berlin Memoirs, vol. 2, for 1746.

Aberration, in Optics, is the deviation or dispersion
of the rays of light, when reflected by a speculum, or re-
fracted by a lens, by which they are prevented from meet-
ing or uniting in the, same point, called the geometrical fo-
cus, but are spread over a small space, and produce a con-
fusion of images. This aberration is either lateral or
longitudinal : the lateral aberration is measured by a
perpendicular to the axis of the speculum or lens, drawn
from the focus to meet the refracted or reflected ray : the
longitudinal aberration is the distance, on the axis, be-
tween the focus and the point where the ray meets the
axis. The aberrations are very amply treated in Smith's
Complete System of Optics, in 2 volumes 4to.

There are two species of aberration, distinguished ac-
cording to their different causes : the one arises from the
figure of the speculum or lens, producing a geometrical
dispersion of the rays, when these are perfectly equal in all
respects : the other arises from the unequal refrangibility
of the rays of light themselves ; a discovery that was
made by Sir Isaac Newton, and for this reason it is often
called the Newtonian aberration. As to the former spe-
cies of aberration, or that arising from the figure, it is well
known that if rays issue from a point at a given distance,
then they will be reflected into the other focus of an ellipse
having the given luminous point for one focus, or directly
from the other focus of an hyperbola ; and will be various-
ly dispersed by all other figures. But if the luminous
point be infinitely distant, or, which is the same, the inci-
dent rays be parallel, then they will be reflected by a para-
bola into' its focus, and variously dispersed by all other fi-
gures. But those figures are very difficult to make, and
therefore curved specula are commonly made spherical,
the figure of which is generated by the revolution of a cir-
culararc, which produces an aberration of all rays, whether
they are parallel or not, and
therefore it has no accurate
geometrical focus which is
common to all the rays.
Let bvf represent a con-
cave spherical speculum,
whose centre is c; and let
ab, ef be incident rays pa-
rallel to the axis cv. Be-
cause the angle of incidence

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is equal to the angle of reflection in all cases, therefore
if the radii c b, cv be drawn to the points of incidence,
and thence b d making the angle cbd equal to the angle
cba, and f« making the angle cfg equal to the angle
cfe; then bd, fg will be the reflected rays, and d,g,
the points where they meet the axis. Hence it appears
that the point of coincidence with the axis, is equally
distant from the point of incidence and the centre: for
because the angle c b t> is equal to the angle cba, which
is equal to the alternate angle b c d, therefore their op-
posite sides cd, db are equal : and in like manner, in
any other, G f is equal to g c. And hence it is evident
that when b is indefinitely near the vertex v, then d is in
the middle of the radius cv ; and the nearer the incident
ray is to the axis c v, the nearer will the reflected ray
come to the middle point d ; and the contrary. So that
the aberration dg of any ray efg, is always more and
more, as the incident ray is farther from the axis, or the
incident point f from the vertex v; till when the distance
v i is 60 degrees, then the reflected ray falls in the vertex
v, making' the aberration equal to the whole length dv.
And this shows the reason why specula are made of a
very small segment of a sphere, namely, that all their re-
flected rays may arrive very near the middle point or fo-
cus d, to produce an image the most distinct, by the least
aberration of the rays. And in like manner for rays re-
fracted through lenses.

In spherical lenses, M. Huygens has demonstrated that
the aberration from the figure, in different lenses, is as
follows.

1. In all plano-convex lenses, having their plane sur-
face exposed to parallel rays, the longitudinal aberration
of the extreme ray, or that remotest from the axis, is
equal to f of the thickness of the lens.

2. In all plano-convex lenses, having their convex sur-
face exposed to parallel rays, the longitudinal aberration
of the extreme ray, is equal to £ of the thickness of the
lens. So that in this position of the same plano-convex
lens, the aberration is but about one-fourth of that in
the former ; being to it only as 7 to 27.

3. In all double convex lenses of equal spheres, the
aberration of the extreme ray is equal to j of the thick-
ness of the lens.

4. In a double convex lens, the radii of whose spheres
are as 1 to 6, if the more convex surface be exposed to
parallel rays, the aberration from the figure is less than
in any other spherical lens ; being no more than -J.J of its
thickness. ,t

But the foregoing species of aberration, arising from
(he figure, is very small, and easily remedied, in compa-
rison with the other, arising from the unequal rcfrangibi-
lity of the rays of light; insomuch that Sir Isaac New-
ton shows in his Optics, p. 84 of the 8vo edition, that if
the object-glass of a telescope be plano-convex, the plane
side being turned towards the object, and the diameter of
the sphere, to which the convex side is ground, be 100
feet, the diameter of the aperture being 4 inches, and the
ratio of the sine of incidence out of glass into air, be to
that of refraction, as 20 to 31; then the diameter of the
Circle of aberrations will in this case be only 7TT^J^5T
parts of an inch : while the diameter of the little circle,
through which the same rays arc scattered by unequal
refrangibility, will be'aboul the 56th part of the aperture
of the object-glass, which here is 4 inches. And there-
lore the error arising from the spherical figure of the

glass, is to the error arising from the different refrangibi-
lity of the rays, as yr o90Va 0 o t0 tt> that is as I to 544y.

So that it may seem strange that objects appear through
telescopes so distinct as they do, considering that the er-
ror arising from the different refrangibility, is almost in-
comparably larger than that of the figure. Newton how-
ever solves the difficulty by observing that the rays, under
their various aberrations, are not scattered uniformly
over all the circular space, but collected infinitely more
dense in the centre than in any other part of the circle ;
and that, in the way from the centre to the circumference,
they grow more and more rare, so as at the circumfe-
rence to become infinitely rare; and, by reason of then-
rarity, they are not strong enough to be visible, unless in
the centre, and very near it.

In consequence of the discovery of the unequal re-
frangibility of light, and the apprehension that equal re-
fractions must produce equal divergencies, in every kind
of medium, it was supposed that all spherical object-
glasses of telescopes would be equally affected by the dif-
ferent refrangibility of light, in proportion to their aper-
ture, of whatever materials they might be constructed :
and therefore, that the only improvement that could be
made in refracting telescopes, was that of increasing their
length. So that Sir Isaac Newton, and other persons
after him, despairing of success in the use and fabric of
lenses, directed their chief attention to the construction
of reflecting telescopes.

However, about the year 1747, M. Euler applied him-
self to the subject of refraction ; and pursued a hint
suggested by Newton, for the design of making object-
glasses with two lenses of glass inclosing water between
them ; hoping that, by constructing them of different ma-
terials, the refractions would balance one another, and so
the usual aberration be prevented. Mr. John Dollond,
an ingenious optician in London, minutely examined this
scheme, and found that M. Euler's principles were not
satisfactory. M. Clairaut likewise, whose attention had
been excited to the same subject, concurred in opinion
that Euler's speculations were more ingenious than useful.
This controv rsyt which seemed to be of great import-
ance in the science of optics, engaged also the attention
of M. Klingenstierna of Sweden, who was led to make a
careful examination of the Sth experiment in the second
part of Newton's Optics, with the conclusions there
drawn from it. The consequence was, that he found (hat
the rays of light, in the circumstances there mentioned,
did not lose their colour, as Sir Isaac had imagined. This
hint of the Swedish philosopher led Mr. Dollond to re-
examine the same experiment : and after several trials it
appeared, that different substances caused the light to di-
verge very differently, in proportion to their general re-
fractive powers. In the year 1757 therefore he procured
wedges of different kinds of glass, and applied them to-
gether so, that, the refractions might be made in Contrary
directions, that he might discover whether the refraction
and divergency of colour would vanish together. The
result of his first trials encouraged him to persevere; for
he discovered a difference tar beyond his hopes in the
qualities of different kinds of glass, with respect to then-
divergency of colours. The Venice glass and English
crown glass were found to be nearly allied in this respect:
the common English plate glass made the rays diverge,
more ; and the English Hint glass most of all. But, with-
out inquiring into the cause of this difference, he pro-

ABE

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A B S

ceeded to adapt wedges of crown glass, and of white flint
glass, ground to different angles, to each other, so as to
refract in different directions; till the refracted light was
entirely free from colours. Having measured the refrac-
tions of each wedge, he found that the refraction of the
white glass, was to that of the crown glass, nearly as 2 to
3: and he hence concluded in general, that any two
wedges made in this proportion, and applied together so
as to refract in contrary directions, would refract the
light without any aberration of the rays.

Mr. Dollond's next object was to make similar trials
with spherical glasses of different materials, with the view
of applying his discovery to the improvement of tele-
scopes : and here he perceived that, to obtain a refraction
of light in contrary directions, the one glass must be con-
cave, and the other convex ; and the latter, which was to
refract the most, that the rays might converge to a real
focus, he made of crown glass, the other of white flint
glass. And as the refractions of spherical glasses are in-
versely as their focal distances, it was necessary that the
focal distances of the two glasses, should be inversely as
the ratios of the refractions of the wedges ; because that,
being thus proportioned, every ray of light that passes
through this compound glass, at any distance from its
axis, will constantly be refracted, by the difference be-
tween two contrary refractions, in the proportion re-
quired ; and therefore the different refrangibility of the
light will be entirely removed.

But in the applications of this ingenious discovery to
practice, Mr. Dollond met with many and great difficul-
ties. At length however, after many repeated trials, by a
resolute perseverance, he succeeded so fur as to construct
refracting telescopes much superior to any that had hi-
therto been made ; representing objects with great di-
stinctness, and in their true colours.

M. Clairaut, who had interested himself from the be-
ginning in this discovery, now endeavoured to ascertain
the principles of Mr. Dollond's theory, and to lay down
rules to facilitate the construction of these new tele-
scopes. With this view he made several experiments, to
determine the refractive power of different kinds of glass,
and the proportions in which they separated the rays of
light: and from these experiments he deduced several
theorems of general use. M. dAlembcrt made likewise
a great variety of calculations to the same purpose; and
he showed how to correct the errors to which these tele-
scopes are subject, sometimes by placing the object-glasses
at a small distance from each other, and sometimes by
using eye-glasses of different refractive powers. But
though foreigners were hereby supplied with the most ac-
curate calculations, they were very defective in practice;
and the English telesco.pes, made, as they imagined, with-
out any precise rule, were greatly superior to the best of
their construction.

M. Euler, whose speculations had first given occasion
to this important and useful inquiry, was very reluctant
in admitting Mr. Dollond's improvements, because they
militated against a prc-conceived theory of his own. At
last however, after several altercations, being convinced of
their reality and importance by M. Clairaut, he assent-
ed ; and he soon after received further satisfaction from
the experiments of M. Zeiher, of Petersburg.

M. Zeiher showed by experiments, that it is the lead,
in the composition of glass, which .ivesit this remarkable
Vol. I.

property, namely, that while the refraction of the mean
rays is nearly the same, that of the extreme rays consi-
derably differs. And, by increasing the lead, he produced
a kind of glass, which occasioned a much greater separa-
tion of the extreme rays than that of the flint glass used
by Mr. Dollond, and at the same time considerably in-
creased the mean refraction. M. Zeiher, in the course
of his experiments, made glass of minium and lead, with
a mixture also of alkaline salts; and he found that this
mixture greatly diminished the mean refraction, while it
made hardly any change in the dispersion : and he at
length obtained a kind of glass greatly superior to the
flint glass of Mr. Dollond, for the construction of tele-
scopes ; as it occasioned three times as great a dispersion
of the rays as the common glass, while the mean refrac-
tion was only as l'6l to 1.

Other improvements were also made on the new or
achromatic telescopes by the inventor Mr. John Dollond,
and by his son Peter Dollond ; which may be seen under
the proper words. For various dissertations also on the
subject of the aberration of light, colours, and the figure
of the glass, see Philos. Trans, vols. 35, 48, 50, 51, 52,
55, 60; Memoirs of the Academy of Sciences of Paris,
for the years 1737, 1746', 1752, 1755, 1756, 1757, 176'2,
1764, i?65, 1767, 1770; the Berlin Acad. 1746, 1762,
1766; Swed. Mem. vol. 16; Com. Nov. Petripol. 1762;
M. Euler's Dioptrics ; M. d'Alembert's Opuscules Math.;
M. de Rochon Opuscules; &c, &c.

ABRIDGING, in Algebra, is the reducing a compound
equation, or quantity, to a more simple form of expres-
sion. This is done either to save room, or the trouble of
writing a number of symbols ; or to simplify the expres-
sion, and ease the memory, or to render the formula more
easy and general:

So the equation j3 — ar'i+ abx — ubc = 0, by putting
p =a, q = ab, and r—abc, becomes x3— px2 ■+■ qx — r = 0.

And the equation x- ■+- (a -t-b) x = 0, by putting

p=a + b, and q = — , becomes x2 -t-/>x — q = 0.

ABSCISS, Abscisse, or Abscissa, is a part or seg-
ment cut off a line, terminated at some certain point, by
an ordinate to a curve; as a 1' or bp.

A P B A P B

So that the absciss may either commence at the vertex
of the curve, or at any other fixed point. And it may
be taken either upon the axis or diameter of the curve,
or upon any other line given in position.

Hence there are an infinite number of variable absciss-
es, terminated at the same fixed point at one end, the
other end being at any point of the given line or diameter.

When an absciss and its corresponding ordinate are
considered together, they are called the co-ordinates of
the curve.

In the common parabola, each ordinate pq has but one
absciss, ap; in the ellipse or circle, the ordinate has two
abscisses, av, bp, lying on the opposite sides of it; and
in the hyperbola, the ordinate pq has also two abscisses,
but they lie both on the same side of it. That is, in ge-
C

A B S

[ 10 ]

ABU

neral, a line of the second kind, or a curve of the first
kind, may have two abscisses to each ordinate. But a

line of the third order may have three abscisses to each
ordinate; a line of the fourth order may have four; and
so on.

The use of the abscisses is, in conjunction with the or-
dinates, to express the nature of the curves, either by
some proportion or equation including the absciss and its
ordinate, with some other fixed invariable line or lines.
Every different curve has its own peculiar equation, or
property, by which it is expressed, and different from all
others : and that equation or expression is the same for
every ordinate and its abscisses, whatever point of the
curve be taken. So, in the circle, the square of any or-
dinate is equal to the rectangle of its two abscisses, or
a r . p b = pq"; in the parabola, the square of the ordi-
nate is equal to the rectangle of the absciss and the para-
meter ; in the ellipse and hyperbola, the square of the or-
dinate is always in a certain constant proportion to the
rectangle of the two abscisses, namely, as the square of
the conjugate to the square of the transverse, or as the
parameter to the transverse axis ; and so other properties
in other curves.

When the natures or properties of curves are ex-
pressed by algebraic equations, any general absciss, as
ap, is commonly denoted by the letter x, and the ordi-
nate pq by the letter y; the other or constant lines being
represented by other letters. Then the equations express-
ing the nature of these curves are as follow ; namely,
for the

circle - - dx — x2 = y2, where d is the diameter ab;
parabola - px = y2, where p is the parameter ;
ellipse - t"1: c2 : : tx—x^-.y2, f where t is the transverse,
hyperbola t" : cl : : tx -+- x"2 :y2, \ and c the conjugate axis.

But if the abscisses are taken from the centre; then in the

circle - - b2 — x- = y'2, ") . , . ^, . .

n- >a ji {a ' -2. . 2 I where h is the semiaxis,

ellipse - b- : c1: : b2 — x :ir, > ,, ■•• .■ . »

. ' , , n t. .. i, J, V and c the semiconiusate.
hyperbola o: c ::xi— tr:y2, J J o"^-

Or we may likewise employ the parameter p in the
equation to the ellipse and hyperbola ; then in the
ellipse - t :p: : tx — x-:y0-,\ when theabscisscs arc taken
hyperbola tip:: x2±lx :y'2,$ from the vertex,
ellipse - f.p:: \&~ xl-.y*, 1 when tire abscisses are taken
hyperbola ( :p :: x2~ -P :f, \ from the centre,
where i is the transverse axis, and p the parameter.
ARSIS, Ai'.sks, AnsiDi.s. Sec Arsis, tkc. -
ABSOLUTE Filiation, in Astronomy, is the sum of
ihe optic and excentric equations. The apparent inequa-
lity of a planet's motion, arising from its not being equally

distant from the earth at all times, is called its optic
equation*; and this would subsist even if the planet's real
motion were uniform. The excentric inequality is caused
by the planet's motion being not uniform. To illustrate
this, conceive the sun to move, or to appear to move, in
the circumference of a circle, in whose centre the earth
is placed. It is manifest, that if the sun move uniformly
in this circle, he must appear to move uniformly to a
spectator at the earth; and in this case there will be no
optic nor excentric equation. But suppose the earth to
be placed out of the centre of the circle: then, though
the sun's motion should be really uniform, it would not
appear to be so, being seen from the. earth; and in this
case there would bean optic equation, without an excen-
tric one. But if we imagine the sun's orbit not to be
circular, but elliptical, and the earth in its focus; it will
be full as evident that the sun cannot appear to have a
uniform motion in such ellipse ; so that his motion will
then be subject to two equations ; that is, the optic equa-
tion, and the excentric equation. See Equation, and
Optic Inequality.

Absolute Number, in Algebra, is that term or mem-
ber of an equation that is completely known, and which
is equal to all the other, or unknown terms, taken toge-
ther ; and is the same as what Vieta calls the homoge-
neum comparationis. ' So, of the equation x'2 -+• l6x —
36 = 0, or x2 -+- l6\r = 36", the absolute number, or known
term, is 36.

Absolute Gravity, Motion, Space, Time, fyc. See the
respective substantives.

ABSTRACT Mathematics, otherwise called pure
mathematics, is that which treats of the properties of
magnitude, figure, or quantity, absolutely and generally
considered, without restriction to any species in particu-
lar: such as Arithmetic and Geometry. In this sense,
abstract, or pure mathematics, is opposed to mixed ma-
thematics, in which simple and abstract properties, and
the relations of quantities, primitively considered in pure
mathematics, are applied to sensible object* ; as in astro-
nomy, hydrostatics, optics, &c.

Abstract Nmnber, is a number, or collection of units,
considered in itself, without being applied to denote a col-
lection of any particular and determinate things. So,
for example, 3 is an abstract number, so far as it is not
applied to something: but when we say 3 feet, or 3 per-
sons, the 3 is no longer an abstract, but a concrete num-
ber.

ABSURD, or Absuudum, a term commonly used in
demonstrating converse propositions ; a mode of demon-
stration, in which the proposition intended is not proved
in a direct manner, by principles before laid down ; but
it proves that the contrary is absurd or impossible; and
so indirectly, as it were, proves the proposition itself. The
4th proposition in the first book of Euclid, is the first in
which he makes use of this mode of proof; where he
shows that if the extremities of two lines coincide, those
lines will coincide in all their parts, otherwise they would
inclose a space, which is absurd or contrary to the 10th
axiom. Most converse propositions are proved in this
way, which mode of proof is called reductio ad absurdum.

ABUNDANT Number, in Arithmetic, is a number
whose aliquot parts, added all together, make a sum
which is greater than the number itself. Thus 12 is an
abundant number, because its aliquot parts, namely 1, 2,

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[ 11 1

\ C A

.3, 4, 6, when added together, make 16, which is greater
than the number 12 itself.

An abundant number is opposed to a deficient one,
which is less than the sum of its aliquot parts taken to-
gether, as the number 14, whose aliquot parts, 1, 2, 7,
make no more than 10; and to a perfect number, which
is exactly equal to the. sum of all its aliquot parts, as the
number (5, which is equal to the sum of 1, 2, 3, its ali-
quot parts.

ACADEMICIAN, a member of a society called an
academy, instituted for the promotion of arts, sciences, or
natural knowledge in general.

ACADEMICS, an ancient sect of philosophers, who
followed the doctrine of Socrates and Plato, as to the
uncertainty of knowledge, and the incomprehensibility of
truth.

Academic, in this sense, amounts to much the same with
Platonist ; the difference between them being only in
point of time. Those who embraced the system of Plato,
among the ancients, were called Academici, academician
or academic; whereas those who did the same since the
restoration of learning, have assumed the denomination
of Platonists.

There are usually reckoned three sects of academics ;
though some make five. The ancient academy was that
of which Plato was the chief. Arcessilas, one of Plato's
successors, introducing some alterations into the philoso-
phy of this sect, founded what is called the second aca-
demy. The establishment of the third, called also the
new academy, is attributed to Lacydes, or rather to Car-
neades. Some authors add a fourth, founded by Philo ;
and a fifth, by Antiochus, called the Antiochan, which
tempered the ancient academy with Stoicism.

The ancient academy doubted of every thing ; and
carried this principle so far as to make it a doubt, whe-
ther or no they ought to doubt. It was a kind of a prin-
ciple with them, never to be certain or satisfied of any
thing; never to affirm or to deny any thing, either for
true or false.

The new academy was somewhat more reasonable ; they
acknowledged several things for truths, but without at-
taching themselves to any with entire assurance. These
philosophers had found that the ordinary commerce of
life and society was inconsistent with the absolute and
universal doubtfulness of the ancient academy : and yet
it is evident that they looked upon things rather as pro-
bable, than as true and certain: by this amendment
thinking to secure themselves from those absurdities into
which the ancient academy had fallen.
ACA.DEMIST, the same as Academician.
ACADEMY, Academia, in Antiquity, a fine villa or
pleasure-house, in one of the suburbs of Athens, about a
mile from the city; where Plato, and the wise men who
followed him, held assemblies for disputes and philoso-
phical conference; which gave the name to the sect of
Academies. The house took its name, Academy, from
one Academus, or Ecademus, a citizen of Athens, to
whom it originally belonged : he lived in the time of The-
seus; and here he used to have gymnastic sports or ex-
ercises.

The. academy was further improved by Cimon, and
adorned with fountains, trees, shady walks, &c, for the
convenience of .the philosophers and men of learning, who
lieie met to^jjjfer and dispute for their mutual improve-
ment. It \te|fturrounded with a wall by Ilipparchus,

the son of Pisistratus; and it was also used as'the bnry-
ing-place for illustrious persons, who had deserved well of
the republic.

It was here that Plato taught his philosophy; and
hence it was that all public places, destined for the as-
semblies of the learned and ingenious, have been since
called Academies. — Sylla sacrificed the delicious walks
and groves of the academy, which had been planted by
Cimon, to the ravages of war; and employed those very
trees in constructing machines to batter the walls of the
city which they had adorned, — Cicero too had a villa, or
country retirement, near Puzzuoli, which he called by
the same name, Academia. Here he used to entertain his
philosophical friends ; and here it was that he composed
his Academical Questions, and his books De Naturd De-
orum.

Academy, among the moderns, denotes a regular so-
ciety or company of learned persons, instituted under the
protection of some prince, or other public authority, for
the cultivation and improvement of arts or sciences.

Some authors confound Academy with University; but
though much the same in Latin, they are very different
things in English. An university is properly a body com-
posed of graduates in the several faculties; of professors,
who teach in the public schools ; of regents or tutors,
and students who learn under them, and aspire likewise
to degrees. Whereas, an academy is not intended to teach,
or profess any art or science, but to improve it : it is not
for novices to be instructed in, but for persons of learning
to confer in, and communicate their lights and discove-
ries to each other, for their mutual benefit and improve-
ment.

The first modern academy we read of, was established
by Charlemagne, by the advice of Alcuin, an English
monk : it was composed of the most intelligent men of
the court, the emperor himself being a member. In their
academical conferences, every person was to give some
account of the ancient authors he had read; and each
one assumed the name of some ancient author, that
pleased him most, or some celebrated person of antiquity.
Alcuin, from whose letters we learn these particulars,
took that of Flaccus, the surname of Horace; a young
lord, named Augilbert, took that of Homer; Adelard,
bishop of Corbie, was called Augustin ; Recluse, bishop
of Mentz, was Dametas; and the king himself, David.

Since the revival of learning in Europe, academics
have multiplied greatly, most nations being furnished with
several, and from their communications the chief im-
provements have been made in the arts and sciences, and
in cultivating natural knowledge. There are now acade-
mies for almost every art, or species of knowledge; but
we shall give a short account only of those institutions of
this kind, which regard the cultivation of mathematical
or philosophical subjects, which are the proper and pecu-
liar objects of our undertaking.

Italy abounds more in academies than any other coun-
try ; there being enumerated by Jarckius not less than
five hundred and fifty; and even to the amount of twenty-
five in Milan itself. These are, however, mostly of a pri-
vate and inferior nature.

The first academy of a philosophical kind was esta-
blished at Naples, in the house of Raptista Porta, about
the j ear 1560, under the name of Academia Secretorum
Naturae; being formed for the improvement of natural
and mathematical knowledge. This was succeeded by the
C 2

A C A

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A C A

-Academy of Lyncei, founded at Rome by prince Fre-
derick Cesi, towards the end of the same century. It was
rendered famous by the notable discoveries made by seve-
ral of its members ; among whom was the celebrated Ga-
lileo Galilei.

Several other academies contributed also to the ad-
vancement of the sciences ; but it was by speculations,
rather than by repeated experiments on the phenomena
of nature: such wire the academy of Bessarian at Rome,
and that of Laurence de Medicis at Florence, in the 15th
century; and in the l6lh those of Infiammati at Padua,
of Vegna Juoli at Rome, of Ortolani at Placentia, and of
Umidi at Florence, were constituted. The first of those
studied fire and pyrotechnia; the second, wine and vine-
yards ; the third, pot-herbs and gardens; the fourth, wa-
ter and hydraulics. To these may be added that of Ve-
nice, called La Veneta, and founded by Frederick Bado-
ara, a noble Venetian; another in the same city, of which
Campegio, bishop of Feltro, appears to have been the
chief; also that of Cosenza, or La Consent! na, of which
Bernadin Telesio, SertOrio Quatromanni, Paulus Aquinas,
Julio Cavalcanti, and Fabio Cicali, celebrated philoso-
phers, were the chief members. The compositions of all
these academies, of the Kith century, were good in their
kind ; but none of them comparable to those of the
Lyncei.

Academy del Cimento, that is, of Experiments, arose
at Florence, some years after the death of Torricelli,
namely in the year l6'57, under the protection of prince
Leopold of Tuscany, afterwards cardinal de Medicis, and
brother to the Grand Duke Ferdinand the Second. Gali-
leo, Toricelh, Aggiunti, and Viviani had prepared the way
for it: and one of its chief members was Paul del Buono,
who in 1657 invented the instrument for trying the in-
compressibility of water, which was a thick globular shell
of gold, having its cavity filled with water; the globe be-
ing compressed by a strong screw, the water came through
the pores of the gold, rather than yield to the strong
compression : there were also, Alphonsus Borelli, well
known for his ingenious treatise DeMotu Animalium, and '
other works; Candide del Buono, brother of Paul ; Alex-
ander Marsilli, Vincent Viviani, Francis Rhedi, and the
Count Laurence Magalotti, secretary of the academy,
who published a volume of their curious experiments in
1667, under the title of Saggi di Naturali Esperienze ; a
copy of which being presented to the Royal Society, it
was translated into English by Mr. Waller, and published
at London, in 4to, 16"84: being A curious collection of
tracts, containing ingenious experiments on the pressure
of the^air, on the compressing of water, on cold, heat,
ice, magnets, electricity, odours, the motion of sound,
projectiles, light, &c, &c. But we have heard little or
nothing more of the academy since that time. It may
not be improper to observe here, that the Grand Duke
Ferdinand, above mentioned, was himself no mean philo-
sopher and chemist, and that he invented thermometers,
pf which the construction and use may be seen in the
collection of the academy del Cimento.

Academy degl' Inquieti, at Bologna, incorporated af-
terwards into that della traccia in the same city, foUowcd
the example of that del Cimento. The members met at
the house, of the abbot Antonio Sampieri ; and here Ge-
miniano Montana™, one of the chief members, made ex-
cellent discourses on mathematical and philosophical sub-
nets, some parts of which were published in lC0'7, under

the title of Pensieri Fisico-Mathematici. This academy
afterwards met in an apartment of Eustachio Manfredi ;
and then in that of Jacob Sandri ; but it arrived at its
chief lustre while its assemblies were held in the palace
Marsilli.

Academy of Rossano, in the kingdom of Naples,
called La Societa Scientijica Rossanese degl' Incuriosi, was
founded about the year 1540, under the name of Navi-
ganti; and was renewed under that of Spensierati, by Ca-
millo Tuscano, about the year 160O. It was then an
academy of belles-lettres ; but was afterwards transformed
into an academy of sciences, on the solicitation of the
learned abbot Don Giacinto Girnma; who, being made
president under the title of promoter-general, in 1695,
gave it a new set of regulations. He divided the acade-
micians into several classes, namely, grammarians, rheto-
ricians, poets, historians, philosophers, physicians, ma-
thematicians, lawyers, and divines; with a separate class
for cardinals and persons of quality. To be admitted a
member, it was necessary that the candidate should have
degrees in some fapulty. Members, in the title of their
books, are not allowed to use the name Academist without
a written permission from the president, which is not
granted till the work has been examined by the censors of
the academy. This permission is the highest honour the
academy can confer; since they hereby, as it were, adopt
the work, and engage to answer for it against any criti-
cisms that may be made upon it. The president himself
is not exempt from this law : nor is any academician per-
mitted to publish any thing against the writings of ano-
ther, without leave obtained from the society.

There have been several other academies of sciences in
Italy, but they have not long subsisted, for want of the
support of the reigning princes. Such were at Naples
that of the Investiganti, founded about the year 1679, by
the marquis d'Arena, Don Andrea Concubletto ; and that
which, about the year 16"£)S, met in the palace of Don
Lewis della Cerda, the duke de Medina, and wceroy of
Naples : at Rome, that of Fisico-Matematici, which in
IfjScj met in the house of Signior Ciampini : at Verona,
that of Aletqfili, founded the same year by Signior Joseph
Gazola, and which met in the house of the count Sercn-
ghi della Cucca: at Brescia, that of Filesotici, founded
the same year for the cultivation of philosophy and ma-
thematics, and terminated the year following: that of F.
Francisco Lana, a Jesuit of great skill in these sciences:
and lastly, that of Fisico-Critici at Sienna, founded in
l6yi, by Signior Peter Maria Gabrielli.

Some other academies, still or lately subsisting in Italy,
repair with advantage the loss of the former. One of the
principal is the academy of Filarmonici at Verona, sup-
ported by the marquis Scipio Maifei, one of the most
learned men in Italy; the members of which academy,
though they cultivate the belles-lettres, do not neglect the
sciences. The academy of Ricovrati at Padua still sub-
sists with reputation; in which, from time to time, learned
discourses arc held on philosophical subjects. The like
may be said of the academy of the Mtiti di Reggio, at
Modena. At Bologna is an academy of sciences, in a
flourishing condition, known by the name of The Institute
of Bologna; which was founded in 1712 by count Mar-
sigli, for cultivating physics, mathematics, medicine, che-
mistry, and natural history. The history of it is written
by M. de Limicrs, from memoirs furnished by the founder
himself. Among the. new academics, thai which ranks

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first, after the Institute of Bologna, is given to that of the
Countess Donna Clelio Grillo Boromeo, one of the most
learned ladies of the age, to whom Signior Ginima dedi-
cates his literary history of Italy, She had lately esta-
blished an academy of experimental philosophy in her
palace at Milan ; of which Signior Vallisnieri was nomi-
nated president, and had already drawn up the regula-
tions for it, though we do not find it has ever taken place.
In the number of these academies may also be ranked
the assembly of the learned, who of late years met at Ve-
nice in the house of Signior Cristino Martinelli, a noble
Venetian, and a great patron of learning.

Academia Cosmogrqfica, or that of the Argonauts,
was instituted at Venice, at the instance of F. Coronelli,
for the improvement of geography ; the design being to
procure exact maps, geographical, topographical, hydro-
graphical, and ichnographical, of the celestial as well as
terrestrial globe, and their several regions ot parts, toge-
ther with geographical, historical, and astronomical de-
scriptions, accommodated to them : to promote which
purposes, the several members oblige themselves, by their
subscription, to take one copy or more of each piece pub-
lished under the direction of the academy; and to ad-
vance the money, or part of it, to defray the charge of
publication. To this end three societies were settled,
namely at Venice, Paris, and Rome ; the first under F.
Moro, provincial of the Minorites of Hungary; the se-
cond under the abbot Laurence au Rue Payenne au Ma-
rais ; the third under F. Ant. Baldigiani, Jesuit, professor
of mathematics in the Roman college ; to whom those
addressed themselves who were willing to engage in this
design. The Argonauts numbered near 200 members, in
the different countries of Europe ; and their device is the
terraqueous globe, with the motto Plus ultra. All the
globes, maps, and geographical writings of F. Coronelli
have been published at the expense of this academy.

The Academy of Aputists, or Impartial Academy,
deserves to be mentioned on account of the extent of its
plan, including universally all arts and sciences. It holds
from time to time public meetings at Florence, where any
person, whether academician or not, may read his works,
in whatever form, language, or subject; the academy re-
ceiving all with the greatest impartiality.

In France there are many academies for the improve-
ment of arts and sciences. F. Mersenne, it is said, gave
the first idea of a philosophical academy in France, about
the beginning of the seventeenth century, by the confe-
rences of mathematicians and naturalists, held occasion-
ally at his lodgings; at which Des Cartes, Gassendus,
Hobbes, Roberval, Pascal, Blonde], and others, assisted.
F. Mersenne proposed to each of them certain problems
to examine, or certain experiments to be made. These
private assemblies were succeeded by more public ones,
lormed by M. Monmort, and M. Thevenot, the cele-
brated traveller. The French example animated several
Englishmen of rank and learning to erect a kind of phi-
losophical academy al Oxford, towards the close of Crom-
well's administration; which after the restoration was
erected, by public authority, into a Royal Society : an
account of which sec under the word. The English ex-
ample, in its turn, animated the French. In 16'60 Louis
xiv, assisted by the counsels of M. Colbert, founded an
academy of sciences at Paris, called The

Academie Royate des Sciences, or Royal Academy of
Sciences, for the improvement of philosophy, mathema-

tics, chemistry, medicine, belles-lettres, &c. Among the
principal members, at the commencement in 1666*, were
the respectable names of Carcavi, Huygens, Roberval,
Frenicle, Auzout, Picard, Buot, Du Hamel the Secretary,
and Mariotte. There was a perfect equality among all
the members, and many of them received salaries from
the king. By the rules of the academy, every class was
to meet twice a week ; the philosophers and geometricians
were to meet, separately, every Wednesday, and then both
together on the Saturday, in a room of the king's library,
where the philosophical and mathematical books were
kept : the history class was to meet on the Monday and
Thursday, in the room of the historical books : and the
class of belles-lettres on the Tuesday and Friday : and
on the first Thursday of every month all the classes met
together, and by their secretaries made a mutual report
of what had been transacted by each, during the preced-
ing month.

In 16S>9; on the application of the president, the abbe
Bignon, the academy received, under royal authority and
protection, a new form and constitution ; by the articles
of w'hich, the academy was to consist of four sorts of
members, namely honorary, pensionary, associates, and
eleves. The honorary class to consist of ten persons, and
the other three classes of twenty persons each. The pre-
sident to be chosen annually out of the honorary class,
and the secretary and treasurer to be perpetual, and of
the pensionary class. The meetings to be twice a week,
on the Wednesday and Saturday; besides two public
meetings in the year.

Of the pensionaries, or those who receive salaries, three
to be geometricians, three astronomers, three mechanists,
three anatomists, three botanists, and three chemists, the
other two being the secretary and treasurer. Of the
twenty associates, of which twelve to be French, and
eight might be foreigners, two were to cultivate geome-
try, two astronomy, two mechanics, two anatomy, two
botany, and two chemistry. Of the twenty eleves, one to
be attached to each pensionary, and to cultivate his pe-
culiar branch of science. The pensionaries and their
eleves to reside at Paris. No regulars nor religious to be
admitted, except into the honorary class: nor any person
to be admitted a pensioner who was not known by some
considerable work, or some remarkable discovery.

In 17l6 the Duke of Orleans, then regent of France,
by the king's authority made some alteration in their con-
stitution. The class of ele\es was suppressed; and in-
stead of them were instituted twelve adjuncts, two to
each of the six classes of pensioners. The honorary mem-
bers were increased to twelve : and a"class of six free as-
sociates was made, who were not under the obligation of
cultivating any particular branch of science, and in this
class only could the regulars or religious be admitted. A
president and vice-president to be appointed annually
from the honorary class, and a director and sub-director
annually from that of the pensioners. And no person to
be allowed to make use of his quality of academician,
in the title of any of his books that he published, unless
such book were first approved by the academy. The
academy had for a device or motto, Invenit et perficit.
And the meetings, which were formerly held in the king's
library, since the year 1699 were held in a fine hall of
the old Louvre.

Finally, in the year 1785, the king confirmed, by let-
ters patent, dated April 23, the establishment of the aca-

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demy of sciences, making the following alterations, and
adding classes of agriculture, natural history, mineralogy,
and physics; incorporating the associates and adjuncts,
and limiting the number of members to six each class,
namely three pensioners and three associates ; by which
the former received an increase of salary, and the latter
approached nearer to becoming pensioners.

By the articles of this instrument it was ordained, that
the academy shall consist of eight classes, 1st geometry,
2d astronomy, 3d mechanics, 4th general physics, 5th
anatomy, 6th chemistry and metallurgy, 7th botany and
agriculture, and' 8th natural history and mineralogy.
That each class shall remain irrevocably fixed at six
members; namely, three pensioners and three associates,
independent however of a perpetual secretary and trea-
surer; of twelve free-associates and of eight associate
strangers or foreigners, the same as before, except that
the adjunct-geographer for the future to be called the as-
sociate-geographer.

The classes at first to be filled by the following persons,
namely, that of geometry by Messieurs de Borda, Jeau-
rat, Vandermonde, as pensioners ; and Messieurs Cousin,
Meusnier, and Charles, as associates : that of astronomy
by Messieurs le Monnier, de la Lande, and le Geutil, as
pensioners ; and Messieurs Messier, de Cassini, and Da-
gelat, as associates : that of mechanics by Messieurs l'abbe
Bossut, l'abbe Rochon, and de la Place, as pensioners ;
and Messieurs Coulomb, le Gendre, and Perrier, as as-
sociates: that of general physics by Messieurs Leroy,
Brisson, and Bailly, as pensioners; and Messieurs Monge,
Mechain, and Quatremere, as associates: that of ana-
tomy by Messieurs Daubluton, Tenon, and Portal, as
pensioners ; and Messieurs Sabatier, Vicq-d'azir, and
Broussonet, as associates : that of chemistry and metal-
lurgy by Messieurs Cadet, Lavoisier, and Beaumc, as
pensioners; and Messieurs Cornette, Berthollet, and Four-
croy, as associates: that of botany and agriculture by
Messieurs Guettard, Fougeroux, and Adanson, as pen-
sioners ; and Messieurs de Jussieu, de la Marck, and Des-
fontaines, as associates : and that of natural history and
mineralogy by Messieurs Desmaretz, Saye, and l'abbe de
Gua, as pensioners ; and Messieurs Darcet, l'abbe Hauy,
and l'abbe Tessicr, as associates. All names respectable
in the commonwealth of letters; and from whom the
world might expect still further improvements in the se-
veral branches of science.

The late M. Rouille de. Meslay, counsellor of the par-
liament of Paris, founded two prizes, the one of 2500
livres, the other of 2000 livres, which the academy dis-
tributed alternately every year: the subjects of the former
prize respecting physical astronomy ; and of the latter,
navigation and commerce.

The world is highly indebted to this academy for the
many valuable works they have executed, or published,
both individually and as a body collectively, especially
by their memoirs, making upwards of a hundred volumes
in 4to, with the machines, indexes, &c. in which may
be found most excellent compositions in every branch of
science. They published a volume of these memoirs every
year, with the history of the academy, and eloges of re-
markable, men lately deceased : also a general index to
the volumes every ten years. An alteration was intro-
duced into the volume for 178.3, which it seems was to
be continued in future, by remitting, in the history, the
minutes or extracts from the registers, containing some

preliminary account of the subjects of the memoirs:
but still however retaining the eloges of distinguished
men, lately deceased.

M. l'abbe Rozier also published, in four quarto vo-
lumes, an excellent index of the contents oi all the
volumes, and the writings of all the members, from the
beginning of their publications to the year 1770; with
convenient blank spaces for continuing the articles in
writing.

Their history also, to the year l6g7, was written bv
M. Du Hamel; and after that time continued from \Wi
to year by M. Fontenelle, under the following titles, Du
Hamel Historia Regias Academias Scientiarum, Paris 4ii>.
Histoirede l'Academie Royale des Sciences, avec les Me-
moires de Mathematique et de Physique, tirez des Reois-
tres de l'Academie, Paris, 4to. Histoire de l'Academie
Royale -des Sciences depuis sou etablissement en 1666
jusqu'en 169*}, en 13 tomes, 4to. A new history* from
the institution of the academy, to the period from when. ■-
M. de Fontenelle commences, has been formed; with a
series of the works published under the name of this aca-
demy, during the first interval.

The French revolution, however, gave occasion to the
new-modeling of the Academy in 1793, as well as an al-
teration in its name. (See Institute.) The memoirs of
the academy are comprehended in 139 volumes, 4to ;
viz. 11 vols, from its foundation in lo'o'f), to its renewal
in 1699; 92 vols, from 1699 to 1790; 11 vols, of me-
moirs presented to the academy, 9 vols, of prizes, 9 of
tables, and 7 of drawings of machines.

Besides the academies in the capital, there were a oreat
many in other parts of France. The

Academie Royale, at Caen, was established by letters
patent in the year 1705; but it had its rise fifty years
earlier, in private conferences, held first in the house of
M. de Brieux. M. de Segrais retiring to this city, to
spend the rest of his days, restored and gave new lustre
to their meetings. In 1707 M. Foucault, intendant of
the generality of Caen, procured the king's letters patent
for erecting them into a perpetual academy, of which
M. Foucault was to be protector for the time, and the
choice afterwards left to the members, the number of
whom was fixed to thirty, chosen for this time by rtl.
Foucault. But besides the thirty original members, leave
was given to add six supernumerary members, from the
ecclesiastical communities in that city.

At Toulouse was the Academic des jeux flcraax, com-
posed of forty persons, the oldest of the kingdom : be-
sides an academy of sciences and belles-lettres, founded
in 1750.

At Monfpelier, the royal society of sciences; which,
after 1708, was united with the royal academy of sciences
at Paris.

Other academies were also established; at Rourdcaux,
founded in 1703; at Soissons in 1674, at Marseilles in
1726, at Lyons in 1700, at Pau in Beam iiv 1721, at
Montauban in 1744, at Angers in 1685, at Amiens in
1750, at Villcfranche in 1(>'79, at Dijon in 1740, at Nimes
in 1.633) at Bcsanc on in 1752, at Chalons in 1775, nt
Rochelle in 1734, at Bcziers in 17-3, at Rouen in 1744,
at Metz in 176*0, at Arras in 1773, &c. The number of
these academies is continually augmenting ; and even in
such towns as have no academies, the. literati form them-
selves into literary societies, having nearly the same ob-
jects and pursuits.

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In Germany and other parts of Europe there are va-
rious academies of sciences, &c. The

Academie Royule des Sciences et dcs Belles- Lettres of
Prussia, was founded at Berlin, in the year 1700, by Fre-
deric i. king of Prussia, of Which the celebrated M. Leib-
nitz was the first president, and its great promoter. The
academy received a new form and a hew set of statutes
in 1710 ; by which it was ordained, that the president
shall be one of the counsellors of state; and that the
members be divided into four classes; the first to culti-
vate physics, medicine, and chemistry ; the second, ma-
thematics, astronomy, and mechanics; the third, the
German language, and the history of the country; and
the fourth, oriental learning, particularly as it may con-
cern the propagation of the gospel among infidels. That
each class elect a director for themselves, who shall hold
his post for life. That they meet in the castle called the
New Marshal, the classes to meet in their turns, one each
week. And that the members of any of ihe classes have
free access into the assemblies of the rest. Several vo-
lumes of their transactions have been published in Latin,
from time to time, under the title of Miscellanea Beroli-
nensia.

In 1743 Frederic n. king of Prussia made great alte-
rations and improvements in the academy. Instead of a
great lord or minister of state, who had usually presided
over the academy, he wisely judged that office would be
better filled by a man of letters; and he honoured the
French academy of sciences by fixing upon one of its
members for a president, namely M. Maupertuis, a dis-
tinguished character in the literary world, and whose con-
duct in improving the academy was a proof of the sound
judgment of the king, who at the same time made new
regulations for the academy, and took the title of its Pro-
tector. From that time the transactions of jhe academy
have been published annually, under the title of Histoire
de l'Academie Royale des Sciences et Belles Lettres a
Berlin, much in the manner of those of the French aca-
demy of sciences, and in, the French language ; particu-
larly enriched with the compositions of the celebrated
Euler, and other learned men. Besides the ordinary pri-
vate meetings of the academy, it has two public ones in
ihe year, in January and May ; at the latter of which is
given a pfize gold medal, of the value of 50 ducats, or
about 20 guineas. The subject of the prize is succes-
sively physics, mathematics, metaphysics, and general li-
terature. For the academy has this peculiar circum-
stance, that it embraces also metaphysics, logic, and mo-
rality; having one class particularly appropriated to these
objects, called the class of Speculative Philosophy.

Imperial Academy of Petersburg. This academy was
projected by the czar Peter 1, commonly called Peter the
Great, who in so many other instances also was instru-
mental in raising Russia from the state of barbarity in
which it had been immerged for so many ages. Having
visited France in 1717, and among other things informed
himself of the advantages of an academy of arts and sci-
ences, he resolved to establish one in his new capital,
whither he had drawn by noble encouragements several
learned strangers, and made other preparations, when his
death prevented him from fully accomplishing that ob-
ject, in the beginning of the. year 1725. Those prepara-
tions and intentions however were carried into execution
the same year, by the establishment of the academy, by
his consort the czarina Catharine, who succeeded him.

And soon after the academy composed the first volume of
their works, published in 1728, under the title of Com-
mentarii Academise Scientiarum Imperialis Pelropolita-
nas; which they continued almost annually till 174-6, the
whole amounting to 14 volumes, which were published in
Latin, and the subjects divided and classed under the fol-
lowing heads, namely mathematics, physics, history, and
astronomy. Their device a tree bearing fruit not ripe,
with the modest motto Paullatim.

Most part of the strangers who composed this academy
being dead, or having retired,- it was rather in a languish-
ing state at the beginning of the reign of the empress Eli-
zabeth, when the count Rasomowski was happily ap-
pointed president, who was instrumental in recovering its
vigour and labours; This empress renewed and altered
its constitution, by letters patent dated July 24, 1747,
giving it a new form and regulations. It consists of two
chief parts, an academy, and a university, having regu-
lar professors in the several faculties, who read lectures
as in our colleges. The ordinary assemblies are held
twice a week, and public or solemn ones thrice in the
year; in which an account is given of what has been
done in the private ones. The academy lias a noble build-
ing for their meetings, &c, with, a good apparatus of in-
struments, a fine library, observatory, &c. Their first
volume, after this renovation, was published for the years
1747 and 1748, and they have been since continued from
year to year, now to a great number of volumes, under
the title of Novi Commentarii Academia: Scientiarum Im-
perialis Petropolitanae. They are printed in the Latin
language, and contain many excellent compositions in all
the sciences, especially the mathematical papers of the
late excellent M. L. Euler, which always made a consi-
derable portion of every volume. The subjects are classed
under heads in the following order, mathematics, physico-
mathematics, physics, which include botany, anatomy,
&c, and astronomy; the whole prefaced by historical
extracts, or minutes, relating to each paper or memoir,
after the manner of the volumes of the French academy;
but wanting however the eloges of deceased eminent men.
Their device is a heap of ripe fruits piled on a table,
with the motto En addit Fructus a?tate recentes. After
the death of Euler, in 178.!, who had been many years
the president, that important literary office was, by the
empress Catharine, conferred on the princess Daschkof.

Imperial and Royal Academy of Sciences and Belles-
Lettres, at Brussels. This academy was founded in the
year 1773; and serveral volumes of their memoirs have
been published.

Royal Academy of Sciences, at Stockholm, was insti-
tuted in 1739, and since that time it has published a great
many volumes of transactions, quarterly, in Svo, in the
Swedish language.

For an account of the Royal Society of London, and
several other similar institutions, see the words Journal,
Society, &c

American Academy of Arts and Sciences, was esta-
blished in 1780 by the council and house of representa-
tives in the province of Massachusct's Bay, for promoting
the knowledge of the antiquities of America, and the
natural history of the country ; for determining the uses
to which its various natural productions may be applied ;
for encouraging medicinal discoveries, mathematical dis-
quisitions, philosophical inquiries and experiments, astro-
nomical, meteorological, and geographical observations,

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and improvements in agriculture, manufactures, and com-
merce ; and, in short, for cultivating every art and sci-
ence, which may tend to advance the interest, honour,
dignity, and happiness, of a free, independent, and virtu-
ous people. The members of this academy are never to
be less than forty, nor more than two hundred.

Academy is also used among us for a kind of colle-
giate school, or seminary ; where youth are instructed in
the liberal arts and sciences in a private way: now indeed
it is used for all kinds of schools.

Frederic i, king of Prussia, established an academy at
Berlin in 1703, for educating the young nobility of the
court, suitable to their extraction. The expence of the
students was very moderate, the king having undertaken
to pay the extraordinaries. This illustrious school, which
was then called the academy of princes, has now lost
much of its first splendour.

The Romans had a kind of military academies esta-
blished in all the cities of Italy, under the name of Campi
Martis. Here the youth were admitted to be trained for
war at the public expense. And the Greeks, besides aca-
demies of this kind, had military professors, called Tac-
tici, who taught all the higher offices of war, &c.

We have three royal academies of this kind in England,
the expenses of which are defrayed by the government;
the one at Woolwich, for the artillery and military engi-
neers; another at Portsmouth, for the navy; and the third
at Sandhurst near Bagshot, called the Royal Military
College, for educating officers for the army in general.
(See College.) The first was established by his majesty
king George n, by warrants dated April the 30th and No-
vember the 18th, 1741, for instructing persons belonging
to the military part of the ordnance, in the several bran-
ches of mathematics, fortification,N&c, proper to qualify
them for the service of artillery and the office of engi-
neers. This institution is under the direction of the ma-
ster-general and board of ordnance for the time being :
and at first the lectures of the masters in the academy
were attended by the practitioner-engineers, with the offi-
cers, Serjeants, corporals, and private men of the artillery,
besides the cadets. At present, however, none are educated
there but the gentlemen cadets, to the number of ISO.
The master-general of the ordnance is always captain of
the cadets' company, and governor of the academy ;
under him are a lieutenant-governor, and an inspector of
of studies. The masters have been gradually increased,
from two or three at first, now (in IS 14) to almost twenty;
namely, a professor of mathematics, and six or seven
other mathematical masters, a professor of fortification,
and two assistants, two drawing-masters, two French ma-
sters, with masters for fencing, dancing, and chemistry.
This institution is of the greatest consequence to the state,
and it is hardly credible that so important an object
should be accomplished at so trifling an expense.

The Royal Naval Academy at Portsmouth was founded
by George i, in 1722, for instructing young gentlemen in
the sciences useful for navigation, to educate them for the
royal navy. The establishment is under the direction of
the board of admiralty, who give salaries to the different
masters, by one of whom the students are bonrded and
lodged, the expense of which is defrayed by their own
faendi, nothing being supplied by the government but
their education.— This academy has been lately new-mo-
deled, and its name changed to the Royal Naval College.
See College.

ACANTHUS, in Architecture, the leaves of a plant
which forms the ornament of the capital of the Corin-
thian order. See Abacus.

ACCELERATED Motion, is that which receives fresh
accessions of velocity; and the acceleration may be ei-
ther equable or variable: if the accessions of velocity
be always equal in equal times, the motion is said to be
equably or uniformly accelerated : but"if the accessions,
in equal times, either increase or decrease, then the mo-
tion is unequably or variably accelerated.

Acceleration is directly opposite to retardation, which
denotes a diminution of velocity.

Acceleration' comes more particularly under consi-
deration in physics, in respect to the descent of heavy
bodies, falling freely towards the centre of the earth.

That bodies are accelerated in their descent is evident .
from various circumstances, which immediately strike
the eye of an attentive observer: thus, the greater the
height from whence a body falls, the greater impression
it makes on the object upon which it impinges; and this
increase of action upon the fixed body, must necessarily
arise from an increase of velocity in the moving one, or
from its acceleration. Before the law of gravitation was
known, which is now generally acknowledged to be the
immediate cause of the acceleration of bodies, various
fantastical hypotheses were advanced to account for this
phenomenon; some ascribing it to the action of the at-
mosphere ; others to a very subtile fluid, which they in-
troduced for that particular purpose; others again, attri-
buted it to a kind of instinct in bodies, by means of which
they accelerated their motion as they approached towards
the centre of the earth, that being considered as the place
appropriated to their element : these and various other
hypotheses of a similar nature are now entirely rejected,
and would be wholly unworthy of being mentioned, did
they not serve to show how much the knowledge of man
is increased, and the solid foundation upon which the
modern philosophy is established, when compared with
the vague and hypothetical theories of the ancients.

Leaving therefore all such visionary causes of accele-
ration, and only admitting the existence of such a force as
gravity, so evidently inherent in all bodies, without re-
gard to what may be the cause of it, the whole mystery
of acceleration will be cleared up. Consider gravity
then, with Galileo, only as a cause or force which acts
continually on heavy bodies; and it will be easy to con-
ceive that the force of gravitation, which determines bo-
dies to descend, must by a necessary consequence acce-
lerate them in falling.

A body then having once begun to descend, by the im-
pulse of gravity will continue to descend by Newton's
first law of nature, even though the first cause of its de-
scent should cease. But besides this determination to
move, impressed upon it by the first cause of motion, and
which would be sufficient to continue to infinity the de-
gree of motion already begun, new impulses are continu-
ally superadded by the same cause ; which continues to
act upon the body already in motion, in the same manner
as if it had remained at rest. There being now then two
causes of motion, acting both in the same direction ; it
necessarily follows, that the motion which they unitedly
produce, must be more considerable than that which either
could produce separately. And as long as the motion is
thuscontinued, the samecausc still operating to increase it,
the descent must necessarily be continually accelerated.

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Supposing then that gravity, from whatever principle it
arises, acts uniformly upon all bodies at the same distance
from the centre of the earth ; and conceive the time
which a heavy body takes in falling to the earth, to be
divided into indefinitely small equal parts; gravity will
impel the body toward the centre, in the first indefinitely
short instant of the descent; and if after this we suppose
the action of gravity to cease, the body will continue per-
petually to advance uniformly toward the earth's centre,
with an indefinitely small velocity, equal to that which
resulted from the first impulse.

But if we suppose that the action of gravity still con-
tinues the same after the first impulse ; in the second in-
stant, or small part of time, the body will receive a new
impulse toward the earth, equal to that which it received
in the first instant; and consequently its velocity will be
doubled; in the third instant, it will be tripled; in the
fourth, quadrupled ; in the fifth, quintupled ; and so on
continually: for the^mpulse made in any preceding in-
stant, is no ways altered by that which is made in the fol-
lowing one ; but they are, on the contrary, always accu-
mulated on each other.

So that the instants of time being supposed indefinitely
small, and all equal, the velocity acquired by the falling
body, will be, at every instant, proportional to the times
from the beginning of the descent ; and consequently the
velocity will be proportional to the time in which it is
produced. So that if a body, by this constant force, ac-
quire a velocity of 32£ feet in one second of time; it
will acquire a velocity of tj4j feet in 2 seconds, 9^i feet
in 3 seconds, 12S| in 4 seconds, and so on. Further,
all bodies, great or small, acquire, by the force of gra-
vity, the same velocity in the same time. For every equal
particle of matter being endued with an equal impelling
force, namely its gravity or weight, the sum of all the
forces, in any compound mass of matter, will be propor-
tional to the sum of all the weights, or quantities of mat-
ter to be moved ; consequently, the forces and masses
moved, being thus constantly increased in the same pro-
portion, the velocities generated will be the same in all
bodies, great or small. That is, a double force moves a
double mass of matter, with the same velocity that the
single force moves the single mass ; and so on. Or other-
wise, the whole compound mass falls all together with the
same velocity, and in the same manner, as if its particles
were not united, but as if each fell by itself, separated
all from one another. And thus all being let go at once,
they would fall together, just as if they were united into
one mass.

The foregoing law of the descent of falling bodies,
namely, that the velocities are always proportional tjo the
times of descent, as well as the following laws concern-
ing the spaces passed over, &c, were first discovered and
taught by the great Galileo, and that nearly in the fol-
lowing manner.

Because the constant velocity with which any body
moves, or the space it passes over in a given time, as sup-
pose one second, being multiplied by the time, or num-
ber of seconds it is in motion, expresses the space passed
over in that time ; and the area or space of a rectangular
figure being denoted by the length multiplied by the
breadth ; therefore the space so run over, may be consi-
dered as a rectangle compounded of the time and velo-
city, that is a rectangle of which the time denotes the
length, and the velocity the breadth. Suppose then a to

Vol. I.

A C C
A,

H

to be the heavy body which
descends, and a b to de-
note the whole time of its
descent; which let be di-
vided into a certain num-
ber of equal parts, denoting
intervals or portions of the t K.

given time, as AC, CD, de, &c; and imagine the body
to descend, during the time expressed by the first of the
divisions ac, with a certain uniform velocity arising from
the force of gravity acting on it, which let be denoted by
A F, the breadth of the rectangle c F ; then the space run
through during the time denoted by AC, with the velo-
city denoted by af, will be expressed by the rectangular
space cf.

Now the action of gravity having produced, in the first
moment, the velocity af, in the body, before at rest; in
the first two moments it will produce the velocity c g,
the double of the former ; in the third moment, to the
velocity c a will be added one degree more, by which
means will be produced the velocity D h, triple of the
first; and so of the rest; so that, during tlie whole time
a b, the body will have acquired the velocity bk. Hence,
taking the divisions of the line ab at pleasure; for ex-
ample, the divisions AC, CD, &c, for the times; the spaces
run through during those times, will be as the areas or
rectangles cf, dg, &c; and so the space described by
the moving body during the whole time a b, will be equal
to all the rectangles, that is, equal to the whole indented
space a b k i h g f. And thus it would happen if the in-
crements of velocity were produced at the end of certain
portions of time; for instance at c, d, &c; so that the
degree of motion remains the same to the instant that a
new acceleration takes place.

By conceiving the divisions of time to be shorter, for
example but half as long as the former, the indentures of
the figure will be proportionably more contracted, and it
will approach nearer to a triangle; and if these divisions
be supposed infinitely small, that is, if increments of the
velocity be supposed to be acquired continually, and at
each indivisible particle of time, which is really the case,
the rectangles so successively produced, will form a tri-
angle, as a bc ; the whole time a b consisting of minute
portions a 1, 1 2,2 3, &c; and the area of the triangle

A ' ?■ 3 4 S 6 7 B

a bc, of all the minute surfaces, or trapeziums, which
answer to the divisions of the times ; the area of the whole
triangle abc, will denote the space run through during
the whole time a b ; and the area of any smaller triangle
a 7 g, denoting the space run through during the corre-
sponding time a 7. But the triangles a 1 a, a 7 g, &c,
being similar, have their areas to each other as the squares
of their like sides a 1, a 7, &c ; and consequently the
spaces gone through, in any times counted from the be-
ginning, arc to each other as the squares of the times.

Hence, in any right-angled triangle, as a b c, the one
side ab represents the time, the other side bc the velo-
city acquired in that time, and the area of the triangle
the space described by the falling body.
D

.\cc

£ is ]

A C C

Prom the preceding demomtration is also drawn this
other general theorem, in motions that are uniformly ac-
celerated; namely, that a body descending with a uni-
formly accelerated motion, describes in the whole time of
its descent, a space, which is exactly the half of that
which it would describe uniformly in the same time, with
the velocity it has acquired at the end of its accelerated
fall. For it has been shown that the whole space which
the falling body has run through in the time a b, is re-
presented by the triangle ABC; the last velocity being
EC; and the space which the body would run through
uniformly in the same time a b, constantly with the said
greatest velocity bc, is represented by the rectangle
a B C D : but it is well known that the rectangle aecd is
douole the triangle abc; and therefore the latter space
run through, is double the former ; that is, the space run
through by the accelerated motion, is just half of that
which the body would describe in the same time, moving
uniformly with its last-acquired velocity.

Hence then, from the foregoing considerations are de-
duced the following general laws of uniformly accelerated
motions, namely, •

1st That the velocities acquired, are constantly pro-
portional to the times ; in a double time a double velo-
city, &c.

2d. That the spaces described in the whole times, each
counted from the commencement of the motion, are pro-
portional to the squares of the times, or to the squares of
the velocities; that is, in twice the time, the body will
describe 4 times the space ; in thrice the time, it will de-
scribe 9 times the space ; in quadruple the time, It) times
the space ; and so on. In short, if the times are proportional
to the numbers - 1, 2, 3, 4, 5, &c,
the spaces will be as 1, 4, 9, 16, 25, &c,
which are the squares of the former, bo that, if a body,
by the natural force of gravity, fall through the space of
KVjJj. feet in the first second of time ; then in the first 2
seconds of time it will fall .through 4 tirhcs as much, or
64| feet ; in the first 3 seconds it will fall 9 times as
much, or 144^ feet; and so on. And as Lhe spaces fallen
through are as the squares of the times, or of the veloci-
ties; therefore the times, or the velocities, are proportion-
al to the square roots of the spaces.

3d. The spaces described by falling bodies, in a series
of equal instants or intervals of time, will be as the odd
numbers - - 1, 3, 5, 7, 9> &c>

which are the differences of7 , , „ ,/• „,. t„

.. u i Mi 4, 9, 16, 25, &c,

the squares or whole spaces 3

that is, the body which has run through 16-^ feet in the

first second, will in the next second run through 48| feet,

in the third second 80TST, and so on.

4th. A body having fallen through any space in a cer-
tain time, it will have acquired a velocity equal to double
that space ; that is, in an equal time, with the last velo-
city acquired, if uniformly continued, it would pass over
just double the space. So, if a body fall through l6T'j
feet in the first second of time, it acquires a velocity of
32-j feet in a second; that is, if the body move uniformly
for one second, with the velocity acquired, it will pass
over 324, feet in ibis one second: and if in any time the
body fall through 100 feet ; then in another equal time,
if it move uniformly with the last acquired velocity, it
will pass over 200 feet. And so on.

But, as the method of demonstration used by Galileo,
by means of infinitely small parts forming a regular tri-

angle, though very ingenious, is not so elegant as might
be wished, the areas of triangles being rather unnatural
representations of the spaces descended by falling u< dies;
the same may be otherwise demonstrated thus: Let the
whole time of a body's free descent be divided into any
number of parts, calling each of these parts I ; and let
a denote the velocity acquired at the end of the first part
of time; then will 2a, 3a, 4«, &c, represent the velo-
cities at the end of the 2d, 3d, 4th, &c, part of time, be-
cause the velocities are as the times; and for the sanre
reason £a, ■§«, \a, &c, will be the velocities at the middle
point of the first, second, third, &c, part of time. But
now as the velocities increase uniformly, the space de-
scribed in any one of these parts of time, may be consi-
dered as uniformly described with its middle velocity, or
the velocity in the middle of that part of time; and there-
fore, multiplying those mean velocities each by their com-
mon time 1, we have the same fractions -§-a> la> ia-> &c>
for the spaces passed over in the s(jec. ssive parts of the
time; that is, the space jam the first time, |a in the se-
cond, |// in the third, and so on: then add these spaces
successively to one another, and we obtain £a, %a, \a,
\£a, &c, for the whole spaces described from the begin-
ning of the motion, to the end of the first, second, third,
&c, portion of time; namely \a space in one time, 4a
in 2 times, -|a in 3 times, and so on : and it is evident
that these spaces are as the numbers 1, 4, 9, l6» &c>
which are as the squares of the times.

And from this mode of demonstration, all the proper-
ties above-mentioned evidently following ; *. e. the whole
spaces - ^a, 4or, \a, &c,

are as the squares of the times - 1, 2, 3, &c,
that the separate spaces - -|a, 4a> ?a> &c,

described in the successive times, 1 „ . c

.u jj . > 1, 3, o, &c,

are as the odd numbers - j

and that the velocity a acquired in any time 1, is double

the space \a described in the same time.

As the laws of acceleration are very important, I shall
here insert the two following propositions, sent me by my
learned friend Mr. Abram Robertson, of Christ Church
College, Oxford, in which those laws are demonstrated in
a manner somewhat different.

"Proposition I. — If from the point p in the straight
line a B, the points m, n begin to move at the same time,
namely, m towards a with a motion uniformly retarded,
and n from rest towards B with a motion uniformly ac-
celerated ; and if the velocity of m decreases as much as
the velocity of n increases in the same time; then the
space m n is generated by a uniform motion, equal to the
velocity with which m begins to move.

M P N
A ! 1 ! B

For, by hypothesis, whatever is lost in the velocity of
M by retardation, is added to the velocity of n by acce-
leration : the joint velocities therefore of m and n, must
always be equal. But it is by the joint velocities of m
and n that the space m n is generated. Consequently
mn is generated by a uniform motion, which is evi-
dently equal to the velocity with which m begins to
move.

"Proposition II. — If a point begins to move in the
direction of a straight line, and continues to raove in the
same direction with a velocity uniformly accelerated; the
space passed over in any given time, will be equal to half

A C C

[ 19 ]

A C C

the space passed over in the same time with the velocity
with which the acceleration ends.

Let the point D begin to move from a towards B, along
the straight line ab, with a motion uniformly accele-
rated ; the space a d passed over, is equal to half the
space which the point would pass over, in the same time
with the acquired velocity ;it d.

D

-B

Let the points itf, n begin to move in the straight line
ge, at the same time, with equal velocities uniformly ac-
celerated ; m beginning to move from a, and n from p;
and at the same time that m comes to the point p, let N
come to h. Then as m and n move with equal veloci-
M P N

G l ! •- H

ties, uniformly accelerated, it is evident that the spaces,
which they pass over in the same time, are equal to one
another ; consequently the space G p is equal to the space
ph. Now as m begins to move from G with a velocity
uniformly accelerated, it will arrive at p with an acquired
velocity. Hence it is evident, if it be supposed to begin
to move from P with this acquired velocity, and proceed
toward G with a velocity uniformly retarded in the same
degree that it was accelerated when it began to move from
g, that it will pass over the same space g p in the same
time. Therefore, supposing the two points M, n to begin
to move from P at the same time, namely the point m be-
ginning to move with the acquired velocity mentioned
above, and proceeding towards G with the velocity uni-
formly retarded, described above; and the point n as be-
fore with the velocity uniformly accelerated : then, as the
acceleration and retardation are uniform, they will be
equal in equal spaces of time. Again; as h is retarded
in the same degree that it was accelerated when it began
to move from g, that is, in the same degree that n is ac-
celerated, by the former prop, mn is generated by an
uniform velocity. But when the point M arrives at G,
its velocity becomes equal to 0 or nothing; and at the
time that M arrives at g, n arrives at h with the acquired
velocity. Therefore, as the velocities of m and n taken
jointly are equal, and consequently uniform, the space
g h is passed over with the velocity of n at 11, in the
same time that ph is passed over by n beginning to move
from p with a velocity uniformly accelerated to H. But
ph is half of g II. " Hence the prop, is manifest."

And hence the other laws of the spaces, before-men-
tioned, easily follow.

Since the spaces descended are as the squares of the
times, and the abscisses of a parabola are as the squares
of the ordinates, therefore the relation of the times and
spaces descended may be very well represented by the
ordinates and abscisses of that figure. Thus, if a b be
the axis of the parabola \bdfh, and AC a tangent at
the veitex perpendicular to the K a C e & C
axis, be divided into any number of
equal parts a a, ac, ce, &c, to re-
present the times; and if there be
drawn ab, cd, ef, &c, parallel to
the axis: then, if ab be the space
descended in the time a a, c d will
be the space descended in the time
a c, and ef the space descended in

the time ac, and so on continually; because, by this pro-
perty of the parabola, ab, cd, ef, &c, are to each other
respectively as a a2, ac2, ac!, &c.

From the properties above demonstrated, are derived
the following practical formulas or theorems. Namely.
if g denote the space passed over in the first second of
time, by a body urged by any constant force, denoted by
1, and t denote the time or number of seconds in which
the body passes over any other space s, and v the velocity
acquired at the end of that time ; then, from the foregoing
laws, we have v == 2 g t, and * = g t- ; and from these
two equations result the following general formulas:

■ = <r fl —

v = 2gt=z— = 2</gS.

. _^_ f_s ,S V s v-

2ff— « ~ g' * — -17 — T5 ~ i7'
And when the constant force 1, is the natural force of
gravity, then the distance g descended in the first second,
in the latitude of London, is 16-^ feet: but if it be any-
other constant force, the value of g will be different, in
proportion as the force is more or less.

The motion of an ascending body, or of one that is im-
pelled upwards, is diminished or retarded by the same
principle of gravity, acting in a contrary direction, after
the same manner that a falling body is accelerated.

A body projected upwards, ascends until it has lost all its
motion; which it does in the same time as, in falling, would
have been necessary for it to have acquired that velocity.
And consequently the heights to which bodies ascend,
when projected upwards with different velocities, are to
each other as the squares of those velocities.

It should be observed that in what has been said re-
specting the ascent and descent of bodies, the resistance of
the atmosphere has not been considered, and therefore,
in practice, the results will not accurately agree with the
above theory. And further, that we have considered the
force of gravity as being constant at all heights, which is
not strictly true, as it varies inversely as the square of the
distance from the earth's centre, (see Attraction).
But as the greatest height, to which we can ascend, is very
small when compared with the radius of the earth, this
assumption may not commonly produce any sensible
error.

Accelerated Motion of Bodies on Inclined Planes.
The same general laws obtain here, as in bodies falling
freely, or perpendicularly ; namely, that the velocities are
as the times, and the spaces descended down the planes as
the squares of the times, or of the velocities. But those
velocities are less, according to the sine of the plane's incli-
nation ; and the spaces less, according to the square of the
sine. See Inclined Plane.

Accelerated Motion of Pendulums. See PenqbTMtM'
Accelerated Motion of Projectiles. See Projectile.
Accelerated Motion of Compressed Bodies, in expan-
ding or restoring themselves. See Dilatation, Com-
pression, and Elasticity.

Accelerating Force, in Physics, is the force that
accelerates the motion or velocity of bodies ; and it is
equal to, or expressed by, the quotient arising from the
motive or absolute force, divided by the mass or weight of
the body that is moved. In treating of physical subjects
respecting forces, where velocities, times, and spaces are
considered, the first inquiry is the accelerating or ac-
celerative farce. This force is greater or less in pro-
D 2

A C C

C 20 ]

A C C

poitioii to the velocity it generates in the same time,
and by this velocity it is measured. All accelerating
forces that have the motive forces directly proportional
to the quantities of matter are equal, and generate
equal velocities; so a double motive force will move
a double quantity of matter, with the same velocity,
a triple motive force a triple quantity, a quadruple
force a quadruple quantity, &c. And this is the reason
why all bodies fall equally swift by the force of gravity;
for the motive force is exactly proportional to their
weight or mass. In general the accelerating force is in
the direct ratio of the motive force, and inverse ratio of
the quantity of matter. When a body is left freely to de-
scend by the force of its natural gravity, it has been found
by experiment that in the latitude of London, it falls
through 16-^ feet in the first second of time, and acquires
a velocity of 32J- feet in that time : but if the quantity of
matter be doubled, and the motive force remain the same
as before, which may be done by connecting the falling
body to another of equal weight by means of a thread, the
latter body being placed on a smooth horizontal plane,
along which it may be drawn without friction by the for-
mer body, which hangs freely over the side of the plane ;
then the accelerating force will be only half of what it was
before, and the space fallen through in one second will be
only S^L. feet, and the velocity acquired lfj-j^- : also, if
the quantity of matter be tripled, or the body drawn
along the plane doubled ; then the accelerating force will
be only one-third of what it was at first, and the space
descended in one second, and the velocity acquired, will
be each one-third of the first : and so on.

But accelerating forces are sometimes variable, and
sometimes constant ; and the variation may be either in-
creasing or decreasing. The nature of constant and vari-
able accelerating forces, may be illustrated in the follow-
ing manner. Let two weights w, w, be connected by a
A B C

thread passing over a pully, at a, b, or c; and let the
weight w descend perpendicularly down, while it draws
the smaller weight id up the line a d, or be, or c f, the
first being a straight inclined plane, and the other two
curves, the one convex and the other concave to the
perpendicular. Then the small weight w will always
make some certain resistance to the free descent of the
large weight w, and that resistance will be constantly the
same in every part of the plane a d, the difficulty to draw
it up being the same in every point of it, because every
part of it has the same inclination to the horizon, or to
the perpendicular; and consequently the accessions to
the. velocity of the descending weight w, will be always
equal in equal times; that is, in this case w descends by
a uniformly accelerating force. But in the. two curves
be, cf, the resistance or opposition of the small weight w
will be constantly varying, as it is drawn up the curves,
because, every part of them has a different inclination to
the horizon, or to the perpendicular. In the former
curve, the direction becomes more and more upright, or

nearer perpendicular, as the small weight w ascends, and
the opposition it makes to the descent of w necessarily
becomes greater ; and consequently the accessions to the
velocity of w will be always less and less in equal times ;
thai is, w descends by a decreasing accelerating force.
But in the latter curve cf, as w ascends, the direction of
the curve becomes less and less upright, and the opposition
it makes to the descent of w, becomes always less and less;
consequently the accessions to the velocity of w will be
always more and more in equal times ; that is, w descends
by an increasing accelerating force. So that though the
velocity continually increases in all these cases, yet while
it increases in a constant ratio to the times of motion, in
the plane a d ; the velocity increases in a less ratio than
the time it ascended up be, and in a greater ratio than
the time increases in the other curve CF.

Now the relations between the times and velocities in
all these cases, may be very well represented by the rela-
tions between the abscisses and ordinates of certain lines.
Thus let ab and ac be two straight lines, making any
angle b a c ; and a d, a e two *
curves, the former concave,
and the latter convex to-
wards ab: divide ab into
any parts a a, a A, &c, repre-
senting the times of motion;
and draw the perpendiculars
acde, bfgh, &c, represent-
ing the velocities. Then in
the rightline ac, theordinates B T) C.

ad, bg, being as the abscisses a a, a b, this represents the
case of uniformly accelerated motion, in which the veloci-
ties are always as the times : but in the curve a d, the or-
dinates ac, 6/increase in a less ratio than the abscisses
ao, a6; and therefore this represents the case of decreas-
ing acceleration, in which the velocities increase in a less
ratio than the times: and in the other curve a e, the or-
dinates ae, bh increase in a greater ratio than the abscis-
ses; and therefore this represents the case of increasing
acceleration, in which the velocities increase in a greater
ratio than the times.

The several algebraic formulas or theorems, respecting
the time, velocity, and space, for constant accelerating
forces, have been delivered above, at the article Accele-
rated Motion, where the value of each is expressed in finite
determinate quantities. But, in the cases of variably ac-
celerated motions, the formulas will require the assistance
of fluxions to express, not those general relations them-
selves, but the fluxions of them ; and consequently, taking
the fluents of those expressions, in particular cases, the re-
lations of time, space, velocity, Sec, are obtained.

Thus, if t denote the time in motion,

v the velocity generated by any force,
s the space passed over,
and Sg the variable force at any part of the motion,
or the velocity the force would generate in one second of
lime, if it should continue invariable, like the force of gra-
vity, during that, time; and therefore the value of this ve-
locity '2g, will be to 32£ feet, as that variable fbl'CO, is to
1 the. force of gravity. Then, because the force may be
supposed constant during the indefinitely small time t,
and that in uniform motions the spaces and velocities are
proportional to the times, we from thence obtain these
two general fundamental proportions,

A C C

[ 21 ]

A C C

l"

2 g : v : : 1" : t, orjp = 2 g t.
From which are derived the four following formulas,
in which the value of each quantity is expressed in terms
of the rest.

s — v t = — .

v s= 2gt ==

2£

And these theorems equally hold good for the destruc-
tion of motion' and velocity, by means of retarding forces,
as for the generation of the same by means of accelera-
ting forces.

Acceleration, in Mechanics, the increase of velocity
in a moving body.

Acceleration, Astron. The Diurnal Acceleration
of the fixed stars, is the time which the stars, in one diur-
nal revolution, anticipate the mean diurnal revolution of
thesun; which is 3m 55s-^ of mean time, or nearly 3m 56s:
that is, a star rises, or sets, or passes the meridian, about
3m 56s sooner each day. This acceleration of the stars,
which is only apparent in them, arises from real retarda-
tion of the sun, owing to his apparent motion in his orbit
towards the east, which is about 59' S"-j^ of a degree every
day. So that the star which passed the meridian yester-
day at the same moment with the sun, is to-day about 59'
8" past the meridian to the west, when the sun arrives at
it; which will take him up about 3m 56s of time to pass
over; and therefore the star passes by 3™ 56s sooner than
the sun each day, or anticipates his motion at thaf rate.
The true quantity of this anticipation, or acceleration, is
found by this proportion, 360° 59' 8"-f- : 5Q'8"-f : : 24 hours:
3n 55s^s, the fourth term of which is the acceleration

The diurnal acceleration serves to regulate the lengths
or vibration of pendulums in clocks. If I observe a fixed
star set or pass behind a hill, steeple, or such like, when
the clock marks for instance 8h 10m ; and the next day,
the eye being in the same place as before, the passage be
at 8h 6m 4s ; I thence conclude that the clock is well re-
gulated, or truly measures mean time.

Acceleration of a Planet. A planet is said to be ac-
celerated in its motion, when its real diurnal motion ex-
ceeds its mean diurnal motion. .And, on the other hand,
the planet is said to be retarded in its motion, when the
mean exceeds the real diurnal motion. This inequality
arises from the change in the distance of the planet from
the sun, which is continually varying; the planet moving
always quicker in its orbit when nearer the sun, and
slower when farther off".

Acceleration of the Moon, is a term used to express
theincrease of the moon's mean motion from the sun, com-
pared with. the diurnal motion of the earth ; by which it
appears that, from some uncertain cause, it is now a little
quicker than it was formerly. Dr. Halley was led to the
discovery, or suspicion, of this acceleration, by comparing
the ancient eclipses observed at Babylon, &c, and those
observed by Albategnius in the ninth century, with some
of his own time; as may be seen in N° 218 of the Philo-
sophical Transactions. He could not however ascertain
the quantity of the acceleration, because the longitudes of
Bagdat, Alexandria, and Aleppo, where the observations
were made, had not been accurately determined. But
mice his time the longitude of Alexandria has been ascer-

tained by Chazelles ; and Babylon, according to Ptolemy's
account, lies 50' east of Alexandria. From these data,
Mr. Dunthorne, vol. 46 Philos. Transactions, compared
the recorded times of several ancient and modern eclipses,
with the calculations of them by his own tables, and
thereby verified the suspicion that had been started by
Dr. Halley ; for he found that the same tables gave the
moon's place more backward than her true place in an-
cient eclipses ; and more forward than her true place in
later eclipses ; and thence he justly inferred that her mo-
tion in ancient times was slower, and in later times quicker,
than the tables give it.

. Not content however with barely ascertaining the fact,
he proceeded to determine, as well as the observations
would allow, the quantity of the acceleration ; and by
means of an eclipse, of which the most authentic account
remains, observed at Babylon in the year 721 before
Christ, he found that the beginning of this eclipse was
about an hour and three quarters sooner than the begin-
ning by the tables ; and that therefore the moon's true
place preceded her place by computation, by about 50'
of a degree at that time. Then, admitting the accelera-
tion to be uniform, and the aggregate of ift as the square
of the time, it will be at the -rate of about 10" or § by
later computations, in 100 years.

M. de la Place announced to the Academy of Sciences
the cause of this very interesting discovery in physical as-
tronomy. Dr. Halley had remarked, as above-mentioned,
that modern observations had given a more rapid motion
to the moon than the ancient ones, and M. Lalande had
inserted some proof of it in the memoirs of the Academy
in 1757- The cause of this acceleration had been for a
long time the discussion of geometers. The academy had
proposed at three different times, in 1 7t>2, 70, and 72, a
premium for the elucidation of this subject, but Euler
himself could find nothing in the theory of attraction,
which could explain this acceleration; and the learned
began almost to conclude, that the difference between
the ancient and modern observations was to be regarded
as doubtful, and it was very possible that the Arabs, who
formed the most substantial proof, might be mistaken ;
but M. Delambre, having calculated in 1783 the observa-
tions made by M. le Paute d'Agelet at the Military Aca-
demy, has found a new proof of this singular phenomenon.
We were impatiently waiting for theory's giving us a satis-
factory account of it, when M. Laplace, whoso ingenious-
ly explained in 1786 the secular equations of Jupiter and
Saturn, succeeded so far, after several attempts, as to re-
duce that of the moon to the laws of universal weight.
It is the diminution which exists for many centuries in
the equation of the solar orbit, that is the cause of this
inequality in the moon: but as the diminution, produced
by the attraction of planets, will become, in the revolu-
tion of ages, a real augmentation, so what appears now to
us an acceleration in the moon's motion, will likewise be-
come a retardment, and it is n 'thing more than a periodi-
cal inequality. M. Laplace finds it eleven seconds for
the first century, and as this increases within a little as
the square of the time, so for the year 720 before our
.Era, the nearest date for the most ancient observations
that have reached us, it produces a difference that per-
fectly agrees with the observation. It is to M. Laplace,
therefore, we owe a confirmation of the attraction of ce-
lestial bodies, which will prove, as useful, as it is ingenious
and learned.

[ 22 ]

A C C

Thus then the moon's mean motion appears to accele-
rate continually ; and the accelerations appear to be in
the duplicate ratio of the times. The acceleration for
the end of the 17th century was about 9"; so that the
whole motion of the moon during the 17th century must
be increased 9", to obtain its motion during the 18th ; and
as much must be taken from it, or added to the computed
longitude, to obtain its motion during the l6th ; and the
double of this must be taken from the motion during the
l6th, to obtain its motion during the 15th, &c. — To ex-
plain the theory, it must be corisidered that the lunar pe-
riod which we observe, is not that which would have ob-
tained, had the moon been influenced by the earth alone.
We should not have known that her natural period was
increased, had the disturbing action of the sun remained
unchanged ; whereas this varies, and that in the inverse
triplicate ratio of the earth's distance from the sun, and is
therefore greater in our winter, when the earth is nearer
to the sun. This is the source of the annual equation, by
which the lunar period in January is made to exceed that
in July nearly 24 minutes.' The angular velocity of the
moon in general is diminished by Ty5, and this numerical
coefficient varies inversely as the cube of the earth's dis-
tance from the sun. If this ratio be expanded into a se-
ries arranged according to the sines and co-sines of the
earth's mean motion, making the earth's mean distance 1,
then the series will contain a term equal to T^ of this
velocity multiplied by \ of the square of the earth's ex-
centricity. Had this excentricity remained constant,
this product would also be constant, and would still be
confounded with the general diminution, making a con-
stant part of it : but it is known that the excentricity of
the earth's orbit diminishes, and its diminution is the re-
sult of the universality of the Newtonian law of the plane-
tary deflections. Though this diminution is exceedingly
small, its effect on the lunar motion becomes sensible by
accumulation in the course of ages. The excentricity di-
minishing, the diminution of the moon's angular motion
must also diminish, that is, the angular motion must in-
crease.

During the 18th century, the square of the earth's ex-
centricity has diminished 0'0000015S25, the mean distance
from the sun being 1. In that time this has increased
the moon's angular motion 0'00000001285. As this aug-
mentation is gradual, we must multiply the angular mo-
tion during the century by the half of this quantity, in
order to obtain its accumulated effect. This will be found
to be 9" very nearly ; which exceeds that deduced from a
most careful comparison of the motion in the last two cen-
turies, only by a fraction of a second ! While the diminu-
tion of the square of the excentricity of earth's orbit
can be supposed proportional to the time, this effect
will be as the squares of the times. When this theory is
compared with observations, the coincidence is wonderful
indeed. The effect on the moon's motion is periodical, as
the change of the solar excentricity is, and its period in-
cludes millions of years. Its effect on the moon's lon-
gitude will amount to several degrees before the se-
cular acceleration change to a retardation. Encyclo-
pedia Brit.

By the solution of this problem, which had for a long
time engaged the attention of mathematicians, M. Laplace
has given the last degree of perfection to theoretical astro-
nomy ; as there is now no anomaly, or irregularity, in
the motion of the heavenly bodies, that is not satisfac-

torily accounted for, on the simple Newtonian laws of
gravitation.

ACCELERATIVE Fo^ice, 8fc, the same as Accele-
rating.

ACCESSIBLE, something that may be approached, or
to which we can come. In Surveying, it is such a
place as will admit of having a distance or length of
ground measured from it; or such a height or depth as
can be measured by actually applying a proper instru-
ment to it. For the means of doing which, see Altim e-

TRY, LONGIMETRY, Or HeJGHTS-AN D-DlSTA N CES.

ACCIDENS, Accident, Philos.

Per Accidens is a term often used among philosophers,
to denote what does not follow from the nature of a thing,
but from some accidental quality of it : in this sense it
stands opposed to per se, which denotes the nature and es-
sence of a thing. Thus, fire is said to burn per se, or
considered as fire, and not per accidtns; but a piece of
iron, though red-hot, only burns per acciderts, by a quality
accidental to it, and not considered as iron.

ACCIDENTAL, something that partakes of the na-
ture of an accident ; or that is indifferent, or not essential
to its subject. — Thus, whiteness is accidental to marble,
and sensible heat to iron.

Accidental Colours, so called by M. Buffon, are
those which depend on the affections of the eye, in contra-
distinction to such as belong to light itself.

The impressions made upon the eye, by looking sted-
fastly on objects of a particular colour, are various ac-
cording to the single colour, or assemblage of colours, in
the object; and they continue for sometime after the eye
is withdrawn, and give a false colouring to other objects
that are viewed during their continuance. M. Buffon
has endeavoured to trace the connection between these ac-
cidental colours, and those that are natural, in a variety
of instances. M. d'Arcy contrived a machine for measu-
ring the duration of those impressions on the eye ; and
from the result of several trials he inferred, that the effect
of the action of light on the eye continued about S thirds
of a minute.

The subject has also been considered by M. dcla Hire,
and M. Aepinus, &c. See Mem. Acad. Paris 174-3, and
176'5; Nov. Com. Petrop. vol. 10; also Dr. Priestley's
Hist, of Discoveries relating to Vision, pa. 6"31.

Accidental Point, in Perspective, is the point in
which a right line drawn from the eye, parallel to another
right line, cuts the picture or perspective plane.

Let ab be the line
given to be put in per-
spective, cfd the pic-
ture or perspective
plane, and e the eye:
draw ef parallel to
ab ; so shall F be the
accidental point of the
line a B, and indeed of
all lines parallel to it,
since only one parallel to them, namely ef, can be
drawn from the' same point e: and in the accidental
point f, there concur or meet the representations of all
the parallels to a b, when produced.

It is called the accidental point, to distinguish it
from the principal point, or point of view, where a
line drawn from the eye perpendicular to the perspec-
tive plane, meet* this plane, and which is the acci-

A C H

[

]

A C H

dental point to all lines that are perpendicular to the
same plane.

ACCLIVITY, the slope or steepness of a line or plane
inclined to the horizon, taken upwards; in contradistinc-
tion to declivity, which is taken downwards. So, the as-'
cent of a hill, is an acclivity ; the descent of the same, a
declivity. Some writers on fortification use acclivity for
tains: though more commonly the word talus is used to
denote the slope, whether in ascending or descending.

ACCOMMODATION, in Physics, or Philosophy, the
application of one thing to another by analogy.

ACCOMPANIMENT, in Music, denotes either the
different parts of a piece of music for the different instru-
ments, or the instruments themselves which accompany a
voice, to sustain it, as well as to make the music more
full. The Accompaniment is used in recitative, as well
as in song; on the stage, as well as in the choir, &c.

The ancients had likewise their accompaniments on the
theatre ; and they had even different kinds of instruments
to accompany the chorus, from those which accompanied
the actors in the recitation. The accompaniment among
the moderns, is often a different part, or melody, trom
the song it accompanies. But it is disputed whether it
was so among the ancients. Organists sometimes apply
the word to several pipes, which they occasionally touch
to accompany the treble ; as the drone, the flute, &cc.

ACCOMPT. See Account.

ACCORD, according to the modern French music, is
the union of two or more sounds heard at the same time,
and forming together a regular harmony.

Accords are divided into perfect and imperfect; and
again into consonances and dissonances. Accord is more
commonly called Concord; which see.

AccoiiD is also spoken of the state of an instrument,
when its fixed sounds have among themselves all the just-
ness that they ought to have.

AQCOUNT, or Accompt, in Arithmetic, &c, a calcu-
lation or computation of the number or order of certain
things; as the computation of time, &c. There are va-
rious ways of accounting; as, by enumeration, or telling
one by one ; or by the rules of arithmetic, addition, sub-
traction, &c.

Account, in Chronology, is nearly synonymous with
style. Thus, we say the English, the foreign, the Julian,
the Gregorian, the Old, or the New account, or style.
We account time by years, months, &c ; the Greeks ac-
counted it by olympiads ; the Romans, by indictions,
lustres, &c.

ACCRETION, in Physics, the growth or increase of
an organical body.

ACHLRNER, or Acharner, in Astronomy, a star of
the first magnitude in the southern extremity of the con-
stellation Eridanus, marked a. by Bayer. Its longitude
for 1761, X 11° 55' 1"; and latitude south 59° 22' 4".

ACHILLES, a name given by the schools to the prin-
cipal argument alleged by each sect of philosophers in
behalf of their system. In this sense we sa\, this is his
Achilles; that is, his master-proof : alluding to the strength
and importance of the hero Achilles among the Greeks.

Zeno's argument against motion is peculiarly termed
Achilles. That philosopher made a comparison between
the swiftness of Achilles and the slowness of a tortoise,
pretending that a very swift animal could never overtake
a slow one that was before it, and that therefore there is
no such thing as motion : for, said he, if the tortoise

were one mile befoic Achilles, and the motion of Achilles
100 times swifter than that of the tortoise, yet he would
never overtake it ; and for this reason, namely, that while
Achilles runs over the mile, the tortoise will creep over
one hundredth part, of a mile, and will be so much the
foremost ; again, while Achilles runs over this T-J^th part,
the tortoise will creep over the 100th part of that -p^th
part, and will still be this last part the foremost ; arid so
on continually, according to an infinite series of 1 00th
parts : from which he concluded that the swifter could
never overtake the slower in any finite time, but that they
must go on approaching to infinity. But this sophism
lay in Zeno's considering as an infinite- time, the sum of
the infinite series of small times in which Achilles could

run over the infinite series of spaces, I -1- _i 1- —^--^ ■+-

-nrssooo &c> not knowing that the sum of this infinite
series is equal to the finite small quantity 1^ of a mile,
and that therefore Achilles will overtake the tortoise when
the latter has crawled over J^th part of a mile.

ACHROMATIC, in Optics, without colour; a term
which, it seems, was first used by Dr. Bevis, to denote
telescopes of a new invention, contrived to remedy aber-
rations and colours. See Aberration and Telescope^

Achromatic Telescope, a singular species of refract-
ing telescope, said to be invented by the late Mr. John
Dollond, optician to the king, and since improved by his
son Mr. Peter Dollond, and others.

Every ray of light passing obliquely from a rarer into
a denser medium, changes its direction towards the per-
pendicular; and every ray passing obliquely from a den-
ser into a rarer medium, changes its direction from the
perpendicular. This bending of the ray, caused by the
change of its direction, is called its refraction ; and the
quality of light which subjects it to this refraction, is
called its refrangibility. Every ray of light, before it is
refracted, is white, though it consists ofanumber of com-
ponent rays, each of which is of a different colour. As
soon as it is refracted, it is separated into its constituent
rays, before mentioned, which, from that time, proceed
diverging from each other, like emanations from a centre:
and this divergency is caused by the different refrangibility
of the component rays, in such sort, that the more the
original or component ray is refracted, the more will the
compounding rays diverge when the light is refracted by
one given medium only. Hence it has been concluded,
that any two different mediums, that can be made to pro-
duce equal refractions, will necessarily produce equal di-
vergencies : whence it should also follow, that equal and
contrary refractions should not only destroy each other,
but that the divergency of the colours caused by one re-
fraction, should be corrected by the other; and that, to
produce refraction which would not be affected by the
different refrangibility of light, is impossible.

But Mr. Dollond proved, by many experiments, that
these conclusions are not well founded : from his ex-
periments it appeared, that a ray of light, after equal and
contrary refractions, was still spread into eomponent rays
differently coloured : in other words, that two different
mediums may cause equal refraction, but different diver-
gency ; and equal divergency, with different refraction.
It follows therefore, that refraction mav be produced,
which is not affected by the different refrangibility of
light; in other words, that, if the mediums be different,
different refractions may be produced, though at the same
time the divergency caused by one refraction., shall be

A C H

C

J

A C H

exactly counteracted by the divergency caused by the
other; and so an object may be seen through medium -
which, together, cause the rays to converge, without ap-
pearing of different colours.

This is the foundation of Mr. Dollond's improvement
of refracting telescopes. By subsequent experiments he
found, that different kinds of glass differed greatly in their
refractive qualities, with respect to the divergency of co-
lours. At the same time he found that crown glass causes
the least divergency, and white flint the most, when they
are wrought into forms that produce equal refractions.
He therefore ground a piece of white flint glass into a
wedge, the angle of which was al#ut 25 degrees; and a
piece of crown glass to another, whose angle was about
29 degrees ; and these he found Tefracted nearly alike,
but that their divergency of -colours was very different.

After this, he ground several other pieces of crown
glass to wedges of different angles, till he got one that
was equal, in the divergency it produced, to that of a
wedge of flint glass of 25 degrees; so that when they were
put together, in such a manner as to refract in contrary
directions, the refracted light was perfectly free from co-
lour; then measuring the refractions of each wedge, he
found that that of the white-flint glass, was to that of the
crown glass, nearly as 2 to 3. And hence any two wedges,
made of these two substances, and in this proportion,
would, when applied together so as "to refract in contrary
directions, refract the light without any effect arising from
the different refrangibility of the component rays.

Therefore, to make two spherical glasses that refract
the light in contrary directions, one must be concave, and
the other convex ; and as the rays, after passing through
both, must meet in a focus, the excess of the refraction
must be in the convex one : and as the convex is to re-
fract most, it appears from the experiment that it must
be made of crown glass ; and as the concave is to refract
least, it must be made of white flint.

And farther, as the refractions of spherical glasses are
in an inverse ratio of their focal distances, it follows, that
the focal distances of the two glasses should be in the ra-
tio of the refractions of the wedges: for, being thus pro-
portioned, every ray of light that passes through this
combined glass, at whatever distance it may be from its
axis, will constantly be refracted by the difference be-
tween two contrary refractions, in the proportion required ;
and therefore the effect of the different refrangibility of
light will be prevented.

The removal of this impediment, however, produced
another: for the two glasses, which were thus combined,
being segments of very deep spheres, the aberrations from
the spherical surfaces became so considerable, as greatly
to disturb the distinctness of the image. Yet, consider-
ing that the surfaces of spherical glasses admit of great
variations, though the focal distance be limited; and that
by these variations their aberration might be made more
or less at pleasure ; Mr. Dollond plainly saw that it was
possible to make the aberrating of any two glasses equal;
and that, as in this case the refractions of the two glasses
were contrary to each other, and their aberrations being
equal, these would destroy each other.

Thus he obtained a perfect theory of making object-
glasses, to the apertures of which he could hardly per-
ceive any limits: for if the practice could answer to the.
theory, they must admit of apertures of great extent, and
consequently bear great magnifying powers.

The difficulties of the practice are, however, still 1 :. y
considerable. For first, the focal distances, as well as the
particular surfaces, must be proportioned with the utmost
accuracy to the densities and refracting powers of the
glasses, which vary even in the same kind of glass, when
made at different times. Secondly, there are four sur-
faces to be wrought perfectly spherical. However, Mr.
Dollond could construct refracting telescopes on these
principles, with such apertures and magnitying powers,
under limited lengths, as greatly exceed any that were
produced before, in forming the images of objects bright,
distinct, and uninfected with colours about the edges,
through the whole extent of a very large field or compass
of view; of which he has given abundant and undeni-
able testimony. See Telescope.

The defects which were, yet unremedied in Mr. Dol-
lond's telescopes, have called forth the exertions of other
philosophers, and many contrivances have been invented.
Father Boscovich, to whom every branch of optics is
much indebted, has displayed much ingenuity in his at-
tempts for this purpose. But the philosopher whose ex-
ertions have been attended with most success, and who
appears to have made the most important discovery, in
this branch of science, since the time of Newton, is Dr.
Robert Blair, regius professor of astronomy in Edinburgh.
By a judicious set of experiments, ably conducted, he
has proved, that the quality of dispersing the rays in a
greater degree than crown glass, is not confined to a few
mediums, but is possessed by many fluids, and by some
of them in a most extraordinary degree. He has shown,
that though the greater refrangibility of the violet rays
than of the red, when light passes from any medium into
a vacuum, may be considered as a law of nature; yet,
in the passage of light from one medium into another, it
depends entirely on the qualities of the mediums as to
which of these rays shall be the most refrangible, or whe-
ther there shall be any difference in their refrangibility.
In order to correct the observation arising from difference
of refrangibility among the rays of light, he instituted a
set of experiments, by which he detected a very singular
and important quality in the muriatic acid. In all the
dispersive mediums hitherto examined, the green rays,
which are the mean refrangible in crown glass, were found
among the less refrangible ; but in the muriatic acid, he
found that these same rays made a part of the more re-
frangible.

This discovery led to complete success in removing the
great defect in optical instruments, viz, that dissipation or
aberration of the rays which arises from their unequal
refrangibility, and has hitherto rendered it impossible to
converge all of them to one point, either by single or op-
posite refractions. A fluid, in which the particles of ma-
rine and metalline particles hold a due proportion, at the
same time that it separates the extreme rays of the spec-
trum much more than crown glass, refracts all the. orders
of the rays in the same proportion that glass does: and
hence rays of all colours, made to diverge by the refrac-
tion of the glass, may either be rendered parallel, by a
subsequent refraction made in the confine of the glass
and this fluid ; or, by weakening the refractive density of
the fluid, the refraction which takes place in the Confine
of it and glass, may be rendered as regular as reflection,
without any the least colour whatever. The doctor, it
seems, has a telescope, not above 15 inches in length, with
a compound object-glass of this kind, which equals in all

A C I

[ 25 ]

A C O

respects, if it does not surpass, the best of Dollond's 42
inches long. Of this object-glass, a figure may be seen
in the 3d volume of the Transactions of the Royal So-
ciety of Edinburgh ; to which volume the reader must be
referred for a full and clear account of the experiments
which led to this discovery, as well as of the important
purposes to which it may be applied. A judicious abridg-
ment of this paper may also be seen in Nicholson's Jour-
nal, vol. i, 4lo. p. 1.

In the Gentleman's Magazine (1790, p. 890) a paper
was published on the refracting telescope, by a person
who signs Veritas, in which the invention is ascribed to
another person, not heretofore mentioned, in these
words: " As the invention has been claimed by M. Eu-
ler, M. Klingenstierna, and some other foreigners, w£
ought, for the honour of England, to assert our right, and
give the merit of the discovery to whom it is due; and
therefore, without farther preface, I shall observe, that the
inventor was Chester More Hall, Esq. of More-hall, in
Essex, who, about 1729, as appears by his papers, consi-
dering the different humours of the eye, imagined they
were placed so as to correct the different refrangibility of
light. He then conceived, that if he could find substances
having such properties as he supposed these humours
might possess, he should be enabled to construct an ob-
ject-glass that would show objects colourless. After many
experiments, he had the good fortune to find those pro-
perties in two different kinds of glass; and making them
disperse the rays ©f light in different directions, he suc-
ceeded. About 1733 he completed several achromatic
object-glasses, though he did not give them this name,
that bore an aperture of more than 2-J inches, though the
focal length did not exceed 20 inches ; one of which is
now in the possession of the Rev. Mr. Smith, of Charlotte
Street, Rathbone Place. This glass has been examined
by several gentlemen of eminence and scientific abilities,
and found to possess the properties of the present achro-
matic glasses.

" Mr. Hall used to employ the working opticians to grind
his lenses; at the same time he furnished them with the
radii of the surfaces, not only to correct the different re-
frangibility of rays, but also the aberration arising from
the spherical figure of the lenses. Old Mr. Bass, who at
that time lived in Bridewell precinct, was one of these
working opticians, from whom Mr. Hall's invention seems
to have been obtained.

" In the trial at Westminster-hall about the patent for
making achromatic telescopes, Mr. Hall was allowed to
be the inventor; but Lord Mansfield observed, that 'It
was not the person that locked up his invention in his
scrutoire that ought to profit by a patent for such an in-
vention, but he who brought it forth for the benefit of the
public' This, perhaps,- might be said with some degree
of justice, as Mr. Hall was a gentleman of property, and
did not look to any pecuniary advantage from his disco-
very; and consequently it is very probable that he might
not have an intention to make it generally known at that
lime.

" That Mr. Ayscough, optician on Ludgate Hill, was in
possession of one of Mr. Hall's achromatic telescopes in
1754, is a fact which at this time will not be disputed"

ACHRONICAL, or Achronycal. See Acron ychal.

ACIDS. All those substances which impress the or-
gans of taste with a sharp, 6dur; and cooling sensation,

Vol. I.

are commonly denominated acids. But as there aresom^
bodies destitute of this property, which are nevertheless
classed by all chemical writers as acids, this popular cha-
racteristic must be abandoned as essential, for one that is
more comprehensive.

The characteristic properties of acids may be enume-
rated as follows :

1. When taken into the mouth they occasion a sour
taste. -

2. They change certain vegetable blue colours to red ;
such as tincture of litmus, or syrup of violets ; and re-
store the blue colours which have turned green, or the
red colours which have been turned blue, by an alkali.

3. They have a stronger affinity for alkalies, than they
have for any other substance.

4. They unite with earths, with alkalies, and metallic
oxyds, forming the numerous and very important classes
of neutral, earthy, and metallic salts.

As this article is entirely chemical, it does not accord
with our plan to extend it to any considerable length ;
we shall therefore content ourselves with giving the most
approved arrangement of acids, referring the reader, for
farther information on this important subject, to Thom-
son's Chemistry. See also Aikin's and Nicholson's Che-
mical Dictionaries, Encycloped. Method., Lavoisier's Ele-
ments of Chemistry, Priestley on Air, Fourcroy Systeme
de Connois. Chimiq. &c. &c.

MODERN ARRANGEMENT OF ACIDS.
States of Oxygenation.
11/ 2nd 3d

Sulphureous Sulphuric

Nitrous Nitric ......

""? Phosphorus Phosphoreous Phosphoric

?™.?1 i Carbon Carbonic

Arsenious

With
simple
radicals

Bases
Sulphur
Amt

Arsenic
Molyldena
^Chrome

("Acetous

With
double
radicals

r Cation and f
; Hydrogen J
] in different |
L proportions I

Arsenic
Molybdic
Chromic
Acetic
Tartaric
Citric
Oxalic
Malic
Gallic
Bonzoic
Succinic
Saccholactic
Formic
Sebacic
f Prussic

'(.Lithic
f Muriatic

JFIoric
(^ Boracic

With f Carbon, Hy- C
triple ra-< drogen, and< ...

dieals L Awl. {_

With f ^Muriatic Oxymuriatic.

unknown <

radicals ^

ACOUSTICS, the doctrine or theory of hearing, and
of sound.

Sturmius, in his Elements of Universal Mechanics,
treating of Acoustics, after examining into the nature of
sounds, describes the several parts of the external and in-
ternal ear, and their several uses and connexions with,
each other ; and from thence deduces the mechanism of
hearing : and lastly, he treats of the means of adding an
intensity of force to the voice and other sounds; and ex-
plains the nature of echoes, otacoustic tubes, and speak-
ing trumpets. See Sound, Ear, Voice, and Echo.

Dr. Hook, in the preface to his Micrography, asserts
that the lowest whisper,' by certain means, may be heard
at the distance of a furlong ; and that he knew a way by
which it is easy to hear dny one speak through a wall of
three feet thick ; also that by mean's of an extended wire,

A C O

[ 2S ]

A C R

sound may be conveyed to a very great distance, almost
in an instant.

The ancients seem to have considered sounds under no
other point of view than that of music ; that is to say,
as affecting the ear in an agreeable manner. It is even
very doubtful whether they were acquainted with any
thing more than melody, and whether they had any art
similar to what we call composition. The modems, by
studying the philosophy of sounds, so much neglected by
the ancients, have given birth to a new science distin-
guished by the name of acoustics, which has for its ob-
ject the nature of sounds, and treats also of the theory
of hearing, and the best means of assisting that sense.
This science is divided by some writers into diacoustics,
which explain the properties of those sounds that come
directly from the sonorous body to the ear ; and cata-
coustics, which treats of reflected sounds ; but such dis-
tinction does not appear to be of any real utility.

On many particulars connected with this science opi-
nions are still much divided; and a sufficient number of
experiments have not been made to determine what is es-
sential to the nature of a vehicle of sound ; how it is
transmitted, reflected, or destroyed, in many instances; or
whether its velocity be uniform or variable..

Thus, with respect to these vehicles ; as a bell or other
sonorous body when struck in vacuo transmits no sound,
it therefore necessarily follows that air is such a vehicle,
or that some subtle fluid is contained in its composition
which performs the office of transmitting sound : this last
seems the more probable, because we know, that air is not
the only vehicle, nor the best, for this transmission ; and
on this supposition it follows, that bodies of any density
or texture will transmit sound more or less perfect, ac-
cording to the quantity of this fluid contained in them.

The aqueducts constructing at Paris have lately given
M. Biot an opportunity of making some interesting ex-
periments on the propagation of sound through solid bo-
dies, being on a larger scale than any hitherto attempted.
The total length of the pipes was 3118 feet. A blow with
a hammer at one extremity was heard at the other, pro-
ducing two distinct sounds, the interval of which, mea-
sured in more than 200 trials, was 2-5". And according
to the experiments of the academy, the time of propaga-
tion of sound to this distance through the air should be
2'79", from which deducting 2,5" the interval observed,
we have '29" for the time that sound was in being propa-
gated through the solid substance to the distance above
stated. These experiments were afterwards varied; the
result however generally agreed with that above stated.

Some have considered the electrical fluid as the agent
by which this office is performed ; it has however been
objected to by others, and many ingenious arguments
have been advanced on both sides, but without coming to
a satisfactory conclusion. One thing however is certain,
that whatever the fluid may be, the sound we hear is pro-
duced by the stroke that the sounding body makes against
it, and is thence conveyed to the organ of hearing. But
the manner in which this conveyance is made is still dis-
puted : whether the sound be diffused into air in circle
beyond circle, like the waves of water when we disturb
the smoothness of its surface by dropping in a stone, or
whether it travels alorg, like rays diffused from a centre,
somewhat in the swift manner that electricity runs along
a rod of iron, are questions which have greatly divided
the opinions of the learned.

Newton was of the former opinion ; but John Bernou-
illi confessed, that he did not understand this part of the
Principia — and attempted to give a more perspicuous de-
monsuation of his own, that might confirm and illustrate
the Newtonian theory ; but the subject seemed to reject
elucidation ; for his theory is obviously wrong, as D'A-
lembert has proved in his treatise, on fluids.

The Newtonian theory has however found an able ad-
vocate in the ingenious Dr. Young, of Trinity College,
Dublin, who has given a clear, explanatory, and able de-
fence of the 47th proposition of the 2d book of the Prin-
cipia, which has been considered the most objectionable ;
and some late -experiments have contributed in a great
degree to remove the difficulties with which this subject
is clouded, and which will be explained in their proper
places. iSee the articles Chord, Echo, Elastic
strin &, Ear, Sound, Transmission, Velocity, Vi-
bration, &c.

ACRE, is a measure of land, containing, by the ordi-
nance for measuring land, made in the 33d and 34th of
Edward I, l6'0 perches or square poles of land ; and as
the statute length of a pole is 54 yards, or lu'-J- feet,
therefore the acre will contain 4840 square yards, or
43560 square feet. The chain with which land is com-
monly measured, and which was invented by Gunter, is
4 poles or 22 yards in length ; and therefore the acre is
just 10 square chains. Further, as a mile contains 1760
yards, or 80 chains in length, therefore the square mile
contains 640 acres.

The acre, in surveying, is divided into 4 roods, and the
the rood is 40 perches.

The French acre, arpent, is equal to li English acre ;
The Strasburg contains about \ an English acre ;
The Welch acre contains about 2 English acres ;
The Irish acre contains 1 ac. 2 r. 19-^ p. English.
The Scotch acre is near \\ English, being equal to
6150 square yards.

Sir William Petty, in his Political Arithmetic, reckons
that England contains 39 million acres: but Dr. Grevc
shows, in the Philos. Trans. No. 330, that England con-
tains not less than 46" million acres. Whence he infers that
England is above 46 times as large as the province of Hol-
land, which it is said contains but about one million of acres.
By a statute of the 31st of Elizabeth, it is ordained,
that if any man erect a cottage, he shall annex four acres
of land to it. But this statute was repealed by the Kith
of Geo. in. cap. 32.

ACROATICS, a name given to Aristotle's lectures in
the more difficult and nice parts of philosophy ; to which
none but his disciples and intimate friends were admitted;
whereas the exoteric were public or open to all.

ACRONYCHAL, or Acronycal, in Astronomy, is
said of a star or planet, when it is opposite to tin' sun.
It is from the Greek axfow^of, the point or extremity of
night, because the star rose at sun-set, or the beginning
of=night, and set at sun-rise, or the end of night ; and so
it shone all the night.

The acronychal is one of the three Greek poetic ri-
sings and settings of the stars; and stands distinguished
from Cosmicafand Heliacal. And by means of which,
for want of accurate instruments, and other observations,
they might regulate the length of their year.

ACUOTERIA, or Acroters, in Architecture, small
pedestals, usually without bases, placed on pediments,
and serving to support statues. Those at the extremities

ACT

c

]

ACT

ought to be half the height of the tympanum; and that
in the middle, according to Vitruvius, one eighth part move.

Acroteria also are sometimes used to signify figures,
whether of stone or metal, placed as ornaments or crown-
ings, on the tops of temples, or other buildings.

It is also sometimes used to denote those sharp pina-
cles or spiry battlements, that stand in ranges about flat
buildings, with rails and balustres.

ACTION, in Mechanics or Physics, a term used to de-
note, sometimes the effort which a body or power exerts
against another body or power, and sometimes it denotes
the effects resulting from such effort.

Action is either instantaneous or continued ; that is,
either by collision or percussion, or by pressure. These
two sorts of action are heterogeneous quantities, and are
not comparable, the smallest action by percussion ex-
ceeding the greatest action of pressure, as the smallest
surface exceeds the longest line, or as the smallest solid
exceeds the largest surface : thus, a man by a small blow
with a hammer, will drive a wedge below the greatest ship
on the stocks, or under any other weight; that is, the
smallest percussion overcomes the pressure of the greatest
weight. These actions then cannot be measured the one
by the other, but each must have a measure of its own
kind, like as solids must be measured by solids, and sur-
faces by surfaces : time being concerned in the one, but
not in the other.

It is one of the laws of nature, that action and reaction
are always equal, and contrary to each other in their di-
rections. Therefore, if a body be urged at the same time
by equal and contrary forces, it will remain at rest. But
if one of these forces be greater than its opposite one, mo-
tion will ensue towards the part which is least urged.

The actions of bodies on each other, in a space that
is carried uniformly forward, are the same as if the space
were at rest; and any powers or forces that act upon
bodies, so as to produce equal velocities in them in the
same, or in parallel right lines, have no effect on their
mutual actions, or relative motions. Thus the motion of
bodies on board of a ship that is carried uniformly for-
ward, are performed in the same manner as if the ship
were at rest. And the motion of the earth about its axis
has no effect on the actions of bodies and agents at its
surface, except so far as it is not uniform and rectilineal.
In general, the actions of bodies upon each other, depend
not on their absolute, but relative motion.

Quantity of Action, in Mechanics, a name given by
Maupertuis, in the Memoirs of the Academy of Sci-
ences of Paris for 1744, and in those of Berlin for 1746,
to the continual product of the mass of a body, by the
space which it runs through, and by its celerity. He lays
it down as a general law, that in the changes made in the
stalo t a body, the quantity of action necessary to pro-
duce such change is the least possible. This principle
he applies to the investigation of the laws of refraction,
and even to what he calls the laws of rest, that
is, of the equilibrium or equipollency of pressures ;
and even to the modes of acting of the Supreme Being.
In this way Maupertuis attempts to connect the meta-
physics of final causes with the fundamental truths of
nuchanics ; to show the dependence of the collision of
both elastic and hard bodies, upon one and the same law,
which before had always been referred to separate laws ;
and to reduce the laws of motion, and those of equili-
brium, to one and the same principle.

But this quantity of motion, or of action, of Mauper-
tuis, which is defined to be the product of the mass, the
space passed over, and the celerity, comes to the same
thing as the mass multiplied by the square of the velo-
city, when the space passed over is equal to that by which
the velocity is measured ; and so the quantity of force
will be proportional to the mass multiplied by the square
of the velocity; since, the space is measured by the velo-
city continued for a certain time.

In the same year that Maupertuis communicated the
idea of his principle, Euler, in the supplement to his
book, intitled Methodus inveniendi lineas curvas maximi vel
minimi proprietate gaudentes, demonstrates, that in the tra-
jectories which bodies describe by central forces, the ve-
locity multiplied by what the foreign mathematicians call
the element of the curve, always makes a minimum:
which Maupertuis considered as an application of his
principle to the motion of the planets.

It appears from Maupertuis's Memoir of 1744, that it
was his reflections on the laws of refractions, that led him
to the theorem above mentioned. The principle which
Fermat, and after him Leibnitz, made use of, in account-
ing for the laws of refraction, is sufficiently known. Those
mathematicians asserted, that a particle of light, in its
passage from one point to another, through two mediums,
in each of which it moves with a different velocity, must
do it in the shortest time possible; and from this principle
they have demonstrated geometrically, that the particle
cannot go from the one point to the other in a right line;
but being arrived at the surface that separates the two
mediums, it must alter its direction in such a manner, that
the sine of its incidence shall be to the sine of its refraction,
as its velocity in the first medium is to its velocity in the
second : whence they deduced the well known law of the
constant ratio of those sines.

This explanation, though very ingenious, is liable to
this pressing difficulty, namely, that the particle must
approach towards the perpendicular, in that medium
where its velocity is the least, and which consequently
resists it the most: which seems contrary to all the me-
chanical explanations of the refraction of bodies, that
have hitherto been advanced, and of the refraction of
light in particular.

Sir Isaac Newton's manner of accounting for it, is the
most satisfactory of any that has hitherto been offered,
and gives a clear reason for the constant ratio of the
sines, by ascribing the refraction to the attractive force
of the mediums; from which it follows, that the densest
mediums, whose attraction is the strongest, should cause
the ray to approach the perpendicular; a fact confirmed
by experiment. But the attraction of the medium could
not cause the ray to approach towards the perpendicular,
without increasing its velocity ; as may easily be demon-
strated. Thus then, according to Newton, the refraction
must be towards the perpendicular, when the velocity is
increased : contrary to the law of Fermat and Leibnitz.

Maupertuis has attempted to reconcile Newton's expla-
nation with metaphysical principles. Instead of suppo-
sing, with Fermat and Leibnitz, that a particle of light
proceeds from one point to another in the shortest time
possible; he contends that a particle of light passes from
one point to another in such a manner, that the quantity
of action shall be the least possible. This quantity of
action, says he, is a real expense, in which nature is al-
ways frugal. In virtue of this philosophical principle he
E2

A C U

[

1

ADD

discovers, that not only the sines are in a constant ratio,
but also that they are in the inverse ratio of the velocities,
according to Newton's explanation, and not In the direct
ratio, as had been asserted by Fermat and Leibnitz.

Now according to this principle, the time, that is, the
space divided by the velocity, should be a minimum ; so
that calling the space run through in the first medium s,
with the velocity v, and the space run through in the se-
cond medium s, with the velocity v, we shall have

- ■+■ - = a minimum : hence — t- s = 0, or - = — -.

V V ' V V ' V V

Now it is easy to perceive, that the sines of incidence and
refraction are to each other, as s to — s; whence it fol-
lows, that those sines are in the direct ratio of the veloci-
ties v, v ; which is exactly what Fermat makes it. But
in order to have those sines in the inverse ratio of the velo-
cities, it is only supposing vs -h vs = 0; which gives
sv + n' or s x v -+- s x v a minimum: which is Mau-
pertuis's principle.

In the Memoirs of the Academy of Berlin, above cited,
may be seen all the other applications which Maupertuis
has made of this principle. And however unfounded his
metaphysical basis of it may be, as also the idea he has an-
nexed to the quantity of action, it will still hold good,
that the product of the space by the velocity is a mini-
mum in some of the most general laws of nature.

ACTIVE, the quality of an agent for communicating
motion or action to some body. In this sense the word
stands opposed to passive: thus we say an active cause,
active principle, &c. — Sir Isaac Newton shows, that the
quantity of motion in the world must be always decreas-
ing, in consequence of the vis inertia?, &c. So that there
is a necessity .for certain active principles to recruit it :
such he takes the cause of gravity to be, and the cause
of fermentation ; adding, that we see but little motion
in the universe, except what is owing to these active prin-
ciples.

ACTIVITY, the virtue or faculty of acting. As the
activity of an acid, a poison, &c : the activity of fire ex-
ceeds all imagination. — According to Sir Isaac Newton,
bodies derive their activity from the principle of attrac-
tion.

Sphere of Activity, is the space which surrounds a
body, as far as its efficacy or virtue extends to produce
any sensible effect. Thus we say, the sphere of activity
of a loadstone, of an electric body, &c.

ACUBENE, in Astronomy, the Arabic name of a star
of the fourth magnitude, in the southern claw of Cancer,
marked a bv Bayer. Its longitude for 176 1, SI 10° IS' 9",
south latitude 5° 5' 56".

ACUTE, or sharp; a term opposed to obtuse. Thus,
Acute, Angle, in Geometry, is that which is less than a
right angle ; and is measured by less than 90", or a quan-
drant of a circle. As the angle abc.

c ; E F H" -

Acute angle Triangle, is that whose three angles are
all acute; and is otherwise called an oxygenous triangle.
As the triangle def.

Acu iz-iungled Cone, is that whose opposite sides make

an acute angle at the vertex, or whose axis, in a right
cone, makes less than half a right angle with the side.
As the cone gh I.

Pappus, in his Mathematical Collections, says, this
name was given to such a cone by Euclid and the ancients,
before the time of Apollonius. And they called an

AcvTE-angled Section of a Cone, an Ellipsis, which was
made by a plane cutting both sides of an acute-angled
cone ; it being not known that such a section could be
generated from any cone whatever, till it was shown by
Apollonius.

Acute, in Music, is understood of a tone, or sound,
which is high, sharp, or shrill, with regard to some other:
in which sense the word stands opposed to grave. And
both these sounds are independent of loudness or force : sa
that the tone may be acute or high, without being loud ;
and loud without being high or acute. For both the af-
fections of acute and grave, depend entirely on the quick-
ness or slowness of the vibrations by which they are
produced. — Sounds considered as grave and acute, that
is, in the relation of gravity .and acuteness, constitute
what is called tune, the foundation of all harmony.

ADAG-IO, in Music, one of the terms used by the Ita-
lians to express a degree or distinction of tinie.
Adagio denotes the slowest time except grave.
Sometimes the word is, repeated, as udugio, adagio, to
denote a still slower time than the former.

Adagio'also signifies a slow movement, when used sub-
stantively., . ,

ADAR, in the Hebrew Chronology, is the 6th month of
their civil year, but the 12th of their ecclesiastical year.
It contains only 29 days ; and it answers to our February ;
but sometimes entering into the month of March, accord-
ing to the course of the moon.

ADDITION, the uniting or joining of two or more
quantities together.

Addition, in Arithmetic, is the first of the four funda-
mental rules or operations of that science; and. consists
in finding a number equal to several others taken together,
or in finding the most simple expression of a number ac-
cording to the. established- made of notation; and the
quantity so found is named their. sum..

The sign or character of addition is -t-, and is called
plus. Thjs character is set between the. quantities to be
added, to denote their sum: thus, 3 -+- 6 = 9> that is, 3
plus 6 are equal to 9-

Simple numbers are either added as above; or 6
else by placing them under one another, as in the 4
margin, and adding them together, one after an- 2

other, beginning at the bottom : thus 2 and 4 make ~[a
6, and 6 make 12. ,,

Numbers consisting of several figures arc added, by
first ranging the numbers in columns under each other,
placing always the numbers of the snme denomination
under each other, that is, units under units, tens under
tens, and so on; and then adding up each column sepa-
rately, beginning at the right hand, selling down the sum
of each column below it, unless it amount to ten or some
number of tens, and in that case setting down only the
overplus, and carrying one for each ten to the next co-
lumn. Thus, to acid 451 and 326,

sstf} tl,at is {

Sum 777 =

400 4- 50 -+- 1
300 -+- 20 h- 6

700 -h 70 ■+- 7

ADD

[ sy ]

ADD

329

1562

20347

712048

734286

26
16
11
3

734286

Also to add the numbers 329 ■+■ 1562 •+-
20347 ■+■ 71204S ; set them down as in the
margin, and beginning at the lowest number
on the right hand, say 8 and 7 make 15,
and 2 make 17, and 9 make 26" ; set down
6, and carry- 2 to the next column, saying 2
and 4 make 6, and 4 make 10, and 6 make
16, and 2 make 18 ; set down 8, and carry
1, saying 1 and 3 make 4, and 5 make 9,
and 3 make 12; set down 2, and carry 1,
saying 1 and 2 make 3, and 1 make 4,
which set down ; then 1 and 2 make 3 ; and
7 is 7 to set down : so the sum of all toge-
ther is 734286. Or it is the same as the
sums of the columns set, under one. another, as in the
margin, and then, these added up in the. same manner.

When a great number, of sep,aratecsums pr numbers are
to be added,, as in, long accounts,' it is easier to break or
separate them into two or. more parcels, which are added
up severally, and then their sums added together for the
total sum.' And thus also the truth of the addition may
be proved, by dividing the numbers into parcels different
ways, as the totals must be the same in both cases when
the operation is right.

Another method of proving addition was given by Dr.
Wallis, in his Arithmetic, as well. as by Lucas de Burgo,
and probably by former authors, also,. by. casting out the
nines, which method of proof extends also to the. o.ther.rules
of arithmetic. The method is ..this. :. add .the figures of each
line of numbers together severally, casting, out always
9 from the sums..as they, arise. in. so, adding, carrying the
overplus to the next figure, and setting at the end of each

£ 5
W 3

line what is overthenineornin.es;. then 339

do the same by thesum-total, as alsoby 1562

the former excesses of 9> so shall the 20347

last excesses be equal when the work is 712048

right. Thus, the former example |-'349Sh'

will be proved as annexed :

When the numbers are of. different denominations; as
pounds, shillings, and pence; or yards,- feet, and inches;
place the numbers of the same kind under, each other, as
pence under pence, shillings under shillings, &c ; then
add each column separately, and carry the overplus as
before, from one column to another. As in the following
examples :

1. s. d. Yards. Feet. Inches.

271 12 3 271 10 3

94 14 7 36 2 7

42 5 10 14 2 5

408 12 8 sums 323" 0~ 3
Addition of Vulvar Fractions, is performed by bring-
ing all the proposed fractions to the same denomination,
when necessary, and all compound fractions and mixed
numbers to simple fractions ; then reduce all the fractions
to a common denominator, and the sum of the numerators
placed over the common denominator will be the answer

MJUght.

S*T * + *= *• And f + f + T = T = If
And £ + \ = % -h £ = £ = l*f,

And4 + fof£ = 4 + -! = 4 = * = lJ.
When the fractions are of different denominations, some
farther operations are necessary : thus,
Add. }o +|l + |s together.
Here | o = f x y =? \l s .

I7S _ 3 73 9 ,. —

Trrs — itc s —

371-0496
25-213
1-704
924-61
•0962

*.■■ =

and Y ■+■ 6T° -+-

17«. 9d- || Ans.

Addition of Decimals, is performed in the same man
ner as that of- whole numbers, placing
the numbers of the same denomination
under each other, in which case the de-
cimal separating points will range straight
in one column.; as in this example, to
add together these numbers 371-0496
•+■ 25-213 ■+- 1-704 -+- 924-61 -t- "0962. 1322-6728

Addition of Circulating. Decimals is performed by
changing each of them into its equivalent vulgar fraction,
and finding the sum of such fractions. — For the method of
changing, recurring or circulating decimals to vulgar frac-
tions, see Circulating Decimals.

ADDITION in Algebra, is the uniting or incorporating
of indeterminate quantities denoted, by letters into one
contracted expression. The operation is performed by
connecting the quantities together.with' their proper signs,
and collecting all similar quantities into one sum ; namely,
by adding their co-efficient's together if their signs are the
same, but subtracting them when different. . Thjus, the
quantity a added to the quantity^, makes a.r-h b; and a
added to — b, gives a — b; also the com pound., quantity
(5a — b) added to (4a + 3b) produces the sum (9a + 26).
See the following examples :

5xy

4ax

— xy

— 4a*

$xy

2ax —
3x* +

5xy ■+■

-~5x2 •+■

Gx-y1
-4x2.y
— 2axy
~3x*y

6x'2y* — Jx'y — 2axy
150 -j- 2** 4y2

4/

2ax
50 -

100 -+-

6x\

33*

lab
4a»62
— 3a*

-20^

iab ■+■ 2a262

— 7xy-h x2
h- 4?xy — is

■+-- 3xy — ax2

— 2ay* ■+■ ax2

4>ax- -+- ixl ■+■ 5xy 7y3 ' -1- 7y* — 2ay2

Addition of Algebraic Fractions, is performed by the
same rule, as has been given for addition of vulgar frac-
tions ; as will be seen by the following examples :
Ex J x~* ' x+3 — ' ?a:~~28 ax+9 _ 10^—19

' ' 3 ''7 • ~" ' ai "*" 'Ji ~ 21 ns'

x.2.y + ^— + 7y - -^— = 8y + ^ Ans.

Addition of Surds. In this rule all the given quan-
tities must be reduced to the same denomination, or to
the same radical, if that can be done ; then addin« or
uniting the rational parts, and subjoining the common
surd. Otherwise connecting them with their own signs.

So y'S + v'lS = ^/(4 x 2) + ^{9 x 2) = 2^2
+ 3^2 = 5v/2 = v/50;

and v/ 12 -t- ^/27 = a/ (4 x 3) + v/(9 x 3) = 2V'3',
-t- 3^/3 = 5^/3 = v/75;

and v^50 + J75 = ^/(25 x 2) •+■ \/(25- x 3) = 5^2
+ 5^/3 = (v/2 + ^3)5;

but of ^5 and ^/6 the sum is set down^/5 -fr y/6, be-
cause the terms are incommensurable, .and not reducible
to a common surd.

Addition of Logarithms. See Logarithms.

Addition of Ratios, the same as composition of ratios;
which see.

ADDITIVE, denotes something to be added to another,
in contradistinction to something to be taken away or

A D H

[ 30

subtracted'. So astronomers speak of additive equations,
and geometricians of additive ratios.

ADELARD, or Athelard, was a learned monk of
Bath, in England, who flourished about the year 1130, as
appears by some manuscripts of his in Corpus Christi, and
Trinity Colleges, Oxford. Vossius says he was universally
learned in all the sciences of his time ; and that, to acquire
all sorts of knowledge, he travelled into France, Germany,
Italy, Spain, Egypt, and Arabia. He wrote many books
himself, and translated others from different languages :
thus he translated, from Arabic into Latin, Euclid's
Elements, at a time before any Greek copies had been
discovered; also Erichiafarim, on the seven planets. He
wrote a book on the seven liberal arts, another on the
astrolabe, another on the causes of natural compositions,
besides several on physics and on medicine.

Though Vossius refers to Oxford for some of these ma-
nuscripts, it would yet seem they were not to be found
there in Wallis's time ; for the Doctor, speaking of this
author, and other English authors and travellers about the
same age, says, " A particular account of these travels of
Shelley and Morley was a while since to be seen in two
prefaces or two manuscript books of theirs in the library of
Corpus-Christi College in Oxford, but hath lately (by
some unknown hand) been cut out, and carried away ;
which prefaces (one or both of them) did also make men-
tion of the travels of Athelardus Bathoniensis, and are, to
that purpose, cited by Vossius out of that manuscript
copy. Whoever hath them, would do a kindness (by
some way or other) to restore them, or at least a copy of
them." Wallis's Algebra, pa. 6.

ADELM, Aldhelmus, or Althelmus, a learned
Englishman, who flourished about the year 680. He was
first abbot of Malmsbury, and afterward bishop of Shir-
burn. He died in the year 709, in the monastery of
Malmsbury. — Adelm was the son of Kenred or Kenten,
who was the brother of Ina, king of the West Saxons in
England. Besides certain books on theology, he compo-
sed several on the mathematical sciences, &c ; as Arith-
metic, and Astrology, and librum de philosophorum disci-
plinis. See Bede's History, lib. 5. cap. 19- He is also
mentioned by Bale and William of Malmsbury.

ADERAIMIN, or Aloeraimin, the Arabic name of
a star of the third magnitude, in the left shoulder of Ce-
pheus marked a by Bayer. Its longitude for 176l, T
9" 30' 8", north latitude 68° 56' 20".
ADFECTED, see Apeected.

ADHESION, Adherence, in Physics and Chemis-
try, a species of attraction which takes place between
the surfaces of bodies, that are either similar or dissimi-
lar, and which in a certain degree connects them to-
gether; thus, water adheres to the fingers, mercury to
gold, two pieces of lead or brass to each other, &c. Hence
arises an important distinction between two words, that in
a loose and popular sense are often confounded, that is,
Adhesion and Cohesion. — The former denotes an union to
a certain point between two similar or dissimilar substan-
ces ; and the latter, that which retains together the com-
ponent particles of the same mass.

The proximate cause of adhesion, has been variously
stated by different philosophers. James Bernoulli attribu-
ted it to the pressure of the air; but this was afterwards
found to be incorrect by some experiments made by Mr.
Hauksbee; and it has since been proved, that the power
of adhesion is proportional to the number of touching

] A D H

points, and this depends on the figure of the particles of
which the body is formed ; and in solid bodies on the
degreein which their surfaces are polished and compressed.
Dr. Brook Taylor first observed, in 1713, the ascent of
water between two planes of glass; this led him to make
several experimeuts on the adhesive power of surfaces ;
from which he concluded that the degree of this force
might be measured by the weight required to separate
them. This idea was taken up and pursued with consi-
derable success by M. de Morveau (now M. Guyton).

He procured cylinders, or rather plates, of different me-
tals in their highest state of purity, which were made per-
fectly circular, highly polished, and furnished with a
small ring fixed to their centre, each plate being exactly
an inch in diameter. Those plates were in turn suspended
to the arm of an assay balance, and being exactly coun-
terpoised by weights placed in the scale attached to
the opposite arm, were applied to mercury in a vessel
about i of an inch below them. After sliding them along
the surface to prevent any air from lodging between them
and the mercury, weights were successively added to the
scale attached to the opposite arm, till the adhesion be-
tween the plate and the mercury was broken : and fresh
mercury was used for each experiment. The results of
which are exhibited in the following table.

Gold adheres to mercury with a force of 446 grains.

Silver 429

Tin 41S

Lead 397

Bismuth 372

Zinc 204

Copper 142

Antimony (regulus) 126

Iron 115

Cobalt 8

These experiments clearly proved, as indeed had been
before done by Hauksbee, that the atmosphere was not
the agent or cause of adhesion ; for had this been the case,
it is evident that, the surfaces being the same in each expe-
riment, the force of adhesion in each must have been
equal also. The above method has been since pursued
to greater lengths, and with more accuracy by M. Ach-
ard and others, who took into their account the tempera-
ture of the fluids. And from the result of these inter-
esting experiments we may conclude — That there exists
a tendency to adhesion between many, and probably
between all substances in nature, absolutely indepen-
dent of atmospherical or any other external pressure;
that the force of this adhesion between solids is in the
order of their chemical affinities ; and between solids
and fluids in an inverse ratio to the thermometrical tem-
perature, and in a direct ratio to the squares of the sur-
faces; also that every solid adheres with a peculiar force to
each fluid ; that this force is truly expressed by the
weight necessary to break the adhesion, in all cases where
the solid comes out clear from the fluid ; but that when
any particles of the fluid adhere to the solid, the. weight of
the counterpoise is then expressive of the mixed force of
adhesion between the surface of the solid and the fluid, and
of the cohesion between the component particles of the
latter.

In order to keep a connected view of this subject in its
progress towards perfection, such experiments only have
been related as most materially tended to the accomplish-
ment of that desirable end; and consequently many iu-

IOL

[ 31 ]

IE 0 L

vious and entertaining experiments have not been men-
tioned. As, however, some of these may afford a degree
of satisfaction and amusement to the reader, we have de-
tailed a few of the most interesting ones, as they are re-
lated by Musschenbroek and Martin. The former, in
his Philosophy, as translated by Colson, relates that two
cylinders of glass, whose diameters were not quite two
inches, being heated to the degree of boiling water, and
joined together by means of melted tallow lightly put be-
tween them, adhered with a force of 130 pounds: lead of
the same diameter, and in similar circumstances, adhered
with a force of 275 pounds, and soft iron with a force of
300 ppunds. And Martin, in his Philosophia Britan-
nica, vol. ], says, that with two leaden balls, not weigh-
ing above a pound each, nor touching on more than —^
of a square inch of surface, he has lifted more. than 130
pounds weight : the balls being first very finely planed
with the edge of a sharp pen-knife, and then equally
pressed together with a considerable force, and a gentle
turn of the hand. The force of adhesion between two brass
plates, of about 44 inches diameter, and smeared with
grease or fat, is so powerful, as to require more than the
.strength of two strong men to separate them. Those who
wish to see more on this subject, may consult to advan-
tage the Supplement to Encyclopedia Britannica, art.
Chemistry, and Rees' Cyclopedia, art. Adhesion ; as also
the two works above mentioned.

ADHIL, in Astronomy, a star, of the 6th magnitude,
on the garment of Andromeda, under the last star in her
foot.

ADJACENT, whatever lies immediately by the side of
another.

Adjacent Angle, in Geometry, is said of an angle
when it is immediately contiguous to another, so that
they have both one common side. And the term is more
particularly used when the two angles have not only one
common side, but also when the other two sides form one
continued right line.

Adjacent Bodies, in Physics, are understood of those
that are near, or next to, some other body.

ADJUTAGE, or rather AJUTAGE; which see.

ADSCRIPTS, in Trigonometry, is used by some ma-
thematicians, for the tangents of arcs. Vieta calls them
also prosines.

ADVANCE-Fosse, In Fortification, a ditch thrown
round the esplanade or glacis of a place, to prevent its
being surprised by the besiegers. — The ditch sometimes
made in that part of the lines or retrenchments nearest
the enemy, to prevent him from attacking them, is also
called the advance-fosse. — The advance-fosse should al-
ways be full of water, otherwise it will serve to cover the
enemy from the fire of the place, if he should become
master of it. Beyond the advance-fosse it is usual to
construct lunettes, redouts, &c.

ADVENT, Adventus, in the Calendar, the time imme-
diately preceding Christmas; and was anciently employed
in pious preparation for the adventus, or coming on, of
the feast of the Nativity.

Advent includes four Sundays, or weeks; commencing
either with the Sunday which falls on St. Andrew's day,
namely the 30th day of November, or the nearest Sunday
to that day, either before or after.

/EOLIIMLE, JEolipita, in Hydraulics, a hollow ball of
metal, with a very small hole or opening ; chiefly used to
show the convertibility of water into elastic steam. The

best way of fitting up this instrument, is with a very slen-
der neck or pipe, to screw on and off, for the convenience
of introducing the water into the inside; for, by un-
screwing the pipe, and immerging the ball in water, it
readily fills, the hole being pretty large ; and then the
pipe is screwed on. But if the pipe do not unscrew, its
orifice is too small to force its way in against the included
air; and therefore to expel most of the air, the ball is
heated red-hot, and suddenly plunged with its orifice into
water, which will then rush in till the ball is about two-
thirds filled with it. The water having been introduced,
the ball is set upon the fire, which gradually heats the
contained water, and converts it into elastic steam, this
rushes out by the pipe with great violence and noise ; and
thus continues till all the water is so discharged ; though
not with a constant and uniform blast, but by sudden
ones : and the stronger the fire is, the more elastic will
the steam be, and the greater the force of the blast. Care
should be taken that the ball be not set upon a violent
fire with very little water in it, and that the small pipe be
not stopped with any thing; for in such case, the included
elastic steam will suddenly burst the ball with a very dan-
gerous explosion.

This instrument was known to the ancients, being men-
tioned by Vitruvius, lib. 1, cap. 6. It is also treated of,
or mentioned, by several modern authors; as Descartes,
in his Meteor, cap. 4; and Father Mersenne, in prop.
29 Phasdom. Pneumat. uses it to weigh the air, by first
weighing the instrument when red-hot, and having no wa-
ter in it ; and afterwards weighing the same when it be-
comes cold. But the conclusion gained by this means,
cannot be quite accurate, as there is supposed to be no
air in the ball when it is red-hot ; whereas it is shown by
Varenius, in his Geography, cap. 19, sect. 6, prop. 10,
that the air is rarefied but about 70 times ; and conse-
quently the weight obtained by the above process, will
be about one-70th too small, or more or less according to
the intensity of the heat.

Descartes, and others, have made use of this instru-
ment, to account for the natural cause and production
of the wind. And hence its name, jEolipila, q. d. pila
.Eoli, iEolus' ball.

In Italy it is said that the jEolipile is often used to
cure smoaky chimneys: for, being hung over the fire,
the blast arising from it carries up the loitering smoke
along with it.

And some have imagined that the aeolipile might be
employed as bellows to blow up a fire, having the blast
from the pipe directed into the fire: but experience would
soon convince them of their mistake; for it would ra-
ther blow the fire out than up, as it is not air, but rarefied
water, that is thus violently blown through the pipe.

jEOLUS, in Mechanics, a small portable machine, in-
vented by Mr. Tidd, for refreshing and changing the air
in rooms which are made too close. The machine is
adapted to supply the place of a square of glass in a sash-
window, where it works with little or no noise, on the
principle of the sails of a mill, or a smoke-jack ; and thus
admitting an agreeable quantity of air, at a convenient
part of the room.

iEoLUs's Harp, or Molian Harp, an instrument so
named, from its producing an agreeable melody, merely
by the action of the wind. Neither the inventor, nor
the date of the invention, of this instrument is accurately
known. Kircher, in his Magia Phenotactica und Phonur-

a;oL

[ 32 ]

A E R

gia, has given the first description of it ; and hence the
invention of it has by some authors been attributed to
him.

Its construction is extremely simple, as it consists of
little more than a number of cat-gut or wire strings, dis-
tended in parallel lines taver a box of thin deal, having a
top of the same thin wood, with sounding-holes. Then
the strings being tuned in unison, and the instrument
placed in a proper situation for receiving a current of
air, as an open window, &c, it produces by the tremulous
motion which the wind communicates to the strings, a
soft murmuring, and pleasing combination of sounds.

The late Rev. W. Jones, constructed an iEolian harp
on an improved plan ; the strings of which, instead of
being on the outside, are fixed .to a sounding-board with-
in the box, and the wind is admitted to them through an
horizontal aperture. In this form the instrument is port-
able and mays be used in the open air. Several attempts to
explain the principles of the wild harmony of this instrument
have been made by different authors : of these Ur. Young's
is considered as the most satisfactory, which is here sub-
joined. The particles of a current of air, which strike
against the middle point of a stretched elastic string, will
move the whole string from its rectilinear position; and
as no breeze continues of the same strength for any con-
siderable time; although it be sufficient to remove the
string from its rectilineal position, yet, unless it be very
rapid and violent, it will not be able to keep it bent; and
consequently a 'vibration will ensue, and pulses in the
air will be excited, which will produce the tone of the
entire. string. But if the current of air be too strong
and rapid, when the string is bent from the rectilineal po-
sition it will not be able to recover that position, but will
continue bent, like the cordage of a ship in a brisk gale.
However, though the whole string cannot perform its vi-
brations, the subordinate aliquot parts may, which will
be of different lengths in different cases according to the
rapidity of the breeze. Thus, when the velocity of the
current of air increases so as to prevent the vibration of
the whole string, those particles which strike against the
middle points of the halves of the string agitate .those
halves ; and as these vibrate in half the time that the
whole string does, notwithstanding the breeze may be too
rapid to admit of the vibration of the whole string, yet
it can have no more effect in preventing the motion of the
halves, than it would have on the whole were its tension
quadruple;' for the times of vibrations of strings of dif-
ferent lengths, agreeing in other circumstances, are directly
as' the lengths ; and in strings of different tensions, and
agreeing in other circumstances, inversely as the square
roots of the tensions (see Chord): and therefore, their
vibrations may become strong enough to excite such
pulses as will affect the drum of the ear : and the same
may be said of the other aliquot divisions of the string.
Those particles which strike against such points of: the
string as are not in the middle or aliquot parts, will inter-
rupt and counteract each other's vibration, as is the case
of sympathetic and secondary tones, and therefore will
not produce a sensible effect. When .ZEolian notes arc
heard, which arc not produced by any submultiples of
the string, they are very transitory, and immediately vary
their pitch, till they coincide with the notes the next
above or below them, which are produced by the exact
aliquot parts of the whole string: and thus the harmony
is never interrupted by long-continued discords.

iERA, in Chronology, is the same as epoch, or epocha,
and means a fixed point of time, from which to begin a
computation of the years ensuing. The word is some-
times also written era in ancient authors. Its origin is
contested, though it is commonly supposed that it had its
rise in Spain, others say from the Arabs. Some imagine
that it is formed from a. er. a. the abbreviations of the
words, annus erat Augusti, or from a. e. r. a. the initials
of the words annus erat regni Augusti, because the Spa-
niards began their computation from the time that their
country came under the dominion of Augustus. Others
derive it from as, brass, the tribute money with which
Augustus taxed the world. It is also said that ara origi-
nally signified a number stamped on money to determine
its current value ; and that the ancients used as or ara
as an article, as we do the word item, to each particular
of an account ; and hence it came to stand for a sum or
number itself.

jEra also means the way or mode of accounting time.
Thus we say, such a year of the Christian aera, &c.

Spanish JEra, otherwise called the year of Caesar, was
introduced after the second division of the Roman pro-
vinces, between Augustus, Anthony, and Lepidus, in the
714th year of Rome, the 4676th year of the Julian pe-
riod, and the 38th year before Christ. In the 447th year
of this aera, the Alani, the Vandals, Suevi, &c, entered
Spain. It is frequently mentioned in the Spanish affairs;
their councils, and other public acts, being all dated ac-
cording to it. Some say it was abolished under Peter iv,
king of Arragon, in the year of Christ 135S, and the
Christian aera introduced instead of it. But Mariana ob-
serves that it ceased in the year ol Christ 1383, under
John 1, king of Castile. The like was afterwards done
in Portugal.

Christian ./Era. ■ It is generally allowed by Chronolo-
gers, that. the. computation of time from the birth of
Christ, was only introduced. in the sixth century in the
reign of Justinian ; and it is commonly ascribed to Dio-
nysius Exiguus. This aera came then into use in deeds,
&c; before which time, either the olympiads, or the year
of Rome, or that of the reign of the emperors, was used
for such purposes.

1 See an account of the other principal aeras under the
word Epoch.

AERIAL Perspective; is that which represents bodies
diminished and weakened, in proportion to their distance
from the eye.

Aerial Perspective chiefly relates to the colours of ob-
jects, whose force and lustre it diminishes more or less,
to make them appear as if more or less remote. It is
founded upon this, that the longer the column of air an
object is seen through, the more feebly do the visual rays
emitted from it affect: the. eye.

AliROGRAPIIY,. a description of the air, or atmo-
sphere, cits limits, dimensions, properties, &c.

AEROLITHS, or. Aerolites, i. e. Air-stonr$, or
stones from the atmosphere, are those compound kind of
masses which sometimes fall from the atmosphere, in all
countries, and which are therefore, often called meteoric
stones. — The descent of any such bodies was for a long
time doubted, the popular opinion that attested the rea-
lity being mostly regarded as a vulgar prejudice ; but the
fact has. been often proved of late, in such a manner as
to leave no reasonable doubt of the existence of the phe-
nomenon. Sec an account of many of these descents,

A E.R

C 33 ]

A E R

described in my paper on this subject, in the Abridg. of these bodies, exhibits a collection of some of the best

Philos. Trans, vol. 6, pa. 100, as well as several other ac- authenticated instances of the falling of stones, and other

counts there referred to. substances, from the atmosphere, thathave been observed,

The following table, drawn up by M. Izaron, a philo- with the times when they fell, and the authorities or

sopher who has paid considerable attention to the history names of the persons on whose evidence the facts rest.

Substan ces fa lien .
Shower of sulphur ....
Shower of stones

Shower of stones

Places where they fell.
Sodom and Gomorrah..
At Rome

At Rome

In Lucania

Inltaly

Near the river Negos,Thrace

In Thrace

Abakaick, Siberia

Ensisheim, Upper Rhine ...

Near Padua, in Italy

Mount Vaiser, Provence ...

Copenhagen

Duchy of Mansfeld

Iceland

Macedonia

Quesnoy

Atlantic Ocean

Brunswick

Niort, Normandy

Near Tabor, Bohemia

Liponas in Bresse

Near Verona

At Luce, in Le Maine

Aire, in Artois

In Le Cotentin

Near Roquefort

Environs of Agen

Sienna, Tuscany

Wold-Cottage, Yorks

In Portugal

Sales, near Ville-Franche...
Sale, dept. of the Rhone ...

Benares, East Indies

America

L'Aigle, Normandy

2d year of 78th Olympiad

Year before J. C. 452

Very old

Nov. 7th, 1492

In 1510

Shower of iron

Shower of Mercury

A very large stone

Three large stones

Mass of iron, 14 quintals...

Large stone, 260lbs

About 1200 stones; one")
120lbs. another of rjOlbs.j

Another, of 59lbs

Sulphurious rain

The same

Shower of viscid matter ....

Stone of 72lbs

Shower of tire

Shower of sand for 15 hours

Shower of sulphur

A stony mass

Shower of stones

Two stones, \vt. 20lbs

Two do. 200 and 300lbs ...

A stone of 74-lbs

A stone

A stone

Shower of stones

Extensive do

About 12 stone

Large stone of 5Glbs

A stone of lOlbs

A stone of 20lbs

A stone of 20lbs

Shower of stones

Mass of iron, 70 cubic ft....
Somestones,froml0tol7lbs

To this list might be added many more instances, and cent fusion. Besides, several of these singular substances
among them several of more recent date. Of these have been very carefully examined and analysed by Four-
masses, the larger kinds have been seen as luminous bo- croy, Howard, Bournois, Lavoisier, Vauquelin, and other
dies moving with very great velocities, descending in ob- eminent chemists and naturalists in Europe; and it has
1-ique directions, commonly with a loud hissing noise, re- been found that all the substances examined agree very
sembliijfl that of a mortar shell, or cannon ball, or ra- nearly in nature and composition, having the same com-
ther that of an irregular hard mass, violently projected ponent materials, and in nearly the same proportions:
Through the air; surrounded by a blaze or flame, tapering thus, the stones examined by Count de Bournon and Mr.
off to a narrow stream in the hinder part; they are heard Howard, were found to consist of four distinct substances,
to explode or burst, and even to fly in pieces, the larger a peculiar martial pyrites, a number of globular and el-
parts going foremost, and the smaller following in succes- liptical bodies, also of a peculiar nature, and an earthy
sion : they arc thus seen to fall on the earth, striking it cement surrounding the other component parts. The na-
•..ilb great violence; and, on examining the place of the ture of the metallic particles was the same in all, being
fall, the paru are found scattered about, being still con- in each an alloy of iron and nikel. In the pyrites, nikel
siderably warm, and mostly entered several inches deep as well as iron was detected ; and the easy decomposition
in the earth. But their most remarkable character is, of the pyrites, by muriatic acid, afforded a distinguishing
that they have a perfect resemblance to one another, character of that substance. The globules contained mag-
They are always found to be different from the neigh- nesia, silica, with oxides of nikel and iron. The earthy
Louring bodies, and show in every case the same appear- cement consisted of the same substances nearly in the
ance of semimetallic matter, coated on the outside with same proportions. With regard to the proportions of
a thin black incrustation, and bearing strong marks of re- these constituents, the celebrated stone which fell at 1 n-
Voi- 1. y

Times of their fall.

Year before J. C. I898

Under Tullius Hostilius ....
Consuls C. Martius & M. ?

Torquatus )

Year bef. defeat of Crassus

Nov. 27, 1627...

In 1646

In 1658

In 1695

January 1706 ....
Jan. 4th, 1717 ...
April 6, 1719....
October 1721 ....

In 1750

July 3d, 1753....
September 1753..

In 1762

Sept. 13, 1768...

In 1768

In 1768

July 1789

July 24, 1790....

July 1794

Dec. 13, 1795 ...
Feb. 19, 1796....
March 12, 1798
March 17, 1798 ,
Dec. 19, 179S...
April 5, 1800....
April 20, 1803...

Testimonies.
Moses
Livy

J. Obsequens

Pliny

Dion

Pliny

Ch. of Count Marcellin

Pallas, Chladni, &c

Butenschoen

Cardin, Varait

Gassendi

Olaus Wormius

Spaugenberg

Muschenbroeck

Paul Lucas

Geoffrey le Cadet

Pere le P'euillee

Siegesber

Lalande

B. de Born

Lalande

Acad, de Bourd

Bachelay

Gurson de Boyaval

Morand

Darcet, jun. Lomet, &c

St. Amand, Baudin, &c

Earl of Bristol

Capt. Topham

Southey

De Dree

Lelievre & De Dree

J. Lloyd Williams

Philos. Magazine

Fourcroy

A E R

[ 34 j

A £ R

sisheim, in Alsace, in 1492, and those which fell at 1'Ai-
gle, in France, in 1803, were found to yield, by the ana-
lyses of Fourcroy and Vauquelin, the proportions follow-
ing, "»»z, »'

Ensisheim. L'Aigle.
56-0 .of silica 54

36
9
3
2

1
105

30-0 oxyd of iron

:i2-0 magnesia ....

2-4 nikel

* 3-5 sulphur

1'4 lime

105-3

It likewise appears that the specific gravities of these
bodies are nearly alike, and such as greatly to exceed all
the known ordinary stones, and approach that of the me-
tallic ores. Thus, the specific gravity of

the Ensisheim stone was 3233

Benares 3352

Sienna 3418

Gassendi's 3456

Yorkshire 3508

Bachelay's - 3535

of such parts as that of water is 1000.

These common and constant characters indicate, with
strong evidence, a common origin. Yet, it is to be re-
marked, that iron is hardly ever found in the metallic
state in terrestrial bodies : volcanic bodies only contain-
ing it in the state of oxyd. Nikel is also very rare, being
never found on the surface of the earth. Whence it ap-
pears natural to conclude, that aerolites have an origin
foreign to our globe, or at least that they are not pro-
duced by phenomena that have been commonly observed.
Various hypotheses have therefore been devised to ac-
count for the origin of these stones. Some philosophers
think that they are nothing else than small planets, that
circulate in space after the manner of the other celestial
bodies ; and which, when found engaged in the earth's at-
mosphere, become there inflamed, by the resistance and
friction they experience ; that they lose little by little
their velocity, and at length fall to the earth by their
gravitation. Mr. King, Sir Wm. Hamilton, and others,
consider these stones as concretions actually formed in
the atmosphere: while others have imagined such phae-
nomena as resulting from electricity. But, in my Abridg.
of the Philos. Trans, (vol 6, p. 100), I have shown not
only that it is possible, but even highly probable, that
they may have been projected on the earth by lunar vol-
canoes. On subjecting this idea to calculation, it has
been found that a projectile force only the triple of that
sometimes given to cannon balls, would be sufficient to
carry a body out of the sphere of the moon's influence,
so that flu terrestrial gravitation would draw it towards
our ph.net. Nor is it at all improbable, that lunar vol-
canoes might impress such a force on a body, since ter-
restrial volcanoes can communicate a much greater im-
pulsion. This opinion acquires a fresh degree of proba-
bility from the recent observations of M. Schroeter on
the mc on, with respect to lh<- great height of some of the
lunar volcanoes, and the frequent variations observed on
the surface of that globe. Thus, then, the probability of
this origin of the aerolites appears to be greatly in fa-
vour of this last opinion ; though, in the present state
of our knowledge of these substances and of meteorology,
it is not possible as yet to decide. For accounts of the
pliaenomena of many aerolites, the inquisitive reader may

consult Mr. Howard's valuable paper in the Philos. Trans,
for the year 1S02, as well as my dissertation above re-
ferred to.

A EROLOGY, the doctrine or science of the air, and
its phajnomena, its properties, good and bad qualities,
&c. It is much the same with the foregoing word,
Aerography.

AEROMETRY, Aeromclria, the science of measuring
the air, its powers and properties; comprehending not
only the quantity of the air itself, as a fluid body, but
also its pressure or weight, its elasticity, rarefaction, con-
densation, &c.

The term is not much used at present; this branch of
natural philosophy being usually called pneumatics, which
see. Wolfius, late professor of mathematics at Hall,
having reduced several properties of the air to geometri-
cal demonstrations, first published at Leipsic his Elements
of Aerometry, in the German language, and afterwards
more enlarged in Latin, which have since been inserted
in his Cursus Mathematicus, in five volumes in 4to.

AERONAUTICA, the pretended art of sailing through
the air, or atmosphere, in a vessel, sustained as a ship in
the sea.

AEROSTATICA, is properly the doctrine of the weight,
pressure, and balance of the air and atmosphere.

AEROSTATION, in its proper and primary sense, de-
notes the science of weights suspended in the air; but in
the modern application of the term, it signifies the art
of navigating or floating in the air, both as to the prac-
tice and principles of it. Hence also the machines which
are employed for this purpose, are called aerostats, or ae-
rostatic machines; and which, on account of their form,
are otherwise called air-balloons. Also aeronaut is the
name given to the person who navigates or floats in the
air by means of such machines.

Principles of Aerostation. The fundamental prin-
ciples of this art have been long and generally known, as
well as speculations on the theory of it ; but the success-
ful application of them to practice seems to be altoge-
ther a modern discovery. These principles chiefly respect
the weight or pressure, and elasticity of the air, with its
specific gravity, and that of the other bodies to be raised
or floated in it : the particular detail of which principles
may be seen under the respective words in this dictionary.
Suffice it therefore in this place to observe, that any body
which is specifically lighter than the atmosphere, will bo
buoyed up by it, and ascend, as wood, or a cork, or a
blown bladder, ascends in water. And thus the body
would continue to ascend to the top of the atmosphere,
if it were every where of the same density as at the sur-
face of the earth. But as the air is compressible and
elastic, its density decreases continually in ascending, on
account of the diminished pressure of the superincumbent
air, at the higher elevations above the earth ; and there-
fore the body will ascend only to that height where the
air is of the same specific gravity with itself; there the
body will float, and move along with the wind or current
of air which it may meet with at that place. This body
then is an aerostatic machine, of whatever form or na-
ture it may be. And an air-balloon is a body of this
kind, the whole mass of which, including its covering and
contents, and the weights annexed to it, is of less weight
than the same bulk of air in which it rises.

We know of no solid bodies however that arc light
enough thus to ascend and float in the atmosphere ; and

AER

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AER

therefore recourse must he had to some fluid -or aeriform
substance.

Among these, that which is called hydrogen gas, or in-
flammable air, is the most proper of any that has hitherto
been discovered. It is very elastic, and from 6 to 10 or
12 times lighter than common atmospheric air at the
surface of the earth, according to the different methods
of preparing it. If therefore a sufficient quantity of this
gas be inclosed in a very thin bag or covering, the weight
of the two together will be less than the weight of the same
bulk of common air; and consequently this compound
mass will rise in the atmosphere, and continue to ascend
till it attain a height at which the atmosphere is of the
same specific gravity as itself; where it will remain or
float with the current of air, as long as the inflammable
air does not escape through the pores of its covering.
And this is an inflammable-air balloon.

Another way is to make use of common air, rendered
lighter by warming it, instead of the inflammable air.
Heat, it is well known, rarefies and expands common air,
and consequently lessens its specific gravity; and the di-
minution of its weight is proportional to the heat applied.
If therefore the air, inclosed in any kind of a bag or co-
vering, be heated, and consequently dilated, to such a de-
gree, that the excess of the weight of an equal bulk of
common air, above the weight of the heated air, be greater
than the weight of the covering and its appendages, the
whole compound mass will ascend in the atmosphere, the
same as in the former case ; till, by the cooling and con-
densation of the included air, the balloon shall gradually
contract and descend again, unless the heat is renewed or
kept up. And such is a heated-air or fire balloon, other-
wise called a Montgolfier, from its inventor.

Now it has been discovered, by various experiments,
that one degree of heat, according to the scale of Fahren-
heit's thermometer, expands the air about one five-hun-
dredth part ; and therefore that it will require about 500
degrees, or nearer 484 degrees of heat, to expand the air
to just double its bulk. Which is a degree of heat far
above what it is practicable to give it on such occasions.
And therefore, in this respect, common air heated, is
much inferior to hydrogen gas.

But, before entering more particularly into this sub-
ject, it will be proper to give a short historical account
of its rise and progress, with the various experiments and
aerial voyages, that have been made of later years by di-
vers aeronauts.

History of Aerostation. Various schemes for rising
in the air, and passing through it, have been devised and
attempted, both by the ancients and moderns, and that
upon different principles, and with various success. Of
these, some attempts have been upon mechanical princi-
ples, or by virtue of the powers of mechanism : and such
are conceived to be the instances related of the flying pi-
geon made by Archytas, the flying eagle and fly by Re-
giomontanus, and various others. Again, some projects
have been formed, by attaching wings to some part of the
human body, which were to be moved either by the hands
or feet, by the help of mechanical powers; so that, stri-
king the air with them, after the manner of the wings of a
bird, the person might raise himself in the atmosphere,
and transport himself through it, in imitation of that ani-
mal. But of these, and various other devices of the like
nature, a particular account will be given under the arti-
cle artificial jlying, as belonging rather to that species or

principle of motion, than to our present subject of aero-
station, which is properly the sailing or floating in the air
by means of a machine rendered specifically lighter than
that element, in imitation of aqueous navigation, or the
sailing upon the water in a ship, or vessel, which is speci-
fically lighter than the water.

The first rational account upon record, for this kind of
sailing, is perhaps that of our countryman Roger Bacon,
who died in the year 1292. He not only affirms that the
art is practicable, but assures us that he himself knew how
to make an engine, in which a man sitting might be able
to carry himself through the air like a bird; and he far-
ther affirms, that there was another person who had tried
it with success. The secret it seems consisted in having a
couple of large thin shells, or hollow globes, of copper,
exhausted of air ; so that the whole being thus rendered
lighter than air, they would support a chair, in which a
person might sit.

Bishop Wilkins too, who died in 1672, in several of his
works, makes mention of similar ideas being entertained
by divers persons. " It is a pretty notion to this purpose,
says he, (in his Discovery of a New World, prop. 14),
mentioned by Albertus de Saxonia, and out of him by
Francis Mendoza, that the air is in some part of it navi-
gable. And that upon this statick principle, any brass or
iron vessel (suppose a kettle), whose substance is much
heavier than that of the water; yet being filled with the
lighter air, it will swim upon it, and not sink." And again,
in his Dedalus, chap. 6, " Scaliger conceives the framing of
such volant automata to be very easy. Volantis columbas
machinulam, cujusautorem Archytam tradunt, vel facil-
lime profiteri audeo. Those ancient motions were thought
to be contrived by the force of some included air: So
Gellius, Ita erat scilicet libramentis suspensum, et aura
spiritus inclusa, atque occulta consitum, &c. As if there
had been some lamp, or other fire within it, which might
produce such a forcible rarefaction, as should give a mo-
tion to the whole frame." From which it would seem
that Bishop Wilkins had some confused notion of such a
thing as a heated-air balloon.

Again, F. Francisco Lana, in his Prodroma, printed in
1670, proposes the same method with that of Roger Ba-
con, as his own thought. He considered that a hollow
vessel, exhausted of air, would weigh less than when filled
with that fluid ; he also reasoned that, as the capacity of
spherical vessels increases much faster than their surface,
if there were two spherical vessels, of which the diameter
of one is double the diameter of the other ; then the ca-
pacity of the former will be equal to 8 times the capacity
of the latter, but the surface of that only equal to 4 times
the surface of this : and if the one sphere have its diame-
ter equal to triple the diameter of the other, then the ca-
pacity of the greater will be equal to 27 times the capa-
city of the less, while its surface is only 9 times greater:
and so on, the capacities increasing as the cubes of the
diameters, while the surfaces increase only as the squares
of the same. And from this mathematical principle, fa-
ther Lana deduces, that it is possible to make a spherical
vessel of any given matter, and thickness, and of such a
size as, when emptied of air, it will be lighter than an
equal bulk of that air, and consequently that it will as-
cend in that element, together with some additional
weight attached to it. After stating these principles, fa-
ther Lana Computes that a round vessel of plate brass, 14
feet in diameter, weighing 3 ounces the square foot, will
F2

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[ 36 ]

A E R

only weigh 1848 ounces ; whereas ^i quantity of air of the
same bulk will weigh 2155f ounces, allowing only one
ounce to the cubic foot ; so that the globe will not only
ascend in the air, but will also carry up a weight of 307f
ounces: and by increasing the bulk of the globe, without
increasing the thickness of the metal, he adds, a vessel
might be made to carry a much greater weight.

Such then were the ingenious speculations of learned
men, and the gradual approaches towards the perfection
of this art. But one thing more was yet wanted : though
acquainted in some degree with the weight of any quan-
tity of air, considered as a detached substance, it seems
they were not aware of its great elasticity, and the uni-
versal pressure of the atmosphere ; by which pressure, a
globe of the dimensions above-described, and exhausted of
its air, would immediately be crushed inwards, fcr want
of the equivalent internal counter pressure, to be sought
for in some element, much lighter than common air, and
yet nearly of equal pressure or elasticity with it; a pro-
perty or circumstance attending iuflammable gas, and also
common air when considerably heated.

It is evident then, that the schemes of ingenious men
hitherto must have gone no farther than mere specula-
tion ; otherwise they could never have recorded fancies,
which, on the first attempt to put in practice, must have
manifested their own insufficiency, b)'an immediate failure
of success. For, instead of exhausting the vessel, it must
either be filled with common air heated, or with some
other equally elastic and lighter fluid. So that upon the
whole it appears., that the art of traversing the air, is an
invention of our own time ; and the whole history of it is
comprehended within a very short period.

Mr. Cavendish was the first who discovered with any
degree of accuracy the specific gravity of inflammable
air ; and his experiments and observations upon it, are
published in the 56th volume of the Philosophical Trans-
actions for the year 1766.

Soon after this discovery of Mr. Cavendish, it occurred
to the ingenious Dr. Black of Edinburgh, that if a blad-
der, or other vessel, sufficiently light and thin, were filled
with this air, it would form altogether amass lighter than
the same bulk of atmospheric air, and consequently that
it would ascend in it. This idea he mentioned in his che-
mical lectures in the year 1767 or 1768; and he further
proposed to exhibit the experiment, by filling the allantois
of a calf with such air. The allantois, however, was not
prepared just at the time when he was at that part of his
lectures, and other avocations afterwards prevented his de-
sign : so that, considering it only as an amusing experi-
ment, and being fully satisfied of the truth of so evident
an effect, he contented himself with barely mentioning the
experiment from time to time in his lectures.

About the year 1777 or 1778 it also occurred to
Mr. Cavallo, that it might be possible to construct a ves-
sel, which, when filled with inflammable air, would as-
cend in tin' atmosphere: and there is no doubt but that
similar ideas occurred to many other persons, of so evi-
dent a consequence of Mr. Cavendish's discovery. But it
seems to have been Mr. Cavallo who first actually at-
tempted the experiment, in which however be succeeded
no further than in being able to raise soap bubbles of two
or three inches diameter : a thing which had been done
by children for their arpusemi nt from time immemorial.
These experiments Mr. Cavallo made, in the beginning of
the year 178?, and an account o( them was read at a

public meeting of the Royal Society on the 20th of June
of that year. From which it appears that he tried blad-
ders and paper of various kinds. But the bladders, how-
ever thin they were made by scraping, &c, were still found
too heavy to ascend in the atmosphere, when iully inflated
with the inflammable air: and in using China paper, he
found that this air passed through its pores, like water
through a sieve. So that, having faded of success by
blowing the same air into a thick solution of gum, thick
varnishes, and oil paint, he was obliged to rest satisfied
with soap-balls or bubbles, which, being filhd with in-
flammable air, by dipping the end of a small glass tube,
connected with a bladder containing the air, into a thick
solution of soap, and gently comprising the bladder, as-
cended rapidly, and broke against the ceiling of the room.

Here, however, it seems the matter might have rested,
had it not been for experiments made in France soon
after, by the two brothers Stephen and Joseph Montgol-
fier, on principles suggested, not by the levity of inflam-
mable air, which probabl) the)' had never heard of, but
by that of smoke ami clouds ascending in the atmosphere.
These two brothers it seems were natives of Annonay, a
town in the Vivarais, about 36 miles from Lyons ; and
that in their youth, Stephen, the elder, had assiduously
studied the mathematics, but the other had applied him-
self more particularly to natural philosophy and che-
mistry. Theywere not intended for any particular way
of business, but the death of a brother obliged them to
put themselves at the. head of a considerable paper manu-
factory at Annonay. In the intervals of time allowed by
their business they amused themselves in several philoso-
phical pursuits, and particularly with the experiments in
aerostation, of which we are now to give some account.
It would be perhaps impossible to know all the particular
steps and ideas which finally produced this discovery : but
it has been said that the real principle, on which the ef-
fect of the aerostatic machine depends, was unknown even
for a considerable time after its discovery: that Montgol-
fier attributed the effect of the machine, not to the rare-
faction of the air, which is the true cause, but t" a cer-
tain gas, specifically lighter than common air, which was
supposed to be developed from burning substances, and
which was commonly called Montgulfier's gas. Be this
however as it may, it is well known that the two brothers
began to think of the experiment of the aerostatic ma-
chine about the middle or the latter part of the year 17S2.
The natural ascension of the smoke and the clouds in the
atmosphere, suggested the first idea; and to imitate those
bodies, or to inclose a cloud in a bag, and let the hitter
be lifted up by the buoyancy of the former, was the first
project of those gentlemen.

Accordinsly, the first experiment was made at Avignon
by Stephen, about the middle of November 1782. Having
prepared a bag of fine silk, in the shape tit a paralhlopi-
pedon, and of about 40 cubic feet in capacity, he applied
burning paper to an aperture in the bottom, which rare-
fied the air, and thus formed a kind of cloud in the bag;
and when it became sufficiently expanded, it ascended ra-
pidly to the ceiling.

Soon afterwards the experiment was repeated with the
same machine at Annonay, by the two brothers, in the
open air; when the bag ascended to '.he height of about
70 feet. Encouraged by this success, they constructed
another machine, of about 650 cubic feet capacity ,
which, when inflated as before, broke the cords which

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confined it, and after ascending rapidly to the height of less than two minutes, to the height of 3123 feet: the
about 600 feet, descended and fell on the adjoining weight of the balloon, when it went up, being 35 pounds
ground. With another larger machine, of 37 feet diame- less than that of an equal bulk of common air. After
ter, they repeated the experiment on the 25th of April, floating in the air for about three quarters of an hour, it
which answered exceedingly well : the machine had such fell in a field at 15 miles from the place of its ascent ;
force ot ascension, that, breaking abruptly from its con- where, as we may easily imagine, it occasioned great
finemerrt of ropes, it rose to the height of more than 1000 amazement to the peasants who found it. Its fall was
feet, and then, being carried by the wind, descended and owing to a rent in the covering, probably occasioned by
fell at a place about three quarters of a mile from the the superior elasticity of the inflammable air, over that
place of its ascension. The capacity of this machine. was of the rare part of the atmosphere to which it had as-
equal to above 23 thousand cubic feet, and, being nearly cended.

globular, when inflated, it measured 1 17 English feet in
circumference. The covering was formed of linen, lined
with paper; and its aperture at the bottom was fixed to
a wooden frame, of about 4 feet square, or 16 feet in sur-
face. When filled with vapour, which it was conjectured

In consequence of this brilliant experiment, number-
less small balloons were made, mostly of gold-beater's
skin, from 6 and y to IS or 20 inches diameter; their
cheapness putting it in the power of almost every family
to satisfy its curiosity relative to the new experiment;

might be about half as heavy as common air, it was ca- and in a few days balloons were seen flying all about Pa

pable of lifting up about 490 pounds, besides its own ris, whence they were soon alter sent abroad,
weight, which, together with that of the wooden frame, M. Joseph Montgolfier repealed an experiment with

was equal to about 500 pounds. With this machine the a machine of his construction before the commissaries of

next experiment was publicly performed at Annonay, on the Academy of Sciences, on the 11th and 12th ot Sep-

the 5th of June 1783, before a great multitude of spec- tember. The machine was about 74 feet high, and 43

tators. -The flaccid bag whs suspended on a pole 35 feet feet in diameter; it was made of canvass, covered with

high; straw and chopped wool were burned* under the paper both within and without, and weighed 1 000 pounds,

opening at the bottom; the vapour, or rather smoke, soon It was' filled with rarefied air in 9 minutes, and in one

inflated the bag, so as to distend it in all its parts; and trial the weight of 8 men was not sufficient to keep it

this enormous mass ascenrled in the air with such velocity, down. It was not suffered to go up, as it had been in-

that in less than ten minutes it reached the height of above tended for exhibition before the Royal Family, a few days

6 thousand feet ; when a breeze carried it in an horizon- after. By the violence of the rain, however, which fell

tal direction to the distance of near a mile and a half, about this time, it was so much spoiled, that he thought

where it descended gently to the ground. proper te construct another for that purpose, in which he

As soon as the news of this experiment reached Paris, used so great dispatch, that it was completed in the short

the philosophers of that cit\, conceiving that a new spe- space of four days. This machine was constructed of

cies of gas, of about half the weight of common air, had cloth made of linen and cotton thread, and painted with

been discovered by Messrs. Montgolfier; and knowing water colours both within and without. Its height was

that the weight of inflammable air was but aoout the 8 60 feet, and diameter 43 feet. Having made the neces-

or 10th part of the weight of common air, they justly sary preparation for inflating it, the operation was begun

concluded that inflammable air would answer the purpose about one o'clock on the 19th of the same month, be-

of this experiment better than the gas of Montgolfier, and fore the king and queen, the court, and the inhabitants of

accordingly they resolved to make trial of it. the place, as well as all the Parisians who could procure

A subscription was opened by M. Faujas de St. Fond, a conveyance to Versailles. The balloon was soon filled,

towards defraying the expense of the experiment. A suf-
ficient sum of money having soon been raised, Messrs.
Roberts were appointed to construct the machine, and
M. Charles, professor of experimental philosophy, to su-
perintend the work. After a considerable time spent,
and surmounting many difficulties in obtaining a sufficient
quantity of this gas, and searching out a substance light

and in 1 1 minutes after the commencement of the ope-
ration, the ropes being cut, it ascended, bearing up with
it a wicker cage, containing a cock, a duck, and a sheep,
the first animals that ever ascended into the atmosphere
with ah aerostatic machine. Its power of ascension, or
the weight by which it was lighter than an equal bulk of
common air, allowing for the animals and their cage, was

enough for the covering, they at length constructed a 696 pounds. The balloon rose to the height of 1440 feet;

globe of the silk called lutestring, which was rendered im- and being driven hy the wind for about 8 minutes, it gra-

pervious to the inclosed air by a varnish of elastic gum or dually descended, in consequence of two large rents made

caeutchouc, dissolved in some kind of spirit or essential in the covering by the wind, and fell in a wood at the dis-

oil. I he diameter pf this globe was about 13 feet; and tance of about two miles from Versailles. The animals

it had only one aperture, like a bladder, to which a stop
cock was adapted : the weight of this covering, when
empty, together with that of the stop cock, was 25 pounds.
On the 23d of August 1783, they began to fill the
globe; but this, being their first attempt, was attended

landed again as safe as when they went up, and the sheep
was found feeding.

The success of this experiment induced M. Pilatre de
Rozier, with a philosophical intrepidity which will be re-
corded with applause in the history of aerostation, to offer

with many obstructions and disappointments, which took himself as the first adventurer in this aerial navigation.
up two or thn < days to overcome. At length, however, For this purpose, M. Montgolfier constructed a new ma-
it was prepared for exhibition, and on the 27th it was chine, of an oval shape, in a garden of the fauxbourg St.
carried to the Champs de Mars, a spacious open ground Antoine; its diameter being about 48 feet, and height 74
in the front of the Military School, where, after intro- feet. To the aperture in the lower part was annexed a
ducing some more inflammable air, and disengaging it wicker gallery about three feet broad, with a ballustrade
from the cords by which it was held down, it rose, in of three feet high. From the middle of the aperture an

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iron grate, or brazier, was suspended by chains, descend-
ing from the sides of the machine, in which a fire was
lighted for inflating the machine; and towards the aper-
ture port-holes were opened in the gallery, through which
anv person, who might venture to ascend, might feed the
fire on the grate with fuel, and regulate at pleasure the di-
latation of the air inclosed in the machine : the weight of
the whole being upwards of 1600 pounds. On the 15th of
October 1783, the fire being lighted, and the balloon in-
flated, M. P. de Rozier placed himself in the gallery, and,
to the astonishment of a multitude of spectators, ascended
as high as the length of the restraining cords would permit,
which was about 84 feet from the ground, and there kept
the machine afloat near 5 minutes, by repeatedly throw-
ing straw and wool upon the fire : the machine then de-
scended gradually and gently, through a medium of in-
creasing density, to the ground ; and the intrepid adven-
turer assured the admiring spectators that he had not ex-
perienced the least inconvenience in this aerial excursion.
This experiment was repeated on the 17th with nearly the
same success; and again several times on the 19th, when
M. Rozier, by a partial ascent and descent, several times
repeated, evinced to the multitude of observers, that the
machine may be made to ascend and descend at the plea-
sure of the aeronaut, by merely increasing or diminishing
the fire in the grate. The balloon having been hauled
down, by the ropes which always confined it, M. Gironde
de Villette placed himself in the gallery opposite to M.
Rozier, and the machine being suffered to ascend, it ho-
vered for about 9 minutes over Paris, in the sight of all
its inhabitants, at the height of 330 feet. And on their de-
scending, the marquis of Arlandes ascended with M. Ro-
zier much in the same manner.

In consequence of the report of these experiments,
signed by the commissaries of the Academy of Sciences,
it was ordered that the annual prize of 600 livres should
be given to Messrs. Montgolfier for the year 1783.

In the experiments above-recited, the machine was al-
ways secured by long ropes, to prevent its entire escape :
but they were soon succeeded by unconfined aerial navi-
gation. For this purpose, the same balloon, of 74 feet in
height, was conveyed to La Muette, a royal palace in the
Bois de Boulogne : and all things being ready, on the 21st
of November 1783, M. Rozier and the marquis d'Ar-
landes took their post in opposite sides of the gallery, and
at 2 o'clock the machine was abandoned to the element,
when it ascended calmly and majestically in the atmo-
sphere. On reaching the height of about 280 feet the in-
trepid aeronauts waved their hats to the astonished multi-
tude : but they soon after rose too high to be distinguish-
ed, their greatest height being estimated at 3000 feet.
At first they were driven, by a north-west wind, horizon-
tally over the river Seine and part of Paris, taking care to
clear the steeples and high buildings by increasing the
fire; and in rising they met with a current of air which
carried them southward. Having thus passed the Boule-
vard, and finally desisting from supplying the fire with
fuel, they descended very gently in a field beyond the
new Boulevard, a little more than 5 miles from the palace
deLa Muette, having been between 20 and 25 minutes in the
air. The weight of the whole apparatus, including that of
the two travellers, was between 160O and 1700 pounds.

Notwithstanding the rapid progress of aerostation in
France, it is remarkable that we have no authentic ac-
count of any experiments of this kind being attempted in

other countries. Even in our own island, where all 3 -
and sciences find an indulgent nursery, and many their
birth, no aerostatic machine was seen before the month of
November 17S3. Various speculations have, been made
on the reasons of this strange neglect of so novel and bril-
liant an experiment. But none seemed to carry any
show of probability except that it was said to be discou-
raged by the leader of a philosophical society, expressly
instituted for the improvement of natural knowledge, for
the reason, as it was said, that it was the discovery of a
neighbouring nation. Be this however as it may, it is a
fact that the first aerostatic experiment was exhibited in
England by a foreigner unconnected and unsupported.
This was a count Zambeccari, an ingenious Italian, who
happened to be in London about that time. He made a
balloon of oiled-silk, 10 feet in' diameter, weighing only
11 pounds : it was gilt, both for ornament, and to render
it more impermeable to the inflammable air with which it
was to be filled. The balloon, after having been publicly
shown for several days in London, was carried to the Ar-
tillery Ground, and there being about three-quarters filled
with inflammable air, and having a direction inclosed in a
tin box for any person by whom it should afterwards be
found, it was launched about one o'clock on the 25th of
November 1783. At half past 3 it was taken up near
Petworlh in Sussex, 48 miles distant from London ; so
that it tfavelled at the rate of near 20 miles an hour. Its-
descent was occasioned by a rent in the silk, the effect of
the rarefaction of the inflammable air, when the balloon
ascended to a rarer part of the atmosphere.

The French philosophers having executed the first
aerial voyage with a balloon inflated by heated air, re-
solved to attempt a similar voyage with a balloon filled
with inflammable air, which seemed to be preferable to
dilated air in every respect, the expense of preparing it
only excepted. A subscription was opened however to
defray that expense, which was estimated at about ten
thousand livres ; and the balloon was constructed by
Messrs. Roberts, of gores of silk, varnished with a solution
of elastic gum. Its form was spherical, and it measured
274- f"eet 'n diameter. The upper hemisphere was covered
by a net, which was fastened to a hoop encircling its mid-
dle, and called its equator. To this equator was suspend-
ed by ropes a car or boat, covered with painted linen,
and beautifully ornamented, which swung a few feet
below the balloon. To prevent the bursting of the ma-
chine by the expansion of the inflammable air in a rarer
medium, or to cause the balloon to descend, it was fur-
nished with a valve, which might be opened by means of a
string descending from it, for discharging a part of the in-
ternal air, without admitting the external to enter : And
the car was ballasted with bags of sand, for the purpose of
lightening it occasionally, and causing it to ascend : so
that by letting some of the air. escape through the valve,
they might descend ; and by discharging some of their
sand ballast, ascend. To this balloon was likewise an-
nexed a long pipe by which it was filled. The apparatus
for this purpose consisted of several casks placed round a
large tub of water, each having a long tin lube, that ter-
minated under a vessel or funnel, which was inverted into
the water of the tub, and communicated with the long
pipe annexed to the lower part of the balloon. Iron
filing; and diluted vitriolic acid being put into the casks,
the inflammable air which was produced from these ma-
terials, passed through the tin tubes, thence through the

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water of the tubs to the inverted funnel, and so through opposite to that of the aeronauts; from winch it is infer-
tile pipe into the balloon. When inflated, the weight of red that there were two currents of air at different heights
the common air which was equal in bulk to the balloon^ above the earth.

was 77H pounds; also the power of ascension, or weight
just necessary to keep it from ascending, was 20 pounds,
and weight of the balloon, with its car, passengers, and all
its appendages, was 6044 pounds, making in all 6'24£

In the month of December this year, several experi-
ments were made at Philadelphia in America with air-
balloons, by Messrs. Rittenhouse and Hopkins. They
constructed and filled a great many small balloons, and

pounds : and this taken from 771-| pounds, the weight of connected them together; by which a man went up se-

common air displaced, leaves 14<7 pounds for the weight veral times, and was drawn clown again ; and finally, the

of the inflammable air contained in the balloon, and ropes being cut, he ascended to the height of 100 feet, and

which is to 77 1-i. pounds, the weight of the same bulk of floated to a considerable distance; but, being afraid, he

common air, nearly as 1 to 5\; that is, the inflammable cut open the balloons with a knife, and so descended,
air used in this experiment was 5£ times lighter than com- About the close of this year small balloons were sent

mon air. up in many places, and were become very common in

The first of December was fixed on for the display of some parts of France and England. In the beginning of

this grand experiment; and every preparation was the year following, their number and magnitude increased

made for conducting it with advantage. The garden of considerably. Some of the more remarkable ones were

the Thuilleries at Paris was the scene of operation ; as follow : — On the 19th of January M. Joseph Montgol-

which was soon crowded and encompassed with a prodi- fier, accompanied by six other persons, ascended from

gious multitude of observers. Signals were given, from Lyons with a rarefied-air balloon, to the height of 1000

time to time, by the firing of cannon, waving of flags, &c: yards. This was the largest machine that had been hi-

and a small montgolfier was launched, for showing the therto made, being 131 feet high, and 104 feet in diame-

direction of the wind, and for the amusement of the people ter: it was formed of a double covering of linen, with

previous to the general display. At a quarter betore 1 three layers of paper between them; and it weighed, when

o'clock, M. Charles and one of the Roberts, having seated it went up, 1600 pounds, including the gallery, passe'n-

themselves in the boat attached to the balloon, and being gers, &c. It was at first intended for six passengers; but

furnished with proper instruments, cloathing, and provi- before it went up, it was not judged safe to freight it with

sions, left the ground, and ascended with a moderate velo-
city to the height of about 600 yards; the surrounding
multitude standing silent with fear and amazement ; while
the aerial navigators at this height made signals of their
safety. When they left the ground, the thermometer, ac-
cording to Fahrenheit's scale stood at 59° ; and the baro-
meter, at 30-18 inches: at the utmost height to which

more than three : however, no authority nor solicitations
could prevail upon any of the six to quit their place, nor
even to cast lots which three should resign their preten-
sions : so that the spectators saw them all ascend with
terror and anxiety ; and to add to their distress, when
the ropes were cut, and the machine had ascended a foot
or two from the ground, a seventh person suddenly

they ascended, the barometer fell to 27 inches ; from leaped into the gallery, and the fire being increased, the

.vhich they deduced their height as above, to be 600 whole ascended together. To add to the terror of the

yards, or one third part of a mile. During the rest of scene, after being in the air about 15 minutes, a Jarge

the voyage the quicksilver in the barometer was generally rent of about 50 feet in length was made by the balloon

between 27 and 27'6'5 inches, rising and falling, as part taking fire, in consequence of which it descended very ra-

of the ballast was thr< wn out, or some of the inflammable pidly to the ground, though fortunately without injury to

air escaped from the balloon. The thermometer generally any of the aeronauts.

stood between 53 and 57 degrees. Soon after their as- On the 22d of February an inflammablo-air balloon

cent, they remained stationary for some time: they then was launched from Sandwich in Kent. It was but a

moved horizontally in a direction n.n.w.: and having small one, being only 5 feet in diameter ; but it was ren-

crossed the Seine, and passed over several towns and vil- dered remarkable by being the first machine that crossed

lages, to the great amazement of the inhabitants, they de-
scended in a field, about 27 miles from Paris, at three-
quarters past 3 o'clock ; so that they had travelled at the
rate of near 15 miles an hour, without feeling the least in-
convenience.

The balloon still containing a considerable quantity of
inflammable air, M. Charles re-ascended alone, and it
was computed he went to the height of 3100 yards, or
near 2 miles, the barometer being then at 20 English
inches : having amused himself in the air about 33 mi-

the sea from England to France. It was found in a field
at Warneton, about 9 miles from Lisle in French Flan-
ders, two hours and a half after it left Sandwich, the dis-
tance being about 74 miles; so that it floated at the rate
of about 30 miles an hour.

The chevalier Paul Andrcani, of Milan, was the first
aerial traveller in Italy. The chevalier himself was at
the sole expense of this machine, but was assisted in the
construction by two brothers of the name of Gerli. They
all three ascended together near Milan on the 25th of

nutes, he pulled the string of the valve, and descended at February, and remained in the atmosphere about 20 mi-
3 miles distance from the place of his ascent. All the in- nutes, when they descended, all their fuel being exhaust-
convenience he experienced in his great elevation, was a ed. This machine was a montgolfier, of a spherical shape,
dry sharp cold, with a pain in one of his ears and a part and about 68 feet in diameter. From calculations made
of his face, which he ascribed to the dilatation of the in- on the power of this, and other machines of the same
ternal air: a circumstance that usunlly happens to per- sort, it appears that the included air is rarefied commonly
sons uho suddenly change the density of their atmosphere, but about one-third, or that the included warm air
either by ascending into a larer, or descending into a den- weighs about two thirds of the same bulk of the external
ser one The small balloiui, launched at the beginning by or common air.
M. Montgolfier, was found to have moved in a direction The next aerial voyage was performed on the 2d oi

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March 178*, by M. Jean Pierre Blanchard, a man who
has since that time made more voyages than any other
person, except M. Garnerin, and who has attained great
celebrity by being the first who has floated in the air over
the channel from England to France. M. Blanchard it
seems had for many years been in pursuit of mechanical
means for flying through the air ; but on hearing of the
late invented air-balloons, he dropped his former pursuits,
and turned his attention to them. He accordingly con-
structed a balloon uf 27 feet diameter, to which a boat
was suspended, with two wings, and a rudder to steer it
by, as also a large parachute spread horizontally between
the boat and the balloon, designed to check the fall in case
the balloon should burst. The machine being filled with in-
flammable air, he ascended, from the Champs de Mars a*
Paris, to the height of near 2 miles ; and after floating in
the air for an hour and a quarter, he descended at Billan-
court, near Seve, having experienced by turns heat, cold,
hunger, and an excessive drowsiness. It appears from
his own account, and as might have been expected, that
the wings and rudder of his boat had little or no power in
turning the balloon from the direction of the wind.

Jn the course of this year, 1784, aerostatic experi-
ments and aerial voyages became so frequent, that the li-
mits of this article will not allow of any thing farther than
mentioning those that were attended with any remarkable
circumstances. On the 25th of April, Messrs. de Mor-
veau and Bertrand ascended from Dijon, with an inflam-
mable-air balloon, to the height of near 2 miles and a
half, where the thermometer marked 25°. They were in
the air an hour and 25 minutes, in which time they floated
18 miles.

On the 20th of May, four ladies and a gentleman as-
cended from Paris, in a large montgolfier, above the
highest buildings, and remained suspended there a consi-
derable time, the balloon being confined by ropes from
flying away.

On the 23d of May, M. Blanchard, with the same bal-
loon as before, ascended from Rouen, to such height that
the mercury in the barometer stood at 20 57 inches,
which on the earth had been at 30* 1 6. It was observed
that in this voyage M. Blanchard's wings or oars could
not turn him aside from the direction of the wind.

On the 4th of June M. Fleurant and Madame Thible,
the first lady who made an aerial voyage, ascended at
Lyons in a machine of 70 feet diameter. They went to
the height of 8500 feet, and floated about 2 miles in 45
minutes.

On the 14th of June, M. Coustard de Massi and M.
Mouchet ascended at Nantes to a great height, with a bal-
loon of 32' feet diameter, filled with inflammable air ex-
tracted from zinc ; and they floated to the distance of 27
miles in 58 minutes.

On the 23d of June, the first aerial traveller M. Rozier,
accompanied with M. Prouts, ascended at Versailles, in
the p'esence of the royal family and the king of Sweden,
with a large montgolfier, whose diameter was 7.9 feet, and
it-, height 91 feet and a half. They floated to the distance
of 36' miles in three- quarters of an hour, when they de-
scended, which is at the rate of 48 miles an hour. Incon-
v quence of this experiment, the king granted to M. de
Rozier a pension of 2000 livres.

On the 15th of July the dukeof Chartres, the two bro-
thers Roberts, and a fourth person, ascended from the.
park of St. Cloud, with an inflammable-air machine, of

an oblong form, its diameter being 34 feet, and its length,
which went in a direction parallel to the horizon, was 55t
feet: they remained in the atmosphere 45 minutes, in the
greatest fear and danger. The machine contained an in-
terior small balloon, filled with common air, by means of
which it was proposed to cause the machine to ascend or
descend without the loss of any inflammable air or ballast :
and the boat was furnished with a helm and oars, which
were intended to guide it. Three minutes after ascending,
the machine was lost in the clouds, and involved in a
dense vapour. A violent agitation of the air, resembling
a whirlwind, greatly alarmed the aeronauts, turned the
machine three times very rapidly, and gave it such shocks
as prevented them from using any of their instruments for
managing the machine. After many struggles, with great
difficulty they tore about 7 or 8 feet of the lower part of
the covering, by which the inflammable air escaped, and
they descended to the ground with great rapidity, though
without any hurt. At the place of departure the baro-
meter stood at 30-12 inches, and at their greatest eleva-
tion it descended to 24 '36 ; so that their ascent was nearly
a mile.

On the 18th of July, M. Blanchard, with a Mr. Boley,
made his third voyage with the same balloon as he had be-
fore, and rose so high as to sink the barometer from 30-l
to 25"34 inches, answering to a height of about 4600 feet.
In 2 hours and a quarter they floated 45 miles, which is
at the rate of 20 miles an hour. In this voyage M. Blan-
chard thought that he was able to turn the machine with
his wings, and to make it ascend and descend at pleasure.
After descending, it is said the balloon remained all the
night at anchor full of air ; and that the next day se-
veral ladies amused themselves by ascending successively
to the height of 80 feet, the length of the ropes by which
it was anchored.

In the course of this summer two persons had nearly-
lost their lives by ascending with machines of heated
air. The one in Spain, on the 5th of June, by the ma-
chine taking fire, was much burnt, and so hurt by the fall
that his life was long despaired of. The latter having as-
cended a few feet, the machinery entangled under the
eaves of a house, which broke the ropes, and the man fell
about twenty feet : the machine presently took fire, and
was consumed. 'Other montgolfiers were also burned
about London, by taking fire, through the defects of their
construction.

The first aerial voyage performed in England was by
one Vincent Lunardi, a native of Italy, who ascended
from the Artillery Ground, London, with an inrlamina-
blc-air balloon on 15th of September. His machine was
made of oiled silk, painted in alternate stripes of blue and
red ; and its diameter was 33 feet. From a net, which
covered about two thirds of the balloon, 45 cords de-
scended to a hoop hanging below it, to which the gallery
was attached. The machine, had no valve; and its neck,
which terminated in the form of a pear, was the aperture
through which the inflammable air was introduced, and
through which it might be let out. (The balloon was
filled with air produced from zinc by means of diluted vi-
triolic acid. And when the aeronaut departed at 2 o'clock
he took up with him a dog, a cat, and a pigeon. After
throwing out some sand, to clear the houses, he ascended
to a considerable height ; ami the direction of his motion
at first was north-west by west ; but as the balloon rose,
higher, it came into another current of air, which carried

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it nearly north. In the course of his voyage the thermo-
meter was as low as 29°, and the drops of water which
had collected round the balloon were frozen. About
half after 3 he descended very near the ground, and
landed the cat, which was almost dead with cold : then
rising, he prosecuted his voyage, till at 10 minutes past
4 o'clock he landed near Ware in Hertfordshire. He
ascribes his descent to the action of his oars ; but as he was
under the necessity of throwing out ballast in order to re-
ascend, his descent was more probably occasioned by the
loss of inflammable air.

The longest and most interesting voyage performed
about this time, was that of Messrs. Roberts and M.
Colin Hullin, who ascended at Paris, at noon on the 19th
of September, with a balloon, filled with inflammable air,
which was 27f feet in diameter, and 46| feet long, the
machine being made to float with its longest part parallel
to the horizon, and having a boat of near 17 feet long at-
tached to it. The boat was fitted up with several wings
or oars, shaped like an umbrella; and they ascended at
12 o'clock with 450 pounds of sand ballast, and after va-
rious manoeuvres finally descended, at 40 minutes past 6
o'clock, near Arras in Artois, 150 miles from Paris,
having still 200 pounds of ballast remaining in the boat.
In one part they found the current of air uniform from
600 to 4200 feet high, which it seems was their greatest
height, and the fall of the barometer had been near 5'6 de-
grees. They found that by means of their oars they
could accelerate their course a little in the direction of
the. wind, when it moved slowly, which may be true ; but
there is great reason to doubt of the accuracy of their ex-
periments by which they pretended to cause their path to
deviate about 22 degrees from the wind, going with a con-
siderable velocity.

The second aerial voyage in England, was performed
by Mr. Blanchard, and Mr. Sheldon, professor of ana-
tomy to the Royal Academy, being the first Englishman
who ascended with an aerostatic machine. They ascended
at Chelsea the loth of October, at 9 minutes past 12
o'clock. Mr. Blanchard having landed Mr. Sheldon at
about 14 miles from Chelsea, re-ascended alone, and
finally landed near Rumsey in Hampshire, about 75 miles
from London, having gone nearly at the rate of 20 miles
an hour. The wings used on this occasion it seems pro-
duced no deviation from the direction of the wind. Mr.
Blanchard sffid that he ascended so high as to feel a great
difficulty of breathing: and that a pigeon, which flew
a\vay from the boat, laboured for some time to sustain it-
self with its wings in the rarefied air, but after wandering
a good while, returned, and rested on the side of the boat.

On the 4th of October, Mr. Sadler, an ingenious trades-
man at Oxford, ascended at that place with an inflamma-
ble-air balloon, of his own construction and filling. And
again on the 12th of the same month he ascended at Ox-
ford, and floated to the distance of 14 miles in 17 minutes,
which is at the rate of near 50 miles an hour.

The 30th of November this year Mr. Blanchard's fifth
aerial voyage, still with his old machine, was performed in
company with Dr. J. Jeffries, a native of America. Their
voyage was about 21 miles; and it does not appcart hat
the greatest action of their oars produced any effect in di-
recting the course of the balloon.

On the 4th of January, 1785, a Mr. Harper ascended at
Birmingham with an inflammable-air balloon, and went to
the distance of 50 milts in an hour and a quarter: he

Vol. 1.

suffered no other inconvenience than a temporary deafness,
and what might be expected from the changes of wet and
cold. The thermometer descended from 40 to 28 degrees.

On the 7th of January, Mr. Blanchard, accompanied
by Dr. Jeffries, performed his sixth aerial voyage, by ac-
tually crossing the British channel from Dover to Calais,
with the same balloon which had five times before carried
him successfully through the air. They ascended with
only 30 pounds of sand ballast, besides their provisions,
some books, instruments, and other necessaries. The ma-
chine parted with the gas very rapidly, and their ballast
was soon all exhausted; after which, from time to time
they threw out every thing else in the boat, to prevent
themselves from dropping into the sea. In this way they
disposed of all their provisions, their books and instru-
ments, and finally with part of their clothes. This how-
ever bringing them near the French coast, they gradually
ascended, cleared the cliffs and houses, and landed in the
forest of Guiennes. In consequence of this voyage, the
king of France presented M. Blanchard with a gift of
12000 livres, and granted him a pension of 1200 livres a
year.

On the 19th of January, Mr. Crosbie ascended at Dub-
lin in Ireland with an inflammable-air balloon to a great
height. He rose so rapidly, that he was out of sight in 3
minutes and a half. By suddenly opening the valve, he
descended just at the edge of the sea, as he was driving to-
wards the channel, being unprovided for properly passing
over to England.

On the 23d of March, Count Zambeccari and Admiral
Sir Edward Vernon ascended at London, and sailed to
Horsham in Sussex, at the distance of 35 miles in less
than an hour. The voyage proved very dangerous, owing
to some of the machinery about the valve being damaged,
which obliged them to cut open some part of the balloon
when they were about 2 miles perpendicular height above
the earth, the barometer having fallen from 30-4 to 20-8
inches. In descending, they passed through a dense
cloud, which felt very cold, and covered them with snow.
The observations they made were, that the balloon kept
perpetually turning round its vertical axis, sometimes so
rapidly as to make each revolution in 4 or 5 seconds ;
that a peculiar noise, like rustling, was heard among the
clouds, and that the balloon was greatly agitated in the
descent.

On May the 5th, Mr. Sadler, and William Windham,
esq. member of parliament for Norwich, ascended at
Moulsey-hurst. The machine took a south-east course,
and the current of air was so strong, that they were in
great danger of being driven to sea. They had the good
fortune however to descend near the conflux of the
Thames and Medway, not a mile from the water's edge.
By an accident they lost their balloon: for while the
aeronauts were busied in securing their instruments, the
country people, whom they had employed in holding
down the machine, suddenly let go the cords, when the.
balloon instantly ascended, and was driven many miles out
to sea, where it fell, and was taken up by a trading vessel.
It was afterwards restored again, and another voyage
made with it from Manchester to Pontefract, in which
Mr. Sadler was still more unfortunate; for no person
being near when he descended, and not being able to con-
fine the balloon by his own strength, he was dragged by
it over trees and hedges ; and at last was forced to quit it
at the utmost peril of his life; after which it rose again,
G

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and was out of sight in a few minutes. It was afterwards
found near Gainsborough.

On the 12th of May, Mr. Crosbie ascended, at Dublin,
as high as the tops of the houses ; but soon descended
again with a velocity that alarmed all the spectators for
his safety. On his stepping out of the car, in an instant
Mr. M'Guire, a college youth, sprang into it, on which
the balloon ascended with him to the astonishment of the
beholders, and presently he was carried with great velo-
city towards the channel in the direction of Holyhead.
This being observed, a crowd of horsemen pursued full
speed the course he seemed to take, and could plainly
perceive the balloon descending into the sea. Lord H.
Fitzgerald, who was among the foremost, instantly dis-
patched a swift-sailing vessel mounted with oars, and all
the boats that could be got, to the relief of the gallant
youth ; whom they found almost spent with swimming,
just time enough to save his life.

The fate of M. Pilatre de Rozier, the first aerial travel-
ler, and his companion M. Romain, has been much la-
mented. They ascended at Boulogne the 15th of June,
with intent to cross the channel to England : for the first
20 minutes they seemed to take the proper direction ;
when presently the whole apparatus was seen in flames,
and the unfortunate adventurers fell to the ground from the
height of more than a thousand yards, and were killed on
the spot, their bones being broken, and their bodies crushed
in a shocking manner. The machine in which they as-
cended,consisted of a spherical balloon, 37 feet in diameter,
filled with inflammable air ; and under this was suspended
a small montgolfier, or fire balloon, of 10 feet diameter;
the gallery which suspended the aeronauts, was attached
to the net of the upper balloon by cords, which were fast-
ened to a hoop rather larger than the montgolfier, and de-
scended perpendicularly to the gallery. The montgolfier
was intended to promote and prolong the ascension, by
rarefying the atmospheric air.

On the 19th of July, at 20 minutes past 2 o'clock, Mr.
Crosbie a third time ascended at Dublin, with intent to
cross the channel to Holyhead in England. The usual
form of the boat had been changed, for a capacious
wicker basket, of a circular form; and round its upper
edges were fastened a great many bladders, which were in-
tended to render his gallery buoyant, in case of a disaster
at sea. About 300 pounds of ballast were put into the
basket, but he discharged half a hundred on his first rise.
At first a current of air carried him due west; but it
soon changed his course to north-east, pointing towards
Whitehaven. At upwards of 40 miles from the Irish
shore, he found himself within sight of both lands, and he
said it was impossible to give an adequate idea of the beau-
ties which the scenery of the sea, bounded by both lands,
presented. He rose at one time so high, that by the in-
tense cold his ink was frozen, and the mercury sunk quite
into the hall of the thermometer. He was sick, and felt a
strong prepulsion on the tympanum of the ears. At his
utmost height he thought himself stationary ; but on libe-
rating some gas, he descended to a current of air blowing
north, and extremely rough. He now entered a thick
cloud, and encountered strong blasts of wind, with thunder
and lightning, which brought him rapidly towards the sur-
face oi the water. Here the balloon made a circuit, but
falling lower, the water entered his car, and he lost his
notes of observation. All his endeavours to throw out
ballast were of no avail ; the force of the wind plunged

him into the ocean ; and with much difficulty he put on
his cork jacket. The propriety of his idea was now very
manifest in the construction of his boat ; as by the admis-
sion of the water into the lower part of it, and the suspen-
sion of his bladders, which were arranged at the top, the
water, added to his own weight, became proper ballast ;
and the balloon maintaining its poise, it became a power-
ful sail, by means of which, and a snatch-block to his
car, he went before the wind as regularly as a sailing ves-
sel. In this situation he found himself inclined to eat,
and he took a little fowl. At the distance of a league he
discovered some vessels crowding after him ; but as his
progress outstripped all their endeavours, he lengthened
the space of the balloon from the car, which gave a check
to the rapidity of his sailing, and he was at length over-
taken and saved by the Dunieary barge, which took him
on board, and steered to Dunieary, towing the balloon
after them.

A similar accident happened to Major (now General)
Money, who ascended at Norwich, on the 22d of July, at
20 minutes past 4 in the afternoon; when meeting with an
improper current, and not being able to let himself down,
on account of the smallness of the valve, he was driven
out to sea, where, after blowing about for near two hours,
he dropped into the water. Here the struggles were as-
tonishing which he made to keep the balloon up, which
was torn, and hung only like an umbrella over his. head.
A ship was once within a mile, but, whether from want of
humanity, or by mistaking the balloon for a sea monster,
they sheered off, and left him to his fate: but a boat
chased him for two hours, till just dark, and then bore
away. He now gave up all hopes, and began to wish
that Providence had given him the fate of Rozier, wither
than such a lingering death. Exerting himself however
to preserve life as long as possible, by keeping the ballcon
floating over his head, to keep himself out of the water,
into which nevertheless he sunk gradually inch by inch,
as it lost its power, till he was at length breast deep in
water, when he was taken up by a revenue cutter, at half
past eleven at night, but so weak that he was obliged to
be lifted out of the car into the ship.

About the latter end of August, the longest aerial
voyage hitherto made, was performed by Mr. Blanchard,
who ascended at Lisle, accompanied by ihe Chevalier de
L'Epinard, and travelled 300 miles in their balloon before
it descended. On this occasion, as on some former ones,
Mr. Blanchard made trial of a parachute, like a large um-
brella, invented to break the fall in case of an accident
happening the balloon: with this machine he dropped a
dog from the car soon after his ascension, which descended
gently and unhurt.

On the Sth of September, Thomas Baldwin, esq. ascend-
ed from the city of Chester, and descended at Rixton-
Moss, at 25 miles' distance, after a voyage of 2- hours and a
quarter. The greatest perpendicular altitude ascended
was about a mile and a half, and the aeronaut computed
that in some parts of the voyage he moved at the rate of
30 miles an hour. Mr. Baldwin published a very circum-
stantial account of his voyage, with many ingenious philo-
sophical remarks relating to aerostation, of which subject
his book may be considered as one of the best treatises yet
given to the public.

October the 5th, Mr. Lunardi made the first aerial
voyage in Scotland. He ascended at Edinburgh, and, after
various turnings, landed near Cupar in Fife, having de-

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scribed a track of 40 miles over the sea, and 10 over the
land, in an hour and a half. The mercury in the baro-
meter sunk as low as 18*3 inches at his greatest elevation.
November the 19th, the celebrated Blanchard ascended
at Ghent to a great height, and after many dangers de-
scended at Delft without his car> which he cut away to
lighten, the machine when he was descending too rapidly,
and slung himself by the cords to the balloon, which
served him then in the nature of a parachute. On his
first ascent, when he was almost out of sight, he let down
a dog, by means of a parachute, which came easily to the
ground.

November the 25th, Mr. Lunardi ascended at Glasgow,
and in two hours it is said he described a track of 125
miles. It is further remarkable that, being overcome
with drowsiness, he slept for about 20 minutes in the bot-
tom of the car, during this voyage.

In order to obviate the danger of a fill, arising from
any accident happening to the balloon, some experiments
were made with a parachute first by Mr. Blanchard, who
performed several other excursions in different parts of
Europe, and made some improvements in the parachute,
wings, &c. An account of the two following voyages
performed by him will be best related in Mr. Blanchard's
own words, taken from his letter, dated Leipsick, Octo-
ber the 9th, 1787, to the editors of the Paris Journal.
" I did not mention," says he, " in your interesting pa-
per, my ascension at Strasburg on the 26th of last Au-
gust: the weather was so horrible, that I mounted only
for the sake of contenting the astonishing crowd of stran-
gers, assembled there from all parts of the country. The
only remarkable thing that occurred at that time, was the
following circumstance : At the height of about 2000
yards, or a mile and half a quarter, I let down a dog
tied to the parachute, which, instead of descending gently,
was forcibly carried by a whirlwind above the clouds. I
met him soon after, bending his course directly down-
wards: when he perceived mc, he began to bark a little,
and I was going to take hold of the parachute, when an-
other whirlwind lifted him again to a great height. I lost
him for about 6 minutes, and perceived him afterwards,
with my telescope, as if sleeping in the cradle or basket
belonging to the machine. Continually agitated, and im-
petuously tossed through every point of the compass, by
the violence of the different currents of air, I determined
to end my voyage on the other side of the Rhine, after
having passed vertically over Zell. I descended at a small
village, with an intention to be assisted a little, and about
thirty men soon came within reach of the balloon very
a-propos, and fixed me to the ground. The wind was so
violent that anchors or ropes would have been of no ser-
vice. I had however added to the large aerostatic globe
a smaller one, of 60 pounds ascensional force, which
would have contributed to fix me, when once I let it
loose; but, notwithstanding this precaution, the men's as-
sistance was very necessary to me. The parachute was
yet wavering in the air, and did not come down till 12
minutes after."

"I perfcrmed my 27th ascension at Leipsick the 2flth of
September, in the midst of an incredible number of specta-
tors, forming one of the most brilliant assemblies I ever be-
held. The sky was as clear and serene as possible, and the air
so calm that many of my friends, and multitudes of others,
could follow me on horseback, and even on foot. I was
•sometimes so near them that they thought they could reach

me, but I could soon find the means of rising ; and once,
when they had actually taken hold of the cords, to see me
float with the strings in their hands, I suddenly cut them,
and mounted again in the air. All these amusing evolu-
tions were in sight of the town and its environs. At length
I yielded to the earnest solicitations of the company, and
entered the town. The next day I emptied the inflamma-
ble air into another globe, with which I intended to try
some experiments ; and I let it off with a cradle, in which
a dog was fixed. The balloon, having reached a considera-
ble height, made an explosion in its under part, as I had
imagined it would, having previously disposed it in a proper
manner for that purpose ; by which means the little ani-
mal fell gently to the ground."

" The day before yesterday having repeated this experi-
ment, I prepared the globe in such a manner as to cause
an explosion in its upper part, and added a parachute
with two small dogs fixed to it. They went so high that,
notwithstanding the serenity of the sky, the. balloon was
lost in its immense expanse. Telescopes of the best sort
became useless, and 1 began to be apprehensive for the
death of the little animals, on account of the severity of
the cold. They descended however about two hours after,
quite safe and well, in the town of Delitzsch, 3 miles from
Leipsick. I went yesterday to claim them, and found
them again over the town in the air with the parachute.
Such experiments had been already tried many times in
the course of the day, and some officers had thrown them
from the top of a steeple,in the sightof all the inhabitants,
of Delitzsch, from which they descended safe."

We shall conclude this historical sketch with a short
account of the aerial voyages performed by M. Garnerin,
in London, in 1802. His first ascent in London was on
June 29th, accompanied by Captain Sowden : they rose
from Ranelagh, and alighted near Colchester, in less than
three quarters of an hour, having in that short period tra-
velled full 60 miles. During this voyage the aeronauts
did not experience any inconvenience from the rapidity of
their motion, until they began to descend, when they were
much affected by the boisterousness of the wind: their
descent was attended with danger, and occupied some
minutes.

On the 2 1st of September, M. Garnerin undertook the
singular and desperate experiment of ascending with an
air balloon, and descending in a parachute. He ascended
from St. George's Parade, North Audley-street, and de-
scended safe into a field near the small-pox hospital at
Pancras. The balloon was as usual made of oiled silk,
covered with a net, from which ropes proceeded that ter-
minated in, or were fixed to, a single rope, a few feet below
the balloon, and to this rope the parachute was fastened
in the following manner.

The parachute was about 30 feet in diameter; and seve-
ral ropes of about 30 feet in length, which proceeded from
its edge, terminated in a common joining, from which
shorter ropes proceeded, and to the extremities of these a
circular basket was attached, in which M. Garnerin
placed himself. Now the single rope that has been said
above to proceed from the balloon, passed through a hole
in the centre of the parachute, and through tin tubes,
which were placed one after another, and was lastly fast-
ened to the basket; so that when the balloon was in the
air, by cutting this rope, the parachute with the basket
would be separated from it, and in falling downwards
would naturally open by the resistance of the air. The
G2

A E R

t 44 ]

AER

use of the tin tubes was to let the rope slip off with the
greater certainty, and to prevent it from getting entangled
with any part of the machinery belonging to the bal-
loon.

About 6 o'clock, every thing being ready, M.Garnerin
placed himself in the basket, and ascended majestically
amidst the acclamations of innumerable spectators. The
weather was the clearest and pleasantest imaginable, the
wind blew a gentle breeze from w by s, and in about
S minutes the balloon and parachute had ascended to an
immense height, when M. Garnerin, in the basket, could
scarcely be perceived. While every spectator was con-
templating the grand sight before them, M. Garnerin cut
the rope, and in an instant he was separated from the
balloon, trusting his safety to the parachute. At first,
viz, before the parachute opened, he fell with a great ve-
locity ; but as soon as it was expanded, which took place
almost immediately after, the descent became very gentle
and gradual.

In this descent a remarkable circumstance was ob-
served, namely, that the parachute with the appendage
of cords and basket, soon began to vibrate like the pen-
dulum of a clock, and the vibrations were so great, that
more than once the basket with M. Garnerin seemed to
lie horizontally, which to the spectators appeared ex-
tremely dangerous: however, the extent of the vibrations
diminished as he approached the ground. On coming to
the earth the aeronaut experienced some pretty severe
shocks, and when he came out of the basket he was much
discomposed; but he soon recovered his spirits, and sus-
tained no material injury from this desperate undertaking.

For several figures of balloons, see plate 1.

Practice of Aerostation. The first consideration in
the practice of aerostation, is the form and the size of the
machine. Various shapes have been tried and proposed,
but the globular, or the egg-like figures, are the most pro-
per and convenient, for all purposes ; and this form also
will require less cloth or silk than any other shape of the
same capacity ; so that it will both come cheaper, and
have a greater power of ascension. The bag or cover of
an inflammable-air balloon, is best made of the silk stuff
called lustring, varnished over. But for a montgolfier, or
heated-air balloon, on account of its great size, linen cloth
has been used, lined within or without with paper, and
varnished. Small balloons are made either of varnished
paper, or simply of paper unvarnished, or of gold-beater's
skin, or such-like light substances.

With respect to the form of a balloon, it will be neces-
sary that the operator remember the common proportions
between the diameters, circumferences, surfaces, and so-
lidities of spheres; for instance, that of different spheres,
the circumferences are as the diameters ; that the surfaces
are as the squares of the diameters ; and the solidities as
the cubes of the same : that any diameter is to its circum-
ference as 7 to 22, or as 1 to 3f ; so that if the diameter
of a balloon be 35 feet, its circumference will be 1 10 feet.
And if the diameter be multiplied by the circumference,
the product will be the surface of the sphere; thus 35
multiplied by 110 givers 3850, which is the surface of the
same sphere in square feet : and if this surface be divided
by the breadth of the stuff, in feet, which the balloon is
to be made of, the quotient will be the number qf feet in
length necessary to construct the balloon ; so if the stuff
be 3 feet wide, then 3850 divided by 3, gives 1283J- feet,
or 428 yards nearly, the requisite quantity of yard-wide

stuff, to form the balloon of 35 feet diameter. Hence
also, by knowing the weight of a given piece of the stuff,
as of a square foot, or square yard, it is easy to find the
weight of the whole bag, namely by multiplying the sur-
face, in square feet or yards, by the weight of a square
foot or yard : so if each square yard weigh 1 pound, then
the whole bag will weigh 428 pounds. Again, the capa-
city, or solid contents, of the sphere, will' be found by
multiplying £ of the surface by the diameter; which gives
22458 cubic feet for the capacity of the said balloon,
that is, it will contain, or displace, 2245S cubic feet of
air. From the content and surface of the balloon, so
found, is to be derived its power of ascension, thus : On
an average, a cubic foot of common air weighs li ounce,
and therefore to the number 22458, which is the content
of our balloon, adding its ypart, we have 26950 ounces,
or l6S4 pounds, for the weight of the common air dis-
placed or occupied by the balloon. From this weight
must be deducted the weight of the bag, namely 428
pounds, and then there remains a power of ] 256 pounds.

If inflammable air be used, as it is of different weights,
from ^ to -j% or ■?? the weight of common air, according
to the modes of preparing it : let us suppose for instance,
that it is ■£ of the weight of common air; then £ of 1684
is 26l pounds, which is the weight of the bag full of that
air; which being taken from 1250, leaves y<J5 pounds for
the power of the balloon when so filled with that inflam-
mable air, or the weight which it will carry up, consisting
of the car, the ropes, the passengers, ballast, &c. But if
heated air be used ; then as it is known from experiment
that, by heat, the contained air is diminished in density
about one-third only, therefore from 1684, take j of it-
self, and there remains 1123 for the weight of the con-
tained warm air ; and this being subtracted from 1256,
leaves only 133 pounds for the power of the balloon in
this case ; which being too small to carry up the car,
passengers, &c, it shows that for those purposes a larger
balloon is necessary, on Montgolfier's principles. But if
now, from the preceding computation, it be required to
find how much the size of the balloon must be increased,
that its power of ascension may be equal to any given
weight, as suppose 1000 pounds; then because the powers
are nearly as the cubes of the diameters, therefore the
diameters will be nearly as the cube roots of the powers;
but 133 and 1000 are nearly as 1 to 8, the cube roots of
which are as 1 to 2, and consequently 1 : 2 : : 35 : 70
feet, which is the diameter of a montgolfier, made of the
same thickness of stuff as the former, capable of lifting
1000 pounds.

On the same principles we can easily find the size of a
balloon that shall just float in air when made of stuff of
a given thickness or weight, and filled with air of a
given density ; the rule for which is this : From the weight
of a cubic foot of common air, subtract that of a cubic
foot of the lighter or contained air ; then divide 6 times
the weight of a square foot of the stuff, by the remainder;
and the quotient will be the diameter, in feet, of the bal-
loon that will just float at the surface of the earth. Sup-
pose, for instance, that the materials are as before, namely,
the stuff 1 pound to the square yard, or '56 ounces to the
square foot, which taken 6 times is 3f; then the cubic
foot of common air weighing ly ounce, and of heated
air § of the same, whose difference is -|; therefore 3T2 di-
vided by |-, gives 2()f feet, which is the diameter of a
montgolfier that will just float : but if inflammable air be

A E R

[ 45 ]

A E R

used of | the weight of common air, the difference be-
tween 1-f and £ of it, is 1 ; by which dividing 3/ or 10-f,
the quotient is the same 10| feet, which therefore is the
diameter of an inflammable-air balloon that will just float.
And if the diameter be more than these dimensions, the
balloons will ascend into the atmosphere.

An estimate of the height to which a given balloon will
rise in the atmosphere, may be made as follows: having
given only the diameter of the balloon, and the weight
which just balances it, or that is just necessary to keep it
from rising: compute the capacity or content of the globe
in cubic feet, and divide its restraining weight in ounces
by that content, then the quotient will
be the difference between the density
or specific gravity of the atmosphere at
the earth's surface, and that at the
height to which the balloon will rise;
therefore subtract that difference or
quotient from 1-f- or T2, the density at
the earth's surface, and the remainder
will be the density at that height: then
the height answering to that density
will be found sufficiently near in the
annexed table. Thus, in the foregoing
examples, in which the diameter of the
balloon is 35 feet, its capacity 22458,
and the power of the first 995 pounds,
or 15920 ounces, the quotient of the
latter number divided by the former, is
•709, which is the density at the utmost
height, and to which in the table answers a little more
than 2| miles, or 2j miles nearly, which therefore is the
height to which the balloon will ascend. And when the
same balloon was filled with heated air, its power was
found equal to only 13 i pounds, or 2128 ounces, and di-
viding this by 22458 the capacity, the quotient -095 taken
from 1-200, leaves 1*105 fur the density; which in the
table corresponds to half a mile, or nearer § of a mile.
And so high would these balloons ascend, if they keep
the same figure, and lost none of the contained air ; or
rather, those are the heights they would settle at : for
their acquired velocity would first carry them above that
height, so far as till all their motion should be destroyed ;
then they would descend and pass below that height, but
not so much as they had gone above ; after which they
would re-ascend, and pass that height again, but not so
far as they had gone below it; and so on for many times,
vibrating alternately above and below that point, but al-
ways less and less every time. The foregoing rule, for
finding the height to which the balloon will ascend, is in-

Height
in miles.

Density.

0

1*200

i

1*14-1

\

1-0S5

if

1031

1

0*980

H

0*932

It

0886

1|

0*842

2

0-800

n

0-761

2t

0 723

•2-1-

0-687

3

0.653

dependent of the different slates of the thermometer at

that highest point, and at the surface of the earth ; but any thing in the nature of sails can be of no use; and all

row, or numerous, they will differ the less from circles
and may be described as such. But more accurately,
the breadths of the slip, at the se-
veral distances from the points to
the middle, where it is broadest,
are directly as the sines of those
distances, radius being the sine of
the half length of the slip, or of
the distance of either point from
the middle of the slip : that is, If
a c b d be one of the slips, a u be-
ing half the circumference, or as
a quadrant conceived to be equal
to ac or ad; then will cd be to
ab, as radius or the sine of ac, to
the sine of A a. So that if the
quadrant a e or AC be divided in-
to any number of equal parts, as
here suppose 9> tnen divide the
quadrant or <)Q degrees by the
number of parts 9, and the quo-
tient 10 is the number of degrees

in each part; and hence the arcs AC, Aa, ac, &c, will
be respectively 90°, SO0, 70°, &c; and, cd being radius,
the several breadths ab, cd, ef, &c, will be respectively
the sines of 80°, 70°, 60°, &c, which are here placed op-
posite to them, the ladius being 1. Therefore, when it is
proposed to cut out slips for a globe of a given diameter;
compute the circumference, and make ab or AC a quar-
ter of that circumference, and c d of any breadth, as 3
feet, or 2 feet, or any other quantity; then multiply each
of the decimal numbers, set opposite the figure, by that
quantity, or breadth of CD, so shall the products be the
several breadths ab, cd, ef, &c.

Various schemes have been devised for conducting bal-
loons in any direction, whether vertical or sideways. As
to the vertical djrections, namely upwards or downwards,
the means are obvious, viz, in order to ascend, the aero-
naut throws out some ballast ; and that he may descend,
he opens a valve in the top of his machine by means of a
string, to let some of the gas escape ; or if it be a mont-
golfier, he increases or diminishes the fire, as he would
ascend or descend. But to direct the machine in a side
or horizontal course, is a very difficult operation, and
what has hitherto not been accomplished, except in a
small degree, and when the current of air is very gentle
indeed. The difficulty of moving the balloon sideways,
arises from the want of wind blowing upon it ; for as it
floats along with the current of air, it is relatively in a
calm, and the aeronaut feels no more wind than if the
machine were at rest in a perfect calm. For this reason,

for greater accuracy, including the allowances depending
on the different states of the thermometer, see under the
word Atmosphere, where the more accurate rules are
given at large.

The best way to make up the whole coating of the bal-
loon, is by different pieces or slips joined lengthways from
end to end, like the pieces composing the surface of a geo-
graphical globe, and contained between one meridian and
another, or like the slices into which a melon is usually
cut, and supposed to be spread flat out. Now the edges

that can be hoped for, is to be attempted by means of
oars ; and how small the effect of these must be, may ea-
sily be conceived from the rarity of the medium against
which they must act, and the great magnitude of the ma-
chine to be forced through it. We can easily assign what
force is necessary to move a given machine in the air with
any proposed velocity. From very accurate experiments
I have determined, that a globe of 6] inches in diameter,
and moving with a velocity of 20 feet per second of time,
suffers a resistance from the air which is just equal to the

of such pieces cannot be exactly described by a pairof weight or pressure of one ounce Averdupois : and further,
compasses, not being circular, but flatter or less round that with different surfaces, and the same velocity, the re-
than circular arches ; but if the slips are sufficiently nar- sistances are directly proportional to the surface nearly, a

AER

[ 46 ]

AER

double surface having a double resistance, a triple sur-
face a tripie resistance, and so on; and also, that with dif-
ferent velocities, the resistances are proportional to the
squares of the velocities nearly, so that a double velocity
produced a quadruple resistance, and 3 times the velo-
city 9 times the resistance, and so on. And hence we can
assign the resistance necessary to move a given balloon,
with any velocity. Thus, take the balloon as before of
35 feet diameter; then by comparison as above it is found
that this globe, if moved with the velocity of 20 feet per
second, or almost 14 miles per hour, will suffer a resist-
ance equal to 371 pounds; to move it at the rate of 7
miles an hour, the resistance will be 68 pounds ; and to
move it 3^ miles an hour, the resistance will be 1 7 pounds ;
and so on: and with such force must the aeronauts act
on the air in a contrary direction, to communicate such a
motion to the machine. But, if the balloon move through
a rarer part of the atmosphere, than that at the surface
of the e.arth, as J-, or £, &c, rarer, and consequently the
resistance be less in the same proportion ; yet the force of
the oars will be diminished as much ; and therefore the
same difficulty still remains. In general, the aeronaut
must strike the air, by means of his oars, with a force
just equal to the resistance of the air on the balloon, and
therefore he must strike that air with a velocity which
must be greater as the surface of the oar is less than the
resisted surface of the globe, but not in the same propor-
tion, because the force is as the square of the velocity.

Now, suppose the aeronaut act with an oar equal to
100 square feet of surface, to move the balloon above
mentioned at the rate of 20 feet per second, or 14 miles
an hour ; then must he move this oar with the great velo-
city of 62 feet per second, or near 43 miles per hour :
and so in proportion for other velocities of the balloon.
Whence it is highly probable, that it will never be in the
power of man to guide such machine with any tolerable
degree of success, especially when any considerable wind
blows, which is almost always the case.

The compositions for varnishing balloons have been va-
riously modified ; but upon the whole, the most approved
appears to be the bird-lime varnish of M. Faujas St.
Fond. Mr. Blanchard used a varnish made of elastic ro-
sin, known commonly by the name of India rubber, dis-
solved in five times its weight of rectified essential oil of
turpentine; which is also held in great esteem. "

The methods of obtaining inflammable air for filling
balloons is generally, by applying acids to certain metals,
by exposing animal, vegetable, and some mineral substan-
ces, in a close vessel to a strong fire, or by transmitting
the vapour of certain fluids through red-hot tubes.

In the first of these methods, iron, zinc, and vitriolic
acid, are the materials most generally used. The vitriolic
acid must be diluted by five or six parts of water. Iron
may be expected to yield in the common way 1700 times
its own bulk of gas; or one foot of inflammable air to
be produced by A\ ounces of iron, the same weight of oil
of vitriol, and 22£ ounces of water. Six ounces of zinc,
an equal weight of oil of vitriol, and 30 ounces of water,
are necessary for producing the same quantity of gas.

And here it should be observed, that the turnings or
chippings of great pieces of iron, as of cannons, &c,
are preferable to the filings of that metal. The weight
of the inflammable air thus obtained, by means of acid
of vitriol, is in the common way of procuring it, gene-
rally about one seventh part of the weight of common

air ; but with the necessary precautions, for philosophical
experiments, less than one tenth of the weight of common
air. Two other sorts of elastic fluids are sometimes ge-
nerated with the inflammable air, which may be sepa-
rated from it, by passing the gas through water, in which
quick lime has been dissolved. The water will absorb
these fluids, cool the inflammable air, and prevent its
over heating the balloon when introduced into it.

Inflammable air may be obtained at a much cheaper
rate by the action of fire upon various substances ; but
the gas that they yield is not so light as that produced by
the effervescence of acids and metals. The substances
proper for this are, pit-coal, asphaltum, amber, rock-oil,
and other minerals ; wood and particularly oak, camphor-
oil, spirits of wine, ether, and animal substances; all of
which yield air in different degrees, and of various specific
gravities : but pit-coal is the preferable substance.

Another method of obtaining inflammable air was dis-
covered by M. Lavoisier, and also by Dr. Priestley;
which is, by making the steam of boiling water pass
through the barrel of a gun, kept red-hot by burning
coal. Dr. Priestley used, instead of the gun barrel, a
red-hot tube of brass, upon which the steam of water has
no effect ; and which he filled with the pieces of iron se-
parated by the boring of cannon ; and by this means he
obtained an inflammable air yyth part the weight of com-
mon air. But of these three methods, the former is the
only one that can be safely depended on for obtaining
proper gas for filling large balloons ; and here it has been
recommended to use zinc instead of iron turnings, be-
cause the white vitriol, the salt produced by the union of
the vitriolic acid and zinc, is much more valuable than
the green sort produced by the union of the same acid
with iron. But though this is undoubtedly the case, it
will certainly be found upon trial, that the superior price
of the zinc, will be more than an equivalent for all the
advantage that can be derived from the additional price
of the white vitriol.

The method of filling balloons, being merely mechani-
cal, is omitted in this place ; but ample information on
this, and many other particulars, may be met with in se-
veral books expressly written on the subject; as in Ca-
vallo's History and Practice of Aerostation, Svo, 1785;
in Baldwin's Airopaidia, Svo, 1786; and the Description
des Experiences, &c, par M. Faujas St. Fond. See also
an ingenious and learned treatise on the mathematical
and physical principles of Air-balloons, by the late Dr.
Damen, professor of philosophy and mathematics in the
University of Leyden, entitled, Physical and Mathema-
tical Contemplations on Aerostatic Balloons, &c; in Svo,
at Utrecht, 1784.

It has often been discussed, says the former of these
gentlemen, whether the preference should be given to ma-
chines raised by inflammable air, or to those raised by
heated air. Each of them has its peculiar advantages and
disadvantages ; a just consideration of which seems to de-
cide in favour of those made with inflammable air. The
principal comparative advantages of the other sort are,
that they do not require to be made of so expensive ma-
terials; that they are filled with little or no expense; and
that the combustibles necessary to fill them are found al-
most every where; so that, when the stock of fuel is ex-
hausted, the. aeronaut may descend and recruit it again,
in order to proceed on his voyage. But then this kind of
machines must be made larger than the other, to take up

A F F

[ 47 ]

AGE

the same weight; and the presence of a fire is a conti-
nual trouble, as well as danger: in fact, among the many
aerial voyages that have been made and attempted with
such machines, very few have succeeded without an in-
convenience, or an accident; and some indeed have been
attended with dangerous and even fatal consequences ;
from which the other kind is in a great measure exempt.
But, on the other hand, the inflammable-air balloon must
be made of a substance impermeable to the subtle gas :
the gas itself cannot be produced without a considerable
expense; and it is not easy to find the materials and ap-
paratus necessary for the production of it in every place.
However, it has been found that an inflammable-air bal-
loon, of 30 feet in diameter, may be made so close as to
sustain two persons, and a considerable quantity of bal-
last, in the air for more than 24 hours, when properly
managed ; and possibly one man might be supported by
the same machine for three days: and it is probable that
the stuff for these balloons may be so far improved, as to
be quite impermeable to the gas, or very nearty so ; in
which case, the machine, once filled, would continue to
float for a long time. At Paris they have already attained
to a great degree of perfection in this point ; and small
balloons have been kept floating in a room for many
weeks, without losing any considerable quantity of their
levity : but the difficulty lies in the large machines : for
in these, the weight of the stuff itself, with the weight
and stress of the ropes and boat, and the folding them up,
may easily crack and rub off the varnish, and make them
leaky.

With regard to the uses to which balloons can be ap-
plied, it must be observed, that the perfection of all arts,
and the application of them to the useful purposes of
man, have always been progressive, and not the immediate
result of a fortunate discovery; if therefore at present no
great advantages have been derived from the art of aero-
station, we ought not to despair, nor discourage any future
attempts, which it is to be hoped may prove more suc-
cessful in this respect. But even in the present incom-
plete state of this art, some advantages have been derived
from it. Thus, the French ascribe to the elevation of a
balloon, and the information obtained in consequence of
thus reconnoitring the army of the enemy, the signal vic-
tory gained in the battle of Fleurus in 1794- Balloons
may likewise serve to explore and ascertain the nature of
the air in the higher regions of the atmosphere. One of
the finest experiments made on this point, is that of Gay-
Lusac, who being elevated in a balloon to the height of
nearly eight miles, the greatest ever attained by any per-
son, brought some atmospheric air from those regions,
which, on being analysed, was found to furnish the prin-
ciples of oxygen, azote, hydrogen, and carbonic acid gas,
in the same proportions as at the surface of the earth. In
fact, the application of these machines to the advance-
ment of our knowledge of the various phaenomena in me-
teorology, stands prominent, as perhaps the only means
of maturing our acquaintance with causes yet known only
by their effects.

.ESTIVAL, see Estivae.

iESTUARY, or Estuary, in Geography, an arm of
the sea, running up a good way into the land. Such as
Bristol channel, many of the friths in Scotland, and such-
like.

^THER, see Ether.

AFFECTED, or Adfected, Equation, in Algebra,

is an equation in which the unknown quantity rises to
two or more several powers or degrees. Such, for exam-
ple, is the equation j3 - p.v'1 ■+- q-c = r, in which there
are three different powers of x, namely, x\ i*, and x.

The term, affected, is also used sometimes in algebra,
when speaking of quantities that have coefficients. Thus,
in the quantity la, a is said to be affected with the co-
efficient 2. It is also said, that an algebraic quantity is
affected with the sign -t- or—, or with a radical si<rn;
meaning no more than that it has the sign -+- or — , or
that it includes a radical sign. The term adfected, or af-
fected, I think, was introduced by Vieta.

AFFECTION, in Geometry, a term used by some an-
cient writers, signifying the same as property.

Affection. Pliys. The affections of a body are cer-
tain modifications occasioned or induced- by motion ; in
virtue of which the body is disposed after such or such a
manner.

The affections of bodies, are sometimes divided into
primary and secondary.

Primary Affections, are those which arise either out of
the idea of matter, as magnitude, quantity, and figure ;
or out of the idea of form, as quality and power; or out
of both, as motion, place, and time.

Secondary, or derivative Affections, are such as arise out
of primary ones, as divisibility, continuity, contiguity,
&c, which arise out of quantity; regularity, irregularity,
&c, which aiise out of figure, &c-

AFFINITY, in natural philosophy, the tendency which
the particles of matter have to be attracted, or united to
each other.

AFFIRMATIVE Quantity, or Positive Quantity,
one which is to be added, or taken effectively ; in contra-
distinction to one that is to be subtracted, or taken de-
fectively.— -The term affirmative was introduced by Vieta.
Affirmative Sign, or Positive Sign, in Algebra,
the sign of addition, thus marked H-, and is ca\\e A plus,
or more, or added to. When set before any single quan-
tity, it serves to denote that it is an affirmative or a po-
sitive quantity; when set between two or more quantities,
it denotes their sum, showing that the latter are to be
added to the former. So •+- 6, and -+- a, and +ab,
are affirmative quantities; also 6 ■+- 8 -+- 10 denote the
sum of 6, 8, and 10, which is 24, and are read thus,
6 plus 8 plus 10. Also a ■+■ b -t- c denote the sum of the
quantities represented by a, b, and c, when added toge-
ther. It seems now not easy to ascertain with certainty,
when, or by whom, this sign was first introduced; but it
was probably by the Germans, as I find it first used by
Stifelius in his Arithmetic, printed in 1544. When quan-
tities are set down without a prefixed sign, as a, or b, or x,
&c, they are understood to be affirmative ones.

The early writers on Algebra used the word plus in La-
tin, or piu in Italian, for addition, and afterwards the ini-
tial p only, as a contraction; like as they used minus, or
meno, or the initial m only, for subtraction : and thus these
operations were denoted in Italy by Lucas deBurgo, Tar-
talea, and Cardan, while the signs ■+- and — were em-
ployed much about the same time in Germany by Stife-
lius, Scheubelius, and others, for the same operations.

AGE, in Chronology, is used for a century, being a
system or period of a hundred years.

Chronologists also divide the time since the creation of
the world into three ages : The first, from Adam to Mo-
ses, which they call the age of nature; the second from

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Moses to Jesus Christ, called the age of the law ; and the
third, or age of grace, from Jesus Christ till the end of
the world.

Age of the Moon, in Astronomy, is the number of days
elapsed since the last new moon. To find the moon's age,
for any time nearly, for ordinary uses; add together the
epact, the day of the current month, and the number of
months from March to the present month inclusive ; the
sum is the moon's age : but if the sum exceed 29, deduct
this number from it in months that have 30 days, or 30
in those that have 31 ; and the remainder will be the age.
— At the end of 19 years the moon's age returns upon
the same day of the month, but falls a little short of the
same hour of the day.

AGENT, Agens, in Physics, that by which a thing is
done or effected ; or any thing having a power by which
it acts on another, called the object.

AGGREGATE, the sum or result of several things
added together. See Sum.

AGITATION, in Physics, a brisk intestine motion, ex-
cited among the particles of a body. Thus, fire agitates
the subtlest particles of bodies. Fermentations, and ef-
fervescences, are produced by a brisk agitation of the par-
ticles of the fermenting body.

AGNESI (Maria Gaetana), an Italian young lady,
highly celebrated for her learning and talents. Signora
Agnesi was born at Milan in 1718, and by her genius and
industry soon became profoundly learned in several lan-
guages, as well as in the mathematical and philosophical
sciences. While very young she composed an excellent
treatise on the doctrine of fluxions, which was published
at Milan 1748, under the title of Analytical Institutions,
in 2 volumes 4to; a very useful work, which was ho-
noured by a French translation, at Paris, by Cousin, in
1775 ; and by an English translation, made many years
since by the late professor Colson, of Cambridge, and
lately published at the expense of Mr. Baron Maseres,
under the care of the Rev. John Holliiis, in 2 vols. 4to,
1801. In just consideration of such eminent qualifica-
tions, S. Agnesi was appointed public professor of mathe-
matics at Bologna in 1760, an office which she discharged
with the greatest credit for some time.

It is with regret however we add, that neither her in-
clination for those honourable intellectual pursuits, nor a
desire of preserving and increasing the celebrity she had
obtained, nor the entreaties of her family and friends,
could prevent her from devoting herself to a monastic
life, among the sisters known by the name of the Blue
Nuns. From the period of her retiring to the cloister, it
may be concluded that her life was not chequered with
any variety of incidents. Accordingly, we only fur-
ther learn that this illustrious ornament of her sex died
about the year 1770.

AGUILON, or Aquilon, (Francis), was a Jesuit of
Brussels, and professor of philosophy at Doway, and of
theology at Antwerp. He was one of the first that in-
troduced mathematical studies into Flanders. He wrote
a large work on Optics, in 6 books, which was published
in folio, at Antwerp, in l6l3 ; and a treatise of Projec-
tions of the Sphere. He promised also to treat upon Ca-
toptrics and Dioptrics; but this was prevented by his
death, which happened at Seville, in the year l6l7.

AIR, in Natural Philosophy and Chemistry, a general
term used to denote such invisible rare fluids as possess a
very high degree of elasticity, and are not condensable

into a visible fluid by any degree of cold; in this respect
air is distinguished from vapour, which may be thus con-
densed.

This term was originally applied to the air of which
our atmosphere is composed, and considered by the an-
cients as a simple element ; but the researches of modern
chemists have shown it to be a compound, and that it
may be resolved into its several component parts, (see
Gas); but at present we shall only consider it as relating
to the atmosphere, without entering into its chemical pro-
perties.

Air, Atmospheric, is an invisible, insipid, inodorous,
heavy, and elastic fluid, possessing great mobility, sus-
ceptible of rarefaction, and of condensation ; it surrounds
the terrestrial globe to a certain height, the. entire mass
constituting the atmosphere. The consideration of this
fluid cannot but be a highly interesting subject, since it is
the depository of the signs of our thoughts and our af-
fections; at the same time, that it serves for the preserva-
tion of life in every animated being.

Mechanical Properties and Effects of Air. Of these
the most considerable are its fluidity, its weight, and its
elasticity.

1. Its Fluidity. — The great fluidity of the air is mani-
fest from the facility with which bodies traverse it ; as in
the propagation of, and easy conveyance it affords to,
sounds, odours and other effluvia and emanations that
escape from bodies : for these effects prove that it is a
body whose parts give way to any force, and in yielding

,are easily moved among themselves ; which is the defi-
nition of a fluid. That the air is a fluid is also proved
from this circumstance, that it is found to exert an equal
pressure in all directions; an effect which could not take
place otherwise than from its extreme fluidity. Neither
has it been found that the air can be deprived of this pro-
perty, whether it be kept for many years together con-
fined in glass vessels, or be exposed to the greatest natural
or artificial cold, or condensed by the most powerful pres-
sure ; for in none of these circumstances has it ever been
reduced to a solid state.

2. Its Weight or Gravity. — The weight or gravity of the
air, is a property belonging to it as a body ; for gravity is
a property essential to matter, or at least a property found
in all bodies. But independent of this, we have many di-
rect proofs of its gravity from sense and experiment :
thus, the hand laid close upon the end of a vessel, out of
which the air is drawn at the other end, soon feels the
load of the incumbent atmosphere : thus also, two hol-
low segments of a sphere, of about 4 inches in diameter,
exactly fitting each other, being emptied of air, are, by
the weight of the ambient air, pressed together with a
force which requires the weight of 1SS pounds to sepa-
rate them ; and that they are thus forcibly held together
by the pressure of the air, is made evident by suspending
them in an exhausted receiver, for then they quickly se-
parate of themselves, and fall asunder. Again, if a tube,
close at one end, be filled with quicksilver, and the open
end be immerged in a bason of the same fluid, and so
held upright, the quicksilver in the tube will be kept raised
up in it to the height of about 30 inches above the sur-
face of that in the bason, being supported and balanced
by the pressure of the external air upon that surface : and
that this is the cause of the suspension of the quicksilver
in the tube, is made evident by placing the whole, appa-
ratus under the receiver of an air-pump ; for then the

A I R

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A I R

fluid descends in the tube in proportion as the receiver is
exhausted of its air; and on gradually letting in the air
again, the quicksilver reascends to its former height : and
this is the common barometer, or what is called, from its
inventor, the Torricellian experiment. Nay farther, air
can actually be weighed like any other body : for a rigid
vessel, full even of common air, by a nice balance is
found to weigh more than when the air is exhausted from
it; and the effect is proportionally more sensible, if the
vessel be weighed full of condensed air, and more still if
it be weighed in a receiver void of air.

But, though we have innumerable proofs of the gravi-
tating property of the air, yet the full discovery of the
laws and circumstances of it are certainly due to the mo-
derns. It cannot indeed be denied, that several of the
ancients had some confused notions about this property :
thus Aristotle says, that all the elements have gravity, and
even air itself; and as a proof of it, says that a bladder
inflated with air, weighs more than the same when empty ;
and Plutarch and Stobseus quote him as teaching that the
air in its weight is between that of fire and of earth ; and
farther, he himself, treating of respiration, reports it as
the opinion of Empedocles, that he ascribes the cause of
it to the weight of the air, which by its pressure forces
itself into the lungs; and much in the same way are the
sentiments of Asclepiades expressed by Plutarch, who re-
presents him as saying, among other things, that the ex-
ternal air, by its weight, forcibly opened its way into the
breast. But nevertheless it is certain, however unreason-
able it may seem, that Aristotle's followers departed in
this instance from their master, by asserting the contrary
for many ages together. Indeed several of the phaeno-
mena arising from this property, have been remarked
from the highest antiquity. Many centuries since, it was
known that by sucking the air from an open pipe, having
its extremity immersed in water, this fluid rises above its
level, and occupies the place of the air. In consequence
of such observations, sucking pumps were contrived, and
various other hydraulic machines; as Heron's syphons,
described in his Spiritalia or Pneumatics, and the Water-
ing-pots known in Aristotle's time under the name of
clepsydrw, which alternately stop or run as the finger
closes or opens their upper orifice. Indeed the reason as-
signed, by philosophers many ages after, for this pheno-
menon, was a pretended horror that nature conceives for
a vacuum, which, rather then endure it, makes a body
ascend contrary to the powerful solicitation of its gravity.
Even Galileo, with all his sagacity, could not for some
time hit upon any thing more satisfactory; for he only
assigned a limit to this dread of vacuity: having observed
that sucking pumps would not raise water higher than lrj
brasses, or 34 English feet, he limited this abhorring force
of nature, to one that was equivalent to the weight of a
column of water 34 feet high, on the same base as the
void space. Consequently he pointed out a way of making
a vacuum, by means of a hollow cylinder, whose piston
is charged with a weight sufficient to detach it from the
close bottom turned upwards : this effort he called the
measure of the force of vacuity, and made use of it for
explaining the cohesion of the parts of bodies.

Galileo, however, was well apprised of the weight of
the air as a body : in his Dialogues he shows two ways of
demonstrating it, by weighing it in bottles : the transition
was easy from one discovery to another : yet still Gali-
leo's knowledge of the matter was imperfect, that is, as to
Vol. I.

the particular instance of the suspension of a fluid above
its level, by the pressure of the external air.

At length, Torricelli conceived the happy idea, that
the counterpoise which keeps fluids above their level,
when nothing presses upon their internal surface, is the
mass of air resting upon the external one. He discovered
it in the following manner: In the year 1643 this disciple
of Galileo, on occasion of executing an experiment on the
vacuum formed in pumps, above the column ©f water,
when it exceeds 34 feet, thought of using some heavier
fluid, such as quicksilver. He conceived, that whatever
might be the cause by which a column of water of 34
feet high is sustained above its level, the same force would
sustain a column of any other fluid, which weighed as
much as that column of water, on the same base; whence
he concluded that quicksilver, being about 14 times as
heavy as water, would not be sustained higher than 29
or 30 inches. He therefore took a glass tube of several
feet in length, sealed it hermetically at one end, and filled
it with quicksilver ; then inverting it, and holding it up-
right, by pressing his finger against the lower or open ori-
fice, he immersed that end in a vessel of quicksilver ;
when removing his finger, and suffering the fluid to run
out, the event verified his conjecture ; the quicksilver,
faithful to the laws of hydrostatics, descended till the co-
lumn of it was about 30 inches high above the surface of
that in the vessel below. And hence Torricelli concluded,
that it was no other than the weight of the air incum-
bent on the surface of the external quicksilver, which
counterbalanced the fluid contained in the tube.

By this experiment Torricelli not only proved, what
Galileo had done before, that the air had weight, but also
that it was its weight which kept water and quicksilver
raised in pumps and tubes, and that the weight of the
whole column of it was equal to that of a like column
of quicksilver of 30 inches in height, or of water 34 or
35 feet high ; but he did not ascertain the weight of any
particular quantity of it, as a gallon, or a cubic foot of it,
nor its specific gravity to water, which had been done by
Galileo, though to be sure with no great accuracy, for he
only proved that water was more than 400 times heavier
than air.

Father Mersenne, who kept up a correspondence with
most of the literati in Italy, was informed of Torricelli's
experiment in 1644, and communicated it to those of
France, who presently repeated the experiment : Messrs.
Pascal and Petit made it first, and varied it several ways;
which gave occasion to the ingenious treatise which Pascal
published at 23 years of age, entitled Experiences Nou-
velles touchant la Vuide. In this treatise indeed he makes
use of the old principle of fuga vacui; but afterwards
getting some notion of the weight of the air, he soon
adopted Torricelli's idea, and devised several experiments
to confirm it. One of these was to procure a vacuum
above the reservoir of quicksilver; in which case, he
found the column sink down to the common level : but
this appearing to him not sufficiently powerful to dissi-
pate the prejudices of the ancient philosophy, he prevail-
ed on M. Perier, his brother-in-law, to execute the cele-
brated experiment of Puy-de-Domme, who found that
the height of the quicksilver half-way up the mountain
was less, by some inches, than at the foot of it, and still
less at the top: so that it was now put out of doubt, that
it was the weight of the atmosphere which counterpoised
the quicksilver.

H

A I R

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Des Cartes too had a right notion of this effect of the
air, to sustain fluids above their level, as appears by some
of his letters about this time, and some years before ; and
in one of those he lays claim to the idea of the Puy-de-
Domme experiment. After having desired M. de Carcavi
to inform him of the success of the experiment, made by
M. Pascal, he adds, " I had reason to expect this from
him, rather than from you, because I first proposed it to
him two years since, assuring him at the same time, that
though I "had not tried it, yet I could not doubt of the
consequence ; but as he is a friend of M. Roberval, who
professes himself no friend to me, I suppose he is guided
by that gentleman's passions." See more of this history
under Barometer.

As to the actual weight of any given portion of com-
mon air, it seems that Galileo was the first who deter-
mined it experimentally ; and he gives two different me-
thods, in his Dialogues, for weighing it in bottles : he did
not however perform the experiment very accurately, as
he stated from the result that the gravity of water was to
' that of air rather above 400 to J, as before said.

A quantity of air was next weighed by Mersenne in a
very ingenious manner. His idea was to weigh a vessel
both when full of air, and when emptied of it: to make
the vacuum for this purpose, he knew no better way
than by expelling the air out of an eolipile by heating it
red-hot : by weighing it both when cold and hot, he found
a certain difference; which however was not the exact
weight of that capacity of air, because the vacuum was
not perfect. For, by plunging the eolipile, when red-liot,
into water, just so much water entered as was equal
in bulk to the air that had been expelled; he then took
it out and weighed it with the water, which gave the
weight of the same bulk of water ; and on comparing
this with the former difference, or weight of air expelled,
he found their proportion to be as 1300 to 1 ; which is
as wide of the truth as Galileo's proportion, namely 400
to ] , but the contrary way. And it is remarkable that
the mean between the two, namely 850 to 1, happens to
be very near the true proportion, as settled by other more
accurate experiments.

Mr. Boyle, by, a more accurate experiment, found the
proportion to be that of 938 to J. And Mr. Hauksbee
found it as 850 to 1, proceeding on the same principles as
Mersenne, with a three-gallon glass bottle, but extracting
the air out of it with the air-pump, instead of expelling
it by fire; the height of the barometer being at that
time 29-7 inches. Also by other experiments made be-
fore the Royal Society by Mr. Hauksbee, Dr. Halley,
Mr. Cotes, and others, the proportion was always between
" 800 and 900 to 1, but rather nearer the latter, namely,
being first found as 840 to 1, then as 852 to 1, and a
third time as 860 to 1 ; the barometer then standing at
29| inches, and the weather warm. Mr. Cavendish de-
termines the ratio 800 to I, the barometer being. 29|,
and the thermometer at 50°; and Sir George Shuckburgh,
by a very accurate experiment, finds it 836 to 1, the ba-
rometer being at that time 29'27> and the thermometer at
51°. The medium of all these is about 832 or 833 to 1,
when reduced to the pressure of 30 inches of the barome-
ter, and the mean temperature 55° of the thermometer.
Upon the whole therefore, it may be safely concluded
that, when the barometer is at 30 inches, and the ther-
mometer at the mean temperature 55°, the density or
gravity of water, is to that of air, as 833^ to 1, that is as

^^2 to 1, or as 2500 to 3 ; and that for any changes in
the height of the barometer, the ratio varies proportion-
ally; also that the density of the air is altered by the
-,-J^th part'for every degree of the thermometer above or
below temperate.

This number, which is a very good medium among
them all, we have chosen with the fraction -£, because it
gives exactly 1\ ounce for the mean weight of a cubic
foot of air, the weight of the cubic foot of water being
just 1000 ounces averdupois, and that of quicksilver equal
to 13600 ounces.

Air, then, having been shown to be a heavy fluid sub-
stance, the laws of its gravitation and pressure must be
the same as those of water and other fluids ; and conse-
quently its pressure must be proportional to its perpendi-
cular altitude. Which is exactly conformable to experi-
ment; for on removing the Torricellian tube to different
heights, where the column of air is shorter, the column
of quicksilver which it sustains is shorter also, and that
nearly at the rate of 100 feet for -j^. of an inch of quick-
silver. And on these principles depend the structure and
use of the barometer.

From the same principle it likewise follows that air,
like other fluids, presses equally in all directions. And
hence it happens, that soft bodies endure this pressure
without change of figure, and hard or brittle bodies with-
out breaking; being equally pressed on all parts : but if
the pressure be taken off, or diminished, on one side, the
effect of it is immediately perceived on the other. See
Atmosphere, for the total quantity of effects and pres-
sure, and the laws of different altitudes, &c.

From the weight and fluidity of the air, jointly consi-
dered, many effects and uses of it may easily be deduced.
By the combination of these two qualities, it closely in-
vests the earth, with all the bodies upon it, constrin»ing
and binding them down with a great force, namely a
pressure equal to about 15 pounds upon every square
inch. Hence, for example, it prevents the arterial vessels
of plants and animals from being too much distended by
the impetus of the circulating juices, or by the elastic
force of the air so copiously abounding in them. For
hence it happens, that on a diminution of the pressure of
the air, in the operation of cupping, we see the parts of
the body grow tumid, which causes an alteration in the
circulation of the fluids in the capillary vessels. And the
same cause hinders the fluids from transpiring through
the pores of their containing vessels, which would other-
wise cause the greatest debility, and often destroy the ani-
mal. To the same two qualities of the air, weight and
fluidity, is owing the mixture of bodies contiguous to one
another, especially fluids ; for several liquids, as oils and
salts, which readily mix of themselves in air, will not mix
at all in vacuo. With many other natural phenomena.

3. Elasticity. Another quality of the air, from whence
arise a multitude of effects, is its elasticity; a quality by
which it yields to the pressure of any other bodies, by
contracting its volume; and dilates and expands itself
again on the removal or diminution of the pressure. This
quality is the chief distinctive property of air, the other
two being common to all fluids.

Of this property we have innumerable instances. Thus,
for example, a blown bladder being squeezed in (he hand,
we find a sensible resistance from the included air; and,
on taking off the pressure, the compressed parts immedi-
ately restore themselves to their former round figure.

AIR

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A I R

And on this property of elasticity depend the structure
and uses of the air-pump.

Every particle of air makes a continual effort to dilate
itself, thus acting forcibly against all the neighbouring
particles, which also exert .the like force in return ; but
if their resistance happen to cease, or be weakened, the
particle immediately expands to an immense extent.
Hence it is that thin glass bubbles, or bladders, filled with
air, and placed under the receiver of an air-pump, will,
on pumping out the air, burst asunder by the force of
the air which they contain. So likewise a close flaccid
bladder, containing only a small quantity of air, being
put under the receiver, swells as the receiver is exhausted,
" and at length appears quite full. And the same thing
happens by carrying the flaccid bladder to the top of a
very high mountain.

The same experiment shows that this elastic property
of the air is very different from the elasticity of solid bo-
dies, and that their dilatation arises from a different
cause. For when air ceases to be compressed, it not only
dilates, but occupies a far greater space, and exists under
a volume immensely larger than before ; whereas solid
elastic bodies only resume their former figure.

It is plain that the weight or pressure of the air does
not at all depend on its elasticity, and that it is neither
more nor less heavy than if it were not at all elastic. But
from its being elastic, it follows, that it is susceptible of a
pressure, which reduces it to such a space, that the force
of its elasticity, which re-acts against the pressing weight,
is exactly equal to that weight. Now the law of the
elasticity is such, that it increases in proportion to the
density of the air, and that its density increases in pro-
portion to the forces or weights which compress it. Now,
there is a necessary equality between action and re-ac-
tion ; that is, the gravity of the air, which effects its com-
pression, and the elasticity of it, which gives it its ten-
dency to expansion, are equal.

So that, the elasticity increasing or diminishing, in the
same proportion as the density increases or diminishes,
that is, as the distances among its particles decrease or
increase; it is no matter whether the air be compressed,
and retained in any space, by the weight of the atmo-
sphere, or by any other cause; as in either case it must
endeavour to expand with the same force. And there-
fore, if such air as is near the earth be inclosed in a ves-
sel, so as to have no communication with the external air,
the pressure of such inclosed air will be exactly equal to
that of the whole external atmosphere. And accordingly
we find that quicksilver is sustained to the same height,
by the elastic force of air inclosed in a glass vessel, as by
the whole pressure of the atmosphere. — And on this prin-
ciple of the condensation and elasticity of the air, de-
pends the structure and use of the air-gun.

That the density of the air isalways directlyproportion-
al to the force or weight which compresses it, was proved
by Boyle and Mariotte, at least as far as their experi-
ments go on this head : and Mr. Mariotte has shown that
The same rule takes place in condensed air. However,
this is not to be admitted as scrupulously exact; for when
air is very forcibly compressed, so as to be reduced to
^lh of its ordinary bulk, the effect does not answer pre-
cisely to the rule ; for in this case the air begins to make
a. greater resistance, and requires a stronger compression,
than is given by the above proportion. And hence it ap-
pears, that the particles of air cannot, by means of any

possible weight or pressure, however great, be brought
into perfect contact, or that it cannot thus be reduced to
a solid mass ; and consequently, that there must be a li-
mit to which this condensation of the air can never ar-
rive. The same remark is true with regard to the rare-
faction of air, namely, that in very high degrees of rare-
faction, the elasticity is decreased rather more than in
proportion to the weight or density of the air : and hence
there must also be a limit to the rarefaction and expan-
sion of the air, by which it is prevented from expanding
to infinity.

To what degree air is susceptible of condensation, by
compression, is not certainly known. Mr. Boyle con-
densed it 13 times more than in its natural state, by this
means : others have compressed it into -^th part of its
ordinary volume ; Dr. Hales made it 38 times more dense,
by means ofa press; but by freezing water in a hollow
cast-iron ball or shell, he reduced it to 1522 times less
space than it naturally occupies; in which state it must
have been of almost double the density or specific gravity
of water: And as water is not compressible, except in a
very small degree, it follows from this experiment, that
the particles of air must be of a nature very different
from those of water ; since it would otherwise be impos-
sible to reduce air to a volume above 800 times less than
in its common state ; there is however some reason to
doubt of the accuracy of this experiment, as has been
asserted by Dr. Halley, from some experiments performed
in London, and others at Florence by the Academy del
Cimento, that it may be safely concluded that no force what-
ever is capable to reduce air into a space 800 times less
than that which it naturally occupies near the surface of
the earth, at least by pressure. But others, and particularly
M. Amontons, in the Memoirs of the French Academy,
maintain, that there is no affixing any limits to its con-
densation ; that greater and greater forces will reduce it
into less and less compass ; and that it is only elastic in
virtue of the fire it contains ; and that, as it is impossible
ever absolutely to drive out all its fire, it is impossible ever
to make the utmost condensation. We may further add,
that the air contained in nitre, is probably condensed
about lrjOO times. See my Tracts, vol. 3, p. 303.

The elasticity of the air exerts its force equally in all
directions; and when it is at liberty, and freed from the
cause which compressed it, it expands equally on all sides,
and in consequence always assumes a spherical figure in
the interstices of the fluids in which it is lodged. This is
evident in liquors placed in the receiver of an air-pump,
by exhausting the air; at first there appears a multitude
of exceeding small bubbles, like grains of fine sand, dis-
persed through the fluid mass, and rising upwards; and
as more air is pumped out, they enlarge in size; but still
they continue round. Also if a plate of metal be im-
merged in the liquor, on pumping, its surface will be seen
covered over with small round bubbles, composed of the
air which adhered to it, now expanding itself. And for
the same reason it is that large glass globes are always
blown up of a spherical shape, by blowing air through
an iron tube into a piece of melted glass at the end of
the pipe.

The expansion of the air, by virtue of its elastic pro-
pert)', when only the compressing force is taken off, or
diminished, is found to be surprisingly great ; and yet we
are far from knowing the utmost dilatation of which it is
capable. In several experiments made by Mr. Boyle, it
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expanded first into 9 times its former space ; then into 31
tiroes ; then into 60, and then into 150 times. After-
wards, it was brought to dilate into S000 times its first
space; then into 10000, and at last even into 13679
times its space ; and all this solely by its own natural ex-
pansive force, by only removing the pressure, but without
the help of fire. And on this principle depends the con-
struction and use of the Manometer.

The elasticity of the air, under one and the same pres-
sure, is still farther increased by heat, and diminished by
cold, and that, by some late accurate experiments made
by Sir George Shuckburgh, at the rate of the 440th part
of its volume nearly, for each degree of the variation of
heat, from that of temperate, in Fahrenheit's thermo-
meter.

Mr. Hauksbee observed, that a portion of air inclosed
in a glass tube, when the temperature was at the freezing
point, formed a volume which • was to that of the same
quantity of air in the greatest heat of summer in England,
as 6 to 7- And it has been found by several experiments,
that air is expanded i of its natural bulk by applying the
heat of boiling water to it.

Dr. Hales found that the air in a retort, when the bot-
tom of the vessel just became red-hot, was dilated into
double its former space; and that in a white, or almost
melting heat, it filled thrice its former space : but Mr.
Robins found that air was expanded, by means of the
white or fusing heat of iron, to 4 times its former bulk.
See also several ingenious experiments on the elasticity
of the air, in the Philos. Trans, for the year 1777, by Sir
George Shuckburgh and Colonel Roy.

This properly explains the common effect observed on
bringing a close flaccid bladder near the fire to warm it ;
when it is presently found to swell as if more air were
blown into it. And on this printiple depends the struc-
ture and office of the thermometer.

M. Amontons first discovered that, with the same heat,
air will expand in a degree proportioned to its density.
And on this foundation the ingenious author has formed
a discourse, to prove " that the spring and weight of the
air, with a moderate degree of warmth, may enable it to
produce even earthquakes, and others of the most vehe-
ment commotions of nature." He computes, that at the
depth of the 74th part of the earth's radius below the sur-
face, the natural pressure of the air would reduce to the
density of gold; and thence infers that all matter below
that depth, is probably heavier than the heaviest metal
that we know of. And hence again, as it is proved that
the more the air is compressed, the more does the same
degree of fire increase the force of its elasticity ; we may
infer that a degree of heat, which in our orb can pro-
duce only a moderate effect, may have a very violent one
in such lower orb ; and that, as there are many degrees
of heat in nature, beyond that of boiling water, it is pro-
bable there may be some whose violence, thus assisted by
the density of the air, may be sufficiently powerful to tear
asunder the solid globe. Mem. de PA cad. 1703.

Many philosophers have supposed that the elastic pro-
perty of the air depends on the figure of its corpuscles,
which they take to be ramous: some maintain that they
are so many minute flocculi, resembling fleeces of wool:
others conceive them rolled up like hoops, and curled
like wires, or shavings of wood, or coiled like the springs
of watches, and endeavouring to expand themselves by
virtue of their texture.

But Sir Isaac Newton (Optics, Qu. 31, &c,) explains
the matter in a different way : such a contexture of parts
he thinks by no means sufficient to account for that ama-
zing power of elasticity observed in air, which is capable
of dilating itself into above a million of times more space
than it occupied before: but, he observes, as it is known
that all bodies have an attractive and a repelling power ;
and as both these are stronger in bodies, the denser, more
compact, and solid they are; hence it follows that when,
by heat, or any other powerful agent, the attractive force
is overcome, and the particles of the body separated so
far as to be out of the sphere of attraction ; the repelling
power, then commencing, makes them recede from each
other with a strong force, proportionable to that with
which they before cohered ; and thus they become per-
manent air.

And hence, he says, it is, that as the particles of air
are grosser, and rise from denser bodies, than those of
transient air, or vapour, true air is more ponderous than
vapour, and a moist atmosphere lighter than a dry one.

And M. Amontons makes the elasticity of air to arise
from the fire it contains; so that by augmenting the de-
gree of heat, the rarefaction will be increased to a far
greater degree than by a mere spontaneous dilatation.

The elastic power of the air becomes the second great
source of the remarkable effects of this important fluid.
By this property it insinuates itself into the pores of bo-
dies, where, by means of this virtue of expanding, which
is so easily excited, it must put the particles of those bo-
dies into perpetual vibrations, and maintain a continual
motion of dilatation and contraction in all bodies, by the
incessant changes in its gravity and density, and conse-
quently its elasticity and expansion.

This reciprocation is observable in several instances,
particularly in plants, in which the trachea? or air-vessels
perform the office of lungs; for, as the heat increases or
diminishes, the air alternately dilates and contracts, and
so by turns compresses the vessels, and eases them again;
thus promoting a circulation of their juices And hence
it is found that no vegetation or germination is carried on
in vacuo.

It is from the same cause too, that ice is burst by the
continual action of the air contained in its bubbles.
Thus too, glasses and other vessels are frequently cracked,
when their contained liquors are frozen ; and thus also
large blocks of stone, and entire columns of marble, some-
times split in the winter season, from some little bubble
of included air acquiring an increased elasticity : and for
the same reason it is that so few stones will bear to be
heated by a fire, without cracking into many pieces, by
the increased expansive force of some air confined within
their pores. From the same source arise also all putre-
faction and fermentation ; neither of which can be car-
ried on in vacuo, even in the best-disposed subjects. And
even respiration, and animal life itself, are supposed, by
many authors, to be conducted, in a great measure, by
the same principle of the air. And as we find such great
quantities of air generated by the solution of animal and
vegetable substances, a good deal must constantly be
raised from the dissolution of these elements in the sto-
mach and bowels. In fact, all natural corruption and al-
teration seem to depend on air; and even metals, parti-
cularly gold, only seem to be durable and incorruptible,
in so far as they are impervious to air.

Composition, &c, of the air. We have already ob-

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served, that for many ages the air we breathe was consi-
dered as a simple homogeneous fluid : various exhalations
and particles of bodies arising from the earth were indeed
observed to mix with it, and on these its different degrees
of salubrity were supposed to depend ; but the aeriform
fluid itself was never, till within a few years, known to be
a compound substance. This point however has been
clearly ascertained by the discoveries of modern che-
mists, particularly those of Priestley, Black, Caven-
dish, Lavoisier, Fourcroy, &c. P'rom the united testimo-
nies of these discoveries we learn, that atmospheric air-is
composed of at least two species of air, or elastic fluids,
which are called vital air, or oxygen gas, and azotic gas.
The former is the great agent in respiration and combus-
tion ; and upon the proper proportion of both depends
the purity of the atmosphere. The latter possesses con-
trary qualities, is noxious to animals, and incapable of
maintaining combustion. The proportion of these in 100
parts of atmospheric air, is commonly about 26 or 27
parts of oxygen gas, and 74 or 73 parts of azotic gas, by
weight ; or about 22 parts of the former, and 78 of the
latter, in bulk.

These two gases however, though they constitute the
principal part of atmospheric air, are not the only ones
that enter i»to its composition ; for beside the numerous
particles of water and other substances, it contains about
1 part in 100 of carbonic acid gas, and a still smaller
quantity of hydrogen gas; and from experiments made
upon air brought from the higher regions of the atmo-
sphere, by means of air-balloons, these constituent parts
of it were found in nearly the same proportion as at the
surface of the earth. For further particulars on this sub-
ject, see Azote, Gas, Oxygen, &c.

For the resistance of the air, see Resistance.

Air-Balloon. See Aerostation.

Air-Gun, in Pneumatics, is a machine for propel-
ling bullets with great violence, by the sole means of con-
densed air.

The first account we meet with of an air-gun, is in the
Elemens d'Artillerie of David Rivaut, who was preceptor
to Louis xifi. of France. He ascribes the invention to
Marin, a burgher of Lisieux, who presented one to
Henry iv. '■

To construct a machine of this kind, it is only neces-
sary to take a strong vessel of any sort, into which the air
is to be thrown or condensed by means of a syringe, or
otherwise, the more the better; then a valve is suddenly
opened, which lets the air escape by a small tube in which
a bullet is placed, and which is thus violently forced out
before the air.

It is evident then that the effect is produced by virtue
of the elastic property of the air; the force of which, as
has been shown in the last article, is nearly proportional
to its condensation ; and therefore the greater quantity
that can be forced into the engine, the greater will be the
effect. Now this effect will be exactly similar to that of
a gun charged with powder, and therefore we can easily
form a comparison between them : for inflamed gun-pow-
der is nothing more than very condensed elastic air; so
that the two forces are exactly similar. It is shown by
Mr. Robins, in bis New Principles of Gunnery, that the
fluid of inflamed gun-powder, has, at the first moment, a
force of elasticity equal to about 1000 times that of
common air; and I have shown (vol. 3 of my Tracts)
that the first force of the inflamed powder i9 often more

than 2000 times that of the common air. Therefore it is
necessary that air should be condensed a 1000 or 2000
times more than in its natural state, to produce the same
effect as gun-powder. But then it is to be considered,
that the velocities with which equal balls are impelled,
are directly proportional to the square roots of the forces;
so that if the air in an air-gun be condensed only 10
times, the velocity it will project a ball with, will be, by
that rule, -j%th of that arising from gun-powder ; and if
the air were condensed 20 times, it would communicate
a velocity of £th of that of gun-powder. But in reality
the air-gun shoots its ball with a much greater proportion
of velocity than as above, and for this reason, namely,
that as the reservoir, or magazine of condensed air, is
commonly very large in proportion to the tube which con-
tains the ball, the density of the air is very little altered
by expanding through that narrow tube, and consequently
the ball is urged all the way by nearly the same uniform
force as at the first instant ; whereas the elastic fluid ari-
sing from inflamed gun-powder is but very small in pro-
portion to the tube or barrel of the gun, occupying at
first indeed but a very small portion of it next the but-
end ; and therefore, by dilating into a comparatively large
space, as it urges the ball along the barrel, its elastic
force is proportionally more weakened, and it acts always
less and less on the ball in the tube. From which cause
it happens, that air condensed into a good large machine
only 10 times, will shoot its ball with a velocity but little
inferior to that given by the gun-powder. And if the
valve of communication be suddenly shut again by a
spring, after opening it to let some air escape, then the
same collection of it serves to impel many balls, one after
another.

In all cases in which a considerable force is required,
and consequently a great condensation of air, it will be
requisite to have the condensing syringe of a small bore,
perhaps not more than half an inch in diameter: other-
wise the force to produce the compression will become
so great, that the operator cannot work the machine: for,
as the pressure against every square inch is about li
pounds, and against every circular inch, or a circle of 1
inch diameter, about 12 pounds, if the syringe be one
inch in diameter, when one injection is made, there will-
be a resistance of 12 pounds against the piston; when
2, of 24 pounds; and when 10 are injected, there will
be a force of 120 pounds to overcome; whereas 10 in-
jections act against the half-inch piston, whose area is
but £ of the former, with | of the forGe only, namely, 30-
pounds; and 40 injections may be made with such a sy-
ringe, as well as 10 with the larger.

Air-guns are of various constructions: an easy and
portable one is represented in Plate 2, fig. 1, whick is a
section lengthways through the axis, to show the inside.
It is made of brass, and has two barrels : the inner barrel
da of a small bore, from which the bullets are dis-
charged ; and a larger barrel escdr, on the outside of
it. In the stock of the gun there is a syringe s»NJSf
whose rod m draws out to take in air; and' by pushing it
in again, the piston sn drives the air before k, through
the valve p E, into the cavity between the two barrels.
The ball k is put down into its place in the small barrel,,
with the rammer, as in another gun. There is another
valve at st, which,, being opened by the trigger o, per-
mits the air to coma behind the ball, so as to drive it out
with great force. If this valve be opened and shut sud-

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C 54 ]

AIR

denly, one charge of condensed air may make several dis-
charges of bullets ; because only part of the injected air
then goes out at a time, and another bullet may be put
into the place k: but if the whole air be discharged on a
single bullet, it will impel it more forcibly. This dis-
charge is effected by means of a lock (fig. 2) when fixed
to its place as usual in other guns; for the trigger being
pulled, the cock will go down, and drive a lever which
opens the valve.

Dr. Macbride (Exper. Ess. p. 81) mentions an improve-
ment of the air-gun, made by Dr. Ellis; in which the
chamber for containing the condensed air is not in the
stock, which renders the machine heavy and unwieldy,
but has five or six hollow spheres belonging to it, of about
3 inches diameter, fitted to a screw on the lpck of the
gun. These spheres are contrived with valves, to confine
the air which is forced into their cavities, so that a ser-
vant may carry them ready charged with condensed air :
and thus the gun of this construction is rendered as light
and portable as one of the smallest fowling-pieces.

Fig. 3 represents one made by the late Mr. B. Martin
of London, and now by several of the mathematical in-
strument and gun-makers of the metropolis ; which, for
simplicity and perfection, perhaps exceeds any other that
has been contrived, a is the gun-barrel, of the size and
weight of a common fowling-piece, with the lock, stock,
and ramrod. Under the lock, at b, is a round steel tube,
having a small moveable pin in the inside, which is push-
ed out when the trigger a is pulled, by the springwork
within the lock ; to this tube b is screwed a hollow cop-
per ball, perfectly air-tight. This copper ball is fully
charged with condensed air by means of a syringe, before
it is applied to the tube b. Hence, if a bullet be rammed
down in the barrel, the copper ball screwed fast at b, and
the trigger a be pulled ; the pin in b then will forcibly
push open a valve within the copper ball, and let out a
portion of the condensed air; which will rush up through
the aperture of the lock, and forcibly act against the
bullet, driving it to the distance of 60 or 70 yards, or
farther. If the air be strongly condensed at every dis-
charge, only a portion of the air escapes from the ball ;
therefore, by re-cocking the piece, another discharge may
be made; and this repeated 15 or 1 6 times. An addi-
tional barrel is sometimes made, and applied for the dis-
charge of shot, instead of the ball above described.

Sometimes the syringe is applied to the end of the bar-
rel c (fig. 4) ; the lock and trigger shut up in a brass case
d; and the trigger pulled, or the discharge made, by pull-
ing the chain b. In this contrivance there is a round
chamber for the condensed air at the end of the spring at
e, and it has a valve acting in a similar manner to that of
the copper ball. When this instrument is not in use, the
brass case d is made to slide off, and the instrument then
beconres a walking-stick : from which circumstance, and
the barrel being made of cane, or brass, &c, it has been
called the Air-cane. The head of the cane unscrews and
takes off at a, where the extremity of the piston-rod in
the barrel is shown. An iron rod is placed in a ring at
the end of this, and the air is condensed in the barrel in
a manner similar to that of the gun as above; but its
force and action is not near so strong as in the gun.

Magazine AiR-Gwn. This is an improvement of the
common air-gun, made by an ingenious artist, L. Colbe.
By his contrivance, ten bullets arc so lodged in a cavity,
near the place of discharge, that they may be succes-

sively drawn into the barrel, and shot so quickly, as to be
nearly of the same use as so many ditTerent guns: the
only motion required, after the air has been injected, be-
ing that of shutting and opening the hammer, and cock-
ing and pulling the trigger. Fig. 5 is a longitudinal sec-
tion of this gun, as large in every part as the gun itself;
and as much of its length is shown as is peculiar to this
construction ; the rest of it being like the ordinary air-
gun, ee is part of the stock; g is the end of the in-
jecting syringe, with its valve h, opening into the cavity
ffff between the barrels, kk is the small or shooting
barrel, which receives the bullets, one at a time, trom the
magazine D e, being a serpentine cavity, in which the bul-
lets b, b, b, &c, are lodged, and closed at the end D ;
whence, by one motion of the hammer, they are brought
into the barrel at I, and thence are shot out by the open-
ing of the valve v, which lets in the condensed air from
the cavity fff into the channel vki, and so along the
inner barrel k k k, whence the bullet is discharged, si si
m k is the key of a cock, having a hole through it ; which
hole, in the present situation, makes part of the barrel
k k, being just of the same bore: so that the air, which
is let in at every opening of the valve v, comes behind
this cock, and taking the ball out of it, carries it forward,
and so out of the mouth of the piece.

To bring in another bullet to succeed I, which is done
in an instant, bring the cylindrical cavity of the key of
the cock, which made part of the barrel kkk, into the
situation ik, so that the part I may be at k; then turn-
ing the gun upside-down, one bullet next the cock will
fall into it out of the magazine, but will go no farther
into this cylindrical cavity, than the two little pieces ss
will permit it; by which means only one bullet at a time
will be taken in to the place I, to be discharged again as
before.

A more particular description of the several parts
may be seen in Desaguliers' Exper. Philos. vol. ii. pa. 399
et seq..

Aiu-Pumf, in Pneumatics, is a machine for exhausting
the air out of a proper vessel, and thus making what is
commonly called a vacuum ; though in reality the air in
the receiver is only rarefied to a certain decree, so as to
take, off the ordinary effects of the atmosphere. By this
machine we learn, in some measure, what our earth would
be without air; and how much all vital, generative, nu-
tritive, and alterative powers depend upon it.

The principle on which the air-pump is constructed, is
the spring or elasticity of the air; as that on which the
common, or water pump is formed, is the effect of its
gravity : the one gradually exhausting the air from a ves-
sel by means of a piston, with a proper valve, working in
a cylindrical barrel or tube ; and the other exhausting
water in a similar manner.

The invention of this instrument, to which the present
age is indebted for so many fine discoveries, is ascribed to
Otto Guericke, the celebrated magistrate of Magdeburg,
who exhibited his first public experiments with it, before
(he emperor and the states of Germany, at the breaking
up of the imperial diet at Ratisbon, in the year l67>4-.
But it was not till the year 1672 that Guericke published
a description of the instrument, with an account of his
experiments, in his Experimenta Nova Magdeburgica de
Vacuo Spatia : though an account of them had been pub-
lished by Schottus in 1057, in his Mechanica Ilydraulico
Pneumatica, by his consent.

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Dr. Hooke and M. Duhamel ascribe the invention of
the air-pump to Mr. Boyle. But that great man frankly
confesses that Guericke was beforehand with him in the
execution. Some attempts, he assures us, he had indeed
made on the same foundation, before he knew any thing
of what had been done abroad : but the information he
afterwards received from the account given by Schottus,
enabled him, with the assistance of Dr. Hooke, after two
or three unsuccessful trials, to bring his design to matu-
rity. The product of their labours was a new air-pump,
much more easy, convenient and manageable, than the
German one. And hence, or rather from the great va-
riety of experiments to which this illustrious author ap-
plied the machine, it was afterwards called Machina
Boyliana, and the vacuum produced by it, Vacuum Boy-
lianum.

Structure of the Air-Pump. Most of the air-pumps that
were first made consisted of only one barrel, or hollow
cylinder, of brass, with a valve at the bottom, opening
inwards; and a moveable embolus or piston, having like-
wise a valve opening upwards, and so exactly fitted to the
barrel, that when it is drawn up from the bottom, by
means of an indented iron rod or rack, and a handle
turning a small indented wheel, playing in the teeth of
that rod, all the air is drawn up from the cavity of the
barrel : there is also a small pipe opening into the bottom
of the barrel, by means of which it communicates with
any proper vessel to be exhausted of air, which is called
a receiver, from its office in receiving the subjects on
which experiments are to be made in vacuo: the whole
being fixed in a convenient frame of wood-work, where
the end of the pipe turns up into a horizontal plate, on
which the receiver is placed, just over that end of it.

The other parts of the machine, being only accidental
circumstances, chiefly respecting conveniency, have been
diversified and improved from time to time, according to
the address and several views of the makers. That of
Otto Guericke was very rude and inconvenient, requiring
the labour of two strong men, for more than two hours,
to extract the air from a glass, which was also placed un-
der water; and which allowed of no change of subjects
for experiments.

Mr. Boyle, from time to time, removed several of these
inconveniences, and lessened others : but still the work-
ing of his pump, which had but one barrel, was labori-
ous, by reason of the pressure of the atmosphere, a great
part of which was to be removed at every lift of the pis-
ton, when the exhaustion was nearly completed. Various
improvements were successively made in the machine, by
the philosophers about that time, and soon after, who
cultivated this new and important branch of pneumatics;
as Papin, Mersenne, Mariotte, and others; but still they
laboured under a difficulty of working them, fr6m the
circumstance of the single barrel, till Papin, in his fur-
ther improvements, removed that inconvenience, by the
use of a second barrel and piston, contrived to rise, as
the other fell, and to fall as that rose; by which means,
and the great improvements made by Mr. Hauksbee, the
pressure of the atmosphere on the descending piston, al-
ways nearly balanced that of the ascending one; so that
the winch, which worked them up and down, was easily
moved by a very gentle force with one hand : and besides,
the exhaustion was hereby made in less than half the time.
Some of the Germans, and others also, made improve-
ments in the air-pump, and contrived it to perform the

counter office of a condenser, in order to examine the
properties of the air depending on its condensation.

Mr. Boyle contrived a mercurial gauge or index to the
air-pump, which is described in his first and second Phy-
sico-Mechanical Continuations, for measuring the degrees
of the air's rarefaction in the receiver. This gauge is si-
rnilar to the barometer, being a long glass lube, having
its lower end immersed in an open bason of quicksilver,
but its other end, which was open also, communicating
with the receiver: which being exhausted, this tube is
equally exhausted of air at the same time, and the ex-
ternal air presses the quicksilver up into the tube, to a
height proportioned to the degree of exhaustion.

Mr. Vream, an ingenious pneumatic operator, made
an improvement in Hauksbee's air-pump, by reducing the
alternate up-and-down motion of the hand and winch, to
a circular one. In his method, the winch is turned quite
round, and yet the pistons are alternately raised and de-
pressed: by which the trouble of shifting the hand back-
wards and forwards, as well as the loss of time, and the
shaking of the pump, are prevented.

The air-pump, thus improved, is represented in plate iii,
fig. 1; where oo is the receiver to be exhausted, ground
truly level at the bottom, set over a hole in the plate,
from which descends the bent pipe hh to the cistern dd,
with which the two barrels a a communicate, where the
pistons are worked by a toothed wheel, by turning the
handle b b; thus the racks cc, with the pistons, are worked"
alternately up and down. II is the gauge tube, immersed
in a bason of quicksilver m at the bottom, and communi-
cating with the receiver at top ; from which however it
may be occasionally disengaged, by turning a cock. And
n is another cock, by turning of which, the air is again
let in to the exhausted receiver ; into which it is heard to
rush with a considerable hissing noise.

Notwithstanding the great excellency of Mr. Hauksbee's
air-pump, it was still subject to inconveniences, from
which it was in a great measure relieved by some contri-
vances of Mr. Smeaton, which are described at large in-
the Philos. Trans, for the year 1752. The principal im-_
provements suggested by Mr. Smeaton, relate to the gauge,
the valves of the piston, and the piston going closer down
to the bottom of the barrel; for his pump has only one.
By the last of these, the air was extracted more perfectly
at each stroke. By the second, he remedied an inconve-
nience arising from the valve hole of the piston being too
wide properly to support the bladder valve which covered
it: instead of the usual circular orifice, Mr. Smeaton per-
forated the piston with seven small and equal hexagonal
holes, one in the centre, and the other six around, forming
together the appearance of a transverse section of a ho-
ney-comb; the bars. or divisions between which served to
support the pressure of the air on the valve. His gauge
consists of a bulb of glass, of a pear-like shape, and ca-
pable of holding about half a pound of quicksilver: it is
open at the lower end, the other terminating in a tube
hermetically sealed : and it has annexed to it a scale, di-
vided into parts of about TV of an inch, and answering to
the 1000th part of the whole capacity. During the ex-
haustion of the receiver, the gauge is suspended in it by a
wire; but when the pump has been worked as much as
necessary, the gauge is pushed down, till the open end be
immersed in a bason of quicksilver placed below. The
air is then let into the receiver again, and the quicksilver
is driven by it from the bason, up into the gauge, till th«

A I R

[ 56 ]

A I R

air remaining in it become of the same density as that
without; and as the air always takes the highest place,
the tube being uppermost, the expansion is determined by
the number of divisions occupied by the air at the top.
This air-pump is made to act also as a condensing engine,
as some German machines had done before, by the very
simple operation of turning a cock.

By means of this gauge, Mr. Smeaton judged that his
machine was incomparably better than any former one,
as it seemed to rarefy the air in the receiver 1000, or even
2000 times, while the best of these only rarefied it about
140 times : and as an implicit confidence was placed in
Mr. Smeaton's accuracy, it was generally understood in
this light, till the fallacy was accidentally detected in the
manner related at large by Mr. Nairne in the Philos.
Trans, for the year 1777. This accurate and ingenious
artist wanting to make trial of Mr. Smeaton's pear-gauge,
executed an air-pump of his improved construction, in
the best manner possible; which, in various experiments
made with it, appeared, by the pear-gauge, to rarefy the
air to an amazing degree indeed, being at times from 4000
to 10000, or 50000, or even 100000 times rarefied. But,
on measuring the same expansion by the usual long and
short tube gauges, which both accurately agreed toge-
ther, he found that these never showed a rarefaction of
more than fjOO times : widely different from the same as
measured by the pear or internal gauge, by experiments
often repeated. ' Finding,' says Mr. Nairne, 'still this dis-
agreement between the pear-gauge and the other gauges,
I tried a variety of experiments ; but none of them ap-
peared to me satisfactory, till one day in April 1776, when
showing an experiment with one of these pumps to the
honourable Henry Cavendish, Mr. Smeaton, and several
other gentlemen of the Royal Society, the two gauges dif-
fered some thousand times from one another, Mr. Caven-
dish accounted for it in the following manner. " It ap-
peared, he said, from some experiments of his father's,
Lord Charles Cavendish, that water, whenever the pres-
sure of the atmosphere on it is, diminished to a certain
degree, is immediately turned into vapour, and is as im-
mediately turned back again into water on restoring the
pressure. This degree of pressure is different according
to the heat of the water : when the heat is 72° of Fahren-
heit's scale, it turns into vapour as soon as the pressure
is no greater than that of three quarters of an inch of
quicksilver, or about l-40th of the usual pressure of the
atmosphere ; but when the heat is only 41°, the pressure
must be reduced to that of a quarter of an inch of quick-
silver before the water turns into vapour. It is true, that
water exposed to the open air, will evaporate at any heat,
and with any pressure of the atmosphere; but that eva-
poration is entirely owing to the action of the air upon
it; whereas the evaporation here spoken of is performed
without any assistance from the air. Hence it follows,
that when the receiver is exhausted to the above-men-
tioned degree, the moisture adhering to the different parts
of the machine will turn into vapour, and supply the
place of the air, which is continually drawn away by the
working of the pump; so that the fluid in the pear-gauge,
as well as that in the receiver, will consist in a great mea-
sure of vapour. Now letting the air into the receiver, all
the vapour within the pear-gauge will be reduced to wa-
ter, and only the real air will remain uncondensed ; con-
sequently the pear-gauge shows only how much real air is
left in the receiver, and not how much the pressure or

spring of the included fluid is diminished ; whereas the
common gauges show how much the pressure of the in-
cluded fluid is diminished, and that equally, whether it
consist of air or of vapour." Mr. Cavendish having ex-
plained so satisfactorily the cause of the disagreement be-
tween the two gauges, Mr. Nairne considered that, if he
were to avoid moisture as much as possible, the two
gauges should nearly agree. And in fact they were found to
do so, each showing a rarefaction of about 600, when all
moisture was perfectly cleared away from the pump, and
the plate and the edges of the receiver were secured by a
cement, instead of setting it upon a soaked leather, as in
the usual way. But, by future experiments, Mr. Nairne
found that the same excellent machine would not exhaust
more than 50 or fjO times, when the receiver was set upon
leather soaked in water, the heat of the room being about
57°. And from hence he concludes, that the air-pump
of Otto Gucricke, and those contrived by Mr. Gratorix
and Dr. Hooke, and the improved one by Mr. Papin, both
used by Mr. Boyle, as also Hauksbee's, Gravesande'sj
Muschenbroeck's, and those of all who have used water
in the barrels of their pumps, could never have exhausted
to more than between 40 and 50, if the heat of the place
was about 57°- And though Mr. Smeaton, with his pump,
where no water was in the barrel, but where leather soak-
ed in a mixture of water and spirit of wine was used on
the pump-plate, to set the receiver upon, may have ex-
hausted all but a thousandth, or even a ten-thousandth
part of the common air, according to the testimony of his
pear-gauge; yet so much vapour must have arisen from
the wet leather, that the contents of the receiver could
never be less than a 70th or 80th part of the density of
the atmosphere. But when nothing of moisture is used
about this machine, it will, when in its greatest perfection,
rarefy its contents of air about 600 times.

It is evident that, by means of these two gauges, we can
ascertain the several quantities of vapour and permanent
air which make up the contents of the receiver, after the
exhaustion is made as perfect as can be ; for the usual ex-
ternal gauge determines the whole contents made up of
the vapour and air, while the pear-gauge shows the quan-
tity of real permanent air; consequently the difference is
the quantity of vapour.

The principal cause which prevents this pump from ex-
hausting beyond the limit above mentioned, is the weak-
ened elasticity of the air within the receiver, which, de-
creasing in proportion as the quantity of the air within is
diminished, becomes at last incapable of lifting up the
valve of communication between the receiver and the bar-
rel ; and consequently no more air can then pass from the
former to the latter.

Several ingenious persons have used their endeavours
to remove this imperfection in the best air-pumps.
Among these it seems that one Mr. Haas has succeeded
tolerably well; having by means of a contrivance to open
the communication valve in the bottom of the barrel,
made his machine so correct, that when every thing is in
the greatest perfection, it rarefies the contents of the re-
ceiver as far as 1000 times, even when measured by the
exterior gauge. The description of this machine, and an
account of some experiments performed with it, are given
by Mr. Cavallo in the Philos. Trans, for the year 1783.

But the imperfections it seems have more recently been
removed by an ingenious contrivance of Mr. Cuthbertson,
a mathematical instrument maker at Amsterdam, now of

A I R

C »7 ]

A I R

London, whose air-pump has neither cocks nor valves, the barrel or cylinder, and therefore always in the same

and is so constructed, that what supplies their place has proportion : so that by thus repeating the operation a

the advantages of both, without the inconveniences of number of times, the air is rarefied to any proposed de-

either. He has also made improvements in the gauges, gree, or till it has not elasticityenough to open the valve of

by means of which he determines the height of the mer- the piston or of the barrel, after which the exhaustion

cury in the tube, by which the degree of exhaustion is in- cannot be any farther carried on : the gauge, in compa-

dicated, to the hundredth part of an inch. And to obvi- rison with the barometer, showing at any time what

ate the inconvenience of the elastic vapour arising from the degree of exhaustion is, according to the particular

the wet leather, upon which the receiver is placed, for nature and construction of it.

common experiments, he recommends the use of leather But supposing no vapour from moisture, &c, to rise in

dressed with alum, and soaked in hog's lard, which he the receiver, the degree of exhaustion, after any number of

found to yield very little of this yapour ; but when the strokes of the piston, may be determined by knowing the

utmost degree of exhaustion is required, his advice is, to respective capacities of the barrel and the receiver, inclu-

dry the receiver well, and set it upon the plate without ding the pipe, &c. For as we have seen above that every

any leather, only smearing its outer edges with hog's lard, stroke diminishes the density in a constant proportion,

or with a mixture of three parts of hog's lard and one of namely as much as the whole content exceeds that of the

oil. But indeed the use of the leather has long been laid cylinder or barrel ; consequently the sum of as many di-

aside by our English instrument-makers, a circumstance minutions as there are strokes of the piston, will show the

which probably had not come to Mr. Cuthbertson's know- whole diminution by all the strokes. So, if the capacity

ledge. An account of this instrument, and of someexpe- of the barrel be equal to that of the receiver, in which the

riments performed with it, was published at Amsterdam in communication pipe is always to be included ; then, the

the year 17S7; from which it appears that, by a coinci- barrel being half the sum of the whole contents, half the

dence of the several gauges, a rarefaction of 1200 times air will be drawn out at one stroke; and consequently the

was shown ; but when the. atmosphere was very dry, the remaining half, being dilated through the whole or first ca-

exhaustionhasbeensocomplete,thatthegaugeshaveshown pacity, will be of only half the density of the first : in

the air in the receiver to be rarefied above 2400 times. like manner, after the second stroke, the density of the

There are made also by different persons, portable, or remaining contents will be only half that of the former,

small air-pumps, of various constructions, to set upon a that is only £ of the original density : continuing this ope-

table, to perform experiments with. In these, the gauge ration, it follows that the density of the remaining air will

is varied according to the fancy of the maker, but com- be i, after 3 strokes of the piston, -^, after 4 strokes, T'T,

rnonly it consists of a bent glass tube, like a syphon, open after 5 strokes, and so on, according to the powers of the

only at one end. The gauge is placed under a small re- ratio ~; that is, such power of the ratio as is denoted by

ceiver communicating, by a pipe, with the principal one the number of the strokes. In like manner, if the barrel

leading from the general receiver to the barrels. The close be j- of the whole contents, that is, the receiver double of

end of the gauge, of 3 or 4 inches long, before the exhaus- the barrel, or f of the whole contents ; then the ratio of

tion, has the quicksilver forced close up to the top by the diminution of density being f, the density of the contents,

pressure of the air on the open end ; but when the ex- after any number of strokes of the piston, will be denoted

haustion is considerably advanced, it begins to descend, by such power of f whose exponent is that number ;

and then the difference of the heights of the quicksilver in namely, the density will be § after one stroke, (f )2 or £

the two legs, compared with the height in the barometri- after two strokes, (f )3 orX after 3 strokes, and in general

cal tube, determines the degree of exhaustion : so, if the it will be (§•)" after n strokes: the original density of the

difference between the two be one inch, when the barome- air being 1. Hence then, universally, if s denote the sum

ter stands at 30, the air is rarefied 30 times ; but if the
difference be only half an inch, the rarefaction is 00 times,
and so on. See Plate Hi. fig./2.

The Use of the Air-Pump. In whatever manner or form
this machine is made, the use and operation of it are al-
ways the same. The handle, or winch, works the rackbar
and piston up and down in the. barrel; by which means
a barrel of the contained air is drawn out at every stroke
of the piston, in the following manner: by pushing the
piston down to the bottom of the barrel, where the air is
prevented from escaping downwards, by its elasticity it
opens the valve of the piston, and escapes upwards ; then
raising the piston up, the external atmosphere shuts down
its valve, and a vacuum would be made below it, but for
the air in the receiver, pipe, &c, which now raising the
valve in the bottom of the barrel, rushes in and fills it
again, till the whole air in the receiver and barrel be of
one uniform density, but less than it was before the stroke,
in proportion as the sum of all the capacities of the recei-
ver, pipe, and barrel together, is to the same sum wanting

of the contents of the receiver and barrel, and r that of
the receiver only without the barrel, and n any number of
strokes of the piston ; then, the original density of the air

being 1, the density after n strokes will be (— ) or—,

namely the n power of the ratio -. So, forexample,if the

capacity of the receiver be equal to 4 times that of the bar-
rel ; then their sum s is 5, and r is 4 ; hence the density of
the. contents after 30 strokes, will be (-f-)30, or the 30th
power of£, which is -jj-y nearly ; so that the air in the re-
ceiver is rarefied 808 times.

See also the Mcmoires de l'Acad. Royale des Sciences
for the years I693 and 1705.

From the same formula, namely (-) = d the density,

we easily derive a rule for finding the number of strokes of
the piston, necessary to rarefy the air any number of
times, or to reduce it to a given density d, that of the na-
tural air being 1. For since (— ) = d, by taking the Io-

dic barrel. And thus the air in the receiver is diminished
at each stroke of the piston, by the quantily or content of garithm of this equation, it is n x log. of— == log. of d;
Vol. I. . *

A I R

Ior. d.

[ 58 ]

A J U

g.r— log.s

; that is, divide the log.

i i 1°G* ^

and hence n = -2— =
l°g--

of the proposed density by the log. of the ratio of the re-
ceiver to the sum of the receiver and barrel together, and
the quotient will show the number of strokes of the piston
requisite to produce the degree of exhaustion required.
So, for example, if the receiver be equal to 5 times the bar-
rel, and it be proposed to find how many strokes of the
piston will rarefy the air 100 times; then r ==;. 5, s .=: 6,

d = -rJ^, whose log. is — 2, and - = -, whose log. is
— •07Q1 8; therefore = 251 nearly, which is thenum-

•* ' -07918 * J

ber of strokes required.

And, farther, the same formula reduced, will give us
the proportion between the receiver and barrel, when the
air is rarefied to any degree by an assigned number of

strokes of the piston. For since (-) = d the density,

therefore, extracting the n root of both sides, it is - = \/d:

that is, the n root of the density is equal to the ratio of
the receiver to the sum of the receiver and barrel. So, if
the density d be -j4a> an^ the number of strokes n = 7 ;
then the 7th root of -j-Jj- is \; which shows that the re-
ceiver is equal to half the receiver and barrel together, or
that the capacity of the barrel is just equal to that of the
receiver.

Some of the principal effects and phenomena of the
air-pump, are the following: That, in the exhausted re-
ceiver, heavy and light bodies fall equally swift : so, a
guinea and feather fall from the top of a tall receiver to
the bottom exactly together. That most animals die in a
minute or two : but however, That vipers and frogs,
though they swell much, live an hour or two ; and after
being seemingly quite dead, come to life again in the open
air. That snails survive about 10 hours; efts, or slow-
worms, two or three days ; and leeches five or six. That
oysters live for 24 hours. That the heart of an eel taken
out of the body, continues to beat for great part of an
hour, and that more briskly than in the air. That warm
blood, milk, gall, &c, undergo a considerable intumescence
and ebullition. That a mouse or other animal may be
brought, by degrees, to survive longer in a rarefied air,
than naturally it does. That air may retain its usual
pressure, after it is become unfit for respiration. That
the eggs of silk-worms hatch in vacuo. That vegetation
stops. That fire extinguishes; the flame of a candle
usually going out in one minute ; and a charcoal in about
five minutes. That red-hot iron, however, seems not to
be affected ; and yet sulphur or gun-powder are not
lighted by it, but only fused. That a match, after lying
seemingly extinct a long time, revives again on re-admit-
ling the air. That a flint and steel strike sparks of fire as
copiously, and in all directions, as in air. That magnets,
and magnetic needles, act the same as in air. That the
Miiokc of an extinguished luminary gradually settles to
the bottom in a darkish body, leaving the upper part of
the receiver clear and transparent; and that on inclining
the vessel sometimes to one side, and sometimes to another,
the fume preserves its surface horizontal, after the nature
of other fluids. That heat may be produced by attrition.
That camphire will not take fire; and that gun-powder,
though sonic of the grains of a heap of it be kindled by a

burning glass, will not give fire to the contiguous grains.
That glow-worms lose their light in proportion as the air
is exhausted, and at length become totally obscure ; but
on re-admitting the air, they presently recover it all.
That a bell, on being struck, is not heard to ring, or very
faintly. That water freezes. But that a syphon will not
run. That electricity appears like the aurora borealis.
With multitudes of other curious and important particu-
lars, to be met with in the numerous writings on this ma-
chine, namely, besides the Philos. Transactions of most
academies and societies, in the writings of Torricelli,
Pascal, Mersenne, Guericke, Schottus, Boyle, Hooke, Du-
hamel, Mariotte, Hauksbee, Hales, Muschenbroeck,
Gravesande, Desaguliers, Franklin, Cotes, Helsham, and
a great many other authors.

Air-Vessel, in Hydraulics, is a vessel of air contained
within some water engines, which being compressed, by
forcing in a considerable quantity of water, will, by its uni-
form spring, force it out at the pipe in a constant uninter-
rupted stream, to a great height.

Air-vessel too, in the improved fire engines, is a me-
tallic cylinder, placed between the two forcing pumps: by
the action of the pistons the water is forced into this ves-
sel, through two other pipes, with valves; then the air,
previously contained in it, is compressed by the water, in
proportion to the quantity admitted, and this, by its
spring, forces the water through a pipe by a constant and
equal stream ; whereas in the common squirting engine,
the stream is discontinued between the several strokes.

AJUTAGE, or Adjutage, in Hydraulics, part of the
apparatus of a jet d'eau, or artificial fountain : being a
kind of tube fitted to the aperture or mouth of the cis-
tern, or the pipe ; through which the water is to be
played in any direction, and in any shape or figure.

It is chiefly the diversity in the ajutage, that makes the
different kinds of fountains. So that, by having several
ajutages, to be applied occasionally, one fountain is made
to have the effect of many.

Mariotte, Gravesande, Desaguliers,andBuat, have written
pretty fully on the nature of ajutages, or spouts for jets
d'eau, and especially the former. He affirms, from expe-
riment, that an even polished round hole, made in the thin
end of a pipe, gives a higher jet than either a cylindrical
or a conical ajutage ; but that, of these two latter how-
ever, the conical is better than the cylindrical figure. See
his Traite du Mouvement des Eaux, part 4.

The quantity of water discharged by ajutages of equal
area, but of different figures, is the same. And for like
figures, but of different sizes, the quantity discharged, is di-
rectly proportional to the area of the ajutage, or to the
square of its diameter, or of any side or other linear di-
mension: so, an ajutage of a double diameter, or side,
will discharge 4 times the quantity of water; of a triple
diameter, 9 times the quantity; and so on; supposing
them at an equal depth below the surface or head of
water. But if the ajutage be at different depths below the
head, then the celerity with which the water issues, and
consequently tlic quantity of it run out in any given time,
is directly proportional to the square root of the altitude
of the head, or depth of the hole : so at 4 times the depth,
the celerity and quantity is double; at <) times the depth,
triple ; and so on.

It has been found that jets do not rise quite so high as
the head of water ; owing chiefly to the resistance of the
air against it, and the pressure of the upper parts of the

ALB

C 59 ]

A L F

jet upon the lower : and for this reason it is, that if the di-
Vection of the ajutage be turned a very little from the per-
pendicular, it is found to spout rather higher, than when
the jet is exactly upright.

It is found by experiment too, that the jet is higher or
lower, according to the size of the ajutage ; that a circular
hole of about an inch and a quarter in diameter, jets
highest ; and that the nearer that size, the better. Expe-
rience also shows, that the pipe leading to the ajutage,
should be much larger than it; and if the pipe be a long
one, that it should be wider the farther it is from the
ajutage.

For the other circumstances relating to jets and the is-
suing of water under various circumstances, see Exhaus-
tion, Flux, Fountain, Jet d'Eau, &c, to which they
more properly belong.

ALBATEGNI, an Arabic prince of Batan in Mesopo-
tamia, who was a celebrated astronomer, about the year
of Christ 880, as appears by his observations. He is also
called Muhammed ben Geber Albatani, Mahomet the son
of Geber, and Muhamedes Aractensis. He made astro-
nomical observations at Antioch, and at Racah or Aracta,
a town of Chaldea, which some authors call a town of
Syria or of Mesopotamia. He is highly spoken of by Dr.
Halley, as a vir admirandi acuminis, ac in administrandis
observationibus exercitatissimus.

Finding that the tables of Ptolemy were imperfect, he
computed new ones, which were long used as the best
among the Arabs : these were adapted to the meridian of
Aracta or Racah. Albategni composed injArabic a work
under the title of The Science of the Stars, comprising all
parts of astronomy, according to his own observations and
those of Ptolemy\ This work, translated into Latin by
Plato of Tibur, was published at Nuremberg in 1537, with
some additions and demonstrations of Regiomontanus; and
the same was reprinted at Bologna in 1645, with this
author's notes. Dr. Halley detected many faults in these
editions: Philos. Trans, for 1693, N° 204.

In this work, Albategni gives the motion of the sun's
apogee since Ptolemy's time, as well as the motion of the
stars, which he makes 1 degree in 70 years. He made the
longitude of the first star of Aries to be 18° 2'; and the
obliquity of the ecliptic 23° 35'. And upon Albategni's
observations were founded the Alphonsine tables of the
moon's motions; as is observed by Nic. Muler, in the
Tab. Frisicae, pa. 248.

ALBERTUS Magnus, a very learned man in the 13th
century, who, among a multitude of books, wrote several
on the various mathematical sciences, as Arithmetic, Geo-
metry, Perspective or Optics, Music, Astrology and As-
tronomy, particularly under the titles, de sphasra, de astris,
de astronomia, item speculum astronomicum.

Albertus Magnus was born at Lawingen on the Danube,
in Suabia, in 1205, or according to some in 1193 ; and he
died at a great age, at Cologn, November lfl, 1280.
Vossius and other authors speak of him as a great genius,
and deeply skilled in all the learning of the age. His
writings were so numerous, that they make 21 volumes in
folio, in the Lyons edition of l6l5. He has passed also
for the author of some writings relating to midwifery, &c,
under the title of De natura rerum, and De secretis mu-
lierum, in which there are many phrases and expressions
unavoidable on such a subject, which gave great offence,
and raised a clamour against him as the supposed author,

and inconsistent with his character, being a Dominican
friar, and sometime bishop of Ratisbon ; which dignity
however he soon resigned, through his love for solitude, to
enter again into the monastic life. But the advocates of
Albert assert, that he was not the author of either of these
two works. It must be acknowledged however, that there
are, in his Comment on the Master of Sentences, some
questions concerning the practice of conjugal duty, in
which he has used some words rather too gross for chaste
and delicate ears : but they allege what he himself used to
say in his own vindication, that he came to the knowledge of
so many monstrous things at confession, that it was impos-
sible to avoid touching upon such questions. Albert was
certainly a man of a most curious and inquisitive turn of
mind, which gave rise to other accusations against him ;
such as, that he laboured to find out the philosopher's
stone ; that he was a magician ; and that he made a ma-
chine in the shape of a man, which was an oracle to him,
and explained all the difficulties he proposed : the com-
mon cant accusations of those times of ignorance and su-
perstition. But having great knowledge in the mathema-
tics and mechanics, by his skill in these sciences he pro-
bably formed a head, with springs capable of articulate
sounds ; like the machines of Boetius and others.

John Matthaeus de Luna, in his treatise De Rerum In-
ventoribus, has attributed the invention of fire-arms to
Albert ; but in this he is refuted by Naude, in his Apolo-
gic des grands hommes.

ALBUMAZAR, otherwise called Abuassar, and
Ja:phar, was a celebrated Arabian philosopher and as-
trologer, of the 9th or 10th century, or according to some
authors much earlier. Blancanus, Vossius, &c, speak of
him as one of the most learned astronomers of his time, or
astrologer, which was then the same thing. He wrote a
work De Magnis Conjunctionibus Annorum Revolutioni-
bus, ac eorum Perfectionibus, printed at Venice in 1515,
at the expense of Melchior Sessa, a work chiefly astrolo-
gical.

He wrote also Introductio in Astronomiam, printed in
the year 1489- And it is reported that he observed a
comet in his time, above the orb of Venus.

ALDEBARAN, the Arabian name of a fixed star, of
the first magnitude, just in the eye of the sign or constel-
lation Taurus, or the bull, and hence it is popularly
called the bull's eye. For the beginning of the year
1800, its

Right Ascension is 66° 6' 5l'-'l0

Annual variation in ar 0 0 51'31

Declination 16 5 52 00 N.

And Annual variat. in Decl. 0 0 8*30

ALDERAIMIN, a star of the third magnitude in the
right shoulder of the constellation Cepheus.

ALDHAFERA, or Aldhaphra, in the Arabian Astro-
nomy, denotes a fixed star of the third magnitude, in the
mane of the sign or constellation Leo, the lion.

ALEMBERT (John le Rond d'). See Dalembert.

ALFECCA, or Alfeta, a name given to the star com-
monly called Lucida Coronas.

ALFRAGAN, Alfercsani, or Faroani, a cele-
brated Arabic astronomer, who flourished about the year
800. He was so called from the place of his nativity,
Fergan, in Sogdiana, now called Maracanda, or Samar-
cand, anciently a part of Bactria. He is also called Ah-
med (or Muhammed) ben-Cothair, or Katir. He wrote
1 2

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the Elements of Astronomy, in 30 chapters or sections.
In this work the author chiefly follows Ptolemy, using
the same hypotheses, and the same terms, and frequently
citing him.

There are three Latin translations of Alfragan's work.
The first was made in the 12th century, by Joannes His-
palensis; and was published at Ferrara in 1493, and at
Nuremberg in 1537, with a preface by Melancthon. The
second was by James Christman, from the Hebrew ver-
sion of James Antoli, and appeared at Frankfort in 1590.
Christman added to the first chapter of the work an am-
ple commentary, in which he compares together the ca-
lendars of the Romans, the Egyptians, the Arabians, the
Persians, the Syrians, and the Hebrews, and shows the
correspondence of their years.

The third and best translation was made by Golius,
professor of mathematics and Oriental languages at Ley-
den : this work, which came out in 166*9, after the death
of Golius, is accompanied with the Arabic text, and
many learned notes upon the first nine chapters ; for this
author was not spared to carry them farther.

ALGAROTI, commonly called Count Algaroti, a ce-
lebrated Italian of the last century, well skilled in Archi-
tecture and the Newtonian philosophy, &c. Algaroti was
born at Padua, but in what year has not been mentioned.
Led by curiosity, as well as a desire of improvement, he
travelled early into foreign countries; and was very young
when he arrived in Fiance in 1736. It was here that he
composed his Newtonian Philosophy for the Ladies, as
Fontenelle had done his Cartesian Astronomy, in the work
entitled The Plurality of Worlds. He was much noticed
Dy the king of Prussia, who conferred on him many marks
of his esteem. He died at Pisa the 23d of May, 1764,
and gave orders for his own mausoleum, with this inscrip-
tion upon it ; Hie jacet Algarotus, sed non omnis. He
was esteemed to be well skilled in painting, sculpture,
and architecture. His works, which are numerous, and
on a variety of subjects, abound with vivacity, elegance,
and wit : a collection of them has lately been made, and
printed at Leghorn ; but that for which he is chiefly en-
titled to a place in this work is his Newtonian Philo-
sophy for the Ladies, a sprightly, ingenious, and popular
work.

ALGEBRA is usually understood to be a general me-
thod of resolving mathematical problems by means of
equations. Or, it is a method of performing the calcu-
lations of all kinds of quantities by means of general signs
or characters. At first, numbers and things were ex-
pressed by their names at full length; butafterwards these
were abridged, and the initials of the words used instead
of them ; and, as the art advanced farther, the letters of
the alphabet came to be employed as general representa-
tions of all kinds of quantities. Other marks were also
gradually introduced, to express all sorts of operations
and combinations ; so as to entitle it to different appella-
tions—as, universal arithmetic, and literal arithmetic, and
the arithmetic of signs.

The etymology of the name, Algebra, is given in vari-
ous ways. It is, however, pretty generally considered,
that the word is Arabian, and that from those people we
had the name, as well as the art itself, as is testified by
Lucas de Burgo, the first European author on this art,
whose treatise was printed, and who also refers to former
authors and masters, from whose writings he had learned

it. The Arabic name he gives it, is Alghebra e Almu-
cabala, which is explained to signify the art of restitution
and comparison, or opposition and restoration, or resolu-
tion and equation, all which agree well enough with the
nature of this art. Some, however, derive it from various
other Arabic words ; as from Geber, a celebrated philo-
sopher, chemist, and mathematician, to whom also they
ascribe the invention of this science : and some derive it
from the word Geber, which, with the particle al, makes
Algeber, which is purely Arabic, and signifies the reduc-
tion of broken numbers or fractions to integers.

But Peter Ramus, in the beginning of his algebra, says,
" the name algebra is Syriac, signifying the art and doc-
trine of an excellent man. For Geber, in Syriac, is a
name applied to men, and is sometimes a term of honour,
as master or doctor among us. That there was a certain
learned mathematician, who sent his algebra, written in
the Syriac language, to Alexander the Great, and he
named it Almucabala, that is, the book of dark or myste-
rious things, which others would rather call the doctrine
of algebra. And to this day the same book is in great
estimation among the learned in the oriental nations, and
by the Indians who cultivate this art it is called aljabra,
and alboret; though the name of the author himself is-
not known." But Ramus gives no authority for this sin-
gular paragraph. It has, however, on various occasions
been distinguished by other names. Lucas Paciolus, or
de Burgo, in. Italy, called it l'Arte Magiore: ditta dal
vulgo la Regola de la Cosa over Alghebra e Almucabala;
calling it lArte Magiore, or the greater art, to distinguish
it from common arithmetic, which is called l'Arte Mi-
nore, or the lesser art. It seems too that it had been
long and commonly known in bis country by the name
Regola de la Cosa, or Rule of the Thing ; whence came
our rule of coss, cosic numbers, and such-like terms.
Some of his countrymen followed his denomination of the
art : but other Italian and Latin writers called it Regula
rei et census, the rule of the thing and the product, or
the root and the square, as the unknown quantity in their
equations commonly ascended no higher than the square
or second power. From this Italian word ce?isus, pro-
nounced chensus, came the barbarous word senzus, used
by the Germans and others, for quadratics; with the se-
veral zenzic or square roots. And hence %>, 3 , {£,
which are derived from the letters r, z, c, the initials of
res, zenzus, cuius, or root, square, cube, came to be the
signs or characters of these words : like as KI and »/,
derived from the letters R, r, became the signs of radi-
cally.

Later authors, and other nations, used some the one of
those names, and some another. It was also called spe-
cious arithmetic by Vieta, on account of the species, or
letters of the alphabet, which he brought into general
use; and by Newton it was called universal arithme-
tic, from the manner in which it performs all arith-
metical operations by general symbols, or indeterminate
quantities.

Some authors define algebra to be the art of resolving
mathematical problems : but this is the idea of analysis,
or the analytic art in general, rather than of algebra,
which is only one particular species of it. Indeed, alge-
bra properly consists of two parts : first, the method of
calculating magnitudes or quantities, as represented by
letters or other characters; and secondly, the manner

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of applying these calculations in the solution of pro-
blems.

In algebra, as applied to the resolution of problems,
the first business is to translate the problem out of the
common, into the algebraic language, by expressing all
the conditions and quantities, both known and un-
known, by their proper characters, arranged in an equa-
tion, or several equations if necessary, and treating the
unknown quantity, whether it be number, or line, or any
other thing, in the same way as if it were a known one :
this forms the composition. Then the resolution, or ana-
lytic part, is the disentangling the unknown quantity from
the several others with which it is connected, so as to re-
tain it alone on one side of the equation, while all the
other, or known quantities, are collected on the other side,
and so giving the value of the unknown one. And as
this disentangling of the quantity sought, is performed by
the converse of the operations by which it is connected
with the others, taking them always backwards in the con-
trary order, it hence becomes a species of the analytic
art, and is called the modern analysis, in contradistinc-
tion to the ancient analysis, which chiefly respected geo-
metry, and its applications.

There have arisen great controversies and sharp dis-
putes among authors, concerning the history of the pro-
gress and improvements of algebra; arising partly from
the partiality and prejudices which are natural to all na-
tions, and partly from the want of a closer examination
of the works of the older authors on this subject. From
these causes it has happened, that the improvements made
by the writers of one nation, have been ascribed to those
of another; and the discoveries of an earlier author, to
some one of much later date. Add to this also, that the
peculiar methods of many authors have been described so
little in detail, that our information derived from such
histories is but very imperfect, and amounting only to
some general and vague ideas of the true state of the arts.
To remedy ibis inconvenience therefore, and to reform
this article, 1 have taken the pains carefully to read over
in succession all the older authors on the subject, which
I have been able to meet with, and to write down dis-
tinctly a particular account and description of their se-
veral compositions, as to their contents, notation, im-
provements, and peculiarities ; from the comparison of
all which, 1 have acquired an idea more precise and ac-
curate than it was possible to obtain from other histories,
and in a great many instances very different from them.
The full detail of these descriptions would employ a vo-
lume of itself, and would be far too extensive for this
place: I must therefore limit this article to a very brief
abridgment of my notes1, remarking only the most mate-
rial circumstances in each author; from which a general
idea of the chain of improvements may be perceived, from
the first rude beginnings, down to the more perfect state;
from which it will appear that the discoveries and im-
provements made by any one single author, are scarcely
ever either very great or numerous; but that, on the con-
trary, the improvements are almost always very slow and
gradual, from former writers, successively made, not by
great leaps, and after long intervals of time, but by gra-
dations which, viewed in succession, become almost im-
perceptible.

Of Diopiiantus's Algebra. As to the origin of
the analytic art, of which algebra is a species, it is doubt-
less as old as any science in the world, being the natural

method by which themind investigates truths, causes, and
theories, trom their observed effects and properties. Ac-
cordingly, traces of it are observable in the works of the
earliest philosophers and mathematicians, the subject of
whose enquiries most of any require the aid of such an
art. And this process constituted their analytics. Of
that part of analytics, however, which is properly called
algebra, the oldest treatise which has come down to us, is
that of Diophantus of Alexandria, who flourished about
the year 150 after Christ, and who wrote, in the Greek
language, 13 books of algebra or arithmetic, as mention-
ed by himself at the end of his address to one Diony-
sius, though only 6 of them have hitherto been printed;
and an imperfect book on multangular numbers, namely,
in a Latin translation only, by Xilander, in the year 1575,
and afterwards in 1621 and l6'70 in Greek and Latin by
Gaspar Bachet. These books, however, do not contain a
treatise on the elementary parts of algebra, but only col-
lections of difficult questions relating to square and cube
numbers, and other curious properties of numbers, with
their solutions. And Diophantus only prefaces the books
by an address to Dionysius, for whose use it was pro-
bably written, in which he just mentions certain precog-
nita, as it were to prepare him for the problems them-
selves. In these remarks he shows the names and gene-
ration of the powers, the square, cube, 4th, 5th, 6th, &c,
which he calls dynamis, cubus, dynamodinamis, dynamo-
cubus, cubocubus, according to the sum or addition of
the indices of the powers; and he marks these powers

with the initials thus Jv, x.", $8", Sk", kyF, &c : the un-
known quantity he calls simply agiSpo;, ?iumerus, the num-
ber; and in the solutions he commonly marks it by the

final thus s ; also he denotes the monades, or indefinite
unit, by jj.". Diophantus then remarks on the multipli-
cation and division of simple species together, showing
what powers or species they produce; declares that minus
(Xsii/i;) multiplied by minus produces plus (uitafav) ;
but that minus multiplied by plus, produces minus; and
that the mark used for minus is /p, namely the ^ inverted
and curtailed; but he uses no mark for plus, but a word
or conjunction copulative.

As to the operations, viz, of addition, subtraction, mul-
tiplication, and division of compound species, or those
connected by plus and minus, Diophantus does not teach,
but supposes his reader to know them. Tie then remarks
on the preparation or simplifying the equations that are
derived from the questions, which we cull reduction of
equations, by collecting like quantities together, adding
quantities that are minus, and subtracting such as are
plus, called by the moderns transposition, so as to bring
the equation to simple terms, and then depressing it to a
lower degree by equal division, when the powers of the
unknown quantity are in every term : which preparation,
or reduction of the complex equation, being now made,
or reduced to what we call a final equation, Diophantus
goes no farther, but barely says what the root or res ignota
is, without giving any rules for finding it, or for the re-
solution of equations; thereby intimating that such rules
were to be found in some other work, done cither by him-
self or others.

Of the body of the work, lib. 1 contains 43 questions,
concerning one, two, three, or four unknown numbers,
having certain relations to each other, viz, concerning

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their sums, differences, ratios, products, squares, sums
and differences of squares, &c, &c; but none of them con-
cerning either square or cubic numbers. Lib. 2 contains
36 questions. The first 5 questions are concerning two
numbers, though only one condition is given in each ques-
tion; but he supplies another by assuming the numbers
in a given ratio, viz, as 2 to 1. The 6th and 7th contain
each two conditions : then in the 8th question he first
comes to treat of square numbers, which is this, to divide
a given square number into two other squares ; and the
9th is the same, but performed in a different way : the
rest, to the end, are almost all about one, two, or three
squares. Lib. 3 contains 24 questions concerning squares,
chiefly including three or four numbers. Lib. 4 begins
with cubes; the first of which is this, to divide a given
number into two cubes whose sides shall have a given
sum : here he has occasion to cube the two binomials
5 ■+- n and 5 — n; the manner of doing which shows that
he was acquainted with the composition of the cube of a
binomial; and many other places manifest the same thing.
Only part of the questions in this book are concerning
cubes; the rest are relating to squares. Two or three
questions in this book have general solutions, and the the-
orems deduced are general, and for any numbers indefi-
nitely ; but all the other questions, in all the four books,
are employed in finding only particular numbers. Lib. 5
is also concerning square and cube numbers, but of a
more difficult kind, beginning with some that relate to
numbers in geometrical progression. Lib. 6 contains 26
propositions, concerning right-angled triangles; such as
to make their sides, areas, perimeters, &c, &c, squares or
cubes, or rational numbers, &c. In some parts of this
book it appears, that Diophantus was acquainted with
the composition of the 4th power of the binomial root,
as he sets down all the terms of it; and, from his great
skill in such matters, it seems probable that he was ac-
quainted with the composition of other higher powers,
and with other parts of algebra, besides what are here
treated of. At the end is part of a book, in 10 proposi-
tions, concerning arithmetical progressions, and multan-
gular or polygonal numbers. Diophantus once mentions
a compound quadratic equation; but the resolution of
his questions is by simple equations, and by means of only
one unknown letter or character, which he chooses or as-
sumes so ingeniously, that all the other unknown quanti-
ties in the question are easily expressed by it, and the final
equation reduced to the simplest form which it seems the
question can admit of. Sometimes he substitutes for a
number sought immediately, and then expresses the other
numbers or conditions by it: at other times he substitutes
for the sum or difference, &c, and thence derives the rest,
so as always to obtain the expressions in the simplest
form. Thus, if the sum of two numbers be given, he
substitutes for their difference ; and if the difference be
given, he substitutes for their sum : in both cases he has
the two numbers easily expressed by adding and subtract-
ing the half sum and half difference ; and so in other cases
he uses other similar ingenious notations. In short, the
chief excellence in this collection of questions, which
seems to be only a set of exercises to some rules which
had been given elsewhere, is the neat mode of substitution
or notation ; which being once made, the reduction to the
final equation is easy and evident: and there he leaves
the solution, only mentioning that the root or agripos is
so much. On the whole, this work is treated in a very

able and masterly manner, manifesting the utmost address
and knowledge in the solutions, and inducing a persuasion
that the author was deeply skilled in the science of alge-
bra, to some of the most abstruse parts of which these
questions or exercises relate. However, as he contrives
his assumptions and notations so as to reduce all his con-
ditions to a simple equation, or at least a simple quadra-
tic, it does not appear what his knowledge was in the re-
solution of compound or affected equations.

But though Diophantus was the first author on algebra
that we now know of, it was not from him, but from the
Moors or Arabians that we received the knowledge of al-
gebra in Europe, as well as that of most other sciences.
And it is matter of dispute who were the first inventors
of it ; some ascribing the invention to the Greeks, while
others say that the Arabians had it from the Persians, and
these from the Indians, as well as the arithmetical method
of computing by ten characters, or digits; but the Ara-
bians themselves say it was invented among them by one
Mahomet ben Musa, or son of Moses, who it seems
flourished about the 8th or 9th century. It is more pro-
bable, however, that Mahomet was not the inventor, but
only a person well skilled in the art; and it is further
probable, that the Arabians drew their first knowledge of
it, either from the Indians, or from Diophantus and other
Greek writers, as they did that of geometry and other
sciences, which they improved and translated into their
own language; and from them it was that Europe re-
ceived these sciences, before the Greek authors were
known to us, after the Moors settled in Spain, and after
the Europeans began to hold communication with them,
and began to travel among them to learn the sciences.
And according to the testimony of Abulpharagius, in the
year 963, the arithmetic of Diophantus had been trans-
lated into Arabic by Mahomet ben-yahya Buziani. But
whoever were the inventors and first cultivators of alge-
bra, it is certain that the Europeans first received the
knowledge, as well as the name, from the Arabians or
Moors, in consequence of the close intercourse which
subsisted between them for several centuries. And it ap-
pears that the art was pretty generally known, and much
cultivated, at least in Italy, if not in Spain, as well as
other parts of Europe also, long before the invention of
printing, as many writings on the art are still extant in the
libraries of manuscripts; and the first authors, presently
after the invention of printing, speak of many former
writers on this subject, from whom they learned the art. .

Or the Indian Algebra. Some notices have lately
been obtained of the science of algebra among the In-
dians; and it is very probable that, through the inter-
course of learned Englishmen with that country, we shall
receive still more considerable information on that head.

There has long existed cause to suspect that the princi-
ples of this art came to Europe through the Arabians and
Moors, as well as the Indian numeration and arithmetic ;
and every extension of our concerns among them, serves
further to increase the probability of that opinion. For
more than a century past, evidence has been received in
Europe, at various times, of the existence of very learned
works on astronomy among the Indians. Such notices
were first imported by certain learned Frenchmen, and
communicated through the Memoirs of the Academy;
whence a very ingenious and learned account of such
works was given in the Astronomic Indienne of the unfor-
tunate M. Bailly. Since then, many other valuable com-

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munications have been made by several of our own learned
countrymen, belonging to the Bengal Society, and other
persons curious in the sciences; as sir William Jones,
Samuel Davis, esq., Edward Strachey, esq., and many
others. Hence the strongest evidence has been obtained,
that, at a period several thousand years (at least three or
four) before the Christian era, the Indians must have
possessed very correct astronomical observations and rules
of calculation ; rules that require a considerable know-
ledge uf geometry and of trigonometry, both plane and
spherical; and even accompanied with regular tables of
sines and versed sines : at a time when all Europe was in
a state of gross barbarity, if it was at all inhabited. See
a valuable paper in the 2d volume of the Asiatic Re-
searches, by Samuel Davis, Esq. on the Astronomical
Computations of the Hindus; also two learned disserta-
tions on the Indian Astronomy and Trigonometry, by
Professor Playfair, in the Edinburgh Philosophical Trans-
actions, vols. ii. and iv.

What we have now, however, particularly to attend to,
is the algebra of that country. It has long been thought
that a people, possessing so much knowledge of many
other branches of mathematical science, could not well
be unacquainted with algebra; and we have now received
incontestable proofs of their very critical skill in that
branch. Several specimens of such works have been seen
in that country, both in the native language and in Per-
sian translations. Some of the latter are also now in the
hands of S. Davis, esq. of Portman Square, one of the
directors of the East-India Company, with a partial
translation into English ; and similar translations of some
others have been sent to England by Mr. Edward Stra-
chey, before mentioned. And as 1 have been favoured
with the perusal of these, I am enabled to give some ac-
count of them.

The first of these communications, by Mr. Strachey,
of the Bengal civil establishment, is a printed account of
some observations on the originality, extent, and import-
ance, of the mathematical science of the Hindoos; with
extracts from Persian translations of the Lilawati or Lee-
lawuttee, and the Beej Gunnit; or the Bija Ganita, as it
is written by Mr. Davis. These two works, Mr. Strachey
informs us, were both written by Bhasker Acharij, a fa-
mous Hindoo mathematician and astronomer, who lived
about the beginning of the 13th century of the Christian
era; the latter of these two treatises being on algebra,
with some of its applications ; and the former on arith-
metic, and algebra, and mensuration or practical geo-
metry. The Beja or Bija Ganita was translated into Per-
sian in 1634, by Utta Ulla Rushcedee, at Agra or Dehli
probably; and the Leelawuttee in 1587, by the ce-
lebrated Fyzee.

It is well known, Mr. Strachey says, that the only Per-
sian science is Arabian, and that the Arabs had much of
their mathematical knowledge from the Greeks ; it is cer-
tain, however, that they had their arithmetic from the In-
dians, and most likely their algebra was drawn from the
same source; but the time, and other circumstances re-
specting the introduction of these sciences among the
Arabs, is unknown. It appears, however, that the first
account of any Indian mathematical science among the
Arabs, was of their astronomy, which was known in the
reign of Al Mamoon. In later times, many Mahomme-
dans have had access to the Hindoo books ; accounts of
several are in the Ayeen Ackbery, and in D'Herbelot.

Abul Fuzl gives a list of Sanscrit books, which were trans-
lated into Persian in Akbar's time; among which the
Leelawuttee is the only mathematical work.

From a comparison of the algebra of the Arabians and
Greeks, and that of the modern Europeans, with the Per-
sian translations of the Beja Ganita and Lelawati, it
would probably appear, that the algebra of the Arabs is
quite different from that of Diophantus, and not taken
the one from the other : that if the Arabs did learn from
the Indians, as is most probable, they did not borrow
largely from them ; that the Persian translations of the
Beej Gunnit and Leelawuttee, contain principles, which
are sufficient for the solution of any propositions in the
Arabian, or in the Diophantine algebra; that these trans-
lations contain propositions, which are not to be solved
on any principles that could be supplied by the Arabian
or the Diophantine algebra; and that the Hindoos were
farther advanced in some branches of this science than
the modern Europeans, with all their improvements, till
the middle of the eighteenth century.

These premises are exemplified by several extracts from
the Lilawati and the Beja Ganita, too long to be repeated
here, but are to be seen in my Tracts, vol. 2, p. 154, &c;
where we trace many instances of the Hindoos' know-
ledge in geometry, and in mensuration, as well as in the
various parts of algebra, as far as compound quadratic
equations inclusively, and that after a more perfect man-
ner, in some respects, than our own practice.
On Indeterminate Problems of the Second Degree, Diophan-
tus, and the Bija Ganita.

The l6th question of the 6th book of Diophantus, is
as follows : —

" Having two numbers given, if one of these drawn into
a certain square, and the other subtracted from the pro-
duct, make a square; it is required to find another square,
greater than the former, which shall do the same thing.
For instance, given the two numbers 3 and 11, and a
certain square 25, which drawn into 3, and II taken from
the product, leaves the square of 8 ; to find another square
greater than 25 having the same property. Put its side
In -*- 5, then its square is 1q + 10n -i- 25; triple of this
diminished by 11 leaves 3Q -f 30n -+• 64 equal a square ;
let its side be 2n — 8, which gives n =62; then 62 + 5 = 67
is the side, and 4489 the square sought."

In the Bija Ganita this problem is solved very generally
and scientifically, by the assistance of another, which was
unknown in Europe till the middle of the 17th century;
and first applied to questions of this nature by Euler, in
the middle of the 18th century. — With the affirmative
sign, the Bija Ganita rule for finding new values of ax" -+- b
■=yl, is this : Suppose ag- ■+- b = h2 a particular case : find
m and n such that air •+- 1 = wj2; then is x = mg 4- nh, and
y = mil ■+■ ang.

Geueral methods, according to the Hindus, for the so-
lution of indeterminate problems of the first and second
degrees, are found in the 4th and 5th chapters of the Bija
Ganita, which diner much from Diophantus's work. It
contains, a great deal of knowledge and skill, which the
Greeks had not; such as, the use of an indefinite number
of unknown quantities, and the use of arbitrary marks to
express them; a good arithmetic of surds; a perfect
theory of indeterminate problems of the first degree; a
very extensive and general knowledge of those of the se-
cond degree; a perfect acquaintance with quadratic equa-
tions, &c. The arrangement and manner of the two works

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are as different as their substance: the one constitutes a any powers raised from multinomial surd roots, some-
regular body of science; the other does not: the Bija thing like our method of finding the number of combina-
Ganita is quite connected and well digested, and abounds tions. Next follow questions about squares; such a*
in general rules, which suppose great learning; the rules (using our own notation), finding QJx°-+- 1, and 6lx" + 1,
are illustrated by examples, and the solutions are perform- and 13a:2— 1, &c, squares ; also a general way or method
edwith skill. Diophant us, though not entirely without me- to make expressions of this kind ax~ + b squares,
thod, gives very few general propositions, being chiefly re- Then questions are treated producing simple equations ;
markable for the dexterity and ingenuity with which he with the application to some questions about triangles,
makes assumptions for the simple solution of his questions. Then more questions about squares ; as, to find x±y and
The former teaches algebra as a science, by treating it syste- xy both squares; also x3+y3 and x2-t-jr both squares,
matically; the latter sharpens the wit, by solving a variety Next, some questions producing quadratics. Some-
of abstruse and complicated problems in an ingenious thing is also said of a cubic; but it seems to hint that it
manner. The author of the Bija Ganita goes deeper into cannot be solved generally : a straight ruler is mentioned,
his subject, treats it more abstractedly, and more me- which it is suspected may allude to some mechanical me-
thodically, though not more acutely than Diophantus. thod of solution, for there is an omission in the transla-
The former has every characteristic of an assiduous and lion. Then more about squares; such as, to find
learned compiler ; the latter of a man of genius in the in- x—y 7 both 7i2
fancy of science. x^-t-y2 j squares ; 7x2— By
Besides the foregoing remarks, derived chiefly from the x—y-t-2\ SQuares . x°~y
printed notices of Mr. Strachey, I have lately been favour- z + y -t- 2 J 4 ' x"-i-y
ed with communications of several other curious particu-
lars relating to the same two books, by S. Davis, esq.
The late Mr. Reuben Burrow collected, in India, many
oriental manuscripts on the mathematical sciences, both
in the Sanscrit and the Persian languages, the latter being
translations only of the former : most of these he bequeath-
ed by will to one of his sons there, but with an injunc

-y 7 both 7x*+By2 7 both r+^1

+y2 1 squares; lx2- 8y* + 1 ) squares ; x +y J scluares>

, J- squares
in whole numbers; - — a whole number, &c, &c.

b

Lastly, more indeterminate questions. In short, a great
part is about indeterminate, and what we call questions of
the Diophantine kind, yet without any one being the very
same as in that author; which alone seems to show a dif-
ferent origin; besides, they are mostly treated in a very

tion not to be delivered to him till he should have learned different way; and several of them are not easy.

those languages and the sciences. But one or two of
these Burrow left to his friend Mr. Dal by, mathematical
professor at the Royal Military College, Farnham. These
are now in Mr. Dal by 's possession, being the Persian
translations of the Bija Ganita and Lilawati, with an at-
tempt at an English translation of some parts of them by
Mr. Burrow; but these attempts being mostly interlinea-

There are three or four questions about light-angled tri-
angles, done algebraically ; and here one might have ex-
pected to have found Eucl. 47, I» quoted ; but, instead of
that, a reference is made to the " figure of wedding chair."
But from these, and many other parts of their writings, we
perceive that they were possessed of, not this property
only, but most others the same as in our geometry; and

tions written with a black-lead pencil, are in danger of here it was probably that Pythagoras acquired his mathe-
being obliterated. Mr. Strachey also, lately arrived matical knowledge, which he carried back with him, and
from India, has an entire English translation of the Bija, taught to his countrymen. The results of the operations
of which he has favoured me with the perusal, and be- by the cipher 0, are the same as we make them : the quo-
sides communicated by letter many descriptive remarks tient" Burrow at first translated "infinite," but after-
of those works, from all which sources I have collected

the following curious particulars, till such time as he may
be pleased to give the whole work to the public.

The first work is called the Beej, or the Beej Gunnit
(as they are pronounced, but written Bija Ganita), and
seems to have been translated into Persian about the
year l6"34. The Persian introduction calls it " Beej
Gunnit, the author Bhasker Acharya, author of the Lila-
wati." It is there also said, that " this excellent method
of computation was translated from the Hindu into Per-
sian, and is called the Book of Composition and Resolu-
tion ;" and " that it is not written in any book, Persian or
Arabic." The two words litre translated " composition
and resolution," arc elsewhere simply called "algebra."

wards crossed it with his pencil, and substituted " cannot
be comprehended :" probably it means unbounded, or unas-
signable.

Respecting this notation in this ms. it appears that un-
known quantities are represented by letters or characters,
which they call " colours." Thus, when only one un-
known is used, it is denoted by a character called " Ma-
jool;'' when two unknowns enter into the computation, the
second is " Aswad" (black); a third is Neelok" (blue);
the fourth yellow, &c. The unknown, or quantity sought,
is called technically the tltingi meaning emphatically the
thing about which the question or enquiry is made. All
their marks, denoting words expressing the operation, arc
Arabian. And it is remarkable that the first Italian wri-

The two Persian words for " Beej Gunnit," are totally
different from them ; so that the 'Persians and Arabians u>1f ™ alSebra: used * word °[ the same import tor the
have adopted the meaning, not the pronunciation, of the unknown quantity, viz, cosa, the thing.; whence, in Lu
Sanscrit. Part of the ms. is a commentary on the origi- roPe' thc sclence came \° beoc'llled the cossic art, an.
nal Hindu work, by the Persian translator.

The work consists of five parts. It commences with
explaining affirmatives and negatives, which he characte-
rises by two terms denoting existing and non-existing, also
properly and debt. Then follow the first rules, as with us.
Next, surds are given at great length; and there seems
here a general method of finding the number of surds in

such quantities cossic numbers, &c.

There is also a character set next an unknown, to de-
note its square; another for the cube, &c: also a mark
for the square root ; and one for the cube root: but there
does not appear to be any thing answering to a vinculum
used in compound quantities. The higher powers are
formed, and named, by repealing and combining these

A L G

[ 65 1

A L G

-3x

■ Qy

-I3.gr

+ «W

-8y

-+- 'ix1

-2xy

+ 2x

_3x

+ 2y |-2

marks; so, the 4th power is called the square square ; the
5th power, the square cube; the 6th power, the cube
cube; and so on, meaning those powers multiplied toge-
ther, i For equality they use the expression " I equalize,"
or " equalizing," and, indeed, all their operations are ex-
pressed in words at length.

When unknowns are multiplied together, they are
placed next one another, in our manner; though there
seems occasionally to be a mark for product. It is ob-
servable too that the affirmative, or the negative sign, and
the coefficient, are always placed on contrary sides of the
unknown, not-i-2xy, as with us, but Ixy-v, oi + xyl; this
latter is in the Persian way, as they read from right to left ;
but the Hindus from left to right, like the Europeans.
But, though the Persians write from right to left, they do
not use, or translate the Sanscrit numerals backwards;
a circumstance which renders the opinion very probable,
that the Persians derived their numeration characters and
calculation from the Indians, but probably at second hand
through the Arabians.

Compounds are multiplied
as we fill the multiplication +4y
table: for example, to mul- _
tiply — 3x->-2y — 2 by-t-iy —
j:+1, they proceed as in the -+- 1
margin, and afterwards collect the products.

In the process of their solutions, there is no distinct ar-
rangement of the different steps, as in the modern alge-
bra; but it is carried on without any breaks in the lines,
something like de Burgo, or Bombelli, &c. ' The language
is written in the first person, as, — I do so and so— I then
equalize — next I multiply, &c.

We have already noticed their method of resolving com-
pound quadratic equations, the same as ours, by comple-
ting the square ; and better than ours in such cases as ax1
-+- bx = c, when the first term has a coefficient a, which
does not divide the two b and c, by which they avoid the
operation in fractions. In imitation of this, also, I see
they have a method of completing the powers in some
cases of cubics and biquadratics: Thus, having the cubic
equation x3 ■+■ 1 2x = 6x2 -+- 35 ; first subtracting 6V,
gives x3 — 6x2-h 12x = 35 ; next subtracting 8, gives x3 —
6x2 + I2x — 8=2T, which completes the binomial cubic,
and the roots are x — 2 = 3, or x = 5. Again, having given
the imperfect biquadratic xA — 400x — 2»°— 9999j a case
which it is not very obvious how to bring it to a complete
power, but which is managed with much address, in this
manner. First add 400x-i-l to both sides, this gives x4 —
2xi ->- 1 = 10000 + 400x, where the first side is a complete
square, and the roots are x1 — 1 = v/(10000+-400x) ; but
as the latter side is not a complete square, the author goes
back again, and tries another course ; thus, to the original
equation he adds 4x1-i-400x+ 1, which gives *4-h2x2-<- 1
= 4x2-h400x + 10000, two complete squares, the roots of
which are x -t- l=2x-t- 100 ; again, subtract 2x, and it
becomes x2 — 2x + 1 = 100, which are. again two complete
squares, the roots of which are x— 1 = 10, and hence x =
1 1. And this process has some resemblance to that which
■was afterwards practised, if notimitated, by Lewis Ferrari.
It appears, however, that the Indians had no general me-
thod for all equations of these two (towers, but only de-
pended on their own ingenuity for artfully managing some
particular cases of them; for, at the conclusion of the
above process, the author emphatically adds, "The solu-

VOL. I.

tion of such questions as these depends on correct judg-
ment, aided by the assistance of God."

At the conclusion of the first book occur some curious
circumstances in the solution of certain problems in the
application of algebra to geometry, particularly some re-
lating to right-angled triangles, by which it appears that
they were well acquainted with the most remarkable pro-
perties in Euclid's Elements, some of which are cited
under names peculiar to themselves, and sometimes, Mr.
Strachey thinks, by numbers of the books and proposi-
tions in Euclid's collection, unless these last references
have been added by the translator, or some transcriber,
which is to be suspected.

In the 3d book the author comes to treat of questions
and equations having several unknown quantities in them.
These he directs to exterminate one after another, much
after our modern way, till they are reduced to one un-
known oidy. The number of independent equations are
stated to be equal to the number of the unknown quan-
tities; but when there are not so many equations, the de-
fect is supplied by assuming values for as many of the un-
known quantities, &c, as with us. The unknown quan-
tities are represented and called by so many different cha-
racters and names, as is our own practice also. We de-
note them usually by the letters x, y, z, &c ; the Hindus
by different, colours, or letters, or other marks also.
" Thus," says our author, " suppose the first unknown, and
the 2d black, and the 3d blue, and the 4th yellow, and
the 5th red, and the 6'th green, and the 7th parti-coloured,
and so on, giving whatever names you please to the un-
known quantities which you wish to discover; and if, in-
stead of these colours, other names are supposed, such as
letters, and the like, it may be done. For what is re-
quired, is to find out the unknown quantities, and the ob-
ject in giving names, is that you may distinguish the
things required."

In this 3d book occur many examples of making cer-
tain analytical 'expressions to be squares and cubes, &c.
These are mostly very curious, many of them difficult, and
often involving several unknown quantities; they are ge-
nerally solved in a masterly way, and that very different
from the manner of Diophantus. The two other books
are on similar indeterminate problems, but gradually more
and more complex and difficult.

Besides the above, there are many curious things in the
Bija Ganita, and It may be presumed, adds Mr. Strachey,
that the Hindu books contain much interesting matter re-
specting this branch of algebra; it is probable that they
have some system of continued-fractions; and perhaps
methods of approximation, and theories of series and equa-
tions. From the rules of the 5th book, and their applica-
tion, there is some ground for an opinion, that the Hindus
may have had a knowledge of curves, and much of the
application of mathematics to natural philosophy.

On the whole, il would seem that the origin of algebra,
as well as of arithmetic, was probably Indian. We have
however as yet but little certain information on this sub-
ject. It has been said that the Arabs ascribe the invention
of algebra to the Greeks ; but the algebra of Diophantus
is widely different from that of the Arabs; and it is very
doubtful whether the Greeks ever had any other than that
of Diophantus. If there be any doubt of Diophantus's
algebra being of Greek origin, it may be worthy of remark,
that at Alexandria he might have had the means of access
K

A L G

[ 66 ]

A L G

to Indian literature. The Bija Ganita is indeed of com-
paratively modern date; but we must not forget that it is
extracted from other books, as stated by Mr. Davis, in
the Asiatic Researches, in a very learned article on the
cycle of 60 years. And it is not unlikely that there are
old Hindu treatises, from which not only the Bija Ganita,
but even the algebra of Diophantus, and that of the Arabs,
may have been derived.

The other work, the Lilawati, as before mentioned, is
on arithmetic, mensuration, &c. In the introduction, we
find lhat " the collector of the book Lilawati was Bhasker
Acharya, of Bidder city," on the northern confines of
Hindustan. And "though the date of it is not known,
yet in another book, made in the year 1 105 of Salbahan,
the reason of making the Lilawati is given." In the Ayeen
Akbery (a Persian work on the manners, history, laws,
&c, of the Hindus, translated by Gladwin), we find " that
Acharya among the followers of Jine, is one who explains
any difficulties that may occur to noviciates;" and there-
fore we may conclude that Bhasker was an instructor in
mathematics. Now, the Salbahan, according to the
Hindu chronology, commenced Anno Dom. SO; hence
this book must have been written in 1 185. But all that
can be inferred from this is, that in 1185 they did not know
when Bhasker lived.

The Lilawati begins with the first rules of arithmetic,
and goes through fractions, the extraction of roots, &c,
with a good deal of what we should call alligation. Like
us, they mark every 3d figure from the place of units, in
extracting the cube root. A part towards the end is upon
" forms," somewhat similar it seems to our permutations.
The numeral figures are nearly the same shape as those
mentioned in p. 8 of Wallis's Algebra.

From which it appears that our present figures have
been derived from them, by gradual and successive alter-
ations in the shape, by different transcribers. And hence
it appears also, that these figures, as well as their use, and
the arts of arithmetic, algebra, &c, have come to us from
India, in their progress through Persia, Arabia, Africa,
Spain, Italy, &c; and that we ought rather to call them
Indian figures, &c, than Arabian.

Translation of Fyzi's Preface to the Lilawati.

" By order of the king Akber, Fyzi translates into Per-
sian, from the Indian language, the book Lilawati, so
famous for the rare and wonderful arts of calculation and
mensuration. He (Fyzi) begs leave to mention, that the
compiler of this book was Bhascara Acharya, whose birth
place, and the abode of his ancestors, was the city of
Biddur, in the country of theDeccan. Though the date of
compiling this work is not mentioned, yet it may be nearly
known from the circumstance, that the author made an-
other book on the Construction of Almanacks, called
Kurrun Kuttohul, in which the date of compiling it is
mentioned to be 1105 years from the date of the Saliuahn,
an era famous in India. From that year, to this, which
is the 32nd Jlahi year, corresponding with the Hejira
year 99$> there have passed 37-3 years." (answering nearly
to 1585 of the Christian era).

" It is said that the composing the Lilawati was occa-
sioned by the following circumstance. Lilawati was the
name of the author's (Bhascara's) daughter, concerning
whom it appeared, from the qualities of the Ascendant at
her birth, that she was destined to pass her life unmarried,
and to remain without children. The father ascertained
a lucky hour for contracting her in marriage, that she

might be firmly connected, and have children. It is said
that when that hour approached, he brought his daughter
and his intended son near him. He left the hour cup on
the vessel of water, and kept in attendance a time-know-
ing astrologer, in order that when the cup should subside
in the water, those two precious jewels should be united.
But, as the intended arrangement was not according to
destiny, it happened that the girl, from a curiosity natural
to children, looked into the cup, to observe the water
coming in at the hole; when by chance a pearl separated
from her bridal dress, fell into the cup, and, rolling down to
the hole, stopped the influx of the water. So the astrolo-
ger waited in expectation of the promised hour. When the
operation of the cup had thus been delayed beyond all
moderate time, the father was in consternation, and ex-
amining, he found that a small pearl had stopped the
course of the water, and that the long-expected hour was
passed. In short, the father, thus disappointed, said to
his unfortunate daughter, I will write a book of your
name, which shall remain to the latest times — for a good
name is a second life, and the ground-work of eternal ex-
istence." Fyzi's preface then proceeds :

" The arrangement of this translation was made with
the assistance of men learned in this science, particularly
the astrologers of the Deccan. Some Indian words, for
which corresponding expressions were not to be found in
books of this science, have been retained, as they were in
the Indian language. The work has been divided into an
Introduction, some Rules, and a Conclusion."

Introduction. In this part are given explanations of
some expressions in the science of calculation; with the
meaning of certain terms employed in operations of the
art of numbers ; the divisions and terms of money, weights,
measures, time, and mensurations; some of which are cu-
rious, as bearing some analogy to our own, of which they
might possibly be the origin. The grain, or barleycorn, is
given by the author, as the element of weights as well as
measures. Thus, two barleycorns make 1 Soorkh. The
breadth of 8 barleycorns is the quantity of 1 finger; 2+
fingers make the Arm or cubit; and 10 cubits make a
Bamboo or rod, nearly the same as our own. The ele-
ment of time, is given thus: The time in which a word of
two letters, as Ta or Ka, can be uttered 10 times, " neither
slowly nor quickly," is called Pran ; then 0" prans make
1 Pul; 6'0 puis I Ghurry; and O'O ghurrys one day and
night. So that it hence appears that,

60 ghurrys are = our 24 hours
1 ghurry -'=•-. .24 minutes
1 pul - ' = - ' 24 seconds

1 pran - = - 4 seconds
1 ka uttered in ■§ of a second.

ON THE ARABIAN ALGEBRA.

Mr. Strachey says, " the Greek algebra may be seen in
Diophantus, who is the only Greek writer on the subject
that we have heard of. — The Indian algebra may be seen in
Bija Ganita, and in the Lilawati, by the author of the
Bija: and as the Persian translations of these works con-
tain a degree of knowledge, which did not exist in any of
the ordinary sources of science, extant in the time of the
translators, they may be safely taken as Indian, and of an-
cient origin. To give some idea of the algebra of the Ara-
bians, by which we may be enabled to judge whether, on
the one hand, it could have been derived from Diophantus,
or, on the other, that of the Hindus could have been

ALG

[ 67 ]

ALG

taken frorri them, the work entitled Khulasat-ul-Hisab,
may be taken as a specimen ; especially because there is
a part of this book which marks the limits of algebraical
knowledge, in the time of the writer.

" We have seen that the first European Algebraists
learnt of the Arabians ; but no account has been given of
the nature, the extent, and the origin of Arabian algebra.
No distinct abstract or translation of any Arabic book,
on the subject, has appeared in print; nor has it been
established beyond controversy, who taught the Arabians.
The Khulasat-ul-Hisab is of considerable repute in India:
'it is thought to be the best treatise on algebra, and it is
almost the only book on the subject read there. I select-
ed it, because I understood that, as well as the shortest, it
was the best treatise that could be procured. Besides ge-
neral report, I was guided by the authority of Maulavi
Roshen Ali, an acknowledged good judge of such matters,
who assured me that, among the learned Moslems, it was
considered as a most complete work, and that he knew of
no Arabian algebra beyond what it contained. In the
Sulafat-ul-Asr, a book of biography, by Nizam-ul-din-
Ahmed, there is this account of Baha-ul-drn, the author
of the Khulasat-ul-Hisab. ' He was born at Balbec, in
the month of D'hi lhaj, 953 Hijri, (anno Chr. 1575) and
died at Isfahan in Shawal, 1031/ (anno Chr. 1653).
Mention is made of many writings of Baha-ul-din, on re-
ligion, law, grammar, &c, a treatise on astronomy, and
one on the astrolabe. In this list of his works, no notice
is taken of his great treatise on algebra, the Behr-ul-
Hisab, which is alluded to in the Khulasat-ul-Hisab.
Mawlavi Roshen Ali tells me the commentators say, it is
not extant. There is no reason to believe that the Arabians
ever knew more than appears in Baha-ul-din's book, for
their learning was at its height long before his time.

" From what has been stated it will appear, that from
the Khalasat-ul-Hisab, an adequate conception may be
formed of the nature and extent of the algebraical know-
ledge of the Arabians; and hence I am induced to hope,
that a short analysis of its contents will not be unaccept-
able here. I deem it necessary here to state that, possess-
ing nothing more than the knowledge of a few words in Ara-
bic, I made the translation, from which the following sum-
mary is abstracted, from the viva voce interpretation into
Persian of Maulavi Roshen Ali, who perfectly understood
the subject and both languages, and afterwards collated
it with a Persian translation, which was made about 60
years after Baha-ul-din's death, (anno Chr. 1713), and
which Roshen Ali allowed to be perfectly correct.

" The work, as stated by the author in his preface, con-
sists of an introduction, 10 books, and a conclusion. The
introduction contains definitions of arithmetic, of number,
which is its object, and of various classes of numbers. The
author distinctly ascribes to the Indian sages the invention
of the nine figures, to express the numbers from 1 to 9-
Book 1 comprises the arithmetic of integers. The rules
enumerated under this head are, Addition, Duplation,
Subtraction, Halving, Multiplication, Division, and the
Extraction of the Square Root. The method of proving
the operation, by casting out the 9's is described under
each of these rules. The author gives the following re-
markable definitions of multiplication and division : viz,
' Multiplication is finding a number such, that the ratio
which one of the factors bears to it, shall be thesame which
unity bears to the other factor; and division is finding a

number, which has the same ratio to unity, as the dividend
has to the divisor.'

" For the multiplication of even tens, hundreds, &c,
into one another, the author delivers the following rule,
which is remarkable in this respect, that it exhibits an
application of something resembling the indices of logari-
thims." ' Take the numbers as if they were units, and
multiply them together, writing down the product, Then
add the numbers of the ranks together, the place of units
being 1, that of tens 2, &c; subtract 1 from the sum,
and call the remainder the number of the rank of the
product.' This is similar to our own practice in such
cases, when we say, Multiply the significant numbers to-
gether, and subjoin to their product all the ciphers that
are in both factors."

There next follow several other ingenious contrivances,
and compendiums, such as, To multiply figures between 5
and 10. To multiply units into numbers between units
and 20. To multiply together numbers between 10 and
20. To multiply numbers between 10 and 20 into com-
pound numbers between 20 and 100. To multiply num-
bers between 20 and 100, where the digits in the place of
tens are the same. To multiply numbers between 10 and
100, when the digits in the place of tens are different. To
multiply two unequal numbers, whose sum is even ; from
the square of the half sum subtract the square of the half
difference. For multiplying numbers consisting each of
several places of figures, the method described by this
author, under the name of Shabacah, or network, and il-
lustrated by the following example, has some resemblance
to the operation by Napier's bones; where, inst«ad of
our way, the method is, to set down the whole of each
product in the alternate chequers, then add up the
columns diagonal wise. On the other rules, nothing is
delivered differing so much from those contained in our
common books of arithmetic, as to require particular
mention.

Book 2d contains the arithmetic of fractions. Book
3d, the rule of three, or to find an unknown number by
four proportionals. Book 4th delivers the rule of position,
both single and double, or to find an unknown number by
assuming a number once or twice, and comparing the
errors. Book 5th gives the method of finding an unknown
number by reversing all the steps of the process described
in the question. This last rule has a near affinity to the
reduction of equations in algebra, and might very naturally
lead to the invention of algebra itself, being easily and
generally applicable to the solution of questions usually
given in the rules of position, and in the reduction of equa-
tions. For instance, taking the first example in Double
Position, in vol. 1, of my Course of Mathematics: viz,
What number is that, which being multiplied by 6, the
product increased by 18, and the sum divided by 9, the
quotient shall be 20 ?

Now, beginning with the last result 20, and performing
in a retrograde order all the reverse operations, they will
be thus : because the last operation was dividing by 9, to
give the result 20; and multiplication being the reverse of
division, therefore multiply the result 20 by the 9, and it
gives 180; then because the next preceding operation was
adding IS, therefore subtracting 18 from the ISO, leaves
16'2 ; lastly, because the next preceding operation was
multiplying by 6, therefore divide the 162 by 6, and it
gives 27, which must be the number sought, or that
K2

A L G

[

first begun with. The proof is thus : 27 * 6 = l62, then
162 -(-18 = 180, and 180-^9=20. Or thus, expressing
all the operations by their respective marks or characters,
— ;now first multiplying by the divisor

it will give —

9, gives27 +6+18; then subtracting the increase 18,
leaves 27 + 6: and lastly, dividing by the divisor 6, must
give the first or number required.

And this natural process is exactly the same as the ge-
neral rule laid down at the beginning of Simple Equations
in the same volume, as the most general and natural
method for the reduction of all equations. Thus, assuming
x for the number sought, in the same example : then by

the question, = 20;

multiply by 9, and it becomes - - 6 .r-i-18 = ISO;
subtract IS, and it is - - - - 6:c = l62:
lastly, divide by 6, and it gives - - x = 27-
So that the mode of operation is exactly the same in all
these ways.

Book 6th treats of Mensuration. The introduction
contains geometrical definitions. Chap. 1 treats on the
mensuration of rectilinear surfaces. Under this head, the
two following articles are deserving of notice. 1st, To
find the point in the base of a triangle, wh'ere it will be
cut by a perpendicular, let fall from the opposite angle.
Call the greatest side the base ; multiply the sum of the
two less sides by their difference; divide the product by
the base, and subtract the quotient from the base ; half
the remainder will show the point on the base, where the
perpendicular falls towards the least side. This property
will easily be recognized as similar to that which we
usually employ in resolving that case in plane trigonome-
try when the three sides are given, probably borrowed
from the orientals ; it is also the same property as that in
the 35th theorem of my Geometry.

2nd, To find the area of an equilateral triangle. Mul-
tiply the square of a quarter of the square of one of the
sides by 3 ; then the square root of the product is the
area. That is, the side being a, its square is or, the quar-
ter of 'this is $a?, the square of which is -^a4, multiplied
by3, it is-j%a4,its root *J -j%a4 is the area. This is naturally
derived from the property of right-angled triangles, and, by
extracting the roots, soon reduces to £a\/3, the same as
our own rule for the same purpose.

Chap. 2nd treats on the mensuration of curvilinear sur-
faces. We here find the same rule for the area of the
circle, as one of those employed by ourselves, for the same
purpose, viz, multiply the square of the diameter by 11,
and divide the product by 14.

Chap. 3d, on the mensuration of solids, contains nothing
sufficiently remarkable to merit particular notice. This
chapter concludes with the following sentence : ' The de-
monstrations of all these rules are contained in my greater
work entitled Bahr-ul-IIisab (the ocean of calculations),
may God grant me grace to finish it.'

Book the 7th treats on Practical Geometry. Of
which, the first chapter is on levelling, for the purpose
of making canals. In this are described the plummet-
level, and the water-level, on the same, principle with our
spirit-level.

The 2nd chapter is on the mensuration of heights, ac-
cessible and inaccessible. Under the former of these heads
arc delivered the common methods, by bringing the top
of a pole (whose height is known) in a line between the

8 ] A L G

eye and the top of the height required ; or by viewing the
image of the top in a horizontal mirror; also by taking
the proportion between a perpendicular stick of known
length, and its shadow; or by taking the length of the
shadow of the height when the sun's altitude is 4-5 de-
grees. The concluding method is this : ' Place the index
of the astrolabe at the mark of 45°, and stand at a place
where the height of the object is seen through the sights;'
then measure from the place where you stand, to the
place where a stone would fall from the top; add your
own height, and the sum is the quantity required.

For the mensuration of inaccessible heights, the follow-
ing rule is delivered : " Observe the top of the object
through the sights, and mark on what shadow-line (di-
vision) the lower end of the index falls. Then move the
index a step forward or backward, and advance or recede
till you see the top of the object again. Measure the
distance between your stations, and multiply by 7 if the
index is moved a Dhil-Kadam, and by 12 if it is moved
a Dhil-Asba, according to the shadow lines on the astro-
labe ; this is the quantity required."

The 3d chapter is on measuring the breadth of rivers
and depth of wells. Stand on the bank of the river, and
through the two sights look at the opposite bank ; then
turn round and look at any thing on the land side, keeping
the astrolabe even : the distance from the observer to the
place is the same as the breadth of the river.

Book 8th is on finding unknown quantities by algebra.
In this book are two chapters. The first is introductory,
chiefly relating to the formation and operation of powers
of the unknown quantity, thus. Call the unknown quan-
tity Shai (thing), its product into itself Mai (possession),
the product of Mai into Shai, Cab (a die or cube), of Shai
into Cab, Mal-Mal ; of Shai into Mal-i-Mal, Mai-Cab ;
Shai into Mal-i-Cab, Cab-i-Cab ; and so on, without end.
For one Cab write two Mais, and of these two Mais one
becomes Cab ; afterwards both Mais become Cab. Thus
the 7th power is Mal-i-Mal-i-Cab, and the 8th Mal-i-Cab-
i-Cab, in the Oth Cab-i-Cab-Cab, and so on. All these
powers are in continued proportion, either ascending or
descending. Thus the ratio of Mal-i-Mal to Cab, is like
(or similar or equal to) the ratio of Cab to Mai, of Mai to
Shai, and of Shai to 1, and of 1 to 1 divided by Shai, and
of 1 divided by Shai to 1 divided by Mai, and of 1 divided
by Mai to 1 divided by Cab, and of 1 divided by Cab to
1 divided by Mal-i-Mal. All this means, that all the
terms in the successive powers are in continued propor-
tion, viz, in our notation,

11 1 1 j 3 . o

— , -ti — ■ — > I) x, x"> x > a > &c>

x r .V x

', x", X3, X*. &c.

or x *, x , x

To multiply one of these powers by another. If they
are both on the same side, (viz, of unity) add the expo-
nents of their powers together ; the product will have the
same denomination as this sum. For example, to multi-
ply Mal-i-Cab by Mal-i-Mal-i-Cab, the first is the 5th
power, and the other is the 7th ; the result then is Cab-i-
Cab-i-Cab-i-Cab, or four Cabs, which is the 12th power.
If the factors are on different sides, the product will be
the excess cm the side of the greater. — So, the product of
1 divided by Mal-i-Mal into Mal-i-Cab, is Shai; and the
product ol I divided by Cab-i-Cab-Cab into Cab-i-Mal-i-
Mal, is 1 divided by Mai. And if the factors are at the
same distance (from 1), the product' is 1. The author

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adds, " The particulars of the methods of division, with
extraction of roots, and other rules, I have given in my
greater book."

" The rules of algebra which have been discovered by
learned men are six, and they relate to number and Shai
and Mai." That is, to n and x and x% including what we
call simple and quadratic equations.

Chapter II. On the Six Rules of Algebra.

To find unknown quantities by algebra depends on
acuteness and sagacity, with an attentive consideration of
the terms of the question, and a successful application of
the invention to such things as may serve to bring out the
quantity required. Call the unknown quantity Shai (the
Thing), and proceed with it according to the terms of the
question^ till the operation ends with an equation. Let
that side where there are negative quantities be made per-
fect, adding the negative quantity to the other side, which
is called restoration, Jebr, (hence doubtless, with the article
Al, comes the name Algebra.) Let those things which are
of the same kind, and equal on both sides, be thrown away
which is called opposition {dlakabalah).

Equality is either of one species to another, which is
of three kinds, called simple (Mufridal), or of one species
to two species, which is of three kinds, called compound
(Muktarinati) .

Case 1st. Mufridal. When number is equal to things
(ax=n); divide the number by the coefficient of the thing,
and the unknown quantity will be found. Example. A
person admitted that he owed to Zaid 1000 and one half
of what he owed Amer; also that he owed Amer 1000 all
but half of what he owed to Zaid. Call Zaid's debt Shai
(x). Then Amer's debt is 1000 wanting half of Shai
(1000 — -ix); and Zaid's is 1500 wanting a fourth of Shai
(1500 — |r). This is equal to Shai (1500 — ix = x).
After Jebr, 1500 is equal to one Shai and a quarter of
Shai (1500= l^x). So for Zaid is 1200, and for Amer
400.

Case 2nd. Multiples of Shai equal to multiples of Mai
(ax = bx'). Divide the coefficient of the thing by that of

Mai; the quotient is the unknown quantity (- = x). Ex,

ample. Some sons plundered their father's inheritance,
which consisted of Dinars. One took 1, another 2, the
third 3, and so on increasing by 1. The ruling power
took back what they had plundered, and divided it among
them in equal shares, by which each received 7. How
many sons were there, and how many Dinars? Suppose
the number of Sons Shai (x), and take the sum of the ex-
tremes, viz, 1 and Shai (1 -)-x). Multiply them by half
of Shai (4*) ; tli is is the number of dinars (■§! -+- ^r ) ; for
the sum of any series of numbers in arithmetical progres-
sion, is equal to the product of the sum of the two ex-
tremes, into half the number of terms. Divide the num-
ber of the Dinars by Shai (x), which is the number of
the sons, the quotient according to the terms of the ques-
tion, will be 7{ix- -+• ^x) _^x = 7). Multiply 7 by Shai
(r), the divisor, 7 Shai is the product, which is equal to ^
Mai and -J- Shai (7x =^x! ■+■ {x). Alter Jebr and Muka-
baluh, 1 Mai is equal to 13 Shai (xl = 13x); Shai then is
13 (x = 13); and this is the number of sons. Multiply
this by 7, and the number of Dinars will be found 91.

It is added, that questions of this kind may be solved by
position. " Thus, suppose the number of sons to be 5,
the first error is 4 in defect ; then suppose it to he 9> the

second error is 2 in defect. The first Mafudh is 10, and
the second is 36; their difference is 26, and the difference
of the errors is 2." — " Another method, which is short, is
this: Double the quotient (7) is 14; subtract 1, and the
result (13) is the number of sons."

Case 3d. Number equal to Mai (n=rax2). Divide the
number by the coefficient of the Mai; the root of the quo-
tient (,/- = x is the unknown quantity. Example. A

person admitted that he owed Zaid the greater of two sums
of money, the sum of which was 20, and the product 96.
Suppose one of them to be 10 and Shai (10 -1- x), and the
other 10 wanting Shai (10 — x). The product, which is
100 all but Ma^lOO — x2) is equal to 96; and after Jebr
and Mukabalah, 1 Mai is equal to 4 (x2 = 4), and Shai
equal to 2 (x = 2). One of the sums then is 8, and the
other 12, which is the debt of Zaid. Here, by substitu-
ting for the half-sum and half-difference the equation,
comes out a simple one.

-First Case of Maktarinal. Number equal to Mai
and Shai (ax2 ->-bx = n). Complete the Mai to unit if
it is deficient, or reduce it to the same if it exceeds, and
reduce the number and Shai in the same ratio, by dividing
all by the coefficient of the Mai. Then square one half
the coefficient of the Shai, and add this square to the
number. From the root of the sum subtract half the co-
efficient of the Shai, and the unknown will remain. An
example follows as usual : and the method is evidently the
same as ours at present.

Case 2nd. Shai equal to number and Mai (bx = x'2 -t- n).
After completing or rejecting, subtract the number from
the square of half the coefficient of Shai ; then add the
root of the remainder to half the coefficient of the Shai, or
subtract the former from the latter, and the result is the
unknown quantity. This also is the same as the present
method, and both the two roots are noticed in this case, by
taking the root of the known quantity either positive or
negative. An example is added as usual.

Case 3d. Mai equal to number and Shai (x2 = n •+- bx).
After completion or rejection, add the square of half the
coefficient of the Shai to the number, and add the root of
the sum to half the coefficient of the Shai ; this is the un-
known quantity. Example. What number is that which
being subtracted from its square, and the remainder addr
ed to its square, is 10? Subtract Shai from Mai and go on
with the operation, than 2 Mai all but Shai is equal to
10 (2x- — x = 10) ; and after Jebr and Radd, Mai is equal
to 5 and half of Shai (x'2 = 5 + ix). The square of half
the coefficient of Shai and 5, is 5 and half an eighth (5-jJg-),
and its root is Si; to this add £, the result 2£ is the num-
ber sought.

Book 9th, contains 12 rules respecting the properties
of numbers. As, 1st. To find the sum of the products
multiplied into itself and into all numbers below it. Add
1 to the number; and multiply the sum by the square of
the number; half the product is the number sought.

2nd. To add the odd numbers in their regular order.
Add 1 to the last number, and take the square of half the
sum.

3d. To add the even numbers from 2 upwards. Multi-
ply half the last even number by a number greater by 1
than that half.

4th. To add the squares of the numbers in order. Add
1 to twice the last number, and multiply a 3d of the sum,
by the sum of the numbers.

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5th. To find the sum of the cubes in succession. Take
the square of the sum of the numbers.

6th. To find the product of the roots of two numbers.
Multiply one by the other, and the root of the product is
the answer.

7th. To divide the root of one number by that of an-
other. Divide one by the other, the root of the quotient
is the answer.

8th. To find a perfect number ; that is, a number which
is equal to the sum of its aliquot parts, (Eucl. book 7, def.
22). The rule is that delivered by Euclid, book 9, prop.
36.

9th. To find a square in a given ratio to its root. Di-
vide the first number of the ratio by the second ; the
square of the quotient is the square required.

10th. If any number be multiplied and divided by ano-
ther, the product multiplied by the quotient, is the square
of the first number.

1 1th. The difference of two squares is equal to the pro-
duct of the sum and difference of the roots.

12th. If two numbers be divided by each other,and the
quotients multiplied together, the result is always 1.

Book 10th contains 9 practical examples, all of which
are capable of solution by simple equations, or by posi-
tion, or by retracing the steps of the operation, and some
of them by simple proportion; so that it is needless to
specify them.

The conclusion, which marks the limits of algebraical
knowledge in the age of the writer, is here given entire, in
the author's words.

" Conclusion. There are many questions in this sci-
ence which learned men have to this time in vain attempted
to solve; and they have stated some of these questions in
their writings, to prove that this science contains difficul-
ties, to silence those who pretend they find nothing in it
above their ability, to warn arithmeticians against under-
taking to answer every question that may be proposed, and
to excite men of genius to attempt their solution. Of
these I have selected seven. — 1st. To divide 10 into two
parts, such, that when each part is added to its square
root, and the sums are multiplied together, the product is
equal to a supposed number. — 2d. What square number
is that, which being increased or diminished by 10, the
sum and remainder are both square numbers? — 3d. A
person said he owed to Zaid 10 all but the square root of
what he owed to Amer, and that he owed Amer 5 all but
the square root of what he owed Zaid. — 4th. To divide a
cube number into two cube numbers. — 5th. To divide 10
into two parts such, that if each is divided by the other,
and the two quotients are added together, the sum is equal
to one of the parts. — 6th. There are three square numbers
in continued geometric proportion, such, that the sum of
the three is a square number. — 7th. There is a square,
such, that when it is increased and diminished by its root
and 2, the sum and the difference are squares. — Know,
reader, that in this treatise I have collected in a small
space the most beautiful and best rules of this science,
more than were ever collected before in one book. Do
not underrate the value of this bride; hide her from the
view of those who are unworthy of her, and let her go to
the house of him only who aspires to wed her."

From the preceding account of this Arabian treatise on
Algebra, the Khulasat-ul-Hisab, of Baha-ul-din, which is
esteemed the best in that language, we pretty clearly per-

ceive what was the state of science in that nation ; that it
was much the same as the first treatises among the Italians,
derived directly from the former ; but that it was much
inferior to the same science among the Indians. It does
not appear that the Arabians used algebraic notation or
abbreviating symbols ; that they had any knowledge of the
Diophantine algebra, or of any but the easiest and ele-
mentary parts of the science. We have si en that Baha-
ul-din ascribes the invention of the numeral figures, in the
decimal scale, to the Indians ; as is done indeed by all the
Arabic and Persian books of arithmetic.

The following is an extract from a Persian treatise of
arithmetic in Mr. Strachey's possession.

" The Indian sages wishing to express numbers conveni-
ently, invented these nine figures, t f Tfil V A1)
The first figure on the right hand they made stand for
units, the second for tens, the third for hundreds, the
fourth for thousands. Thus, after the third rank, the
next following is units of thousands, the second tens of
thousands, the third hundreds of thousands, and so on.
Every figure therefore in the first rank is the number of
units it expresses; every figure in the second, the num-
ber of tens which the figure expresses; in the third, the
number of hundreds; and so on. When in any rank a
figure is wanting, write a cipher like a small circle (°) to
preserve the rank. Thus, ten is written 10, a hundred
100, five thousand and twenty-five 5025."

In short, of the Indian algebra, in its full extent, the
Arabians seem to have been ignorant; but most likely
they had their algebra from the same source as their
arithmetic. The Arabian and Persian treatises on algebra,
like the first and old European ones, begin with arith-
metic, called in those treatises the arithmetic of the Indi-
ans, and have a second part on algebra; but.no notice is
taken of the origin of the latter. Most likely their alge-
bra, being numeral, was considered by the authors as a
part of arithmetic.

Though part only of the Khulasat-al-Hisab, Mr. Stra-
chey says, is concerning algebra, the rest, relating to arith-
metic and mensuration, must be thought not wholly un-
connected with the subject. It is to be hoped that ere long
we shall have either translations from the Sanscrit of the
Bija Ganita and Lilawati, or perfect accounts from the
originals; and that other Hindu books of algebra will be
found, and made known to the world. But in the mean
time the Persian translations will be found well deserving
of attention ; observing carefully to distinguish between
what is interpolated, and what is original.

From the preceding account of this Arabic treatise, also,
is clearly seen the origin of the name of Algebra, being an
Arabic compound, viz, of the article «/, and jebr, which
denotes one of the modes of reducing the equations, viz,
by transposing or adding the negative terms, to make them
all affirmative.

For a fuller account of the Indian and Arabian algebra,
see my Tracts, vol. 2, p. 151, &c.

Of algebra in italy and other parts of Eu-
rope. We have seen that algebra had probably its rise
in Hindustan, as well as our present numeration and arith-
metic, all the rules of it having been (bund in the ancient
Mss. of that country, much the same, in matter and form,
as they appear in the first Italian authors; and among
these, the two rules of false position, which are nearly al-

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lied to algebra, and the extension of which probably led
to, and ultimately became, the art of algebra itself. Hence
it appears that these arts passed successively into Persia,
Arabia, Africa, and Europe. In this last quarter it seems
doubtful, whether their introduction was first into Spain
or into Italy: the probability would appear to be on the
side of the former, as it would most likely be introduced
by the Moors on their settlement in the peninsula in the
Sth, 9th, and 10th centuries ; whence it might be com-
municated to Gaul and Germany and England, &c. This
route is rendered the more probable by the circumstance,
that the early state of the algebraic art in these countries
was very different, and even more perfect, in several re-
spects, than that in Italy, in the contemporary stages.
And yet, on the other hand, against this probability of
the first introduction of the art into Spain by means of
the Moorish conquest, it may reasonably be objected,
that it does not appear that the Arabians themselves had
cultivated it at a ti me so early as the date of that conquest ;
and besides that we have not heard of any early works on
Algebra having ever been found in the Spanish peninsula.
It appears, however, at any rate, that Italy received the
art by a different route, though probably at a later date,
viz, immediately from the eastern Arabians themselves,
without the knowledge of any other source or communi-
cation : and hence it was natural for the Italians to as-
cribe the invention of the art to that people.

It has usually been thought, that the first introduction
of algebra into Italy, was about the 14th century. But
we have lately, viz, in 1797, been favoured with a new
and very diffuse history of algebra, by Sig. Cossali, in two
large volumes, 4to, by which it appears, that the art was
first imported into Italy, from the east, by Leonard Bo-
nacci, of Pisa, who composed his arithmetic in the year
1202, and again in 1228; adding the algebra at the end,
as a part of it.

In ^his work of Cossali's, accounts are given of several
other old authors, as well Italian as Arabian and Persian,
&c, among which some were much earlier than Leonard:
as, a Mohamed ben Musa or Moses, called also Mohamed
of Buziani, a place in Corasan, near the south-east point
of the Caspian Sea. This Mohamed, it seems, in the year
959, travelled eastward to the confines of India, to learn
the mathematical sciences, and afterwards to teach them ;
and who, according to Abulfaragio,in the year 9°9i wrote
a Commentary on Diophantus ; he wrote also Demonstra-
tions of the Propositions in the same; and another work
on the Universal Logistic Art, in three books.

It seems, however, that it was Leonard Bonacci, of Pisa,
who first introduced the art in Italy, as before mentioned.
Leonard's work, it appears, was a very orderly and regu-
lar treatise on arithmetic and algebra, as far as it was then
known; teaching and demonstrating all the rules, and il-
lustrating them with many examples; being also much
occupied with questions about square and cube numbers,
like those of Diophantus, or rather like what we have
seen and described among the Indians and Arabians. — In
algebra, Leonard distinguished three kinds of numbers,
viz, the absolute known number in any question; then
the unknown number, which he calls radice, the root;
and its square, which lie calls census in Latin, or censo in
Italian; for his algebra extended only to the solution of
equations of tin; 1st and 2d degree, the same as that ofi
the Indians and Arabians. The language in which the
work was written, was in barbarous Latin, or something

i

between Latin and Italian, when the language of the
country was changing from the one to the other.

In treating his subject, Leonard had no such notation as
is used by modern authors; on the contrary, he expressed
every thing, both the quantities and the several operations,
by their names, or words at full length. Those equations
he treated of in six different forms, which are as follow,
when expressed in the modern notation. 1st, x* = ax;
2d, x2 = ra; 3d, ax=zn; 4th, x2 -+- ax = n; 5th, x2 = ax
s-n; 6th, x2 -t-rc = ax ; where x denotes the radice, or
quantity sought, x2- its censo or square, a the number mul-
tiplying x, and n the absolute known number; and where
it is remarkable that he places the terms, more or less, on
the one side or the other, so as they may be all affirma-
tive : all which cases and forms are exactly the same as
before given by their masters the Arabians, &c.

Now omitting the simple forms, as including but little
remarkable, we shall only consider the compound ones, or
the solution of quadratic equations, by the method of
completing the square, which Leonard founds on geome-
trical demonstration, as follows: — In the first case, if
x2 -+- ax = n, then the rule is x = ^/(|a'2 •+■ n) — ^a; which
he thus demonstrates.

Demonslr. If upon a right line bc, a. ar t>

greater than ^a, the square a BCD be
constructed, on the sides of which
there are taken the parts cg, ch, df,
be, each equal to \a, and drawing
the right lines eh, fg, intersecting in
1, they form the square igch = i«2. " '*"
Supposing ei = af=ea = if=bg = dh, to denote the
required quantity x; then will the square ai orof ei = x3,
the rectangle bi = ^ax = also to the rectangle id. There-
fore thewhole squareABCD ~x2-t-ax-i- i«2: but x2 ■+■ ax
= n; therefore is the same square a BCD = n -i-ia2: con-
sequently the side bc = */{n ■+■ ia2); and the quantity
sought x = bc — gc = bc — \a = */(ii -+- ^az)—ia.

2ndly. If xz = ax ■+■ n; then will x = \/(^o'2 -+- n) ■+• •£«•

Demonstr. Let the right line BC = x,
and the square upon it abcd = x1. If
ce be taken = a, and the perp. ef = x.
Then the rectangle f.d = ax. Therefore
the remaining rectangle ae=b. But
ae = ab x be = bc x be; therefore n
= bc x be == be2 -(-BEX EC. If EC be
bisected in G, then will n ■+- eg2 = be1 ■+-
Hence y/{n -+- eg2) = be -+- eg ; and gc -+- */{n •+- eg2)
= GC -+- bg. So that, it being GC = EG=-£a, and BC
= x, it will be \a ■+- \/{n -t- ±ar) = x.

3dly. Let x- ■+■ n = ax. — Here he says, if £<r < n, the
equation is impossible. — If ^a'L=n, then isx = -Ja. — If
£a* > n, then is x = \a — v^Ci"2 ~n)i or =ia + v/d^1
-n).

Demonstr. Draw ab representing
a, bisected in g, and unequally di-
vided in d; and upon one of the
unequal parts, supposed = x,if there
be made a square = x2. Now, if,
in the first place, it be formed on
the less part db, supposed = x; let fe be produced to r,
till fi be = ab, and let ai be drawn. Then the whole
rectangle af = ab x bf= ax; hence, taking away the
square be s= x*, there remains the rectangle ae = ad x
de = ad x db=)i; and adding to both the square gd2,
gives ad x db +gd1 = b+gd". But, by Eucl. 11. 5, it

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is ad x db + gd2= bg1; therefore bg8 = ?j ■+■ gd2;
hence bc'-ji =gd!, and v/(bg2 — n) = gd; then bg —

■v/(bG- — n) =BG-GD= DB, Or ■§» — v/(^2 — «) = X.

— If the square x"' be constituted on the greater part ad,
considered as = x; then by a like process it will be
•v/(ag- -b)=gd; hence AG -+- ^/(ag* — n) = AG ■+■
gd = ad, that is ■£« -t- is/(±a7' — n) = x.

Thus we see that Leonard derived the rules for quadra-
tic equations from geometrical considerations, and even
the double values in the possible case of x" -hn = ax; as
it is probable his predecessors had done, and as we find
his successors Lucas de Burgo and others did also. And
this method shows the reason why these writers never ad-
mitted those values of x, in which n is greater than ±ar.
We farther find, that neither Leonard, nor the Arabians,
nor Indians, made any use of the form x* ■+■ ax ■+- n = 0,
nor of the negative roots of the two forms x% + ax = n,
and x* = ax •+• n : it being Harriot who first departed
from the usual custom (of equalizing the positive terms to
the positive in an equation), by placing all the terms on
one side, with their signs ■+■ or — , and made = 0 on the
other side, thus x2 ± ax ± n = 0 ; and it was Cardan who
had the honour of showing that, of the two forms x~ ■+- ax
= n, and x* = ax -+- n, the negative root of the one is the
positive root of the other.

It appears that Leonard Pisanus was well skilled in the
various ways of reducing equations to their final simple
state, by all the usual methods; as addition, subtraction,
multiplication, division, powers and roots, to free them
from radicals, &c. He was also well acquainted with the
modes of substitution, so as to bring out the equation in
the lowest degree ; such as, in cases of two unknown quan-
tities, instead of finding either of them separately, he would
first search out their sum or difference. So, for instance,
in this problem, " To divide the number 10 into two such
parts, that from the greater part taking its own square-
root, and to the less part adding its square root, the two
results should be equal." Now, in this case, if in the
common way x be made to denote one of the parts, as the
greater suppose, then 10 — x will denote the less part, and
the equation will be x — ^/x = 10 — x ■+■ y'C10 ~ x)i tnc
reduction of which leads to a complete equation of the 4th
degree. But, instead of this method, Leonard employs
the way of the half-sum and half-difference, as used by
Diophantus and the Arabians and Hindus, which has
been accounted by some persons an artifice of the modern
algebraists. Thus, if x be put to denote the half-difference
of the two parts; then 5 being their half-sum, 5 +a; will
be the greater, and 5 — x the less, and the equation will
be 5 + x — */ {5 -h x) =z 5 — x + */ (5 — x), which re-
duces to a final quadratic, as is done by this author.

From Pisa, as from a centre, it seems the art gradually
spread through Tuscany and all Italy; in consequence of
which, many other authors in that country had respectable
names before the period of the art of printing. So, we
read that Raffaello Canacci, a Florentine mathematician,
was author of a " Ragionamentodi Algebra," who praises
another that preceded him, named Guglielmo di Lunis;
of whom Canacci writes, at the beginning of his Ragiona-
mento, " La regola dell' algibra,la quale regolaGhughelmo
di Lunis la translato d'Arabico a nostra lingua;" whence
»ome have thought, though without probability, that the
honour of having made Italy acquainted with algebra was
due to Guglielmo di Lunis, rather than to Leonardo Bo-
unce! di Pisa. Many other early authors are mentioned.

Bombelli, in the preface to his book, writes that a work of
Mohamed ben Musa had been shown, but it was of little
value. And it is said that Lucas de Burgo, the first au-
thor in print, was instructed in this science at Venice, by
Domanico Bragadini, successor in the public chair to the
learned Paolo della Pergola, his preceptor, who died in
1366.

Proceed we now to the consideration of the books of
Lucas de Burgo, and other authors, whose works we are
possessed of in print.

LUCAS PACIOLUS, OR DE BURGO.

It was chiefly among the Italians that this art was first '
cultivated in Europe. And, the first author whose works
were in print, was Lucas Paciolus, or Lucas de Burgo, a
cordelier, or minorite friar. He wrote several treatises of
arithmetic, algebra, and geometry, which were printed in
the years 1470, 1476, 14S1, 1487, and in 1494 his prin-
cipal work, entitled Summa de Arithmetica, Geometria,
Proportioni, et Proportionalita, which is a very masterly
and complete treatise on those sciences, as they then stood.
In this work he mentions various former writers, as Eu-
clid, St. Augustine, Sacrobosco or Halifax, Boetius, Pro-
docimo, Giordano, Biagio da Parma, and Leonardus Pi-
sanus, from whom he learned those sciences.

The order of the work is, 1st arithmetic, 2d algebra,
and 3d geometry. Of the arithmetic, the contents, and
the order of them, are nearly as follow. First, of num-
bers figurate, odd and even, perfect, prime and composite,
and many others. Then of common arithmetic in seven
parts, namely, numeration or notation, addition, subtrac-
tion, multiplication, division, progression, and extraction
of roots. Before him, he says, duplation and mediation,
or doubling and halving, were accounted two rules in
arithmetic; but that he omits them, as being included in
multiplication and division. He ascribes the present no-
tation and method of arithmetic to the Arabs ; and savs
that according to some the word Abaco is a corruption of
Modo Arabico, but that according to others it was from a
Greek word. This, however, must be a mistake; for
though the Italians had those arts from the Arabs, these
latter had them probably from the Indians. All those
primary operations he both performs and demonstrates in
various ways, many of which are not in use at present;
proving them not only by what is called casting out the
nines, but also by casting out the sevens, and otherwise.
In the extraction of roots. he uses the initial 1*2 for a root;
and when the roots can be extracted, he calls them dis-
crete or rational ; otherwise surd, or indiscrete, or irra-
tional. The square root is extracted much the same way
as at present, namely, dividing always the last remainder
by double the root found ; and so he continues the surd
roots continually nearer and nearer in vulgar fractions.
Thus, for the root of 6, he first finds the nearest whole
number 2, and the remainder 2 als6; then*£. or •§ is the
first correction, and 2i the second root: its square is 6^-,
therefore ^ divided by 5, or -j1^, is the next correction, and
2^ minus T%, or 2^ is the Sd root: its square is 6^^,
therefore ^g divided by 4^, or -r^o; is the 3d correc-
tion, which gives 2-I~F85'5 for the 4th root, whose square
exceeds 6 by only T84{(;05: and so on continually : and
this process he calls approximation. He observes that
fractions, which he sets down the same way as we do at
present, are extracted, by taking the root of the denomi-
nator, and of the denominated, for so he calls the nume-
rator : and when mixed numbers occur, he directs to re-

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ducc the whole to a fraction, and then extract the roots of
its two terras as above: as if it be 12|-; this he reduces to
■tj9, and then the roots give £ or 3| : in like manner he
finds that 4-§ is the root of 20| ; 5|- the root of 30| ; " and
so on (he adds) in infinitum;" which shows that he knew
how to form the series of squares by addition. He then
extracts the cube root, by a rule much the same as that
which is used at present ; from which it appears that he
was well acquainted with the co-efficients of the binomial
cubed, namely, 1, 3, 3, 1; and he directs how the opera-
tion may be continued " in infinitum" in fractions, like as
in the square root. After this, he describes geometrical
methods for extracting the square and cube roots instru-
mentally : he then treats professedly of vulgar fractions,
their reductions, addition, subtraction, and other opera-
tions, much the same as at present : then of the rule-of-
three, gain-and-loss, and other rules used by merchants.

Paciolus next enters on the algebraical part of this
work, which he calls " L'Arte Magiore; ditta dal vulgo
la Regola de la Cosa, over Alghebra e Almucabala :"
which last name he explains by restauratio et oppositio, and
assigns as a reason for the first name, that it treats of
things above the common affairs in business, which make
the arte minore. Here he mistakenly ascribes the inven-
tion of algebra to the Arabians ; and he says that the Ara-
bian algebra means in Italian position, or rather opposi-
tion. He denominates the series of powers, with their
marks or abbreviations, as n°. or numero, the absolute or
known number; co. or coxa, the thing or 1st power of the
unknown quantity; ce. or censo, the product or square;
cu. or cubo, the cube, or 3d power; ce. ce. or censo de
censo, the square-squared, or 4-th power ; p°. r". or primo
reluto, or 5th power ; ce. cu. or censo de cubo, the square
of the cube, or 6th power; and so on, compounding the
names or indices according to the multiplication of the
numbers 2, 3, &c, and not according to their sum or ad-
dition, as used by Diophantus, with the Arabians and In-
dians. He describes also the other characters made use
of in this part, which are for the most part no more than
the initials or other abbreviations of the words themselves,
after the manner of the Indians; as for T$L radici, the
root; §£. §£. radici de radici, the root of the root; §!. u.
radici universale, or radici legata, or radici unita ; VjL cu.

radici cuba; and q3 quantita, quantity; p for piu or
plus, and m for meno or minus; and he remarks that the
necessity and use of these two last characters are for con-
necting, by addition or subtraction, different powers to-
gether; as 3 co. p. 4 ce. m. 5 cu. p. 2 ce. ce. m. 6 n'. that
is, 3 cosa, piu 4 censa, meno 5 cubo, piu 2 censa-censa,
mono 6" numeri, or, as we now write the same thing, 3x -i-
4r2 — 5x3 ■+■ 2x4 — 6".

He first treats very fully of proportions and proportio-
nalities, both arithmetical and geometrical, accompanied
with a large collection of questions concerning numbers
in continued proportion, resolved by a kind of algebra.
He then treats of el Cataym, which he says, according to
some, is an Arabic or Phenician word, and signifies the
Double Rule of False Position : but he here treats of both
single and double position, as we do at present, dividing
the el Cataym into single and double, as the Arabians and
Indians did. He gives also a geometrical demonstration
of both the cases of the errors in the. double rule, namely,
when the errors are both plus or both minus, and when
the one error is plus and the other minus; and adds a
large collection of questions, as usual. He then goes

Vol. I.

through the common operations of algebra, with all the
variety of signs, as to plus and minvts ; proving that, in
multiplication and division, like signs give plus, and unlike
signs give minus. He next treats of different foots in in-
finitum, and the extraction of roots ; giving also a copious
treatise on radicals or surds, as to their addition, subtrac-
tion, multiplication, and division, and that both in square
roots and cube roots, and in the two together, much the
same as at present. He makes here a digression concern-
ing the 15 lines in the 10th book of Euclid, treating them
as surd numbers, and teaching the extraction of the roots
of the same, or of compound surds or binomials, such as
of 23 p g! 448, or of gt IS p fy 10 ; and gives this rule,
among several others, namely : Divide the first term of
the binomial into two such parts, that their product may
be i of the number in the second term ; then the roots of
those two parts, connected by their proper sign p or m, is
the root of the binomial ; as in this 23 p VjL 448, the two
parts of 23 are 7 and 16, whose product, 112, is \ of 448,
therefore their roots give 4 p §J 7 for the root §£ u. 23
p |Z 448.

He next treats of equations both simple and quadratic,
or simple and compound, as he calls it ; and this latter
he performs by completing the square, and then extract-
ing the root, just as usual. He also resolves equations
of the simple 4th power, and of the 4th combined with
the 2d power, which he treats the same way as quadra-
tics; expressing his rules in a kind of bad verse, and giv-
ing geometrical demonstrations of all the cases, the same
as those of Leonard of Pisa, before described. He uses
both the roots or values of the unknown quantity, in that
case of the quadratics which has two positive roots; but
he takes no notice of the negative roots in the other two
cases. As to any other compound equations, such as the
cube and any other power, or the 4th and 1st, or 4th and
3d, &c, he gives them up as impossible, or at least he
says that no general rule has yet been found for them,
any more, he adds, than for the quadrature of the circle.
— The remainder of this part is employed on rules in trade
and merchandise, such as Fellowship, Barter, Exchange,
Interest, Composition or Alligation, with various other
cases in trade. And in the third part of the work, he
treats of Geometry, both theoretical and practical.

From this account of Lucas de Burgo's book, we may
perceive what was the state of algebra about the year 1500,
in Europe ; and probably it was much the same in Africa
and Arabia, whence the Europeans had it. It appears
that their knowledge extended no farther than quadratic
equations, of which they used only the positive roots ; that
the)' used only one unknown quantity ; that they had no
marks or signs for either quantities or operations, except-
ing only some few abbreviations of the words or names
themselves; and that the art was only employed in resolv-
ing certain numeral problems. So that either the orientals
had not carried algebra beyond quadratic equations, or
else the Europeans had not learned the whole of the art,
as it was then known to the former. And indeed it is not
impossible but this might be the case : for whether the art
was brought to us by an European, who, travelling into
the East, there learned it; or whether it was brought to
us by those people ; in either case we might receive the art
only in an imperfect state, and perhaps far short of the
degree of perfection to which it had been carried by their
best authors. And this suspicion is rendered rather pro-
bable by the circumstance of an Arabic manuscript, said
L

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to be on cubic equations, deposited in the library of the
university of Leyden by the celebrated Warner, bearing a
title, which in Latin signifies Omar Ben Ibrahim al'Gha-
jamcei Algebra cubicarum cequationum, sire de problematum
solidorum resolutions At any rate, however, we have
found that the Hindus, if not the Arabians, treated of
problems including several unknown quantities, with ap-
plications of algebra to geometrical and to indeterminate
problems.

FERREUS AND CARDAN.

After the publication of the books of Lucas de Burgo,
the science of algebra became more generally known, and
improved, especially by many persons in Italy ; and about
this time, or soon after, namely, about the year 1505, the
first rule was there found out by Scipio Ferreus, for re-
solving one case of a compound cubic equation. But
this science, as well as other branches of mathematics,
was most of all cultivated and improved there by Hier-
onymus Cardan of Bononia, a very learned man, whose
arithmetical writings were the next that appeared in print,
namely, in the year 1539, i" nine books, in the Latin lan-
guage, at Milan, where he practised physic, and read pub-
lic lectures on mathematics; and in the year 1545 came
out a 10th book, containing the whole doctrine of cubic
equations, which had been in part revealed to him by Tar-
talea, about the time of the publication of his first nine
books. And as it is only this 10th book which contains
the new discoveries in algebra, I shall here confine myself
to it alone, as it will also afford sufficient occasion to speak
of his manner of treating algebra in general. This book
is divided into 40 chapters, in which the whole science of
cubic equations is most amply and ably treated. Chap. 1,
treats of the nature, number, and properties, of the roots
of equations, and particularly of single equations that have
double roots. He begins with a few remarks on the in-
vention and name of the art : calls it Ars Magna, or Cosa,
or Rules of Algebra, after Lucas de Burgo and others :
says it was invented by Mahomet, the son of one Moses,
an Arabian, as is testified by Leonardus Pisanus; and that
he left four rules or cases, which only included quadratic
equations: that afterwards three derivatives were added by
an unknown author, though some think by Lucas Pacio-
lus; and after that again three other derivatives, for*the
cube and 6th power, by another unknown author; all
which were resolved like quadratics : that then Scipio
Ferreus, professor of mathematics at Bononia, about 1505,
found out the rule for the case cubum et rerum numero
aqualium, or, as we now write it, xl ■+■ bx = c, which he
speaks of as a thing admirable: that the same thing was
next afterwards found out, in 1535, by Tartalea, who re-
vealed it to him, Cardan, after the most earnest entreaties :
that finally, by himself and his quondam pupil Lewis Fer-
rari, the cases were greatly augmented and extended,
namely, by all that is not here expressly ascribed to
others; and that all the demonstrations of the rules are his
own, except only three adopted from Mahomet for the
quadratics, and two of Ferrari for cubics.

Cardan then delivers some remarks, showing that all
square numbers have two roots, the one positive, and the
other negative, or, as he calls them, vera et ficla, true and
fictitious or false ; so the ivstimatio rei, or root, of <), 's ci-
ther 3 or — 3 ; of i6 it is 4 or — 4; the 4th root of 81 is
3 or —3; and soon for all even denominations ox powers.
And the same is remarked on compound cases of even
powers that arc added together; as if x'1 + 3x'J = 28, then

the sestimatio x is = 2 or — 2 ; but that the form x* ■+ 12
= 7x2 has four answers or roots, in real numbers, two plus
and two minus, viz, 2 or — 2, and */3 or — -y/3; while
the case x4 -+- 12 = 6x2 has no real roots ; and the case
x1 = 2x'2 -+- 8 has two, namely, 2 and — 2; and in like
manner for other even powers. So that he includes both
the positive and negative roots; but rejects what we now
call imaginary ones. I here express the cases in our mo-
dern notation, for brevity sake, as he commonly expresses
the terms by words at full length, calling the absolute or
known term the numero, the 1st power the re*, the 2d the
quadratum, the 3d the cubum, and so on, using no mark for
the unknown quantity, and only the initials p and m for
plus and minus, and §£ for radix or root. The res he
sometimes calls positio (as derived from the like quantity
in the rule of position), and quanlitas ignota; and in sta-
ting or setting down his equations, he as well as Lucas de
Burgo before him, sets down the terms on that side where
they will be plus, and not minus.

On the other hand, he remarks that the odd denomina-
tions, or powers, have only one ffistimatio, or root, and
that true or positive, but none fictitious or negalhe, and
for this reason, that no negative number raised to an odd
power, will give a positive number; so of 2x = 16, the
root is S only; and if 2x3 = 16, the root is 2 only : and if
there be ever so many odd denominations, added together,
equal to a number, there will be only one asstimalio or
root; as if x3 + 6x = 20, the only root. is 2. But that
when the signs of some of the terms are different, as to
• plus and minus, they may have more. roots; and he shows
certain relations of the coefficients, when they have two
or more roots : so the equation x3 + 10' = 12x has two
Kstimatios, the one true or 2, and the other fictitious or

— 4, which he observes is the same as ihe true ajstimatio
of the case x3 = 12x -i- \6, having only the sign of the ab-
solute number changed from the former, the 3d root 2
being the same as the first, which therefore he does not
count. He next shows what are the relations of the co-
efficients when a cubic equation has three roots, ot which
two are true, and the 3d fictitious, which is always equal
to the sum of the other two, and also equal to the true
root of the same equation with the sign of the absolute
number changed: thus, in the equation x3 + 9 = I2x,
the two true roots are 3 and */5^ — !■£, and the fictitious
one is — *J b\ — 1t> which last is the same as the true
root of x3 = 12x +9, viz, y/ 5% + !•£; and he here in-
fers generally that the fictitious asstimatio of the case x3 •+■
c = bx, always answers to the true root of x3 = bx -+- c.
Cardan also shows what the relation of the coefficients is,
when the case has no true roots, but only one fictitious
root, which is the same as the true root of the reciprocal
case, formed by changing the sign of the absolute number.
Thus, the case x3 -t- 21 = 2x has no true root, and only
one false root, viz, — 3, which is the same as the true root
of x3 = 2x -t- 21 : and he shows in general, that changing
the sign of the absolute number in such cases as want the
2d term, or changing the signs of the even terms when it
is not wanting, changes the signs of all the three roots,
which he also illustrates by many examples; thus, the
roots of x3 -h 1 lxa = 72, are -+- y^40 - 4, and —3, and

— ^40 — 4; and the roots of x' h- 72 = 11. r% are —
^/40 -+- 4, and -+- 3, and ■+■ ^/40 + 4.

And he further observes, that the sum of the three
roots, or the difference between the true and fictitious .
roots, is equal to 11, the coefficient of the 2d term, lie

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also shows how certain cubic cases have one, or two, or
three roots, according to circumstances : that the case
x4-t-d = bxz has sometimes four roots, and sometimes none
at all, that is, no real ones : that the case x3 ■+■ bx = axa ■+■ c
may have three true aestimatios, or positive roots, but no
fictitious or negative ones ; and for this reason, that the
odd powers of minus being minus, and the even powers
plus, the two terms x3 -+- bx would be negative, and equal
to a positive sum axz ■+• c, which is absurd : and further,
that the case x3 ■+■ ax2 -+- bx = c has three roots, one plus
and two minus, which are the same, with the signs changed,
as the roots of the case x3 -+• bx -+- c = ax\ He also shows
the relation of the coefficients when the equation has only
one real root, in a variety of cases : but that the case
x3 ■+- c = ax2, -+- bx has always one negative root, and either
two positive roots, or none at all ; the number of roots
failing by pairs, or the impossible roots, as we now call
them, being always in pairs. Of all these circumstances
Cardan gives a great many particular examples in numeral
coefficients, and subjoins geometrical demonstrations of
the properties here enumerated ; such as, that the two
corresponding or reciprocal cases have the same root or
roots, but with different signs or affections ; and how
many true or positive roots each case has.

On the whole, it appears from this short chapter, that
Cardan had discovered most of the principal properties of
the roots of equations, and could point out the number
and nature of the roots, partly from the signs of the terms,
and partly from the magnitude and relations of the co-
efficients. He shows in effect, that when the case has all
its roots, or when none are impossible, the number of its
positive roots is the same as the number of changes in the
signs of the terms, when they are all brought to one side :
that the co-efficient of the 2d term is equal to the sum of
all the roots positive and negative collected together, and
consequently that when the 2d term is wanting, the posi-
tive roots are equal to the negative ones : and that the
signs of all the roots are changed, by changing only the
signs of the even terms: with many other just remarks
concerning the nature of equations.

In chap. 2, Cardan enumerates all the cases of com-
pound equations of the 2d and 3d order, namely 3 qua-
dratics, and 19 cubics; with 44 derivatives of these two,
that is, of the same kind, with higher denominations.

In chap. 3 are treated the roots of simple cases, or sim-
ple equations, or at least that will reduce to such, having
only two terms, the one equal to the other. He directs to
depress the denominations equally, as much as they will,
according to the height of the least; then divide by the
number or coefficient of the greatest ; and lastly extract
the root on both sides. So if 20x3=lSOxs, then 20= ISO*2,
and i = x2, and x = \.

Chap. 4 treats of both general and particular roots, and
contains various definitions and obsorvations concerning
them. It is here shown, that the several cases of quadra-
tics and cubics have their roots of the following forms or
kinds, namely, that the case
x1 = ax ■+■ b has its root of this kind a/ 19 -+■ 3,
x2 ■+■ ax = b has its root of this kind v/)9 — 3,
x1 -+- b = ax has two roots like 3-f-.y/2 and 3 — y/2,
x3 = bx ■+- c has its root of this kind ?/4 ■+■ \/2,
x3 -+- bx = c has its root of this kind */4 — 1/2,
#3 = axz -+. c has this kind of root i/ 1 6 -+- 2 •+- 1/ 4,
x3 •*- ax"- = c has this kind of root l/\6 — 2 -+- (/4,
where the three parts l/\ 6, 2, ?/4, are in continual pro-
portion.

Chap. 5 treats of the asstimatio of the lowest degree of
compound cases, that is, affected quadratic equations ;
giving the rule for each of the three cases, which consists
in completing the square, &c, as at present, and which
was the method given by the Arabians and Hindus; and
proving them by geometrical demonstrations from Eucl. i,
43, and ii, 4 and 5 ; in which he makes some improve-
ment on the demonstrations of Mahomet. And hence it
appears that the work of this Arabian author was in be-
ing, and well known in Cardan's time.

Chap. 6, on the methods of finding new rules, contains
some curious speculations concerning the squares and
cubes of binomial and residual quantities, and the propor-
tions of the terms of which they consist, shown from geo-
metrical demonstrations, with many curious remarks and
properties, forming a foundation of principles for investi-
gating the rules for cubic equations.

Chap. 7 is on the transmutation of equations, showing
how to change them from one form to another, by taking
away certain terms out of them; as x3 ■+- ox2 — c, to
x3= bx -+- d, &c. The rules are demonstrated geometri-
cally ; and a table is added, of the forms into which any
given cases will reduce ; which transformations are ex-
tended to equations of the 4th and 5th order. And hence
it appears that Cardan knew how to take away any term
out of an equation.

Chap. 8 shows generally how to find the root of any such
equation as this, axm = x" ■+■ b, where m and n are any ex-
ponents whatever, but n the greater ; and the rule is, to
separate or divide the coefficient a into two such parts z
and a — z, as that the absolute number b shall be equal to

(a — z).zn — m, the product of the one part a — z, and the
— — power of the other part: then the root x is =

i
z"—m. The rule is general for quadratics, cubics, and all
the higher powers : and could not have been formed with-
out the knowledge of the composition of the terms from
the roots of the equation.

Chap. 9 and 10 contain the resolution of various ques-
tions producing equations not higher than quadratics.

Chap. 1 1 is on the case or form x3 -+- 3bx = 2c. Cardan
now comes to the actual resolution of the first case of cu-
bic equations. He begins with relating a short history of
the invention of it, observing that it was first found out,
about thirty years before, by Scipio Ferreus, of Bononia,
and by him taught to Antonio Maria Florido, of Venice,
who having a contest afterwards with Nicolas Tartalea, of
Brescia, it gave occasion to Tartalea to find it out him-
self, who after great entreaties taught it to Cardan, but
suppressed the demonstration. By help of the rule alone,
however, Cardan of himself discovered the source or geo-
metrical investigation, which he gives here at large, from
Euc. ii, 4. In this process he makes use. of the Greek
letters a, S, y, $, &c, to denote certain indefinite numbers
or quantities, to render the investigation general ; which
may be considered as the first instance of such literal no-
tation in algebra. He then gives the rule in words at
length, which comes to this, i=|/[v'(c! + iJ) + (i]
—i/[\/(c2' ■+■ b3)— c]; illustrating it by a variety of ex-
amples ; in the resolution of which, he extracts the cubic
roots of such of the binomials as will admit of it, by some
rule which he had for that purpose ; such as x3 -+- 6x = 20,
where x =1/(^108 + 10) - 1/ (^108 - 10) = (</Z
+ ])-(v/3- 1) = 2.

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Chap. 12, on the case x3 =t 3bx +■ 2c. Thishe treats ex-
actly as the last, and finds the rule to be x = l/[c-*- */(<?

— J3)] ■+- l/[c— v/(c2 — bz)] ; which he illustrates by
many examples, as usual. But when 63 exceeds c°, which
has since been called the irreducible case, he refers to
another following book, called Aliza, for other rules of
solution, to overcome this difficult}', about which he took
infinite pains.

Chap. 13, on the case x3 ■+■ 2c = 3bx. This case, by a
geometrical process, he reduces to the case in the last
chapter : thus, find the sestimatio y of the case y3 = 3by
•+■ 2c, having the same coefficients as the given case
x3 + 2c = 3bx ; then is x =±y ± W(l2b ~ 3/)> giv-
ing two roots. He shows also how to find the second
root, when the first is known, independent of the foregoing
case. From this relation of these two cases he deduces
several corollaries, one of which is, that the sestimatio or
root of the casey3 = 3by ■+■ 2c, is equal to the sum of the
roots of the case x3 -+- 2c = 3bx. As in the example
y =s l6y -+- 21, whose aestimatio is \/9i ■+■ li> which is
equal to the sum of 3 and y^i ~ ]tj tne tw0 roots of
the case x' -t- 21 == l6*x;

In chapters 14, 15, and 16, he treats of the three cases,
which contain the 2d and 3d powers, but wanting the 1st
power, according to all the varieties of the signs; which
he performs by exterminating the 2d term, or that which
contains the 2d power of the unknown quantity x, by sub-
stituting^ ± | the coefficient of that term for x, and so
reducing these cases to one of the former. In these chap-
ters Cardan sometimes also gives other rules ; thus, for the
case x3 + 3axz = 2c, find first the sestimatio y of the case
y> == 3ay +n/2c, then is x = y* — 3a : also for the case
x3 + 2c =z Sax1, first find the two roots of yz ■+- 2c,

— 3ay$/2c, then is x = ^— - the two values of x according

to the two values of y. He here also gives another rule,
by which a second aestimatio or root is found, when the
first is known, namely, if e be the first aestimatio, or value
of x in the case x3 •+- 2c = 3ax", then is the other value of
x = Wl(3a ~ e) ■ (3a -*- 3(0] ■+- i(3a ~ 0-

In chapters 17, 1 8, 19, 20, 21, 22, 23, Cardan treats of
the cases in which all the four terms of the equation are
present ; and this he always effects by taking away the 2d
term out of the equation, and so reducing it to one of the
foregoing cases which want that term, giving always geo-
metrical investigations, and adding a great many examples
of every case of the equations.

Chap. 24 is on the 44 derivative cases; which are only
higher powers of the forms of quadratics, and cubics.

Chap. 25, on imperfect and special cases ; containing
many particular examples when the coefficients have cer-
tain relations among them, with easy rules for finding the
roots ; also 8 other rules for the irreducible case a3 = bx
■+■ c.

Chap. 26, in like manner, contains easy rules for biqua-
dratics, when the coeflicicnts have certain special rela-
tions.

Then the following chapters, from chap. 27 to chap. 38,
contain a great number of questions and applications of
various kinds, the titles of which are these: De transitu
capituli specialis in capitulum speciale; De operationibus
radicum pronicarum sen mixtarum et Allellarum ; De regula
modi ; De regula Auria; De regula Magna, or the method
of finding out solutions to certain questions; De regula
mjuulis positionis, being a method of substituting for the
half sum and half difference of two quantities instead of

the quantities themselves ; De regula inaqualiter ponendi,
scu proportionis ; De regula medii ; De regula aggregati ;
De regula libera positionis; De regula falsum ponendi, in
which some quantities come out negative ; Quomodo exci-
dant partes et denominations multiplicando. Among the
foregoing collection of questions, which are chiefly about
numbers, there are some geometrical ones, being the ap-
plication of algebra to geometry; such as, In aright-
angled triangle, given the sum of each leg and the ad-
jacent segment of the hypotenuse, made by a perpendicu-
lar from the right angle, to determine the area, &c ; with
other such geometrical questions, resolved algebraically.
Chap. 39, De regula qua pluribus posilionibus invenimus
ignotam quantitatem ; which is employed on biquadratic
equations. After some examples of his own, Cardan gives
a rule of Lewis Ferrari's, for resolving all biquadratics,
namely, by means of a cubic equation, which Ferrari in-
vestigated at his request, and which Cardan here demon-
strates, and applies in all its cases. The method is very
general, and consists in forming three squares, thus: first,
complete one side of the equation up to a square, by add-
ing or subtracting some multiples or parts of some of its
own terms on both sides, which it is always easy to do :
2d, supposing now the three terms of this square to bebut
one quantity, viz, the first term of another square to
which this same side is to be completed, by annexing the
square of a new and assumed indeterminate quantity,
with double the product of the roots of both ; which evi-
dently forms the square of a binomial, consisting of the
assumed indeterminate quantity and the root of the first
square: 3d, the other side of the equation is then made
to become the square of a binomial also, by supposing the
product of its 1st and 3d terms to be equal to the square
of half its 2d term; for it consists of only three terms, or
three different denominations of the original unknown
quantity: then this equality will determine the value of
the assumed indeterminate quantity, by means of a cubic
equation, and from it, that of the original ignota, by the
equal roots of the 2d and 3d squares. Here we have a
notable example of the use of assuming a new indetermi-
nate quantity to introduce into an equation, long before
Des Cartes was born, who made use of a like assumption
for a similar purpose. And this method is very general,
and is here applied to all forms of biquadratics, either
having all their terms, or wanting some of them. To il-
lustrate this rule, I shall here set down the process in one
of his examples, which is this,!1 -1- 4x -+- 8 = 10.i'-. Now
first subtract 2x* ■+■ 4x -t- 7 from both sides, then the first
becomes a square, viz, xA — 2x2 -+- 1 or (x* — 1)"= 8x2
— 4x — 7. Next assume the indeterminate y, and sub-
tract 2y (x- — 1) — y1 from both sides, making the first
side again a square, viz, (x — I)2 — 2y (x — 1) -t-y3 or
(x* - 1 - yY = (8 - 2j0 x- - 4x +• ys -1- ty — 7, Of
this latter side, make the product of the 1st and 3d terms
equal to the square of half the 2d term, that is, (8 — 2y) .
(y1 ■+■ 2y — 7) = 2% which reduces toy3 + 30 = 2$r ■+-
loy ; the positive roots of which are y = 2 or ^15 : and
hence, using 2 for y, the equation of equal squares be-
comes (x1 — 1 — y/2or(x'2 — 3)a= 4xs — 4x -+- 1, the roots
of which give x2 — 3 = 2x ~ 1 ; and hence x2 = 2x •+• 2, or
x ■+■ 2x = 4; the two positive roots of which are ^/5 ■+■ 1
and ^/ 5 — 1, which are two of the values of x in the
given equation x' -1- 4x ■+■ 8 = 10x2. The other roots he
leaves to be tried by the reader.

The 40th, or last, chap, is entitled, On modes of general
supposition relating to this art; with some rules of an un-

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usual kind; and aestimatios or roots of a nature different
from the foregoing ones. Some of these are as follow : If
x3 =r ax2 ■+■ c, and x — a = y, and x : y : : c : d; then
isy3 ■+- uyl-= d.

Secondly, if xz -+■ ax1 = c, and y3 = ay2 •+■ c,
then is x ■+■ a : y — a : : y2 : x2.

Thirdly, when x3 -+- c = ax2, the square will be taken
away, by putting x = y + ^a ; and then the equation be-
comes y3 -t- c — 2(jO)3 = \(^y.

Cardan adds some other remarks concerning the solu-
tions of certain cases and questions, all evincing the accu-
racy of his skill, and the extent of his practice ; he then
concludes the book with a remark concerning a certain
transformation of equations, which quite astonishes us, to
find that the same person who, through the whole work,
has shown such a profound and critical skill in the na-
ture of equations, and the solution of problems, should
yet be ignorant of one of the most obvious transmutations
attending them, namely, increasing or diminishing the
roots in any proportion. Cardan having observed that the
form x3 = bx -+■ c may be changed into another similar one,

viz, y3 = - y ■+■ \/-, of which the coefficient of the term

y is the quotient arising from the coefficient of x divided by
the absolute number of the first equation; and that the
absolute number of the 2d equation, is the root of the
quotient of \ divided by the said absolute number of the
first ; he then adds, that finding the aestimatio or root of
the one equation from that of the other is very difficult,
valde difficilis.

It is matter of wonder that Cardan, among so many
transmutations, should never think of substituting, instead
of x in such equations, another positio or root, greater or
less than the former in any indefinite proportion, that is,
multiplied or divided by a given number; for this would
have led him immediately to the same transformation as
he makes above, and that by a way which would have
shown the constant proportion between the two roots.
Thus, instead of x in the given form x3 =s bx ■+- c, substi-
tute dy, and it becomes d3y3 = bdy -+- c; and this divided

b c

by d3 becomes y3 = j2y -+- •» ; and here if d be taken =

i/c, it becomes y3 = -y ■+- ^/ -; the very transformation
in question, and in which it is evident that x is = y^/c,

and y :

Instead of this, Cardan gives the following

strange way of finding the one root x from the other y,
when this latter is by any means known; viz, Multiply

the first given equation by y2x ■+■ 1, then add — 3 to both

sides, and lastly extract the roots of both, which can al-
ways be done, as they will always be both of them squares ;
and the roots will give the value of x by a quadratic
equation.

Thus, x3 = bx •+■ c multiplied byy2x + 1 gives

then

y2x* ■+- x3 ss by"x2 -h (cy1 -+- b)
a?

c; add

t = by2 h ;. i' + (i + cy2) x -+- c; and

y'x- ■+■

the roots are yx2 -+- — = n/[(by2 -h -^5) x2 ■+- (b -+- cy2)

x -+■ c] ; and this 2d side of the equation, he says, will al-
ways have a root also. It is indeed true, that it will
have an exact root; but the reason of it is not obvious,

which is, because^ is the root of the equation y3 = -y ■+•

\/-. Cardan has not shown the reason why this happens ;

but I apprehend he made it out in this manner, viz, simi-
lar to the way in which he forms the last square in the
case of biquadratic equations, namely, by making the pro-
duct of the 1st and 3d terms equal to the square of half

the 2d term : thus, in the present case, it is 4c (by2 -+■ — ;)

= (b + cy2)2, which reduces to y3 = ~y ■+■ */- the equa-
tion in question. Therefore taking y the root of the
equation^3 = -y + „J ' c; and substituting its value in the

(b •+- cy*) x ■+■ c, this becomes

quantity {by2 + ±) x2

a complete square.

Of Cardan's Libellus de Aliza Regula. Subjoined to
the above treatise on cubic equations, is this Libellus de
Aliza Regula, or the algebraic logistics, in which the au-
thor treats of some of the more abstruse parts of arithme-
tic and algebra, especially cubic equations, with many
more attempts on the irreducible case x3 = bx ■+■ c. This
book is divided into 60 chapters; but I shall only set
down the titles of some few of them, whose contents re-
quire more particular notice.

Chap. 4. De modo redigendi quantitates omncs, qua di-
cuntur latera prima, ex decimo Euclidis in compendium.- He
treats here of all Euclid's irrational lines, as surd numbers,
and performs various operations with them.

Chap. 5. De consideration binomiorumet recisorum, Ifc;
ubi de aslimatione capitulorum. Contains various opera-
tions of multiplying compound numbers and surds.

Chap. 6. De operationibus p: et m: (i. e. ■+■ and — )
secundum communem usum. Here it is shown that, in mul-
tiplication and division, plus always gives the same signs,
and minus gives the contrary signs. So also in addition,
every quantity retains its own sign ; but in subtraction
they change the signs. That the \/ -+-, or the square root
of plus, is -1- ; but the \/ — , or the square root of minus
is nothing as to common use : (but of this below.) That
1/ — is — , as \/ — 8 is — 2. That a residual, composed
of ■+• and — may have a root also composed of -+-
and - : so ^/(5 - ^/24) is = ^3 - </2. The rules
for the signs in multiplication and division are illus-
trated by this example; to divide 8 by 2,-)- \/6 or 4/6
■+■ 2. Take the two corresponding residuals 2 — ,^/6and
\/6 — 2, and by these multiply both the divisor and divi-
dend; then the products are ■+■ and — respectively, and
the quotients still both alike. Thus,
Divid. Divis.

8 ^/6 + 2

y/6-2 y/6 - 2
\/384 — 16 divide -+- 2
Quot. ^96 — 8.

Divid.

2-v/6

Divis.
2 + ^/6
2- ^5

16 — v/384div. -2

Quot. y/96 — 8.

And this method of performing division of compound
surds, was fully taught before him, by Lucas de Burgo,
namely, reducing the compound divisor to a simple quan-
tity, by multiplying by the corresponding quantity, having
the sign changed.

In chap. 1J and 18, and elsewhere, Cardan makes a
general notation of a, b, c, d, e,f, for any indefinite quan-
tities, and treats of them in a general way.

Chap. 2. De contemplatione p: et m : (or ■+■ and — ) et
quod m : in m : facit p : et de causis horumjuxta veritatem.
Cardan here demonstrates geometrically that, in mullipli-

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cation and division, like signs give plus, and unlike signs
give minus. And he illustrates this numerically, by
squaring the quantity S, or 6 ■+■ 2, or 10 — 2, which
must all produce the same thing, namely, 64.

Among many of the chapters which treat of the irredu-
cible case x3 = bx -+- c, there is a peculiar kind of way-
given in chap. 31, which is entitled De cestimatione generali
x3 = bx ■+• c solida vocata, et operationibus ejus ; in which
he shows how to approximate to the root of that case, in
a manner similar to approximating the square root and
cube root of a number. The rule lie uses for this purpose,
is the 3d in chap. 25 6f the last book, and it is this : Di-
vide b into two parts, such that the sum of the products
of each, multiplied by the square root of the other, may be
equal to \c; then the sum of the roots of 9 1
these parts is the aestimatio or value of j>£
x required. So,of this equation x3 = 10a: 3 1
-i- 24; the two parts are 9 and 1, andtheir 3+9or 12=^c,
roots 3 and 1 , and their sum 4 = x, as 1 = 3 + 1 = 4,
in the margin. Again, take x3 = 6x -+- 1. Here he in-
vents a new notation to express the root or radix, which
he calls solida, viz, x = ^/solida 6 in i, that is, the roots
of the two parts of 6, so that each part multiplied by the
root of the other, the two products may be i or \c. Then
to free this from fractions, and make the operation easier,
multiply that root by some number, as suppose 4, that is,
the square part 6 by the square of 4, and the solid part •£
by the cube of 4; then x = \*/ solida 96 in 32. Now,
by a few trials, it is found that the parts are

nearly 95-| and i, which give too much,
or 95t£j and ^, which give too little,
and thereof 95f£ and T\ are still nearer. Divide both by
42 or 16, then 5-J-f£ and -^ are the quotients. And the
sum of their roots, or x = >/^fH + V'xfs 's near'y trie
value of the root x.

Chap. 42. De duplici cequatione comparanda in capitulo
cubi et numeri cequalium rebus. Treats of the two positive
roots of that case, neglecting the negative one; and show-
ing, not only that the case has two such roots, but that
the same number may be the common root of innumera-
ble equations.

Chap. 57. De tractatione astimationis generaliscapitidi x3
= bx ■+- c. Cardan here again resumes the consideration
of the irreducible case, making ingenious observations
upon it, but still without obtaining the root by a general
rule. In this place also, as well as elsewhere, he shows
how to form an equation in this case, that shall have a
given binomial root, as suppose ^/m ■+- n, where the equa-
tion will be x* = (m ■+■ 3ra2) x ■+■ 2n(m — n2), having
^/m -+- n for one root, namely the positive root. From
which it appears that he was well acquainted with the
composition of cubic equations from given roots.

Chap. 59. De ordine et exemplis in binomiis secundo et
quinto. Contains a great many numeral forms of the
same irreducible case x3 = bx ■+■ c, with their roots ; from
which are derived these following cases, with many cu-
rious remarks. As, when

x3 = ( c -+- 1) x + c, then x = ^/( c -+■ £) + i-
x3 = (|c + 4) x + c, then x = 1/ (£c -+- $)-»-$•
x3 = (|c •+• 9) x -+- c, then x = \/(|c ■+■ |) p*- 1.
x3 = (Jc ■+• 16) x *■ c, then x = </\%c + V6) -t- %.

x 2= (-c -+• na) x •+- c, then x = \/(-c h ) h — •

Chap.GO. Demonstratio generalis capituli cubi aqualis re-
bus et numero. This demonstration of the irreducible case

is geometrical, like all the rest. Some more ingenious re-
marks are again added, as if he reluctantly finished the
book, without perfectly overcoming the difficulty of the ir-
reducible case. Cardan here also uses the letters a and b
for any two indefinite numbers, in order to show the form

and manner of the arithmetical operations : thus - is the

fraction for theirquotient, also ^/y or ^ the square root
b ^/b ^

of that quotient, and 1/ - or'j-rr the cube root of it, &c.

Having considered the chief contents of Cardan's alge-
bra, it will now be proper to sum them up, and set down
a list of the improvements made by him, as collected from
his writings.

And 1st, Tartalea having only communicated to him
the rules for resolving these three cases of cubic equa-
tions, viz,

x3 -t- bx = c,~l he thence raised a very large and complete
x3 = bx ■+■ c, > work, laying down rules for all forms and
x3 -t- c 3= br,) varieties of cubic equations, having all their
terms, or wanting any of them, and having all possible
varieties of signs ; demonstrating all these rules Geometri-
cally ; and treating very fully of almost all kinds of trans-
formations of equations, in a manner before unknown.

2d, It appears that he was well acquainted with all
the roots of equations that are real, both positive and ne-
gative; or, as he calls them, true and fictitious; and
thathe made use of them both occasionally. Healso showed
that the even roots of positive quantities, are either posi-
tive or negative ; that the odd roots of negative quantities,
are real and negative ; but that the even roots of them are
impossible, or nothing as to common use. He was also
acquainted with the

3d, Number and nature of the roots of an equation,
and that partly from the signs of the terms, and partly
from the magnitude and relation of the coefficients. He
also knew,

4th, That the number of positive roots is equal to the
number of changes of the signs of the terms.

5th, That the coefficient of the second term of the
equation, is the difference between the positive and nega-
tive roots.

6th, That when the second term is wanting, the sum
of the negative roots is equal to the sum of the posi-
tive roots.

7th, How to compose equations that shall have given
roots.

8th, That, changing the signs of the even terms,
changes the signs of all the roots.

9th, That the number of roots failed in pairs ; or what
we now call impossible roots were always in pairs.

10th, To change the equation from one form to another,
by taking away any term out of it.

11th, To increase or diminish the roots by a given
quantity. It appears also,

12th, That he had a rule for extracting the cube root
of such binomials as admit of extraction.

13th, That he often used the literal notation a, b, c, d,
&c.

14th, That he gave a rule for biquadratic equations,
suiting all their cases; and that, in the investigation of
that rule, he made use of an assumed indeterminate quan-
tity, and afterwards found its value by the arbitrary as-
sumption of a relation between the terms.

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15th, That he applied algebra to the resolution of geo-
metrical problems. And,

lf>th, That he was well acquainted with the difficulty of
what is called the irreducible case, viz, r3 = bx -+- c, upon
which he spent a great deal of time, in attempting to
overcome it. And though he did not fully succeed in this
case, any more than other persons have done since,
he nevertheless made many ingenious observations about
it, laying down rules for many particular forms of it, and
showing how to approximate very nearly to the root in all
cases whatever.

Cardan died at Rome in the year 1575.

OJ? TARTALEA.'

Nicholas Tartalea, or Tartaglia, of Brescia, was con-
temporary with Cardan, and was probably older than
he was ; but I do not know of any book of algebra
published by him till the year 1546, the year after the
date of Cardan's work on Cubic Equations, when he
printed his Quesiti et Invention! diverse, at Venice, where
he residi d as a public lecturer on mathematics. This
work is dedicated to our king Henry the 8th of England,
and consists of nine books, containing answers to various
questions which had been proposed to him at different
times, concerning mechanics, statics, hydrostatics, &c ;
but it is only the 9th, or last book, that we shall have
occasion to take notice of in this place, as it contains all
those questions which relate to arithmetic and algebra.
These are all set down in chronological order, forming a
pretty collection of questions and solutions on those sub-
jects, with a short account of the occasion of each of
them. Among these, the correspondence between him
and Cardan forms a remarkable part, as we have here
the history of the invention of the rules for cubic equations,
which he communicated to Cardan, under the promise,
and indeed oath, to keep them secret, on the 25th of
March, 1539- But, notwithstanding his oath, finding
that Cardan published them in 1545, as above related, it
seems Tartalea published the correspondence between
them, in revenge for his breach of faith ; and it elsewhere
appears, that many other sharp bickerings passed between
them on the same account, which only ended with the
the death of Tartalea, in the year 1557-
• It seems it was a common practice among the mathe-
maticians, and others, of that time, as in most other times,
to send to each other curious and difficult questions, as
trials of skill; and to this circumstance it is that we owe
the principal questions and discoveries in this collection,
as well as many of the best discoveries of other authors.
The collection now before us contains questions and solu-
tions, with their dates, in a regular order, from the year
1521, and ending in 154-1, in 42 dialogues, the last of
which is with an English gentleman, namely, Mr. Richard
Wentworth, who it seems was no mean mathematician,
and who learned some algebra, &c, of Tartalea, while he
resided at Venice. The questions at first are mostly very
easy ones in arithmetic, but gradually become more diffi-
cult, and exercising simple and quadratic equations, with
complex calculations of radical quantities : all showing
that he was well skilled in the art of algebra as it then
stood, and that he was very ingenious ill applying it to the
solution of questions. Tartalea made no alteration in the
notation or forms of expression used by Lucas de Burgo,
calling the first power of the unknown quantity, iri his
language, cosa, the second power censa, the third cubo, &c,
and writing the names ofall the operationsin words atlcngth,

without using any contractions, except the initial ]£, for
root or radicality. So that the only thing remarkable in
this collection, is the discovery of the rules for cubic equa-
tions, with the curious circumstances attending the same.

The first two of these were discovered by Tartalea in
the year 1530, namely, for the two cases x2 H- ax* = c,
and x3 = ax'z -+- f, as appears by Quest. 14 and 25 of this
collection, on occasion of a question then proposed to him
by one Zuanni de Tonini da Coi or Colle, John Hill, who
kept a school at Brescia. And from the 25th question we
learn, that he discovered the rules for the other two cases
x3 -+- bx = c, and x3 — bx -+- c, on the 12th and 13th of
February, 1535, at Venice, where he had come to reside
the year before. And the occasion of it was this : There
was then at Venice one Antonio Maria Fiore or Florido,
who, by his own account, had received from his preceptor
Scipio Farreo, about thirty years before, a general, rule for
resolving the case x3 -t- bx = c. Being a captious man,
and presuming on this discovery, Florido used to brave his
contemporaries, and by his insults provoked Tartalea to
enter into a wager with him, that each should propose to
the other 30 different questions; and that he. who soonest
resolved those of his adversary, should win from him as
many treats for himself and friends. These questions
were to be proposed on a certain day at some weeks dis-
tance ; and Tartalea made such good use of his time, that
eight days before the time appointed for delivering the pro-
positions, he discovered the rules both for the case x3 -+- bx
= c, and the case x3 = bx ■+■ c. He therefore proposed
several of his questions, so as to fall either on this latter
case, or on the cases of the cube and square, expecting
that his adversary would propose his in the former. And
what he suspected fell out accordingly ; the consequence
of which was, that on the day of meeting Tartalea resolved
all his adversary's questions in the space of two hours,
without receiving one answer from Florido in return ; to
whom, however, Tartalea generously remitted the forfeit
of the 30 treats won of him.

Question 31 first brings us acquainted with the corre-
spondence between Tartalea and Cardan. This corre-
spondence is very curious, and would well deserve to be
given at full length in their own words, if it were not too
long for this place. I may enlarge farther upon it under
the article Cubic Equations; but must here be content with
a brief abstract only. Cardan was then a respectable phy-
sician, and lecturer in mathematics, at Milan ; who having
nearly finished the printing of a large work on arithmetic,
algebra, and geometry, and having heard of Tartalea's dis-
coveries in cubic equations, he was very desirous of draw-
ing those rules from him, that he might add them to his
book before it was finished. For this purpose, he first ap-
plied to Tartalea, by means of a third person, a bookseller,
whom he sent to him, in the beginning of the year 1539,
with many flattering compliments, and offers of his ser-
vices and friendship, &c, accompanied with some critical
questions for him to resolve, according to the custom of
the times. Tartalea however refused to disclose his rules
to any one ; as the knowledge of them gained him great
reputation among all people, and gave him great advan-
tage over his competitors for fame, who were commonly
afraid of him on account of those very rules. He only
sent Cardan therefore, at his request, a copy of the
30 questions which had been proposed to him in the con-
test with Florido.

Not to be rebuffed so easily, Cardan next applied, in

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the most urgent manner, by letter, to Tartalea ; which
however procured from him onty the solution of some
other questions proposed by Cardan, and of a few of the
questions that had been proposed to Florido, but none of
their solutions. Finding he could not thus prevail, with
all his fair promises, Cardan then fell upon another
scheme. There was at Milan a certain Marquis dal
Vasto, a great patron of Cardan, and, it was said, of
learned men in general. Cardan conceived the idea of
making use of the influence of this nobleman to draw
Tartalea to Milan, hoping that then, by personal entrea-
ties, he should succeed in drawing the long-concealed
rules from him. Accordingly he wrote a second letter
to Tartalea, much in the same strain with the former,
strongly inviting him to come and spend a few days in his
house at Milan, and representing that, having often com-
mended him in the highest terms to the marquis, this no-
bleman desired much to see him ; for which reason Car-
dan advised him, as a friend, to come to visit them at Mi-
lan, as it might be greatly to his interest, the marquis be-
ing very liberal and bountiful ; and he besides gave Tar-
talea to understand, that it might be dangerous to offend
such a man by refusing to come, who might, in that case,
take offence, and do him some injury. This manoeuvre
had the desired effect : Tartalea on this occasion laments
to himself in these words, " By this I am reduced to a
great dilemma ; for if I go not to Milan, the marquis may
take it amiss, and some evil may befall me on that ac-
count; I shall therefore go, though very unwillingly."
When he arrived at Milan, however, the marquis was
gone to Vigeveno ; and Tartalea was prevailed on to stay
three days with Cardan, in expectation of the marquis re-
turning ; at the end of which time he set out from Milan,
with a letter from Cardan, to go to Vigeveno to that no-
bleman. While Tartalea was at Milan the three days,
Cardan plied him by all possible means, to draw from
him the rules for the cubic equations ; and at length, just
as Tartalea was about to depart from Milan, on the 25th
of March, 1539, he was overcome by the most solemn
protestations of secrecy that could be made. Cardan
says, " I shall swear to you on the holy evangelists, and
by the honour of a gentleman, not only never to publish
your inventions, if you reveal them to me; but I also
promise to you, and pledge my faith as a true Christian,
to note them down in cyphers, so that after my death no
other person may be able to understand them." To this
Tartalea replies, " If I refuse to give credit to these as-
surances, I should deservedly be accounted utterly void
of belief. But as I intend to ride to Vigeveno, to see his
excellency the marquis, as I have been here now these
three days, and am weary of waiting so long; whenever
I return therefore, I promise to show you the whole."
Cardan answers, " Since you determine at any rate to go
to Vigeveno to the marquis, I shall give you a letter for
his excellency, that he may know who you are. But now
before you depart, I entreat you to show me the rule for
the equations, as you have promised." " I am content,"
says Tartalea : " but you must know, that to be able on
all occasions to remember such operations, I have brought
the rule into rhyme; for if I had not used that precau-
tion, I should often have forgot it; and though my rhymes
are not very good, I do not value that, as it is sufficient
that they serve to bring the rule to mind as often as I re-
peat them. I shall here write the rule with my own hand,
that you may be sure I give you the discovery exactly."

X3 -+- bx = C,~l

x3 = bx -+- c, >
I3 ■+- c = bx, J

These rude verses contain, in rather dark and enigmatical
language, the rule for these three cases, viz,

bx = c,T which differ however only in the sign of
one quantity, an'l the rule amounts to
this: Find two numbers, z and y, such
that their difference in the first case, and their sum in the
2d and 3d, may be equal to c the absolute number, and
their product equal to the cube of -i of b the coefficient
of the less power; then the difference of their cube roots
will be equal to x in the first case, and the sum of their
cube roots equal to x in the 2d and 3d cases : that is,
taking z —y = c in the first case, or z -+- y = c in the 2d
and 3d, and zy = (|A)3; then i =£/z — \/y in the first
case, and x = %/z -+- %/y in the other two. At parting,
Tartalea, fails not again to remind Cardan of his obliga-
tion : " Now your excellency will remember not to break
your promised faith, for if unhappily you should insert
these rules either in the work you are now printing, or in
any other, though you should even give them under my
name, and as of my invention, I promise and swear that
I shall immediately print another work that will not be
very pleasing to you." " Doubt not," says Cardan, " but
that I shall observe what I have promised : Go, and rest
secure as to that point; and give this letter of mine to the
marquis." It should seem, however, that Tartalea was
much displeased at having suffered himself to be worried
as it were out of his rules ; for as soon as he quitted Mi-
lan, instead of going to wait upon the marquis, he turned
his horse's head, and rode straight home to Venice, saying
to himself, " By my faith I shall not go to Vigeveno, but
shall return to Venice, come of it what will."

After Tartalea's departure, it seems Cardan applied him-
self immediately to resolving some examples in the cubic
equations by the new rules, but not succeeding in them ;
for indeed he had mistaken the words, as it was very easy
to do in such bad verses, having mistaken (j-i)3 for \b3,
or the cube of -| of the coefficient, for | of the cube of
the coefficient; accordingly we find him writing to Tarta-
lea in 14 days after the above, blaming him much for his
abrupt departure without seeing the marquis, who was so
liberal a prince, he said, and requesting Tartalea to re-
solve him the example x3 -t- 3x = 10. This Tartalea did
to his satisfaction, rightly guessing at the nature of his
mistake ; and concludes his answer with these emphatical
words, " Remember your promise." On the 12th of May
following, Cardan returns him a letter of thanks, together
with a copy of his book, saying, " As to my work, just
finished, to remove your suspicion, I send you a copy,
but unbound, as it is yet too fresh to be beaten. But as
to the doubt you express lest 1 may print your inventions,
my faith, which I gave you with an oath, should satisfy
you; for as to the finishing of my book, that could be
no security, as I could always add to it whenever I please.
But on account of the dignitv of the thing, I excuse you
for not relying on that which you ought to have done,
namely, on the faith of a gentleman, instead of the finish-
ing of a book, which might at any time be enlarged by
the addition of new chapters; and there are besides a
thousand other ways. But the security consists in this,
that there is no greater treachery than to break one's
faith, and to aggrieve those, who have given us pleasure.
And when you shall try me, you will find whether I be
your friend or not, and whether I shall make an ungrate-
ful return for your friendship", and the satisfaction you
have given me."

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It was within less than two months after this, however,
that Tartalea received the alarming news of Cardan's
showing some symptoms of breaking the faith he had so
lately pledged to him ; this was in a letter from a quon-
dam pupil of his, in which he writes, " A friend of mine
at Milan has written to me, that Dr. Cardano is composing
another algebraical work, concerning some lately-disco-
vered rules ; hence I imagine they may be those same
rules which you told me you had taught him ; so that I
fear he will deceive you." To which Tartalea replies,
" I am heartily grieved at the news you inform me of,
concerning Dr. Cardano of Milan ; for if it be true, they
can be no other rules but those I gave him ; and there-
fore the proverb truly says, ' That which you wish not to
be known, tell to nobody.' Pray endeavour to learn more
of this matter, and inform me of it."

Tartalea, after this, kept on the reserve with Cardan,
not answering several letters he sent him, till one written
on the 4th of August the same year, 1539, complaining
greatly of Tartalea's neglect of him, and farther request-
ing his assistance to clear up the difficulty of the irredu-
cible case x3 = bx ■+■ c, which Cardan had thus early
been embarrassed with : he says that when (|6)3 exceeds
(-|c)2, the rule cannot be applied to the equation in hand,
because of the square root of the negative quantities. On
this occasion Tartalea turns the tables on Cardan, and
plays his own game back upon him ; for being aware of
the above difficulty, and unable to overcome it himself,
he wanted to try if Cardan could be encouraged to ac-
complish it, by pretending that the case might be done,
though in another way. He says thus to himself, " I
have a good mind to give no answer to this letter, no more
than to the other two. However I will answer it, if it
be but to let him know what I have been told of him.
And as I perceive that a suspicion has arisen concerning
the difficulty or obstacle in the rule for the case x3 = bx
-+- c, I have a mind to try if he can alter the data in
hand, so as to remove the said obstacle, and to change
the rule into another form, though I believe indeed that
it cannot be done ; however there is no harm in trying."
— " M. Hieronime, I have received your letter, in which
you write that you understand the rule for the case x3 =
bx ■+■ c; but that when (\b)3 exceeds {\c)2, you cannot
resolve the equation by following the rule, and therefore
you request me to give you the solution of this equation
x3 = 9.c -)- 10. To which I reply, that you have not
used a good method in that case, and that your whole
process is entirely false. And as to resolving you the
equation you have sent, I must say that I am very sorry
that I have already given you so much as I have done ;
for I have been informed, by a credible person, that you
are about to publish another algebraical work, and that
you have been boasting through Milan of having disco-
vered some new rules in algebra. But, take notice, that
if you break your faith with me, I shall certainly keep
my word with you, nay, I even assure you to do more
than I promised." In Cardan's answer to this he says,
" You have been misinformed as to my intention to pub-
lish more on algebra. But I suppose you have heard
something about my work De Mysteriis .flitcrnitatis,
which you take for some algebra I intend to publish. As
to your repenting of having given me your rules, 1 am not
to be moved from the faith J promised you for any thing
you say." To this, and many other things contained in
the tame letter, Tartalea returned no answer, being still
Vol. I.

suspicious of Cardan's intentions, and declining any more
correspondence with him.

This however did not discourage C. for we find him
writing again to T. on the 5th of January, 1540, to clear
irp another difficulty which had occurred in this business,
namely, to extract the cube root of the binomials, ot
which the two parts of the rule always consisted, and for
.which purpose it seems C. had not yet found out a rule.
On this occasion he informs T. that his quondam compe-
titor Zuanne Colle had come to Milan, where, in some
contests between them, Colle gave Cardan to understand
that he had found out the rules for the two cases x3 -t-
bx = c, and x3 = bx ■+■ c, and further that he had disco-
vered a general rule for extracting the cube roots of all
such binomials as can be extracted ; and that, in parti-
cular, the cube root of ./108 -+- 10 is ,/3 * 1, and that
of y' 108 — 10 is ./3—1, and consequently that ^/(./lOS
+ 10) - ^/(v/lOS - 10) is = 0/3 + 1) - (,/3 - 1)
= 2. He then earnestly entreats T. to try to find out
the rule, and the solution of certain other questions which
had been proposed to him by Colle. By this letter T. is
still more confirmed in his resolutions of silence ; so that,
without returning any answer, he only sets down among
his own memorandums some curious remarks on the
contents of the letter, and then concludes to himself,
" Wherefore I do not choose to answer him again, as I
have no more affection for him than for M. Zuanne, and
therefore I shall leave the matter between them." Among
those remarks he sets down a rule for extracting the cube
root of such binomials as can be extracted, and that is
done from either member of the binomial alone, thus :
Take either term of the binomial, and divide it into two
such parts that one of them may be a complete cube, and
the other part exactly divisible by 3 ; then the cube root
of the said cubic part will be one term of the required
root, and the square root of the quotient arising from the
division of •£ of the 2d part by the cube root of the first,
will be the other member of the root sought. This rule
will be better understood in characters thus : let m be one
member of the given binomial, whose cube root is sought;
and let it be divided into the two parts «3end $b, so that

a3 ■+■ 3b be = m ; then is a ■+- </~ the cube root required,

if it has one. Thus, in the quantity ,/108 -t- 10, taking
the term 10 for m, then 10 divides into 1 and 9, where
a3 = 1 or a = 1, and 3b = 9 °r 6 = 3: therefore a ■+-

\/~ becomes 1 -t- ./3 for the cube root of ./108 ■+■ 10.

And taking the other member ,/ 108, this divides into
the two equal parts ,/27 and ^/27, making a3 = ,/27,
and 36 = ./27; hence a = ,/3, and b = ,/3 also; con-
sequently a ■+• ,/- is = a/3 ■+■ ./f or \/3 ■+" * f°r tne
cube root of the binomial sought, the same as before.
"And thus," headds, "we may know whetherany proposed
binomial or residual be a cube or a noncube ; for if it be
a cube, the same two terms for the root must arise from
both the given terms separately; and if the two terms of
the root cannot thus be brought to agree both ways, such
binomial or residual will not be a cube." And thus ends
the correspondence between them, at least for this time.
But it seems they had still more violent disputes when C.
in violation of his faith, so often pledged to the contrary,
published his work on cubic equations 4 years afterwards,
viz. in the year 1545, of which we have before given au
M

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account, which disputes, it appears, continued till the
death of Tartalea in the year 1557.

The last article in the volume contains a dialogue on
some other forms of cubic equations, in the year 1541,
between T. and Mr. Eichard Wentworth, the English
gentleman before-mentioned, who it seems had resided
some time at Venice, on some public service from England,
as T. in the dedication of the volume to Henry vin, king
of England, makes mention of him as " a gentleman of
his sacred majesty." Mr. Wentworth had learned some
mathematics of T. and being about to depart for England,
requests T. to show him his newly discovered rules for
cubic equations, as a farewell lesson ; and it is worth
while to note a few particulars in this conference, as they
show pretty well the limited knowledge of T. at that time,
as to the nature and roots of such equations. T. had be-
fore, it seems, showed Mr. W. the rules for the cases of
the 3d and 1st powers, and now the latter desires him to
do the same as to the three cases in which the 3d and 2d
powers only are concerned. On this T. professes great
gratitude to Mr. W. for many obligations, but desires to
be excused from giving him the rules for these, because
he says he intends soon to compose a new work on Arith-
metic, Geometry, and Algebra, which he intends to de-
dicate to him, and in which he means to insert all his
new discoveries. On Mr. W. urging him further, how-
ever, T. gives him the roots of some equations of that
kind, as for instance :
If x3 ■+- 6x2 = 100, then

z=£"(42 -+- .v/17000) +^/(42-v/l7000)-2.
If x3 -+- 9x2 = 100, then x = ^/24 - 2.
If x3 -+- 3x2 = 2, then x = ^3-1.
If x3 + 4 = 5x2, then x = ^/8 -+■ 2.
If x3 ■+- 6 = 7«2, then x =y/15 -+■ 3.
But he does not mention the rules for finding them.

In the course of the conversation T. tells him that "all
such equations admit of two different answers, and per-
haps more ; and hence it follows that they have, or admit
of, two different rules, and perhaps more, the one more
difficult than the other." And on Mr. W. expressing his
wonder at this circumstance of a plurality of roots, T.
replies, " It is however very true, though hardly to be
believed, and indeed if experience had not confirmed it,
I should scarcely have believed it myself." He then com-
mits a strange blunder in an example which he takes to
illustrate this by, namely the equation x3 -+■ 3x = 14,
which, he says, it is evident has the number 2 for one of
its roots; and yet, he adds, "Whoever shall resolve the
same equation by my rule, will find the value of x to be
\/(7 + \/50) -♦- i/(7 — \/50), which is proved to be a
true root by substituting it in the equation for x. And
therefore," continues he, " it is manifest that the case x3 ■+■
bx = c admits of two rules, namely, one (as in the above
example) which ought to give the value of x rational,
viz, 2, and the other is my rule, which gives the value of
x irrational, as appears above ; and there is reason to
think that there may be such a rule as will give the value
of x = 2, though our ancestors may not have found it

out." " And these two different answers will be found

not only in every equation of this form x3 ■+■ bx = C,
when the value of x happens to be rational, as in the ex-
ample x3 ■+■ 3x = 14 above, but the same will also happen
in all the other five forms of cubic equations: and there-
fore there is reason to think that they also admit of two
different rules; and by certain circumstances attending

some of them, I am almost certain that they admit of
more than two rules, as, God willing, I shall soon demon-
strate."

Now all this discourse shows a strange mixture of
knowledge and ignorance. It is very probable that he had
met with some equations which admit of a plurality of
roots ; indeed it was hardly possible for him to avoid it:
but it seems he had no suspicion what the number of roots
might be, nor that his reasoning in this instance was
founded on an error of his own, mistaking the root x =
3/(7 ■+■ \/50) + 3/(7 — \/50), of the equation x3 ■+■
3x = 14, for a different root from the number or root 2,
when in reality it is the very same, as he might easily
have found, if he had extracted the cube roots of the bi-
nomials by the rule which he himself had just given above
for that purpose: for by that rule he would have found
j/(7 + v/50) = 1 + ^/2,and^/(7 - v^O) =1-^2,
and therefore their sum is 2 = x, the same root as the
Other, which T. thought had been different. And besides
this root 2, the equation in question, x3 •+- 3x = 14, ad-
mits of no other real roots. Nor indeed does any equa-
tion of the same form, i3 ■+■ bx =s c, admit of more than
one real root.

It seems also they had not yet discovered that all cases
belong to the rules and forms for quadratic equations,
which have only two powers in them, in which the expo-
nent of the one is just double of the exponent of tlie
other, as x2n ■+• bxa = c; but some particular cases only
of this sort they had as yet ventured to refer to quadra-
tics, as the case x4 -+• bx" = c. But, in the conclusion of
this dialogue, T. informs W. of another case of this sort
which he had accomplished, as a notable discovery, in these
words : " I well remember," says he, " that in the year
1536, on the night of St. Martin, which was on a Satur-
day, meditating in bed when I could not sleep, I disco-
vered the general rule for the case x6 ■+■ bx3 = c, and also
for the other two, its accompanying cases, in the same
night." And then he directs that they are to be resolved
like quadratics, by completing the square, &c. And in
these resolutions it is remarkable that he uses only the
positive roots, without taking any notice of the negative
ones.

Tartalea also published at Venice, in 1556, &c, a very
large work, in folio, on Arithmetic, Geometry, and Alge-
bra. This is a very complete and curious work on the
first two branches; but that of Algebra is carried no far-
ther than quadratic equations, called book the first, with
which the work terminates. It is evidently incomplete,
owing to the death of the author, which happened before
this latter part of the work was printed, as appears by the
dates, and by the prefaces. It appears also, from several
parts of this work, that the author had many severe con-
flicts with Cardan and his friend Lewis Ferrari : and par-
ticularly, there was a public trial of skill between them,
in the year 1547; in which it would seem that Tartalea
had greatly the advantage, his questions mostly remaining
unanswered by his antagonists.

OF MICHAEL STIFELIUS.

After the foregoing analysis of the works of the first
algebraic writers in Italy, it will now be proper to con-
sider those of their contemporaries in Germany; where
it is remarkable that, excepting for the discoveries in
cubic equations, the art was in a more advanced state,
and of a form approaching nearer to that of our modern
algebra; the states and circumstances indeed being so

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different, that one would almost be led to suppose they
had derived their knowledge of it from a different origin.

Here Stifelius and Scheubelius were writers of the same
time with Cardan and Tartalea, and even before their dis-
coveries, or publication, concerning the rules for cubic
equations, Stifelius's Arithmetica Integra, being published
at tyorimberg in 1544, the year before Cardan's work on
c.ubic equations, and is an excellent treatise, both on
arithmetic and algebra. The work is divided into three
books, and is prefaced with an Introduction by the fa-
mous Melanchthon. The first book contains a complete
and ample Treatise on Arithmetic, the second an Ex-
position of the 10th book of Euclid's Elements, and the
third a Treatise on Algebra, being therefore properly the
part with which we are at present concerned. In the de-

from the nature of arithmetical and geometrical progres-
sions. It is remarkable that these compound denomina-
tions of the powers are formed from the simple ones ac-
cording to the products of the exponents, while those of
Diophantus are formed according to the sums of them ;
thus the 6th power here is 5C£ 01" quadrato-cubi, but with
Diophantus it is cubo-cubi ; and so of others. Which is
presumptive evidence that the Europeans had not taken
their Algebra immediately from him, independent of other
proofs.

Chap. 4. On the extraction of the roots of cossic num-
bers. He here treats of quadratic equations, which he
resolves by completing the square, from Euclid ii, 4, &c.
Also quadratics of the higher orders, showing how to re-"
solve them in all cases, whatever the height may be, pre-

dication of this part, he ascribes the invention of algebra vided the exponents be but in arithmetical progression, as
to Geber, an Arabic astronomer ; and mentions besides, 2,

the authors Campanus, Christ. Rudolph, and Adam Ris,
Risen, or Gigas, whose rules and examples he has chiefly
given. In other parts of the book he speaks, and makes
use also, of the works of Boctius, Campanus, Cardan
(i. e. his Arithmetic published in 1539, before the work
on cubic equations appeared), de Cusa, Euclid, Jordan,
Milichius, Schonerus, and Stapulensis. So that he ap-
pears to have been very little acquainted indeed with any
besides the German authorsi

Chap. I. On the Rule of Algebra, and its parts. Stifelius
here describes the notation and marks of powers or deno-
minations as he calls them, which marks for the several
powers are thus :

1st, '2d, 3d, 4th, 5th, 6th, &c,
%, 3, &, .33, «, 3 ft, &c,
being formed from the initials of the barbarous way in
which the Germans pronounced and wrote the Latin and
Italic names of the powers, namely, res or cosa, zensus,
cubo, zensi-zensus, sursolid, zensi-cubo, &c. And the
coss or first power fy, he calls the radix or root, which is
the first time that we meet with this word in the printed
authors. He also here uses the signs or characters, -f-
and — , for addition and subtraction, and the first of any
that I know of: for in Italy they used none of these cha-
racters for a long time after. He has no mark however
for equality, but makes use of the word itself.

Chap. 2. On the Parts of the Rule of Geber or Alge-
bra: teaching the various reductions by addition, sub-
traction, multiplication, division, involution, and evolu-
tion, &c.

Chap. 3. On the Algorithm of Cossic numbers ; teach-
ing the usual operations of addition, subtraction, multi-
plication, division, involution, and extraction of roots,
much the same as they are at present. Single terms, or
powers, he calls simple quantities; but such asl 5 ■+■ 1%
a composite or compound, and 2%, — 8 a defective one.
In multiplication and division, he proves that like signs
give h-, and unlike signs — . He shows that the powers
]> 9p> 5> > &c, form a geometrical progression from
unity; and that the natural series of numbers 0, 1,2, 3,
&c, from 0, are the exponents of the cossic powers ; and
he, for the first time, expressly calls them exponents : thus,
Exponents, 0, 1, 2, 3, 4, 5, 6, &c.
Powers, 1, %,, 5, ft, 33, ss, 3 ft, &c.
And he shows the use of the exponents, in multiplication,
division, powers, and roots, as we do at present; viz, ad-
ding the exponents in multiplication, and subtracting them
in division, &c. And these operations he demonstrates

2, I, 0") *
4, 2, o( fc\

8 40 tki

where it is plain that he always counts 0
the exponent of the unknown quantity in
absolute term.

Chap. 5. Of irrational cossic numbers, and of surd or
negative numbers. In this treatise of radicals or irra-
tionals, he first uses the character */ to denote a root, and
sets after it the mark of the power whose root is intended;
as ^/ 3 20 for the square root of 20, and ^/ $.20 for the
cube root of the same, and so on. He treats here also of
negative numbers, or what he calls surd or fictitious, or
numbers less than 0. On which he takes occasion to ob-
serve, that when a geometrical progression is continued
downwards below 1, then the exponents of the terms, or
the arithmetical progression, will go below 0 into nega-
tive numbers, and will yet be the true exponents of the
former ; as in these,

Expon. —3, —2, —1, 0, 1, 2, 3,
Pow. *, i, i, 1, 2, 4, S,

And he gives examples to show that these negative expo-
nents perform their office the same as the positive ones, in
all the operations.

Chap. 6. On the perfection of the Rule of Algebra, and
of Secondary Roots. In the reduction of equations, he
uses a more general rule than those who had preceded
him, who detailed the rule in a multitude of cases; in-
stead of which, he directs to multiply or divide the two
sides equally, to transpose the terms with -+- or — , and
lastly to extract such root as may be denoted by the ex-
ponent of the highest power.

As to secondary roots, Cardan treated of a 2d- ignota
or unknown, which he called quantitas, and denoted it by
the initial q, to distinguish it from the first. But here
Stifelius, for distinction sake, and to prevent one root from
being mistaken for others, assigns literal marks to all of
them, as a, b, c, d, &c, and then performs all the usual
operations with them, joining them together as we do
now, except that he subjoins the initial of the power, in-
stead of its numeral exponent : thus,
3a into 9b makes 27ab,
33 into 4b makes 123b,
2?? into 4a5 makes 8 #A,
1a squared makes 1a5>
6 into 3c makes 18c,
2a3 into 5a makes 10am, &c, &c,
8ff a5 divided by 4ft makes 2a3, &c,
The square root of 25a5 is 5a, &c,
Also 2a added to 2% makes 2% + 2a,
and 2a subtr. from 'Zcif makes 2 % — 2a.
M2

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And lie shows how to use the same, in questions concern-
ing several unknown numbers ; where he puts a different
character for each of them, as ty, a, b, c, &c ; he then
makes out, from the conditions of the question, as many
equations as'there are characters ; from these he finds the
value of each letter, in terms of some one of the rest;
and so, expelling them all but that one, reduces the whole
to a final equation, as we do at present.

The remainder of the book is employed with the solu-
tions of a great number of questions, to exercise all the
rules and methods; some of which are geometrical ones.

From this account of the state of Algebra in Stifelius,
it appears that the improvements made by himself, or
other Germans, beyond those of the Italians, as contained
in Cardan's book of J 539) were as follow :

1st. He introduced the characters ■+-, — , */, for plus,
minus, and root, or radix, as he calls it.

2d. The initials <^, 5, f£, &c, for the powers.

3d. He treated all the higher orders of quadratics by
the same general rule.

4th. He introduced the numeral exponents of the
powers, —3, — 2, — 1, 0, 1, 2, 3, &c, both positive and ne-
gative, so far as integral numbers, but not fractional ones;
calling them by the name exponens, exponent : and he
taught the general uses of the exponents, in the several
operations of powers, as we now use them, or the loga-
rithms.

5th. And lastly, he used the general literal notation
a, b, C, D, &c, for so many different unknown or general
quantities.

OF SCHEUBELIUS.

John Scheubelius published several books on Arithme-
tic and Algebra. The one now before me, is entitled Al-
gebra? Compendiosa Facilisque Descriptio, qua depromun-
tur magna Arithmetices miracula. Authore Johanne
Scheubelio Mathematicarum Professore in Academia Tu-
bingensi. Parisiis 1552. But at the end of the book it
is dated 1551. The work is most beautifully printed, and
is a very clear though succinct treatise; and both in the
form and matter much resembles a modern printed book.
He says that the writers ascribe this art to Diophantus,
which is the first time that I find this Greek author men-
tioned by the modern algebraists: he further observes,
that the Latins call it Regula Rei et Census, the rule of
the thing and the square, or of the 1st and 2d power;
and the Arabs, Algebra. His characters and operations
are much the same as those of Stifelius, using the signs
and characters +, — , a/, and the powers &, fy, 3jpt,
&c, where the character & is used for 1 or unity, or a
number, or the 0 power ; prefixing also the numerical co-
efficients; thus 44«3 ■+• 11 3 ■+■ 31 & — 53 p£. He uses
also the exponents 0, I, 2, 3, &c, of the powers, the
same way as Stifelius, before him. He performs the al-
gebraical calculations, first in integers, and then in frac-
tions, much the same as we do at present. Then of equa-
tions, which lie says may be of infinite degrees, though
he treats only of two, namely the first and second orders,
or what we call simple and quadratic equations, in the
usual way, taking however only the positive roots of
these; and adverting to all the higher orders of quadra-
tics, namely,

x\ ax1, b ; and ,t", «.r3, b ; and x", ax\ b ; &c.

Next follows a tract on surds, Imth simple and com-
pound, quadratic, cubic, binomial, and residual. Here
he first marks the notation, observing that the root is ci-

ther denoted by the initial of the word, or, after some
authors, by the mark a/:, viz, the sq. root */:, the cube
root vv^/ :, and the 4th root, or root of the root thus v1/ :,
which latter method he mostly uses. He then gives the
Arithmetic of surds, in multiplication, division, addition,
and subtraction. In these last two rules he squares the
sum or difference of the surds, and then sets the root to
the whole compound, which he calls radix collecti, what
Cardan calls radix universalis. Thus a/12 4- a/ 20 is ra.
col. 32 ± 4/96O. But when the terms will reduce to a
common surd, he then unites them into one number; as
4/27 h- ♦/ 12 is equal a/75. Also of, cubic surds, and
4th roots. In binomial and residual surds, he remarks
the different kinds of them which answer to the several
irrational lines in the 10th book of Euclid's elements ;
and then gives this general rule for extracting the root of
any binomial or residual a± b, where one or both parts
are surds, and a the greater quantity, namely, that the
square root of it is ^a+«A'--'-) ± >/? ~*/'°'-*%

which he illustrates by many examples. This rule will
only succeed however, so as to come out in simple terms,
in certain cases, namely, either when a'2 — b" is a square,
or when a and a/ (a2 — b2) will reduce to a common surd,
and unite : in all other cases the root is in two compound
surds, instead of one. He gives also another rule, which
comes however to the same thing as the former, though
by the words of them they seem to be different.

Scheubelius wrote much about the time of Cardan and
Stifelius. And as he takes no notice of cubic equations,
it is probable he had neither seen nor heard any thing
about them ; which might very well happen, the one living
in Italy, and the other in Germany. And besides, I know
not if this be the first edition of Scheubel's book: it is
rather likely it is not, as it is printed at Paris, and he
himself was professor of mathematics at Tubingen in
Germany.

ROBERT KECOItDE.

To this ingenious man we are indebted for the first trea-
tise on algebra, then named the Cossic Art, in the English
language ; but his meritorious labours, like those of the
greatest benefactors of mankind, appear to have been ill
requited, since, after removing to the capital, he died un-
der confinement for debt in the Fleet-prison. In his book
on Arithmetic he is styled " teacher of mathematics and
practitioner in physic at Cambridge." It was for many
ages the custom to unite the title as well as the practice
of medicine with those of chemistry, alchymy, mathema-
tics, and astrology, by the .Moors, and after them by the
Europeans, and is still continued among the almanac-
makers. And it is remarkable, that as the Moors were
not less famed in Europe for their skill in medicine, than
their dexterity in calculation, the term Physician and Al-
gebraist appear at first to have been regarded as almost
synonymous. Thus, it is curious to remark, that in the
celebrated romance of Don Quixote, published about
this time, the bachelor Samson Carrasco, who in his ren-
counter with the knight was thrown from his horse, and
had his ribs broken, sent in quest of an Algebrista to heal
his bruises.

The first part of his Arithmetic was published in 1552;
and the second part in ISSJlj under the title of " The
Whetstone of Witte, which is the seconde pane of Arith-
metike: containing the Extraction of Ilootes: The Cossikc
Practise, with the llulc of Equation : and the Workes of

A L G

[ Si ]

A L G

Surde Nombers." The work is in dialogue between the
master and scholar; and is nearly after the manner of
the Germans, Stifelius and Scheubelius, but especially
the latter, whom he often, quotes, and takes examples
from. The chief parts of the work are, 1st, The proper-
ties of abstract and figurate numbers. 2d, The extrac-
tion of the square and cube roots, much the same as at
present. Here, when the number is not an exact power,
but having some remainder over, he either continues the
root into decimals as far as he pleases, by adding to the
remainders always periods of ciphers ; or else makes a
■vulgar fraction for the remaining part of the root, by
taking the remainder for the numerator, and double the
root for the denominator, in the square root ; but in the
cube root he takes, for the denominator, either the triple
square of the root, which is Cardan's rule, or the triple
square and triple root, with one more, which is Scheu-
bel's rule. 3d, Of Algebra, or " Cossike Nombers." He
uses the notations of powers with their exponents the
same as Stifel, with all the operations in simple and com-
pound quantities, or integers and fractions. And he gives
also many examples of extracting the roots of compound
algebraic quantities, even when the roots are from two to
six terms, in imitation of the same process in numbers,
just as. we do at present ; which is the first instance of
this kind that I have observed. As of this quantity :
Square Root.

255ff I- 80.5-2655- 144^ + 815 (5 tf + 85-9?..
4th, " The Rule of Equation, commonly called Alge-
ber's Rule." He here, first of any, introduces the charac-
ter =, for brevity sake. His words are, "And to avoid
the tediouse repetition of these woordes : is equalle to : I
will sette as I doe often in woorkeuse, apaire of paralleles,
or gemowe lines of one lengthe, thus : =, bicause noe 2
thynges can be moare equalle." He gives the rules for
simple and quadratic equations, with many examples. He
gives also some examples in higher compound equations,
with a root for each of them, but gives no rule how to find
it. 5th, "Of Surde Nombers." This is a very ample
treatise on surds, both simple and compound, and surds of
various degrees, as square, cubic,and biquadratic, marking
the roots in Scheubel's manner, thus: */, ^/, vy/- He
here uses the names bimedial, binomial, and residual; but
says they have been used by others before him, and indeed
by Lucas de Burgo, as in cart. 120, edit. 1494. — Hence it
appears that the things which chiefly are new in this author,
are these two, viz,

1. The extraction of the roots of compound algebraic
quantities.

2. The use of the- sign of equality, or ==.

OF PELETARIUS.

The first edition of this author's algebra was printed in
4to at Paris, in 1558, under this title, Jocobi Peletarii Ce-
nomani, de occulta parte Numerorum, quam Algebram vocant.
Lib. duo.

In the preface he speaks of the supposed authors of Al-
gebra, namely Geber, Mahomet the son of Moses, an Ara-
bian, and Diophantus. But he thinks the art older, and
mentions some of his contemporary writers, or a very little
before him, as Cardan, Stifel, Scheubel, Chr. Januarius ;
and a little earlier again, Lucas Paciolus of Florence, and
Stephen Villafrancus a Gaul.

Of the two books, into which the work is divided, the
first is on rational, and the second on irrational or surd

quantities; each being divided into many chapters. It
will be sufficient to mention only the principal articles.

He calls the series of powers humeri creati, or derived
numbers, or also radicals, because they are all raised from
one root or radix. He names them thus, radix, quadratus,
cubus, quadrato-quadratus, or biquadratus, supersolidus,
quadrato-cubus, &c; and marks them thus fy, q, $, qq,
ss> 9pt) bss, fyc. Of these he gives the following series in
numbers, having the common ratio 2, with their marks set
over them, and the exponents set over these again, in an
arithmetical series, beginning at 0, thus:

0 12 3 4 5 6 7 8

1 $ 9 K- qq ^ q& bss qqq

1 2 4 8 16 32 64 128 256 &c.
And he shows the use of the exponents, the same as Stifel
and Scheubel; like them also he prefixes coefficients to
quantities of all kinds, as also the radical ^/. But he
does not follow them in the use of the signs ■+- and — , but
employs the initials p and m for the same purpose. After
the operations of addition, &c, he performs involution,
and evolution also, much the same way as at present:
thus, in powers, raise the coefficient to the power required,
and multiply the exponent, or sign, as he calls it, by 2,
or 3, or 4, &c, for the 2d, 3d, 4th, &c, power; and the
reverse for extraction : and hence he observes, if the
number or coefficient will not exactly extract, or the sign
do not exactly divide, the quantity is a surd.

After the operations of compound quantities, and frac-
tions, and reduction of equations, namely simple and
quadratic equations, as usual, in chap. 16, De Invcniendis
generatim Radicibus Denominatorum, he gives a method of
finding the roots of equations among the divisors of the
absolute number, when the root is rational, whether it be
integral or fractional ; for then, he observes the root al-
ways lies hid in that number, and is some one of its di-
visors. This is exemplified in several instances, both of
quadratic and cubic equations, and both for integral and
fractional roots. And he here observes, that he knows
not of any person who has yet given general rules for the
solution of cubic equations ; which shows that when he
wrote this book, either Cardan's last book was not pub-
lished, or else it had not yet come to his knowledge.

Chap. 17 contains, in a few words, directions for bring-
ing questions to equations, and for reducing these. He
here observes, that some authors call the unknown num-
ber res, and others the positio; but that he calls it ra-
dix, or root, and marks it thus §J : hence the term,
root of an equation. But it was before called radix by
Stifelius.

Chap. 21 if seq. treat of secondary roots, or a plurality
of roots, denoted by a, is, c, &c, after Stifelius.

The 2d book contains the like operations in surds, or
irrational numbers, and is a very complete work on this
subject indeed. He treats first of simple or single surds,
then of binomial surds, and lastly of trinomial surds. He
gives here the same rule for extracting the root of a bino-
mial and residual as Scheubelius, viz, \/ (a ± b) =

a + ^/iar-lr) a-^{a'-l?) .

4/ ^—t ;± t/ — ■ -. In dividing by a

binomial or residual, he proceeds as all others before
him had done, namely, reducing the divisor to a simple
quantity, by multiplying it by the same two terms with
the sign of one of them changed, that is by the binomial
if it be a residual, or by the same residual if it be a bino-

ALG [86

mial ; and -multiplying the dividend by the same thing :
thus — t =— -. x =-7 = — =3*/5+o.

^/5— 2 ^/5 — 2 v/5+2 5 — 4

And, in imitation of this method, in division by trinomial
surds, he directs to reduce the trinomial divisor first to
a binomial or residual, by multiplying it by the same trino-
mial with the sign of one term changed, and then to re-
duce this binomial or residual to a simple nominal as
above; observing to multiply the dividend by the same
quantities as the divisor. Thus, if the divisor be 4 -f- »/1
— ^/3 ; multiplying this by 4 -+- 1/2 -+- 1/3, the product is
15 -+- 8»/2; then this binomial multiplied by the residual
15 — 8^2, gives 225 — 12S or 97 for the simple divisor:
and the dividend, whatever it is, must also be multiplied
by the two 4 -i- */2 -+- ,y/3 and 15 — 8^/2. Or in general
if the divisor bea + ^/b — */c;
multiply it by a -+- ^/b -+- */c,
which gives (a -+- \/b)2 — c = a? -t- b — c + 1a*Jb;
then multiply this by - - or -v b — c — Za^/b,
and it gives ----- (a2 -+- b — c)s — 4a2 Z>, which
will be rational, and will all collect into one single term.
But Tartalea must have been in possession of some such
rule as this, as one of the questions he proposed to Florido
was of this nature, namely to find such a quantity as mul-
tiplied by a given trinomial surd, shall make it rational :
and it appears, from what is done above, that, the given
trinomial being a -+- ^/b — y'c, the answer will be (a ■+-
*/b ■+- ,/c) x (a2 + b - c ■+- 2y/b).

Chap. 24 shows the composition of a cube of a binomial
or residual, and thence remarks on the root of the case or
equation 1 (t p 3§t equal to 10, which he seems to know
something about, though he had not Cardan's rules.

Chap. 30, which is the last, treats of certain precepts
relating to square and cubic numbers, with a table of such
squares and cubes from all numbers to 140 ; also showing
how to compute them both, by adding always their dif-
ferences.

He then concludes with remarking that there are many
curious properties of these numbers, one of which is this,
that the sum of any number of the cubes, taken from the
beginning, always makes a square number, the root of
which is the sum of the roots of the cubes ; so that the
series of squares so formed, have for their roots 1, 3, 6, 10,
15, 21, &c, whose diff. are the natural nos 1, 2, 3, 4, 5,
6, &c. Namely, 13=12; 13h-23 = 3!; l3 + 23 + 33=62,
&c. Or in general, 13-+- 23 -+■ 33 - - - n3 = (1 ■+■ 2 ■+■ 3--- n)2
= [•£«. (n -+- 1)Y = £ra1 ty ■+■ !)*•

This work of Peletarius is a very ingenious and masterly
composition, treating in an able manner of the several
parts of the subject then known, excepting the cubic
equations. But his real discoveries, or improvements,
may be reduced to these three, viz.

1st. That the root of an equation is one of the divisors
of the absolute term.

2d. He taught how to reduce trinomials to simple terms,
by multiplying them by compound factors.

3d. He taught curious precepts and properties concern-
ing square and cube numbers, and the method of con-
structing a series of each by addition only, namely by ad-
ding successively their several orders of differences.

PETER RAMUS.

Peter Ramus wrote his arithmetic and algebra about
the year 1560. His notation of the powers is thus, /, (/,
e, bq, being the initials of latus, quadratus, cubus, biqua-

J

ALG

dratus. He treats only of simple and quadratic equations.
And the only thing remarkable in his work, is the first ar-
ticle, pn the name and invention of Algebra, which we
have noticed at the beginning of this history.

OF PEDRO NUGNEZ OR NUNEZ, OR IN LATIN NOSIUS.

Peter Nunez or Nonius, was a very ingenious and emi-
nent physician and mathematician, for the time in which he
lived. He was born in 1497, at Alcazar in Portugal, and
died in 1577, at 80 years of age. He was professor of
mathematics in the university of Coimbra, where he pub-
lished several ingenious and useful pieces on different bran-
ches of the mathematics, as may be seen by the account of
his life given in the 2d vol. of this Dictionary; but it is
only with his Algebra that our business is at present.

This work he had composed in Portuguese, but trans-
lated it into the Castilian tongue, when he resolved on
making it public, which he thought would render his
book more useful, as this language was more generally
known than the former. The dedication, to his former
pupil, prince Henry, was dated from Lisbon, Dec. 1, 1564;
and the work contains 341 leaves, equal to 682 closely
printed pages, in the Antwerp edition of 1567, inSvo;
the folios being numbered only on one side.

The work is very methodically and plainly treated ;
being divided into regular and distinct chapters or sec-
tions; leading the reader gradually through the several
operations of computation, in integers and fractions, in
powers and roots, in surds and in proportions, &c. The
rules for dignities or powers are given; and these he de-
nominates from the product of their indices; thus, for
the powers of 2, with their names, and denominations
under them :

2 4 S 16 32 64

Co. Ce. Cu. Ce. Ce. Re. p°. Ce. Cu. or Cu. Ce. &c.
1-234 5 6

where the 6th denomination is called Ce. Cu. or Cu, Ce.
that is Censo-Cubo or Cubo-Censo, meaning the square
cubed or the cube squared, the index 6 denoted by 2 x 3,
the product of the indices of the powers; after the man-
ner of the former European authors, Lucas de Burgo, Tar-
talea, and Cardan. And exactly after the manner of these
also is his practice in every other part, with little or no
variation, as far as he goes, which is to quadratic equa-
tions; without treating on cubics, further than giving
some account of the dispute between Tartalea and Cardan
concerning their invention; and that in such a manner as
shows he did not very well understand them. Like those
authors are his names and rules for Raizes, roots or radi-
cals, which he sometimes calls Sorda, surds. In their
marks or signs also ; as R for root ; p for -+- ; m for minus ;
R. u. for root universal, instead of the vinculi used by the
moderns; also L for ligature or composition, such as L.
R. 9 with R 4, composed of 3 which is R 9, and of 2
which is R 4, making 5. He places likewise the name of
the root after the R; as R. cu. 25. tii. R. cu. 15. p. R. cu.
9; that is, $/25 -$/l5 +$/9.

After all the usual preparatory rules, Nunez then treats
of equations, simple and quadratic, in the common way,
and giving geometrical demonstrations of the rules, as had
been done before. He then applies these equations in the
solution of a great number of examples, of questions or
problems, first in numbers or arithmetic, and then in geo-
metrical problems or figures; in which he proceeds or-
derly through the several kinds; as squares, rectangles,

ALG

C 87 3

ALG

triangles, rhombs, rhomboids, trapeziums, pentagons, &c:
all which he calls by the same names as at present. The
content of a figure too he calls its Area, as, Do Burgo did
before him ; and after the same author also he gives the
geometrical demonstration of the common rule for finding
the area of a triangle from the three sides given. He treats
on the inscription of circles and squares in triangles of va-
rious kinds ; and the division of triangles into several
parts in different ways.

In an address to his readers, at the end of the book,
Nunez informs them what are the authors whose books, on
his subject, are to be found in Spain, which consist only
of De Burgo, Tartalea, and Cardan ; stating his ideas on
the merits of their works, with critical remarks on many

world ; and he says that they had already translated fiVe
books, of the seven which were then extant, being as yet
hindered by other avocations from completing the work.
He then adds the following strange circumstance, viz, that
they had found that in the said work the Indian authors are
often cited; by ivhich he learned that this science was known
among the Indians before the Arabians had it : a paragraph
the more remarkable as I have never understood that any
other person could ever find, in Diophantus, any reference
to Indian writers : and I have examined his work with
some attention, for that purpose. Probably the copy
which Bombelli saw, contained marginal remarks by Pla-
nudes, or some other scholiast, making mention of the In-
dian works on the science, or some such remarks ; which

parts of them, in which he more particularly approves of might be mistaken for part of the text of Diophantus.
those of Tartalea. Upon the whole, the merit of Nunez, Bombeili's work is divided into three books. In the

in this art, consists chiefly, or wholly, in having given a first, are laid down the definitions and operations of

very neat and orderly treatise on it, after the manner of powers and roots, with various kinds of radicals, simple

those authors; but without having made any improve- and compound, binomial, residual, &c ; mostly after the

ments or inventions of his own, in the art.

On this occasion it is very remarkable to observe, that
Nunez appears not to have been at all acquainted with
any of the Germanic authors, several of whom were con-
temporaries of Tartalea and Cardan, and who treated the
subject in a better manner, in some respects, than these
did. Another thing may be here noticed on this occa-
sion, as remarkable ; not only that we have never heard of
any other early writers on this subject, in Spain or Portu-
gal ; but that we have never heard of writings on it by
the Moors, who occupied great part of that peninsula
during several centuries, by whom we have always been
taught to believe that the arts of arithmetic and algebra
were brought into that country. — Had that people left any
such works in the country, would not some of them have
been found in some of the great cities or the universities, or
could ihey yet be in existence there ? — Or could the peo-
ple have carried all their books away with them when
they were expelled from the country ?

BOMBELLI.

Raphael Bombeili's Algebra was published at Bologna
in the year 1572, in the Italian language. In a short, but
neat, introduction, he first adverts, in a few words, to the
great excellence and usefulness of arithmetic and algebra.
He then laments that it had hitherto been treated in so
imperfect and irregular a way; and declares it his intention
to remedy all defects, and to make the science and prac-
tice of it as easy and perfect as may be. And for this
purpose he first resolved to procure and study all the
former authors. He then mentions several of these, with
a short history or character" of them ; as Mahomet the
son of Moses, an Arabian; Leonard Pisano; Lucas de
Burgo, the first printed author in Europe ; Oroncius ;
Scribelius; Boglione Francesi ; Stifelius in Germany; a
certain Spaniard, perhaps meaning Nunez or Nonius; and
lastly Cardan, Ferrari, and Tartalea; with some others
since, whose names he omits. He then adds a curious
paragraph concerning Diophantus: he says that some
years since there had been found, in the Vatican library,
a Greek work on this art, composed by a certain Dio-
phantus, of Alexandria, a Greek author, who lived in the
time of Antoninus Pius; which work having been shown
to him by Mr. Antonio Maria Pazzi Reggiano, public lec-
turer on mathematics at Rome; and finding it to be a
good work, these two formed the design of giving it to the

is the same as (p + q)2; and p ot^/(v/(- + !

rules and manner of former writers, excepting in some few
instances, which I shall here take notice of. And first of
his rule for the cube root of binomials or residuals, which,
for the sake of brevity, may be expressed in modern no-
tation as follows: let ^/b ■+- a be the binomial, the term
*/b being greater than a ; then the rule for the cube root
of 4/b + a comes to this, p — <j 4- ^/[(p— q)2+;£/(J_

a2)]; where p = %, W{£ + tll + 1), and , =

v/(/v/(i"4 + ~6i~)~f )■ Whicu isarule that can beof lit-
tle or no use. For in the first place, (r — q)2 + */(b — a-)

!)+~)

' 8 '

= i^/i.V'b -t- a); there-
fore the whole p — q -+- ^/(p - q)2 -+- */tb — af reduces
t0p-Q + p + Q = 2p = 2 x il/{y/b -+- a) =l/Wb
-+- a), the original quantity first proposed.

The next thing remarkable in this 1st book, is his me-
thod for the square roots of negative quantities, and his
rule for the cube roots of such imaginary binomials as
arise from the irreducible case in cubic equations. His
words, translated, are these : " I have found another sort
of cubic root, very different from the former, which arises
from the case of the cube equal to the first power and a
number, when the cube of the |d part of the (coef. of
the) 1st power, is greater than the square of half the ab-
solute number, which kind of square root hath, in its al-
gorism, names and operations different from the others;
tor in that case, the excess cannot be called either plus or
minus; I therefore call it plus of minus when it is to be
added, and minus of minus when it is to be subtracted."
He then gives a set of rules for the signs when such roots
are multiplied, and illustrates them by a great many ex-
amples. His rule for the cube roots of such binomials,
viz, such as a + ^/ — b, is this: First find ^/(d1 -+- b);
then, by trials search out a number c, and a sq. root, ^/d,
such, that the sum of their squares cs ■+■ d may be =■.$/((?
-+- b), and also c3 — 3cd = a; then shall c-h^/ — d be
=\/{a + \/ — b) sought. Thus, to extract the cube root
of 2 +,/-121: here £/(a2 4- b) =$/l25= 5; then
taking c=2, and d = 1, it is c* ■+■ d=5 =^/(a2 -t- b,) and
c3— 3cd = 8 - 6 = 2 = a, as it ought; and therefore
2 ■+■ y/~ 1 is„= the cube root of 2 -+- ^ — 121, as re-
quired.

A L G

[

J

A L G

The notation in this book, is the initial R for root, with
q or c, &c, after it, for quadrate or cubic; &c root. Also
p for plus, and m for minus.

In the 2d book, Bombelli treats of the algorism with un-
known quantities, and the resolution of equations. He
first gives the definitions and characters of the unknown
quantity and its powers; in which he deviates from the
former authors, but professes to imitate Diophantus. He
calls the unknown quantity tanto, and marks it
thus i»

Its square or 2d power potenza, *,-,
Its cube - cubo, 3 ,

and the higher names are compounded of these, and mark-
ed 4 , 5 , 6 , 7 , &c ; so that he denotes all the powers by
their'exponents set over the common character _. And
all these powers he calls by the general name dignita, dig-
nity. He then performs all the algorism of these powers,
by means of their exponents, as we do at present, viz,
adding them in multiplication, subtracting in division,
multiplying them by the index in involution, and dividing
by the same in evolution.

In equations he goes regularly through all the cases,
and varieties of the signs and terms; first all the simple
or single powers, and then all the compound cases ; de-
monstrating the rules geometrically, and illustrating them
by many examples.

In compound quadratics, he gives two rules : the first
is by freeing the potenza or square from its coefficient by
division, and then completing the square, &c, in the usual
way : and the 2d rule, when the first term has its coeffi-
cient, may be thus expressed ; if ax2 -+■ bx = c, then x =
y/(ac + jtf) - li^ be-ng a way tjiat was practised by the In-
dians. He takes only the positive root or roots; and in
the case ax- ■+■ c = bx, which has two, he observes that
the nature of the problem must show which of the two is
the proper one.

~~ In the cubic equations, he gives the rules and transfor-
mations, &c, after the manner of Cardan ; remarking that
some of the ca.ses have only one root, but others two or
three, of which some are true, and others false or nega-
tive. And in one place he says that by means of the case
x3 = bx ■+• c, he trisects or divides an angle into three equal
parts.

When he arrives at biquadratic equations, and particu-
larly to this case x4 -t- ax = b, he says, " Since I have seen
Diophantus's work, I have always been of opinion that
his chief intention was to come to this equation, because
I observe he labours at finding always square numbers,
and such, that adding some number to them, may make
squares ; and I believe that the six books, which are lost,
may treat of this equation, &c."— " But Lewis Ferrari,"
he adds, " of this city, also laboured in this way, and
found out a rule for such cases, which was a very fine in-
vention, and therefore I shall here treat of it the best I
can." This he accordingly does, in all the cases of bi-
quadratics, both with respect to the number of terms in
the equalion, and the signs of the terms, except I think
this most general case only rx — qxz -+- px3 — x* = s; fully
applying Ferrari's method in all cases. Which concludes
the 'Zd book.

The 3d book consists only of the resolution of near 300
practical questions, as exercises in all the rules and equa-
tions, many of which arc taken from other authors, espe-
cially from the first five books of Diophantus.

Upon the whole it appears, that this is a plain, explicit,
and very orderly treatise on algebra, in which are very-
well explained the rules and methods of former writers.
But Bombelli does not produce much of improvement or
invention of his own, except his notation, which varies
from others, and is by means of one general character,
with the numeral indices of Stifelius. He also first re-
marks that angles are trisected by a cubic equation.

GULIELMI GOSSALINI, &C.

De Arte Magna, seu de Occulta Parte Numerorum qua: et
Algebra et Almucabala vulgo dicitur, libri quatuor. In
quibus explicantur wquationes Diophanti, Regulce Siuanti-
tatis Simplicis, et Sluantitatis Surdce.—- Paris, 1577, in
small 8vo, 86 leaves, numbered only on one side.
In the dedication, this author notices the term Cossic
Numbers; and mentions many of the former writers; as,
Diophantus, Forcadel, Scheubel, Peletare, Stifel, Cardan,
Lucas, Villafranc, Nunez, and Tartaglia, whose work he
says he had translated and published in French. The
present work seems meant as an elementary book of in-
struction, being regularly divided into chapters, and treat-
ed methodically, though not deeply. The chapters are,
on quantity, on the methods of numeration, on the object
of algebra, on the names of numbers with respect to their
powers, which are thus denoted : L the latus or number,
Q the quadrant or square, c the cube, q q the square-
squared or 4th power, r p the relatus-primus or 5th power
(called by Diophantus quadratocubus), rs the relatus-
secundus or 7th power, qqq quadrato-quadrato-quadra-
tus or 8th power, cc the cubocubus or yth power, &c. —
Chap. 7, gives a long account of the composition of the
cube of a binomial, and the extraction of the cube root,
from the theorem c3 = (a 4-6)3=: a3-i- b3-i- Sab . a ■+- 3ab
. b, from Tartaglia. — Chap. 8, &c, treats on proportion,
arithmetical, geometrical, and harmonical, with the arith-
metic of ratios. — Chap. 15, \6, on the single and double
rule of Hypothesis, or Position. — Chap. 17, on finding
several mean proportionals, by the double rule of Hypo-
thesis, with its application to the celebrated problem of
doubling the cube; also to its tripling, and quadru-
pling, &c.

Book ii. Chap. 1. On the values of the powers, (or what
are now called their exponents), being the numbers of
their situation in the scale of powers, 1, 2, 3, 4, &c. —
Chap. 2, on the addition of the powers, by means of their
signs p and m, that is, plus and minus. — Chap. 3, 4, on the
multiplication and division of powers, by adding and sub-
tracting their values or indices. — Chap. 5, 6", addition and
subtraction of complex quantities, with mixed signs, p and
vi, and various powers. — Chap. 7, 8, multiplication and
division of mixed quantities. Here the author states that
in these operations, like signs produce/) (plus), but unlike
signs m (minus) ; and he says he has proved these in his
Arithmetic, which no one has done before him. — Chap. Q,
10, &c, on finding the sides or roots of simple and com-
pound quantities.

Book in. On Equations. Chap. 1, 2, 3, on simple equa-
tions.— Chap. 4, 5, 6, 7, 8, 9, on quadratic equations, by
the usual method of completing the square, &c. In the
case x'-h- n = ax, the author assigns the two roots. In the
other two cases, he finds one root only, without noticing
the negative roots. — In chap. 10, he speaks of the cubic
equations treated of by Cardan and Tartaglia, but without
stating the rules for their solution. In chap. 11, 12, 13,
some Diophantinc problems are treated of.

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Book iv, titled De 2uantitate Absoluta et Surda,
contains two chapters. These include various problems
or questions, in which several unknown quantities are re-
quired to be found, from as many proposed independent
jelations. In the solution of these questions, the au-
thor uses the capitals a,b,c, d,&c, for the unknown
quantities to be found, which he employs in the same
way as we do now the letters x, y, z, &c, making out with
them as many independent equations as are the unknown
letters, a, b, c, &c; by the reduction of which equations,
in the usual way, the values of these letters are determi-
ned.— But Stifelius, before him, had used a, b, C, &c, for
the unknown quantities.

CLAVIUS.

Christopher Clavius wrote his Algebra about the year
1580, though it was not published till l608, at Orleans.
He mostly follows Stifelius and Scheubelius in his nota-
tion and method, &c, having scarcely any variations from
them ; nor does he treat of cubic equations. He mentions
the names given to the art, and the opinions about its
origin, in which he inclines to ascribe it to Diophantus,
from what Diophantus says in his preface to Dionysius.

STEVINUS.

The Arithmetic of Simon Stevin of Bruges, was pub-
lished in 1585, and the same with his Algebra in 1605, in
the Flemish dialect, and containing a free translation of
the first 4 books of Diophantus. They were also printed
in a French edition of his works at Leyden in 1634, with
some notes and additions of Albert Guard, who it seems
died the year before, this edition being published for the
benefit of Girard's widow and children. This edition con-
tains all the 6 books of Diophantus.

The Algebra is an ingenious and original work. He de-
notes the res, or unknown quantity, in a way of his own,
namely by a small circle O, within which he places the
numeral exponent of the power, as ©, (?, @, (J), &c,
which are the 0, 1, 2, 3, &c, power of the quantity Oj
where @ , or the 0 power, is the beginning of quantity, or
arithmetical unit. He also extends this notation to roots
or fractional exponents, and even to radical ones.
Thus ®, (J), Q, 6ec, are the square root, cube root, 4th

root, &c ;
also ® 's 'he cube root of the square ;
and © is the square root of the cube. And so of others.

The first three powers, @, ©, ©, he also calls coste
(side) , qna> re (square), cube (cube) ; and the first of them,
®, the prime quantity, which he observes is also meta-
phorically called the racine or root, (the mark of which
is also */,) because it represents the root or origin from
which all other quantities spring or arise, called the polences
or powers of it. He condemns the terms sursolids, and
numbers absurd, irrational, irregular, inexplicable, or
surd, and shows that all numbers are denoted the same
way, and are all equally proper'expressions of some length
or magnitude, or some power of the same root. He also
rejects all the compound expressions of square-squared,
cube-squared, cube-cubed, &c; and observes that it is best
to name them all by their exponents, as the 1st, 2d, 3d,
4th, 5th, 6th, &c, power or quantity in the series. And
on his extension of the new notation he justly observes,
that what was before obscure, laborious, and tiresome, will
by these marks be clear, easy, and pleasant. He also
makes the notation of algebraic quantities more general in
their coefficients, including in them not only integers, as
3 ©, but also fractions and radicals, as £ ©, and i/2Q),

Vol. I.

&ci He has various other peculiarities in his notations ;
all showing an original and inventive mind. A quantity of
several terms, he calls a multinomial, and also binomial,
trinomial, &c, according to the number of the terms. He
uses the signs ■+- and — , and sometimes : for equality ;
also x for division of fractions, or to multiply crosswise
thus, f x f : 44.

He teaches the generation of powers by 2

means of the annexed table of numbers, 3 3

which are the coefficients of all the terms, 464
except the first and last. And he makes 5 10 10 5
use of the same numbers also for extract- 6 15 20 15 6
ing all roots whatever : both which things &c.

had first been done by Stifelius. In ex-
tracting the roots of non-quadrate or non-cubic numbers,
he has the same approximations as at present, viz, either
to continue the extraction indefinitely in decimals, by add-
ing periods of ciphers, or by making a fraction of the re-
mainder in this manner, viz, ^/u = n -

and l/s =;

— nearly,

- r ■"

nearly ; where n is the nearest

An -t- An + 1

exact root of n ; which is Peletarius's rule, and which
differs from Tartalea's rule, as this wants the 1 in the de-
nominator. And in like manner he goes on to the roots
of higher powers.

He then treats of equations, and their inventors, which
according to him are thus :

Mahomet, son of Moses, an Arabian, )■»."• ^ .

. j ., ' ' < its derivatives,

invented these - - - )_ , ..-.v^ '

(_ © egale a @, © .

And some unknown author, the derivatives of this.

o 1 .u . j ..i S ©egale a ® @,

borne unknown author invented these < x 1 ■ ^ ^

I ©egale a © ©.

But afterwards he mentions Ferreus, Tartalea, Cardan,
&c, as being also concerned in the invention of them.

Lewis Ferrari invented - (£. egale d © © ® 0'-

He says also that Diophantus once resolves the case ©
egale 08®. In his reduction of equations, which is
full and masterly, he always places the highest power on.
one side alone, equal to all the other terms, set in their
order, on the other side, whether they be -+- or—. And
he demonstrates all the rules both arithmetically aud geo-
metrically. In cubics, he gives up the irreducible case,
as hopeless: but says that Bombelli resolves it by plus of
minus, and minus of minus:

thus, if 1 © = 30 © ■+■ 36, then 1® =(/(18 + of -26)
•+■2/(18 -of- 26), that is, 1® ={/(lS + 26y/-l)
4-^/(18 — 26^—1). He resolves biquadratics by
means of cubics and quadratics. In quadratics, he takes
both the two roots, but looks for no more than two in cu-
bics or biquadratics. He gives also a general method of
approaching indefinitely near, in decimals, to the root of
any equation whatever ; but it is very laborious, being
little more than trying all numbers, one after another,
finding thus the 1st figure, then the 2d, then the 3d, &c,
among these ten characters 0, 1,2, 3, 4, 5, 6, 7, 8, 9- And
finally he applies the rules, in the resolution of a great
many practical questions.

Though a general air of originality and improvement
runs through the whole of Stevinus's work, yet his more
remarkable or peculiar inventions, may be reduced to
these few following : viz,

1st. He invented not only a new character for the un-
known quantity, but greatly improved the notation of
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powers, by numeral indices, first given by Stifelius as to
integral exponents; which Stevinus extended to fractional
and all other sorts of exponents, thereby denoting all kinds
of roots the same way as powers, by numeral exponents.
A circumstance hitherto thought to be of much later in-
vention.

2d. He improved and extended the use and notation of
coefficients, including in them fractions and radicals, and
all sorts of numbers in general.

3d. A quantity of several terms, he called generally a
multinomial; and he denoted all nomials whatever by
particular names expressing the number of their terms, bi-
nomial, trinomial, quadrinomial, &c.

4th. A numeral solution of all equations whatever, by
one general method.

Besides which, he hints at some unknown author as the
first inventor of the rules for cubic equations; by whom
may perhaps be intended the author of the Arabic manu-
script treatise on cubic equations, given to the library at
Leyden by the celebrated Warner.

VIETA.

Most of Vieta's algebraical works were written about or
before the year lfjOO, but some of them were not publish-
ed till after his death, which happened in the year l603,
in the 63d year of his age. And his whole mathematical
works were collected together by Francis Schooten, and
elegantly printed in a folio volume in l6i6. Of these,
the algebraical parts are as follow :
<" 1. Isagoge in Artcm Analyticam.

2. Ad Logisticen Speciosam Notas Priores.

3. Zeteticorum libri quinque.

4. De iEquationum Recognitione, et Emendatione.

5. De Numerosa Potestatum ad Exegesin Resolutione.
Of all these I shall give a particular account, especially

in such parts as contain any discoveries, as we here meet
with more improvements and inventions on the nature of
equations, than in almost any former author. And first
of the Isagoge, or Introduction to the Analytic Art. In
this short introduction Vietalays down certain praecognita
in this art ; as definitions, axioms, notations, common pre-
cepts or operations of addition, subtraction, multiplication,
and division, with rules for questions, &c. From which
we find, 1st, That the names of his powers are latus, qua-
dratum, cubus, quadrato-quadratum, quadrato-cubus,
cubo-cubus, &c; in which he follows the method of Dio-
phantus, and not that derived from the Arabians. 2d,
That he calls powers pure or adfected, and first here uses
the terms coefficient, affirmative, negative, specious lo-
gistics or calculations, homogeneum comparationis, or the
absolute known term of an equation, homogeneum, adfcc-
tionis, or the 2d or other term which makes the equation
adfected, &c. 3d, That he uses the capital letters to de-
note the known as well as unknown quantities, to render
his rules and calculations general ; namely the vowels
a, e, I, o, v, Y for the unknown quantities, and the con-
sonants, b, c, d, &c, for the known ones. 4th, That he
uses the sign -t- between two terms for addition ; — for
subtraction, placing the greater before the less ; and when
it is not known which term is the greater, he places = be-
tween them for the difference, as we now use ~ ; thus
a = n is the same as a ~ b ; that he expresses division
by placing the terms like a fraction, as at present ; though
he was not first in this. But that he uses no characters
for multiplication or equality, but writes the words them-
selves, as well as the names of all the powers, as he uses

no exponents, which causes much trouble and prolixity
in the progress of his work ; and the numeral coefficients
set after the literal quantities, have a disagreeable effect.

II. Ad Logisticen Speciosam Nota Prioi'es. These con-
sist of various theorems concerning sums, differences, pro-
ducts, powers, proportionals, &c, with the genesis of
powers from binomial and residual roots, and certain pro-
perties of rational right-angled triangles.

III. Zeteticorum libri quinque. The zetetics or ques-
tions, in these 5 books, are chiefly from Diophantus, but re-
solved more generally by literal arithmetic. And in these
questions are also investigated rules for the resolution of
quadratic and cubic equations. In these also Vieta first
uses a line drawn over compound quantities, as a vincu-
lum.

IV. De JEquationum Recognitione, et Emendatione. These
two books, which contain Vieta's chief improvements in
algebra, were not published till the year l6l5, by Alex-
ander Anderson, a learned and ingenious Scotchman, with
various corrections and additions. The 1st of these two
books consists of 20 chapters. In the first six chapters,
rules are drawn from the zetetics for the resolution of
quadratic and cubic equations. These rules are by means
of certain quantities in continued proportion, but in the
solution they come to the same thing as Cardan's rules.
In the cubics, Vieta sometimes changes the negative roots
into affirmative, as Cardan had done, but he finds only the
affirmative roots. And he here refers the irreducible case
to angular sections for a solution, a method which had
been mentioned by Bombelli.

Chap. 7 treats of the general method of transforming
equations, which is done either by changing the root in
various ways, namely by substituting another instead of
it which is either increased or diminished, or multiplied
or divided, by some known numbeF, or raised or depressed
in some known proportion ; or by retaining the same root,
and equally multiplying all the terms. Which kinds of
transformation, it is evident, are intended^to make the
equation become simpler, or more convenient for solu-
tion. And all or most of these reductions and transfor-
mations were also practised by Cardan.

Chap. 8 shows what purposes are answered by the fore-
going transformations; such as taking away some of the
terms out of an equation, and particularly the 2d term,
which is done by increasing or diminishing the root by the
coefficient of the 2d term, divided by the index of the
first : by which means also the affected quadratic is re-
duced to a simple one. And various other effects are
produced.

Chap. 9. shows how to deduce compound quadratic
equations from pure ones, which is done by increasing or
diminishing the root by a given quantity, being one appli-
cation of the foregoing reductions.

Chap. 10, the reduction of cubic equations affected
with the 1st power, to such as are affected with the 2d
power ; by the same means.

In chap. 11, by the same means also the 2d term is re-
stored to such cubic equations as want it.

In chap. 12, quadratic and cubic equations are raised to
higher degrees, by substituting for the root, the square or
cube of another root divided by a given quantity.

In chap. 13, affected biquadratic equations arc deduced
from affected quadratics in this manner, when expressed
in the modern notation . If a2 h- ba sa z,
then shall a4 -+■ b3 -t- 2bz . a = zl -t- b5z.

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Fot since a* -t- ba = z, therefore a2 = z — 6a,

and its square is a4 = z2 — 2baz -t- b2a2:

but b3a2 = b2z — B3A,

therefore a4 = z2 — 2baz -+- b2z — b3a,

or a4 ■+■ b3 + 2bz . a = z2 ■+- B2Z.

And in like manner for the biquadratic affected with its

other terms. And in a similar manner also, in chap. 14,

affected cubic equations are deduced from the affected

quadratics.

In chap. 15 it is shown that the quadratic ba — a2 = z
has two values of the root a, or has ambiguous roots, as he
calls them ; and also that the cubics, biquadratics, &c,
which are raised or deduced from that quadratic, have
also double roots.

Having in the foregoing chapters, shown how the coef-
ficients of equations of the 3d and 4th degree are formed
from those of the 2d degree, of the same root ; and that
certain quadratics, and others raised from them, have
double roots j then in the l6th chap. Vieta shows what
relation those two roots bear to the coefficients of the two
lowest terms of an equation consisting of only three
terms.

Chap. 17 contains several theorems concerning quan-
tities in continued geometrical progression. Which are
preparatory to what follows, concerning the double roots
of equations, the nature of which he expounds by means
of such properties of proportional quantities.

Chap. 18, JEquationum ancipitum constitutive/, ; treating
of the nature of the double roots of equations.

Next follows the 2d of the pieces published by Alexan-
der Anderson, namely,

De Emendalionc Mquationum, in 14 chapters.

Chap. 1. Of preparing equations for their resolution in
numbers, by taking away the 2d term ; by which affected
quadratics are reduced to pure ones, and cubic equations
affected with the 2d term are reduced to such as are af-
fected with the 3d only. Several examples of both kinds
of equations are given. He here too remarks on the
method of taking away any other term put of an equa-
tion, when the highest power is combined with that other
term only ; and this Vieta effects by means of the co-
efficients, or, as he calls them, the uncias of the power of
a binomial. All which was also performed by Cardan
for the same purpose.

Chap. 2 De transmutatiene II §wtov — zkcctov, qua reme-
dium est adversus vitium negationis. Concerning the trans-
formations by changing the given root a for another root i,
which is equal to the homogencum comparationis divided
by the first root a ; by which means negative terms are
changed to affirmative, and radicals are taken out of the
equation when they are contained in the homogeneum
comparationis.

Chap. 3. De Anastrophe, showing the relation between the
roots of correlate equations ; whence, having given the
root of the one equation, that of the other becomes
known.

Chap. 4. De Isomccria, adversus vitium fractionis. To
take away fractions out of an equation. Thus,

it a3 -*- —a = z. Put a = — ; then e3 ■+■ bde =zd3.

Chap. 5. De Symmetrica Climactismo, adversus vitium
asymmctrice. To take away radicals or surds out of equa-
tions, by squaring, &c, the other side of the equation.

Chap. 6. To reduce biquadratic equations by means of
cubics and quadratics, by methods which are small varia-
tions from those of Ferrari and Cardan.

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Chap. 7. The resolution of cubic equations by rules
which are the same with Cardan's.

Chap. 8. De Canonica Mquationum transmutatione, ut co-
efficientes subgraduales sint quce prascribuntur. To trans-
mute the equation so that the coefficient of the lower
term, or power, may be any given number, he changes the
root in the given proportion, thus : Let A be the root of
the equation given, e that of the transmuted equation,
B the given coefficient, and x the required one ; then

take a ==— , which substitute in the given equation, and

it is done. — He commonly changes it so, that x may be
1 ; which he does, that the numeral root of the equation
may be the easier found ; and this he here performs by
trials, by taking the nearest root of the highest power
alone : and if that does not turn out to be" the root of
the whole equation, he concludes that it has no rational
root.

Chap. 9- To reduce certain peculiar forms of cubics to
quadratics, or to simpler forms, much the same as Cardan
had done.

Chap. 10. Similium reductionum continuatio. Being
some more similar theorems, when the equation is affected
with all the powers of the unknown quantity a.

Chap. 11, 12, 13 relate also to certain peculiar forms of
equations, in which the root is one of the terms of a cer-
tain series of continued proportionals.

Chap. 14, which is the last in this tract, contains in
four theorems, the general relation between the roots of an
equation and the coefficients of its terms, when all its
roots are positive.

And from these last four theorems itappears, that Vieta
was acquainted with the composition of these equations,
that is, when all their roots are positive, for he never ad-
verts to negative roots; and from other parts of the work
it appears, that he was not aware that the same proper-
ties will obtain in all sorts of roots whatever. "But it is
not certain in what manner he obtained these theorems,
as he has not given any account of the investigations,
though that was usually his way on other occasions ; but
he here contents himself with barely announcing the theo-
rems, and for this strange reason, that he might at length
bring his work to a conclusion.

To this piece is added, by Alexander Anderson, an
Appendix, containing the construction of cubic equa-
tions by the trisection of an angle, and a demonstration of
the property referred to by Vieta for this purpose.

De Numerosa Potestatum Purarum Resolutione. Vieta
here gives some examples of extracting the roots of pure
powers, in the way that had been long before practised,
by pointing the number into periods of figures according
to the index of the root to be extracted, and then proceed-
ing from one period to another, in the usual way.

De Numerosa Potestatum adfectarum Resolutione. And
here, in close imitation of the above method for the roots
of pure powers,' Vieta extracts those of adfected ones ; or
finding the roots of affected equations, placing always the
homogeneum comparationis, or absolute term, on one
side, and all the terms affected with the unknown quan-
tity, and their proper signs, on the other side. The
method is very laborious, and is but little more than what
was before done by Stevinus on this subject, depending
not a little on trials. The examples he uses are such as
have either one or two roots, and indeed such as are af-
fected commonly with only two powers of the unknown
quantity, and which therefore admit only of these two

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varieties as to the number of roots, namely according as
the higher of the two powers is affirmative or negative, the
homogeneum comparationis, on the other side of the equa-
tion, being always affirmative ; and he remarks this general
rule, if the higher power be negative, the equation has two
roots, otherwise only one ; that is, affirmative roots ; for
as to negative and imaginary ones, Vieta knew nothing
about them, or at lea6t he takes no notice of them. By the
foregoing extraction, Vieta finds both the greater and less
root of the two that are contained in the equation, and
either of them that he pleases ; having first, for this pur-
pose, laid down some observations concerning the limits
within which the two roots are contained. Also, having
found one of the roots, he shows how the other root may
be found by means of another equation, which is a degree
lower than the given one; though not by depressing the
gi-yen equation, by dividing it as is now done ; but from
the nature of proportionals, and the theorems relating to
equations, as given in the former tracts, he finds the terms
of another equation, different from that last mentioned,
from the root &c of which, the 2d root of the original
equation may be obtained.

In the course of this work, Vieta makes also some ob-
servations on equations that are ambiguous, or have three
roots; namely, that the equation lc — 6q-i-11n = 6,
or as we write it x3 — 6x~ + 1 lx = 6, is ambiguous,
when the 2d term is negative, and the 3d term affirma-
tive, and when J of the square of 6, the coefficient of the
2d term, exceeds 11, the coefficient of the 3d term, and
has then three roots. Or in general, if x3 — ax'2 ■+■ bx = c,
and \a?> b, the equation is ambiguous, and has three
roots. He shows also, from the relation of the coefficients,
how to find whether the roots are in arithmetical progres-
sion or not, and how far the middle root differs from the
extremes, by means of a cubic equation of this form x3 —
bx = c. In all or most of which remarks he was pre-
ceded by Cardan. — Vieta also remarks that the case x3 —
9xi |t- 24x = 20, has three roots by the same rule, viz,
2, 2, 5, but that two of them are equal. And further,
that when §«* is = b, then all the three roots are equal,
as in the case x3 — 6x2 ■+• 12x = 8, the three roots of
which are 2, 2, 2. But when -j-a' is less than b, the case
is not ambiguous, having but one root. And when ab =sc,
then a = x is one root itself.

Many curious notes are added at the end, with remarks
on the method of finding the approximate roots, when
they are not rational, which is done in two ways, in imi-
tation of the same thing in the extraction of pure powers,
viz, the one by forming a fraction of the remainder after
all the figures of the homogeneum comparaiionis are ex-
hausted ; the other by increasing t In- root of the equation
in a 10 fold, or 100 fold &c, proportion, and then di-
viding the root which results by 10, or 100, &c: and this
is a decimal approximation. And Vieta observes, that
the roots will be increased 10 or 100 fold, &c, by adding
the corresponding number of ciphers to the coefficient! of
the 2d term, double that number to the 3d, triple the
same number to the 4th, and so on. So if the equation
were lc h- 4q ■+- 6n = 8, then lc -t- 40q -+- 600 n =
8000 will have its root 10 fold, and lc •+- 400Q + 00000 N
= 8000000 will have it 100 fold.

Besides the foregoing algebraical works, Vieta gave va-
rious constructions of equations by means of circles and
fight lines, and angular sections, which may be consi-
dered as an algebraical tract, or a method of exhibiting
the roots of certain equations having all their roots affir-

mative, and by means of which he resolved the celebrated
equation of 45 powers, proposed to all the world by Adri-
anus Romanus.

Having now delivered a particular analysis of Vieta's
algebraical writings, it will be proper, as with other au-
thors, to collect into one view the particulars of his more
remarkable peculiarities, inventions, and improvements.

And first it may be observed, that his writings show
great originality of genius and invention, and that he made
alterations and improvements in most parts of algebra;
though in other parts and respects his method is inferior
to some of his predecessors ; as, for instance, where he
neglects to avail himself of the negative roots of Cardan;
the numeral exponents of Stifelius, instead of which he
uses the names of the powers themselves; or the frac-
tional exponents of Stevinus; or the commodious way of
prefixing the coefficient before the quantity or factor;
and such like circumstances ; the want of which gives
his Algebra the appearance of an age much earlier than
its own. But his real inventions of things before not
known, may be reduced to the following particulars.

1st. Vieta introduced the general use of the letters of
the alphabet, to denote indefinite given quantities; which
had only been done on some particular occasions before
his time. But the general use of letters, for the unknown
quantities, was before pretty common with Stifelius and
his successors. Vieta uses the vowels a, e, i, o, u, y for
the unknown quantities, and the consonants, b, c, d, &c,
for known ones.

2d. He invented, and- introduced, many expressions or
terms, several of which are in use to this day : such as
coefficient, affirmative and negative, pure and adfected or
affected, uncis, homogeneum adfectionis, homogeneum
comparationis, the line or vinculum over compound quan-
tities, thus a -+- b. And his method of setting down his
equations, is to place the homogeneum comparationis, or
absolute known term, on the right-hand side alone, and
on the other side all the terms which contain the unknown
quantity, with their proper signs.

3d. In most of the rules and reductions for cubic and
other equations, he made some improvements, and varia-
tions in the modes.

4th. He showed how to change the root of an equation
in a given proportion.

5. lie derived or raised the cubic and biquadratic, &c,
equations, from quadratics ; not by composition in Har-
riot's way, but by squaring and otherwise multiplying
certain parts of the quadratic. And as some quadratic
equations have two roots, therefore the cubics and others
raised from them, have also the same two roots, and no
more. And hence he comes to know what relation these
two roots bear to the coefficients of the two lowest terms
of cubic and other equations, when they have only 3
terms, namely, by comparing them with similar equations
so raised from quadratics. And, on the contrary, what
the roots are, in terms of such coefficients.

6". He made some observations on the limits of the two
roots of certain equations.

7. He stated the general relation between the roots of
certain equations and the coefficients of its terms, when
the terms are alternately plus and minus, and none of
them arc wanting, or the roots all positive.

8. He extracted the roots of affected equations, by a
method of approximation similar to that for pure powers.

0. He gave the construction of certain equations, and

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[ 93 ]

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exhibited their roots by means of angular sections; be-
fore adverted to by Bombelli.

OF ALBERT GIRARD.

Albert Girard was an ingenious Dutch or Flemish ma-
thematician, who died about the year 16'33. He pub-
lished an edition of Stevinus's Arithmetic in l625, aug-
mented with many notes; and the year after his death
was published, by his widow, an edition of the whole
works of Stevinus, in the same manner, which Girard had
left ready for the press. But the work which entitles him
to a particular notice in this history, is his " Invention
Nouvelle en l'Algebre, tant pour la solution des equa-
tions, que pour recognoistre le nombre des solutions qu'el-
les recoivent, avec plusieurs choses qui sont necessaires a
la perfection de ceste divine science;" which was printed
at Amsterdam l6"29, in small quarto, in 6'3 pages, viz, 49
pages on arithmetic and algebra, and the rest on the
measure of the superficies of spherical triangles and po-
lygons, by him then lately discovered.

In this work, Girard first premises a short tract on
Arithmetic; in the notation of which he has something
peculiar, viz, dividing the numbers into the ranks of mil-
lions, billions, trillions, &c.

He next delivers the common rules of algebra, not only
in integers and fractions, but radicals also; with the no-
tation of the quantities and signs. In this part he uses
sometimes the letters a, b, c, &c, after the manner of
Vieta, but more commonly the characters of Stevinus,
viz, ©, ®, ©, , &c, for the powers of the unknown
quantity, with their roots (J), @, ©, , ®, &c, used by
Stevinus ; and some times the more usual marks of the
roots, as «/ or \J , y/, \/ , &c; prefixing the coefficients,
as 6 -- , or 3^/3", or 2ti.-. In tfifi signs he follows his pre-
decessors so far as to have ■+■ for plus, — or -f- for mi-
nus, = for general or indefinite difference, a ■+■ b for the
sum, a — b or a = b for the difference, ab the product,

and - for the quotient of a and B. He lises the paren-
theses ( ) for the vinculum or bond of compound quan-
tities, as is now commonly practised ; as a(ab ■+■ Bq), or
%/ (a. cub. — 3Aqn); and he introduces the new characters
ff for greater than, and § for less titan ; but he uses no cha-
racter for equality, only the word itself.

Girard gives a new rule for extracting the cube root of
binomials, which however is in a good measure tentative,
and which he explains thus: To extract the cube root of
72 + v/5120.

The squares of the terms < e,on

their difference 64, and its cube

root 4. Which shows that the difference between the

squares of the terms required is 4; and the rational part

72 being the greater, the greater term of the root will be

rational also ; and further, that the greater 9 .,.

terms of the power and root arc commen- „ ,.

surable, as also the two less terms. Then . ,.n

u j t li u i 4 ■+■ V 12

having made a table as in the margin, where , /o\

the square of the rational term always ex- ' ^ ~
ceeds that of the other, by the number 4 above mention-
ed, one of these binomials must be the cubic root sought,
if the given quantity have such a root ; and it must be
pne of these four forms; for it is known to be carried far
enough by observing that the cube root of 72 is less than
5, and the cube root of 5120 less than 21; indeed, this
being the case, the last binomial is excluded, as evidently

too great; and the first is excluded because one of its
terms is 0; therefore the root must be either 3 ■+■ */ 5 or
4 ■+■ ^/12. And to know whether of these two it must
be, try which of them has its two terms exact divisors of
the corresponding terms of the given quantity; then it is
found that 3 and 4 are both divisors of 72, but that only
5, and not 12, is a divisor of 5120; therefore 3 ■+■ */5
, is the root sought, which upon trial is found to answer.
It is remarkable here, that Girard uses 4 ■+■ y'SO instead
of 4 h-,^/12, and 5 -f- ,^20. instead of 5 -t- ,y/20, con-
trary to his own rule.

Girard then gives distinct and plain rules for bringing
questions to equations, and for the reduction- of those
equations to their simplest form, for solution, by the usual
modes, and also by the way called by Vieta Isomcria,
multiplying the terms of the equation by the terms of a
geometrical progression, by which means the roots are
altered in the proportion of 1 to the ratio of the progres-
sion. He then treats of the methods of finding the roots
of the several kinds of equations, quadratic, cubic, &c;
and adds remarks on the proper number of conditions or
equations for limiting questions. The quadratics are re-
solved by completing the square, and both the positive
and negative roots are taken; and he observes that some-
times the equation is impossible, as © equ. 6® — 25,
whose roots, he adds, are 3 -+- y/ — l6 and 3 — */ — 16.

The cubic equations he resolves by Cardan's rule, ex-
cept the irreducible case, which he the first of any re-
solves by a table of sines; the, other cases he also re-
solves by tables of sines and tangents ; and adds geome-
trical constructions by means of the hyperbola or the tri-
seclion of angles. He next adds a particular mode of
resolving all kinds of equations, that have rational roots,
on the principle of the roots being divisors of the last or
absolute term, as before mentioned by Peletarius ; and
then gives the method of approximating to other roots
that are not rational, much in the same way as Stevinus.

Having found one root of an equation, by any of the
former methods, by means of it he depresses the equation
one degree lower, then finds another root, and so on till
they are all found ; for he shows that every algebraic
equation, admits of as many solutions or roots, as there
are units in the index of the highest power, which roots
may be either positive or negative, or imaginary, or, as he
calls them, greater than nothing, or less than nothing, or
involved ; so the roots of the equation 1 © equ. 7® — 6,
are 2, 1, and — 3; and the roots of the equation l©equ.
4® — 3 are, 1, and 1, and — 1 ■+- s/ — 2, and — 1 —
v/-2.

Jn depressing an equation to lower degrees, he does not
use the method of resolution of Harriot, but that which
is derived from the general relation of the roots and coeffi-
cients of the terms, which he here fully and universally
states, viz, that the coefficient of the 2d term is equal to
the sum of all the roots ; that of the 3d iterm equal to
the sum of all the products of the roots, taken two by
two ; that of the 4th term, the sum of the products, taken
three by three; and so on, to the last or absolute term,
which is the continual product of all the roots; a property
which was before stated by Vieta, as to the equations that
have all their roots positive ; and here extended by Girard
to all kinds of roots whatever: but how either Vieta or
he came, by this property, no where appears that I know
of. From this general property, among other deductions,
Girard shows how to find the sums of the powers of the

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roots of an equation; thus, let a, e, c, tj, &c, be the
1st, 2d, 3d, 4th, &c, coefficient, after the first term, or
the sums of the products taken one by one, two by two,
three by three, &c ; then, in all kinds of equations,
a ) 2 2 f roots,

Aq-2B (IZ J squares,

a cub. — 3ab +3c C-Sj cnbes>

Aqq — 4AqB -+- 4ac -t- 2Bq — 4dJ s2 (. biquadrates.

Girard next explains the use of negative roots in geo-
metry, showing that they represent lines only drawn in a
direction contrary to those representing the positive roots;'
and he remarks that this is a thing hitherto unknown. He
then terminates the algebra by some questions having
two or more unknown quantities. And he subjoins to the
whole a tract on the mensuration of the surfaces of sphe-
rical triangles and polygons, by him lately discovered.

From the foregoing account it appears that,

1st, He was the first person who understood the general
doctrine of the formation of the coefficients of the powers,
from the sums of their roots, and their products, &c.

2d, He was the first who understood the use of negative
roots in the solution of geometrical problems.

3d, He was the first who spoke of the imaginary roots,
and understood that every equation might have as many
roots, real and imaginary, and no more, as there are units
in the index of the highest power. And he was the first
who gave the whimsical name of quantities less than no-
thing to the negative ones.

4th, He was also the first who discovered the rules for
summing the powers of the roots of any equation.

OF HARRIOT.

Thomas Harriot, an excellent astronomer, philosopher,
and mathematician, was born at Oxford in 1500. After
taking the degree of bachelor of arts in 1579, he accom-
panied the famous Sir Walter Raleigh in an expedition to
Virginia, where the first English establishment was made.
Harriot here drew the map of the country, and in 1588
gave a relation of the voyage. Being returned to his
country, it appears that he gave himself up entirely to
the study of the mathematics, and especially to that of
algebra. He remained not long unknown to the earl of
Northumberland, a great encourager of the sciences, who
maintained several learned men, such as Rob. Hues, Wal-
ter Warner, and Nathaniel Tarporley. This enlightened
nobleman entertained Harriot in his house, with a salary
of <£300 sterling a-year, a great sum in those times; and
in this situation it was that Harriot finished his days, in
the year 1621, at 6l years of age. It appears by Kepler's
letters, that he held a correspondence with this astrono-
mer, chiefly on the theory of the rainbow. Harriot's ma-
nuscripts, lately discovered in the castle of the earl of
Egremont, inform us of many of his astronomical obser-
vations, and particularly of those of the spots in the sun
as early as the beginning of December lfjlO, while the
first of those of Galileo were not made before the prece-
ding month. So that Harriot must then either have made
himself a telescope, or procured one from Holland. He
made in the same year also, observations on Jupiter's sa-
tellites, and on the remarkable comets of the years 1607'
and 1618.

His Algebra was left behind him unpublished, as well
as those other papers, at his death, which happened in
the year l6'21, as before mentioned, and but 6" years after
the first publication of the principal parts of Victa's Al-
gebra, by Alexander Anderson ; so that it is probable that

[ 94 ] A L G

Harriot's Algebra was written long before this time, and
indeed that he had never seen these pieces.

Harriot's Algebra was published by his friend Walter
Warner, in the year 1631 : and it would doubtless be
highly grateful to the learned in these sciences, if his
other curious algebraical and astronomical works were
published, from his original papers in the possession of the
earl of Egremont, to whom they have descended from
Henry Percy, the earl of Northumberland, that noble
Maecenas of his day. The book is in folio, and entitled
Artis Analyticas Praxis, ad JEquationes Algebraicas nova
expedita, et general! methodo, resolvendas ; a work in all
parts of it showing marks of great genius and originality,
and is the first instance of the modern form of algebra,
in which it has ever since appeared. It is prefaced by
18 definitions, which are these : 1st, I.ogistica Speciosa ;
2d, Equation; 3d, Synthesis; 4, Analysis; 5, Composi-
tion and Resolution ; 6, Forming an Equation ; 7, Re-
duction of an Equation; 8, Verification; 9, Numerosa
et Speciosa; 10, Excogitata ; 1 1, Resolution ; 12, Roots;
13 and 14, The kinds and generation of equations by mul-
tiplication, from binomial roots or facturs, called original

equations :

as a -t- h\ =

: + l-a
— ca — Zjc,

a + i\ = naa + laa + tea
a H- c| -f- caa — Ida

a — d\ — daa — (da — bed;

where he puts a for the unknown quantity, and the small
consonants, b, c, d, &c, for its literal values or roots.
15,The first form of canonical equations, which aie derived
from the above originals, by transposing the hoinoge-
neum, or absolute term, thus aa -+- ba

— ca = -+- be, &C.
16", The secondary canonicals, formed from the primary
by expelling the 2d term, thus aa = -+- bb,
or aaa — bba

— bca

— cca = -+- bbc

-+- bcc.

17, That these are called canonicals, because they are ad-
apted to canons or rules for finding the numeral roots, &c.

18, Reciprocal equations, in which the homogeneum
is the product of the coefficients of the other terms, and
the first term, or highest power of the root, is equal to
the product of the powers in the other terms, as aaa —
caa -+- bba = ■+■ bbc.

After these definitions, the work is divided into two
principal parts; 1st, of various generations, reductions,
and preparations of equations for their resolution in the
2d part. The former is divided into 6 sections as follows.

Sect. 1. Logistices Speciosa; exemplified in the 4 ope-
rations of addition, subtraction, multiplication, and divi-
sion ; as also the reduction of algebraic fractions, and the
ordinary reduction of irregular equations to the form pro-
per for the resolution of them, namely, so that all the un-
known terms be on one side of the equation, and the known
term on the other, the powers in the terms ranged in or-
der, the greatest first, and the first or highest power made
positive, and freed from its coefficient; as aa ■+- ba =cd;
or aaa •+■ baa — cda = — c'd.

In this part he explains some unusual characters which
he introduces, namely, = for equality, as a = b.
> for majority, as a > b,
< for minority, as « < b;
but the first had been before introduced by Robert Re-
corde.

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[ 95 ]

A L G

Sect. 2. The generation of original equations from bi-
nomial factors or roots, and the deducing of canonicals
from the originals. He supposes that every equation has
as many roots as dimensions in its highest power; then
supposing the values of the unknown letter a in any equa-
tion to be b, c, d,f, &c, that is a = b, and a == c, and
a = d, &c; by transposition, or equal subtraction, these
become a — b = 0, and a — c = 0, and a — d = 0, &c,
or the same letters with contrary signs for negative values
or roots; then two of these binomial factors, multiplied
together, give a quadratic equation, three of them a cu-
bic, four of them a biquadratic, and so on, with all the
terms on one side of the equation, and 0 on the other
side, since, every binomial factor being = 0, the conti-
nual product of all of them must also be = 0.

And these operations are carried through all the cases
of the 2d, 3d, and 4th powers, as to the varieties of the
signs ■+■ and —, and the proportions of the roots as to
equal and unequal, with the reciprocals, &c. From which
are made evident, at one glance of the eye, all the rela-
tions and properties between the roots of equations, and
the coefficients of the terms.

Sect. 3. JEquationum canonicarum secundariarum a pri-
mariis reductio per qradus alicujus parodici sublationem ra-
dice supposilitiainvar/ata manente. Containing a great many
examples of preparing equations, by taking away the 2d,
3d, or any other of the intermediate terms, which is done
by making the positive coefficients in that term, equal to
the negative ones, by which means the whole term va-
nishes, or becomes equal to nothing. They are extended
as far as equations of the 5th degree ; and at the end are
collected, and placed in regular order, all the secondary
canonicals, so reduced; so that by the uniform law which
is visible through them all, the series may be continued
to the higher degrees as far as we please.

Sect. 4. JEquationum canonicarum tarn primnriarum ,
quam secundariarum, radicum des/gnalio. A great many
literal equations are here set down, and their roots as-
signed from the form of the equation, that is, all their po-
sitive roots; for their negative roots are not noticed here;
and it is every w here proved that they cannot have any
more positive roots than these, and consequently the rest
are negative. That those are roots, he proves by substi-
tuting them instead of the unknown letter a in the equa-
tion, when they make all the terms on one side come to
the same thing as the homogeneum on the other side.

Sect. .5. In qua aquationum communium per canonicarum
mquipollentium, radicum numerus determinatur. On the
number of the roots of common equations, that is the po-
sitive roots. This Harriot determines by comparing them
with the like cases found among his canonical forms,
which two equations, having the same number of terms
with the same signs, and the relations of the coefficients
and homogeneum correspondent, he calls equipollents.
And whatever was the number of positive roots used in
the composition of the canonical, the same, he infers, is
the number in the proposed common equation. It is re-
markable that, in all the examples here used, the number
of positive roots is just equal to the number of the changes
in the signs from ■+■ to — and from — to +, which is a
circumstance, though not here expressly mentioned, that
could not escape the observation, or the eye, of any one,
much less of so clear and comprehensive a sight as that of
Harriot.

In this section are contained many ingenious disquisi-

tions, concerning the limits and magnitudes of quantities,
with several curious lemmas laid down to demonstrate
the propositions by, which lemmas are themselves de-
monstrated in a pure mathematical way, from the magni-
tudes themselves, independent of geometrical figures ;
such as, 1, If a quantity be divided into any two unequal
parts, the square of half the quantity will be greater than
the product of the two unequal parts. 2, In three con-
tinueoVproportionals, the sum of the extremes is greater
than double the mean. 3, In four continued proportio-
nals, the sum of the extremes is greater than the sum of
the two means. 4, In any two quantities, one-fourth the
square of the sum of the cubes, is greater than the cube
of the product of the two quantities. 5, Of any two
quantities q and r, then ^T{qq -+• qr -t- rr)3 > \{qqr -+-
qrr)2. 6, If any quantity be divided into three unequal
parts, the square of £ of the whole quantity is greater
than y of the sum of the three products made of the
three unequal parts. 7, Also the cube of the j part of
the whole, is greater than the solid or continual product
of the three unequal parts.

Sect. 6. JEquationum communium reductio per gradus ali-
cujus parodici exclusionem el radicis supposititia mutationem.
Here are a great many examples of reducing and trans-
forming equations of the 2d, 3d, and 4th degrees; chiefly
either by multiplying the roots of equations in any pro-
portion, as was done by Vieta, or increasing or diminish-
ing the root by a given quantity, after the manner of Car-
dan. The former of these reductions is performed by
multiplying the terms of the equation by the correspond-
ing terms of a geometrical progression, the first term be-
ing 1, and the 2d term the quantity by which the root is
to be multiplied. And the other reduction, or trans-
forming to another root, which may be greater or less
than the given root by a given quantity, is performed com-
monly by substituting e -+- or — b for the given root a,
by which the. equation is reduced to a simpler form.
Other modes of substitution are also used ; one of which

is this, viz, substituting — — — ore + — for the root a in

the given equation aaa ^ 3.bba =s 2.ccc, by which it re-
duces to this quadratic form efi =j- 2cV = — bc\ from
which Cardan's forms are immediately deduced; namely,

e = ^/3) c3 — */cs ± !»", and therefore a or e ± — =

•v/3) e3 + v^c6 ±b6± v/3) c3 - v/c6 ± b6; where he de-
notes the cubic or 3d root thus, \/o), but without any
vinculum over the compound quantities.

In this section, Harriot makes various remarks as they
occur: thus, he remarks, and demonstrates, that eee —
3.bb = — ccc — 2bbb is an impossible equation, or has
no affirmative root. He remarks also that the three cases
of the equation aaa — S.bba = -t- 2.ccc are similar to the
three conic sections; namely to the hyperbola when
c -> b, to the parabola when c = b, or to the ellipsis when
c< b, and for which reason this case is not generally re-
soluble in species.

Having thus shown how to simplify equations, and pre-
pare them for solution, Harriot enters next upon the se-
cond part of his work, being the

Exegetice Numerosa,
or the numeral resolution of all kinds of equations by a
general method, which is exemplified in a great number
of equations, both simple and affected, as far as the 5th
power inclusive; and they are commonly prepared, by

ALS

t

]

A L G

the foregoing parts, by freeing them from their 2d term,
&c. These extractions are explained and performed in a
way different from that of Vieta ; and the examples are
first in perfect or terminate roots, and afterwards for irra-
tional or interminate ones, to which Harriot approxi-
mates, by adding always periods of ciphers to the given
number or resolvend, as far as necessary in decimals,
which are continued and set down as such, but with their
proper denominator 10, or 100, or 1000, &c.
He then concludes the work with

Canones Directorii,
which form a collection of the cases or theorems for
making the foregoing numeral extractions, ready arranged
for use, under the various forms of equations, with the
factors necessary to form the several resolvends and sub-
trahends.

And from a review -of the whole work, it appears that
Harriot's inventions, peculiarities, and improvements in
algebra, may be comprised in the following particu-
lars.

1st. He introduced the uniform use of the small let-
ters, a, b, c, d, &c, viz, the vowels a, e, &c for unknown
quantities, and the consonants b,c,d,f, &c for the known
ones; which he joins together like the letters of a word,
t» represent the multiplication or product of any number
of these literal quantities, and prefixing the numeral co-
efficient as we do at present, except only separated by a
point, thus 5.bbc. For a root, he set the index of the
root after the mark ^/; as */3) for the cube root. He
also introduced the characters -> and <, far greater and
less; and in the reduction of equations, he arranged the
operations in separate steps or lines, setting the explana-
tions in the margin on the left hand, for each line. By
which, and other means, he may he considered as the in-
troducer of the modern state of algebra, which quite
changed its form under his hands.

2d. He showed the universal generation of all the com-
pound or affected equations, by the continual multiplica-
tion of as many simple ones, or binomial roots ; thereby
plainly exhibiting to the eye the whole circumstances of
the nature, mystery and number of the roots of equa-
tions ; with the composition and relations of the coeffi-
cients of the terms ; and from which many of the most
important properties have since been deduced.

3d. He greatly improved the numeral exegesis, or ex-
traction of the roots of all equations, by clear and expli-
cit rules and methods, drawn from the foregoing genera-
tion or composition of affected equations of all degrees.
of oughtred's clavis.
Oughtred was contemporary with Harriot, but lived a
long time after him, as he died only in 1660, at S7 years
of age. His Clavis was first published in l6'3I, the same
year in which Harriot's Algebra was published by his
friend Warner. In this work, Oughtred chiefly follows
Vieta, in the notation by the capitals a, b, c, d, &c, in
the designation of products, powers, and roots, though
with some few variations. His work may be compre-
hended under the following particulars.

1. Notation. This extends to both algebra and arith-
metic, vulgar and decimal. The algebra chiefly after the
manner of Vieta, as abovesaid. And he separates the de-
cimals from the integers thus 21\5<7, which is the first time
I have observed such a separation, and the decimals set
down without their denominator.

2. The commou rules or operations of arithmetic and

algebra. In algebraic multiplication, he either joins the
letters together like a word, or connects them by the mark
x , which is the first introduction of this character of
multiplication: thus a x a or aa or .\q. But omitting
the vinculum over compound factors, used by Vieta. He
introduces here many neat and useful contractions in
multiplication and division of decimals : as that common
one of inverting the multiplier, to have fewer decimals,
and abridge the work; that of omitting always one figure
at a time, of the divisor, for the same purpose ; dividing
by the component factors of a number instead of the num-
ber itself; as 4 and 6 for 24; and many other neat con-
tractions. He states his proportions thus 7-9 ■'■ 28.36,
and denotes continued proportion thus -ff ; which is the
first time I have observed these characters.

3. Invents and describes various symbolical marks or
abbreviations, which are not now used.

4. The genesis and analysis of powers. Denotes powers
like Vieta, and also roots, thus \/q6, */c20, i/qq2±, &c;
and muchrin his manner too performs the numeral ex-
traction of roots. He here gives a table of the powers of
the binomial a ■+■ e as far as the 10th power, with all
their terms and coefficients, or uncia? as he calls them,
after Vieta. .

5. Equations. He here gives express and particular di-
rections for the several kinds of reductions, according as
the form of the equation may require. He uses the letter
u after ,y/, for universal, instead of the vinculum of Vieta.
And he observes that the signs of all the terms of the
powers of a ■+■ e are positive, but those of a — e are al-
ternately positive and negative.

6. Next follow many properties of triangles and other
geometrical figures ; and the first instance of applying
algebra to geometry, so as-to investigate new geometri-
cal properties ; also, after the algebraical resolution of
each problem, he commonly deduces and gives a geome-
trical construction adapted to it. He gives also a good
tract on angular sections.

7. The work concludes with the numeral solution of
affected equations, in which he follows the manner of
.Vieta, though he is more explicit.

OF DESCARTES.

Descartes's Geometry was first published in 1637, be-
ing six years after the publication of Harriot's Algebra.
That work was rather an application of algebra to geo-
metry, than the science either of algebra or geometry
itself, purely and properly so called. And yet he made
improvements in both. We must observe, however, that
all the properties of equations, &c, which he sets down,
are not to be considered as even meant by himself for new
inventions or discoveries ; but as statements and enume-
rations of properties, before known and taught by other
authors, which he is about to make some use or applica-
tion of, and for which reason it is that he mentions those
properties.

Descartes's Geometry consists of three books. The
first of these is, De Problematibus, qua; construi possunt,
adhihendo tantum rectas linens et circulos. He here ac-
commodates or performs arithmetical operations by geo-
metry, supposing some line to represent unity, and then,
by means of proportionals, showing how to multiply, di-
vide, and extract roots, by lines. He next describes the
notation he uses, but not because it is a new one, for it is
the same as had been used by former authors, viz, a -t- b
for the addition of a and b, also a — b for their subtrac-

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tion, ab multiplication, — division, an or a2 the square

of a, a3 its cube, &c: also l/.iP ■+- b* for the square root
of a2 -i- 62, and ^/c.a^ — fc3 ■+- abb for the cube root, &c,
with a bar over the quantities. He then observes, after
Stifelius, that there must be as many equations as there
are unknown lines or quantities; and that they must be
reduced all to one final equation, by exterminating all the
unknown letters except one ; when the final equation will
appear like these,
2 30 b, or

Si; 30 — az ■+■ b1, or
s3 30 ■+■ az- -+- tfz — c3, or
s' 30 ■+- az* h- 6-z2 - <?z + d\ &c.
Where he uses 30 for = or equality, setting the highest
term or power alone on one side of the equation, and all
the other terms on the other side, with their proper signs.
Descartes next defines plane problems, namely, such as
can be resolved by right lines and circles, described on a
plane superficies; and then the final equation rises only
to the 2d power of the unknown letter. He then con-
structs such equations, viz, quadratics, by the circle, thus
finding geometrically the root or roots, that is, the posi-
tive ones. But when the lines, by which the roots are
determined, neither cut nor touch, he observes that the
equation has then no possible root, or that the problem is
impossible. He then concludes this book with the alge-
braical solution of the celebrated problem, before treated
of by the ancients, namely, to find a point, or the locus
of all the points, from which a line being drawn, to meet
any number of given lines in given angles, the product of
the segments of some of them shall have a given ratio to
that of the rest.

Book 2. De Natura Linearum Curvarum. This is a good
algebraical treatise on curve lines in general, and the first
of the kind that has been produced by the moderns. Here
the nature of the curve is expressed by an equation con-
taining two unknown or variable lines, and others that

But,

cry
30 cy r2 ■+- ay — ac.

are known or constant, as y

not relating to pure algebra, the particulars will be most
properly placed under the article of curve lines, and other
terms relating to them. Only one discovery, among many
ingenious applications of algebra to geometry, may here
be particularly noticed, as it may be considered as the
first step towards the arithmetic of infinites; and that is
the method of tangents, here given, or, which comes to
the same thing, of drawing aline perpendicular to a curve
at any point; which is an ingenious application of the
general form of an equation, generated in Harriot's way,
that has two equal roots, to the equation of the curve.
Of which a particular account will be given at the article
Tangents.

Book 3. De Constructions Problematum Solidorum, et So-
lida excedentium. Descartes begins this book with re-
marks on the nature and roots of equations, observing,
that they have as many roots as dimensions, which he
shows, after Harriot, by multiplying a certain number of
simple binomial equations together, as x — 2 30 0, and
■f - 3 30 0, and x — 4 30 0, producing xJ — Qxx ■+■ 26x
— 24 X 0. He here remarks that equations may some-
times have their roots false, or what we call negative,
which he opposes to those that are positive, or as he calls
them true, as Cardan had done before. As a natural de-
duction from the generation or composition of equations.

Vol.1.

by multiplication, he infers their resolution, or depression,
or decomposition, namely, dividing them by the binomial
factors which were multiplied to produce the equation :
and he observes, that, by this operation it is known that
this divisor is one of the binomial roots, and that there
can be no more roots than dimensions, or than those
which form with the unknown lettej x, binomials that
will exactly divide the equation, as Harriot had shown
before.

Descartes adverts to several other properties, mostly
known before, which he has occasion to make use of in
the progress of his work ; such as, that equations may
have as many true roots as the terms have changes of the
signs -+- and — , and as many false ones as successions of
the same signs ; which number and nature of the roots
had before been partly shown by Cardan and Vieta, from
the relation of the coefficients, and their signs, aud the
same thing appears more fully by Harriot's °5th section.
And hence Descartes infers the method of changing the
true roots to false, and the false to true, namely, by
changing the signs of the even terms only, as Cardan had
taught before. Descartes then adverts to other reduc-
tions and transmutations which had been taught by Car-
dan, Vieta, and Harriot, such as, To increase or diminish
the roots by any quantity ; To take away the 2d term ;
To alter the roots in any proportion, and thence to free
the equation from fractions and radicals.

Descartes next remarks that the roots of equations,
whether true or false, may be either real or imaginary ;
as in the equation *3 — 6xx ■+■ I3x — 10 30 0, which has
only one real root, namely 2. The imaginary roots were
first noticed by Albert Girard, as before mentioned. He
then treats of the depression of a cubic equation to a
quadratic, or plane problem, that it may be constructed
by the circle, by dividing it by some one of the binomial
factors, which, in Harriot's way, compose the equation.
Peletarius having shown that the simple root is one of the
divisors of the known term of the equation, and Harriot
that that term is the continual product of all the roots,
Descartes therefore tries all the simple divisors of that
term, till he finds one of them which, connected with the
unknown letter*, by + or — ,will exactly divide the equa-
tion. And the process is the same for higher powers than
the cube. But when a divisor cannot be thus found, for
depressing a biquadratic equation to a cubic, he gives
another rule, which is a new one, for dissolving it into
two quadratics, by means of a cubic equation, in this
manner:

Let the given biqu. be -+- x\ . pxx .
Which suppose}

equal to the f+xx-yx + iyy

product of (

these two ) +«+J«f + fcy

qx . r 30 0 ;

■ iP

T » °>

where the sign of ■£/> in the two quadratics must be the
same as the sign of p in the given equation, and in the 1st
quadratic the sign of -|- must be the same as the sign of

q, but in the 2d quadratic the contrary. Then if there
be found tho root yy of this cubic equation y6 . 2py* -+-
*lr W — 11 33 °> where the sign 2p is the same as ofp
in the given biquadratic, but the sign of 4r contrary to
that of r in the same; then the valueof y, hence deduced
being substituted for it in the two quadratic equations'
and their two pairs of roots taken, they will be the four
roots of the proposed biquadratic. And thus also he
O

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hints, may equations of the 6th power be reduced to
those of the 5th, and those of the 8th power to those of
the 7th, and so on. Descartes does not give the investi-
gation of this rule; but it has evidently been done, by
assuming indeterminate quantities, after the manner of-
Ferrari and Cardan, as coefficients of the terms of the two
quadratic equations, and after multiplying the two toge-
ther, determining their values by comparing the resulting
terms with those of the proposed biquadratic equation.

After these reductions, which are only mentioned for
the sake of the geometrical constructions which follow,
by simplifying and depressing the equations as much as
they will admit, Descartes then gives the construction of
solid and other higher problems, or of cubic and higher
equations, by means of parabolas and circles; where he
observes, that the false roots are denoted by the ordinates
to the parabola lying on the contrary side of the axis to
the true roots, as before mentioned by Girard. Finally,
these constructions are illustrated by various problems
concerning the trisecting of an angle, and the finding of
two or four mean proportionals; which concludes this in-
genious work.

From the foregoing analysis may easily be collected the
real inventions and improvements made in algebra by Des-
cartes. His work, as has been observed before, is not al-
gebra itself, but the application of algebra to geometry,
and the algebraical doctrine of curve lines, expressing
and explaining their nature by algebraical equations ;
and, on the contrary, constructing and explaining equa-
tions by means of the curve lines. What respects the
geometrical parts of this tract we shall have occasion to
advert to elsewhere ; and therefore shall here only enu-
merate the circumstances which belong more peculiarly
to the science of algebra, which I shall distinguish into
the two heads, of improvements, and inventions. And

1st. Of his improvements. — That he might fit equations
the better for their application in the construction of pro-
blems, Descartes mentions, as it were by-the-by, many
things concerning the nature and reduction of equations,
without troubling himself about the first inventors of
them, stating them in his own terms and manner, which
is commonly more clear and explicit, and often with im-
provements of his own. And under this head we find that
he chiefly followed Cardan, Vieta, and Harriot, but espe-
cially the last, and explains some of their rules and dis-
coveries more distinctly, and varies but a little in the no-
tation, putting the first letters of the alphabet for the
known, and the latter letters for the unknown quantities;
also r* for aaa, &c; and 30 for =. But Herigone used
the numeral exponents in the same manner some years
before, as well as several older authors. Descartes ex-
plained or improved most parts of the reductions of equa-
tions, in their various transmutations, the number and na-
ture of their roots, true and false, real and what he deno-
minates imaginary, called involved by Girard; and the
depression of equations to lower degrees.

2d. As to his inventions and discoveries in algebra, they
may he com prised in these particulars : namely, the ap-
plication of algebra to the georiietry of curve lines, the
constructing equations of the higher orders, and a rule
for resolving biquadratic equations by means of a cubic
and two quadratics.

Having now traced the science of algebra from its ori-
gin and rude state, down to its modem and more polished
form, in which it has ever since continued, with very

little variation ; having analysed all or most of the prin-
cipal authors, in a chronological order, and deduced the
inventions and improvements made by each of them ;
from this time the authors both become too numerous,
and their improvements too inconsiderable, to merit a
detail in the same minute and circumstantial way: and
besides, these will be better explained in a particular
manner under the word or article to which each of them
severally belongs. It may therefore now suffice to enu-
merate, or announce only in a cursory way, the chief im-
provements and authors on algebra down to the present
time.

After the publication of the Geometry of Descartes-, a
great many other ingenious men followed the same course,
applying themselves to algebra and the new geometry, to
the mutual improvement of them both ; which was done
chiefly by reasoning on the nature and forms of equations,
as generated and composed by Harriot. Before proceed-
ing upon these, however, it is but proper to take notice
here of Fermat, a learned and ingenious mathematician,
who was contemporary and a competitor of Descartes,
for his brightest discoveries, which he was in possession
of before the geometry of Descartes appeared. Namely,
the application of algebra to curve lines, which he ex-
pressed by an algebraical equation, and by them con-
structed equations of the 3d and 4th orders; also a me-
thod of tangents, and a method de maximis et minimis,
which approach very near to the method of fluxions or
increments, which they strikingly resemble, both in the
manner of treating the problems, and in the algebraic no-
tation and process. The particulars of which, see under
the proper heads. Besides these, Fermat was deeply
learned in the Diophantine problems, and the best edition
of Diophantus's Arithmetic, is that which contains the
notes of Fermat on that ingenious work.

But to return to the successors of Descartes. His geo-
metry having been published in Holland, several learned
and ingenious mathematicians, of that country, presently
applied themselves to cultivate and improve it; as Schoo-
ten, Hudde, Van-Heuraet, De Witte, Slusius, Huygens,
&c; besides M. de Beaune, and perhaps some others in
France.

Francis Schooten, professor of mathematics in the uni-
versity of Leyden, was one of the first cultivators of the
new geometry. He translated Descartes's geometry out
of French into Latin, and published it in 10*4y, with his
commentary on it, as also Brief Notes of M. de Beaune;
both of them containing many ingenious and useful things.
And in 10*59 he gave a new edition of the same, in two
volumes, with the addition of several other ingenious
pieces: as, two posthumous tracts of de Beaune, the one
on the nature and constitution, the other on the limits of
equations, showing how to assign the limits between which
are contained the greatest and least roots of equations,
extended and completed by Erasmus Bartholine : two
letters of M. Hudde on the reduction of equations, and
on the maxima and minima of quantities, containing
many ingenious rules; among which are some concerning
the drawing of tangents, and on the equal roots of equa-
tions, which he determines by multiplying tin' terms of
the equation by the terms of any arithmetical progression,
0 being one of the terms, tin equation is commonly de-
pressed one degree lower; also a tract of Van Heuraet
on the rectification of curve lines; the elements of curves
by Dc Witte ; Bertholinc's principles of universal mathc-

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matics, or introduction to Descartes's geometry, which
had before been published by itself in 1651 ; and to the
end of the work is added a posthumous piece of Schoo-
ten's (for he died while the 2d vol. was printing) entitled
Tractatus de concinnandis demonstrationibus geometricis
ex calculo algebraico. Schooten also published, in 1657,
Exercitationes Mathematics;, in which are contained
many curious algebraical and analytical pieces, among
others of a geometrical nature.

An elaborate commentary on Descartes's . geometry
was .also published by F. Rabuel, a Jesuit ; and James
Bernoulli, enriched with notes, an edition of the same,
printed at Basil in 169 — .

The celebrated Huygens also, among his great discove-
ries, very much cultivated the algebraical analysis : and
he is often cited by Schooten, who relates divers inven-
tions of his, while he was his pupil.

Slusius, a canon of Liege, published in 165$, Mesola-
bum, seu duas mediae propor. per circulum et ellips. vel
hyperb. infinitis modis exhrbita:; by which, any solid pro-
blem may be constructed by infinite different ways. And
in 1668, he gave a second edition of the same, with the
addition of the analysis, and a miscellaneous collection of
curious and important problems, relating to spirals, cen-
tres of gravity, maxima and minima, points of inflexion,
and some Diophantine problems ; all showing him deeply
skilled in algebra and geometry.

There have been a great number of other writers and
improvers of algebra, of which it may suffice slightly to
mention the chief part, as in the following catalogue.

In 1619 several pieces of Van Collen, or Ceulen, were
translated out of Dutch into Latin, and published at Ley-
den by W. Sneli ; among which are contained a particular
treatise on surds, and his proportion of the circumference
of a circle to its diameter.

In Irj21 Bachet published, in Greek and Latin, an edi-
tion of Diophantus, with many notes. And another edi-
tion of the same was published in ]670, with additions
by Fermat.

In lrj24 Bachet's Problemes Plaisans et Delectables,
being curious problems in mathematical recreations.

In l6'34 Herigone published, at Paris, the first course
of mathematics, in 5 vols. 8vo; in the 2d of which is con-
tained a good treatise on algebra; in which he uses the
notation by small letters, introduced by the algebra of
Harriot, which was published three years before, though
the rest of it does not resemble that work, and one would
suspect that Herigone had not seen it. The whole of this
piece bears evident marks of originality and ingenuity.
Besides ■+- for plus, he uses *o for minus, and 2|2 for
equality, with several other useful abbreviations and marks
of his own. In the notation of powers, he does not re-
peat the letters like Harriot, but subjoins the numeral
exponents to the letter, as Descartes did two years after-
wards. And Herigone uses the same numeral exponents
for roots, as ^/3 for the cube root.

In 16.35 Cavalcrius published his Indivisibles; which
proved a new aera in analytics, and gave rise to other new
modes of computation in analytics.

About lfi40, et seq. Roberval made several notable im-
provements in analytics, which are published in the early
volumes of the Memoirs of the Academy of Sciences; as,
I, A tract on the composition of motion, and a method
of tangents. 2, De recognitionc cequationum. 3, De gco-

metrica planarum et cubicarum asquationum resolutione.
4, A treatise on indivisibles, &c.

In 1643 De Billy published Nova Geometries Clavis
Algebra. And in 1670 Diophantus Redivivus. He was
an author particularly well skilled in Diophantine pro-
blems.

In ) 644. Renaldine published, in 4to,Opus Algebraicum,
both ancient and modern, with mathematical resolution'"
and composition. And in l665, in folio, the same, greatly
enlarged, or rather a new work, which is very heavy and
tedious. In this work Renaldine uses the parentheses
(a? 4- h?) as a vinculum, instead of the line over, as
u~ -+- I)-.

In 1655 was published Wallis's Arifhmelica Infinite-
rum, being a new method of reasoning on quantities, or a
great improvement on the indivisibles of Cavalcrius, and
which in a great measure led the way to infinite series,
the binomial theorem, and the method of fluxions. Wallis
hear, treats ingeniously of quadratures and many other
problems, and gives the first expression for the quadrature
of the circle by an infinite series. Another series is here
added for the same purpose, by the Lord Brouncker, the
first pres. of the R. S.

In 1659 was published, at Zurich in Switzerland, Alge-
bra Rhonii Germanice ; which was, in 1 668, translated
into English by Mr. Thomas Branckcr, with additions and
alterations by Dr. John Pell.

In 1661 was published in Dutch, a neat piece of Al-
gebra by Mr. Kinckhuysen ; which Sir Isaac Newton,
while he was professor of mathematics at Cambridge,
made use of and improved, and he meant to republish it,
with the addition of his method of fluxions and infinite
series ; but was prevented by the accidental burning of
some of his papers.

In 1665 or 1666 Sir Isaac Newton made several of his
brightest discoveries, though they were not published till
afterwards : such as the binomial theorem ; the method
of fluxions and infinite series ; the quadrature, rectifica-
tion, &c, of curves ; to find the roots of all kinds of equa-
tions, both numeral and literal, in infinite converging se-
ries; the reversion of series, &c. Of each of which a
particular account may be seen in their proper articles.

In 1666 M. Frenicle gave several curious tracts con-
cerning combinations, magic squares, triangular numbers,
&c ; which were printed in the early volumes of the Me-
moirs of the Academy of Sciences.

In 1668 Thomas Brancker published a translation of
Rhonius's Algebra, with many additions by Dr. John Pell,
who used a peculiar method of registering the steps in
any algebraical process, by means of marks and abbre-
viations in a small column drawn down tlie margin, by
which each line, or step, is clearly explained, as was be-
fore done by Harriot in words at length. This book con-
tains a valuable table of prime numbers.

In 1668 Mercator published his Logarithmotechnia, or
method of constructing logarithms; in which he gives the
quadrature of the hyperbola, by means of an infinite series
of algebraical terms, found by dividing a simple algebraic
quantity by a compound one, and for the first time that
this operation was given to the public, though Newton had
before that expanded all sorts of compound algebraical
quantities into infinite series.

In the same year was published James Gregory's'
Exercitationes Geometricte, containing, among other
02

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things, a demonstration of Mercator's quadrature of the
hyperbola, by the same series.

And in the same year was published, in the Philoso-
phical Transactions, Lord Brouncker's quadrature of the
hyperbola, by another infinite series of simple rational
terms, which he had been in possession of since the year
1657, when it was announced to the public by Dr. Wallis.
Lord Brouncker's series for the quadrature of the circle
had been published by Wallis in his Arithmetic of Infi-
nites, as before mentioned.

In 1669 Dr. Isaac Barrow published his Optical and
Geometrical Lectures, abounding with profound re-
searches on the dimensions and properties of curve lines;
but particularly to be noticed here for his method of tan-
gents, by a mode of calculation similar to that of fluxions,
or increments, from which these differ but little, except
in the notation. Of these lectures, the 13th merits the
most special notice, being entirely employed on equa-
tions, and delivered in a very curious way. He there
treats of the nature and number of their roots, audi the
limits of their magnitudes, from the description of lines
accommodated to each, viz, treating the subject as a
branch of the doctrine of maxima and minima, which in
the opinion of some persons is the right way of consider-
ing them, and as far preferable to the so much boasted in-
vention of the generation of equations from each other,
explained by Harriot and Descartes.

In 1673 was published, in 2 vols, folio, Elements of
Algebra, by John Kersey ; a very ample and complete
work, in which Diophantus's problems are fully ex-
plained.

In 1675 were published Nouveaux Elcmens des Mathe-
matiques, par J. Prestet, pretre : a prolix and tedious
work, which he presumptuously dedicated to God Al-
mighty.

About 1677 Leibnitz discovered his Methodus Diffc-
rentialis, or else made a variation in Newton's Fluxions,
or an extension of Barrow's method, for it is not certain
which. He gave the first instance of it in the Leipsic
Acts for the year 1684. He also improved infinite series,
and gave a simple one for the quadrature of the circle, in
the same acts for 1682. But this series had before been
found by Mr. James Gregory, viz, in l6'71.

In 1 682 Ismael Bulliald published, in folio, his Opus
Novum ad Arithmeticam Infinitorum, being a large am-
plification of Wallis's Arithmetic of Infinites.

In 1683 Tschirnausen gave a memoir, in the Leipsic
Acts, concerning the extraction of the roots of all equa-
tions in a general way ; in which he promised too much,
as the method did not succeed.

In 1684 came out, in English and Latin, 4to, Thomas
Baker's Geometrical Key, or Gate of Equations Unlocked ;
being an improvement of Descartes's construction of all
equations under the 5th degree, by means of a circle and
only one and the same parabola for all equations, using
any diameter instead of the axis of the parabola.

In J 685 was published, in folio, Wallis's Treatise of
Algebra, both Historical and Practical, with the addition
of several other pieces ; exhibiting the origin, progress,
anil advancement of that science, from time to time. It
cannot be denied that, in this work, Wallis has shown too
much partiality to the Algebra of Harriot. Yet, on the
other hand, it is as true, that M. de Gua, in his account
of it, in the Memoirs of the Academy of Sciences for 1741,

has run at least as far into the like extreme on the con-
trary side, with respect to the discoveries of Vieta ; and
both these I believe from the same cause, namely, the
want of examining the works of all former writers otf
algebra, and specifying their several discoveries; as has
been done in the course of this article.

In 1687 Dr. Halley gave, in the Philos. Trans, the con-
struction of cubic and biquadratic equations, by a para-
bola and circle; with improvements on what had been
done by Descartes, Baker, &c. Also, in the same Trans-
actions, a memoir on the number of the roots of equations,
with their limits and signs.

In 169O was published, in 4to, by M. Rolle, Traite
d'Algebre; in 1699, Une Methode pour la Resolution d(^
Problemes indetermines; and in 1704, Memoires sur l'in-
verse des tangents ; and other pieces.

In 1690 Joseph Raphson published Analysis iEqua-
tionum Universalis ; being a general method of -approxi-
mating to the roots of equations in numbers. And in
1715 he published the History of Fluxions, both in En-
glish and Latin.

In I69O was also published, in 4 vols 4to, Dechale's
Cursus seu mundus mathematicus; in which is a piece of
algebra, of a very old-fashioned sort, considering the time
when it was written.

About 1692, and at different times afterwards, De
Lagny published many pieces on the resolution of equa-
tions in numbers, with many theorems and rules for that
purpose.

In 1693 was published, in a neat little volume, Synopsis
Algebraica, opus posthumum Johannis Alexandri.

In 1694 Dr. Halley gave, in the Philos. Trans, an in-
genious tract on the numeral extraction of all roots, with-
out any previous reduction. And this tract is also added
to some editions of Newton's Universal Arithmetic.

In 1695 Mr. John Ward, of Chester, published, in 8vo,
A Compendium of Algebra, containing plain, easy, and
concise rules, with examples in an easy and clear way.
And in 1706 he published the first edition of his Young
Mathematician's Guide, or a plain and easy introduction
to the mathematics: a book which is still in great request,
especially with beginners, and which has been ever since
the ordinary introduction of the greatest part of the ma-
thematicians of this country.

In 1696 the Marquis de l'Hopital published his Analyse
des infiniment petits ; and gave several papers to the
Leipsic Acts and the Memoirs of the Academy of Sciences.
He left behind him also an ingenious treatise, which was
published in 1707, entitled Traite analytique des Sections
Coniques, et de la construction des lieux geometriques.

In 1697, .and several other years, M. Ab. Demoivre
gave various papers in the Philos. Trans, containing im-
provements in algebra; viz, in 1697, a method of raising
an infinite multinomial to any power, or extracting any
root of the same. In 16*98, The extraction of the root of
an infinite equation. In 1707, Analytical solution of
certain equations of the 3d, 5th, 7th, &c, degree. In
1722, Of algebraic fractions and recurring series. In
1738, The reduction of radicals into more simple forms.
Also in 1730, he published Miscellanea analytica de se-
riebus et quadraturis, containing great improvements in
series, &c.

In 16'9S Mr. Richard Sault published, in 4to, A Now
Treatise of Algebra, apply'd to numeral questions and

A L G

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A LG

geometry. With a converging series for all manner of
Adfectcd Equations. The scries here alluded to, is Mr.
Raphson's method of approximation, which had been lately
published.

In 1699 Hyac. Christopher published at Naples, in 4to,
De constructione Eequationum.

In 1702 was published Ozanam's Algebra; which is
chiefly remarkable for the Diophantine analysis. He
had published his mathematical dictionary in 1691, and
in l6'93 his course of mathematics, in 5 vols. 8vo, con-
taining also a piece on algebra.

In 1704, Dr. John Harris published his Lexicon Tech-
nicum, the first dictionary of arts and sciences : a very
plain and useful book, especially in the mathematical
articles. And in 1705, a neat little piece on algebra and
fluxions.

In 1705 M. Guisnee published, in 4to, his Application
de l'algebre a la geometrie : a useful book.

In 1706 Mr. William Jones published his Synopsis
Palmariorum Matheseos, or a new introduction to the
mathematics : a very useful compendium in the mathe-
matical sciences. And in 1711 he published, in 4to, a
collection of sir Isaac Newton's papers, entitled Analysis
per quantitatum series, .fluxiones, ac differentias : cum
enumeratione linearum tertii ordinis. .

In 1707 was published, by Mr. Whiston, the first edi-
tion of sir Isaac Newton's Arithmetica Universalis: sive
de compositione et resolutione arithmetica liber : and
many editions have been published since. This work, it
seems, was the text-book used by our great author in his
lectures, while he was professor of mathematics in the
university of Cambridge. And though it was never in-
tended tor publication, it contains many and great im-
provements in analytics ; particularly in the nature and
transmutation of equations; the limits of the roots of
equations; the number of impossible roots; the invention
of divisors, both surd and rational ; the resolution of
problems, arithmetical and geometrical ; the linear con-
struction of equations ; approximating to the roots of all
equations, &c. To the later editions of the book is com-
monly subjoined Dr. Halley's method of finding the roots
of equations. As the principal parts of this work are not
adapted to the circumstance of beginners, there have been
published commentaries upon it by several persons, as
s'Gravesande, Castillion, Wilder, &c.

In 1708 M.Reyneau published his Analyse Dernontree,
in 2 vols, 4to. And in 1714, La Scieenc du Calcul, &c.
In 1709 was published an English translation of Alex-
ander's Algebra. With an ingenious appendix by Hum-
phry Ditton.

In 1715 Dr. Brooke Taylor published his Methodus
Incrementorum : an ingenious and learned work. And
in the Philos. Trans, for 1718, An improvement of the
method of approximating to the roots of equations in
numbers.

In 1717 M. Nicole gave, in the memoirs of the aca-
demy of sciences, a tract on the calculation of finite dif-
ferences. And in several following years, he gave various
other tracts on the same subject, and on the resolution
of equations of the 3d degree, and particularly on the ir-
reducible case in cubic equations : those tracts are ground-
ed on a production of M. Leibnitz in the year l6'77-

Also in 1717 was published a Treatise on Algebra by
Philip Ronayne.

Also in 1717 Mr. James Sterling published Lineae

tertii Ordinis; an ingenious work, containing good im-
provements in analytics. And in 1730, Methodus Diffe-
rentialis : sive tractatus de summations et interpolatione
serierum infinitarum : with great improvements on infi-
nite series.

In 1726 and 1729 Maclaurin gave, in the Philos. Trans,
tracts on the imaginary roots of equations. And after-
wards was published, from his posthumous papers, his
Treatise on Algebra, with its application to curve lines.

In 1727 came out s'Gravesande's Algebra, with a speci-
men of a commentary on Newton's universal arithmetic.

In J 728 Mr. Campbell gave, in the Philos. Trans, an
ingenious paper on the number of impossible' roots of
equations.

In 1732 was published Wolfius's Algebra, in his course-
of mathematics, in 5 vols, 4to.

In 1735 Mr. John Kirby published his arithmetic and
algebra. And in 1748 his doctrine of ultimators.

In 1740 were published Mr. Thomas Simpson's Essays;
in 1743 his Dissertations, and in 1757 his Tracts ; in all
which are contained several improvements in series and
other parts of algebra. As also in his Algebra, first print-
ed in 1745, and in his Select Exercises, in 1752.

Also in 1740 was published professor Saunderson's Ele-
ments of Algebra, in 2 vols, 4to; being a very useful
work for beginners.

In 1741 M. de la Caille published Lecons de mathe-
matiqucs; ou elcmens d'algebre et de geometrie.

Also in 1741, in the Memoirs of the academy of sci-
ences, were given two articles by M. deGua, on the num-
ber of positive, negative, and imaginary roots of equations.
With an historical account of the improvements in al-
gebra; in which he severely censures Wallis for his par-
tiality: a circumstance in which he himself is not less
faulty.

In 1746 M. Clairaut published his Elemens d'algebre,
in which are contained several improvements, especially
on the irreducible case in cubic equations ; chiefly after
the manner of Leibnitz and Nicole. He has also several
good papers on different parts of analytics, in the Memoirs
of the academy of sciences.

In 1747 M. Fontaine gave, in the Memoirs of the aca-
demy of sciences, a paper on the resolution of equations.
Besides some analytical papers in the memoirs of other
years.

In I76l M. Castillion published, in 2 vols, 4to, Newton's
universal arithmetic, with a large commentary.

In 1763 Mr. Emerson published his Increments. In
1764 his Algebra, &c.

In 1764 Mr. Landen published his Residual Analysis.
In 176'5 his Mathematical Lucubrations. And in 1780
his Mathematical Memoirs. All containing good improve-
ments in infinite series, &c.

In 1770 was published, in the German language, Ele-
ments of Algebra by M.Euler, which havebcensince trans-
lated into French and English. And in 1774 a French
translation of the same. The memoirs of Berlin and Pe-
tersburg academies also abound with various improve-
ments in series and other branches of analysis, by this ce-
lebrated mathematician.

In 1775 was published at Bologna, in 2 vols, 4to, Com-
pendio d'Analisi di Girolamo Saladin.

In 1797 was published, by Signor Cossali, in the Italian
language, in 2 vols, 4to, a very elaborate history of tho
origin of the first introduction of algebra into Italy, and

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A L G

of its progress and more early improvements among his
countrymen. In this work, the author clearly shows, what
however was well known long before, that the science was
imported into Italy from the Arabians ; but he seems
erroneously to ascribe the very invention of algebra to
these latter people, from his ignorance of the discove-
ries lately made of the writings on this science among the
Hindus.

LE GENDRE.

M. Le Gendre published in 1798, at Paris, in 1 vol 4to,
his Essai sur la Theorie des Nombres, in which all the
most interesting properties of numbers are recorded, and
faithfully attributed to their respective authors, which is
I believe the first book of any importance that was dedi-
cated to this particular subject. The work is divided into
four parts, each containing several theorems besides an
Introduction at considerable length ; and at the end of
the book are added several tables relating to the particu-
lar forms of divisors which belong exclusively to certain
algebraical formula?, the use and application of which are
explained in the body of the work.

The Introduction treats of the particular forms under
which prime numbers are contained, and the number of
the latter is proved to be infinite. It is also demonstrated
that no algebraical formula can express prime numbers
only. Lastly, the number of prime numbers b, contained
between unity and any given number a, is shown to be

very nearly expressed by b = — : .

Parti Contains an explanation of Continued Fractions,
and their application to various cases in the indetermi-
nate analysis ; among others, a general solution is given in
integer numbers to every possible indeterminate equation
of the second degree, and a method ofjudgingof the pos-
sibility, or impossibility of such equations; this solution,
however, is due to La Grange.

The second part commences with the demonstration of
several general properties of numbers, most of which are
mentioned under that article in this Dictionary. The rest
of this part is employed in discovering the forms of divi-
sors which belong exclusively to certain algebraical for-
mulae, and the application of them for finding prime num-
bers, with an explanation of the law of reciprocity be-
tween the latter.

Part 3. Here numbers are considered as decomposed
into three squares, and many properties relating to them
are demonstrated. Every number is shown td be the sum
of one, two, or three triangular numbers, which is the first
demonstration that was given to this part of the celebrated
theorem of Fermat.

The fourth part contains the demonstration of some
other properties, in which the possibility and impossibi-
lity of various formulae are demonstrated with regard to
their being particular powers, thus x3± y3 = z3 is impos-
sible, as also x1 ±y = z*, and generally xn ±yn = 2n is
impossible, when n is greater than 2, but this last has
never yet been demonstrated.

In the course of this work some new symbols are occa-
sionally introduced, which however our limits will not
permit us to explain.

Several other works have been published by the same
author, but as they do not properly fall under this ar-
ticle, they are omitted.

GAUSS.

Soon after the above Essays appeared, M. Gauss pub-

lished a similar work in Latin, afterwards translated into
French in 1 vol, 4to, under the title of Recherches Arith-
metique. The subject of which is much the same as the
former, these two able mathematicians having undertaken
the same task at nearly the same time, without knowing
each other's intentions ; but here the subject is carried to
a greater length than in the former. M. Gauss has by
an extension of his analysis been able to express the side
of a polygon of 17 sides inscribed in a circle by means of
a quadratic equation, which was before thought to be
impossible ; and the same method may be applied to other
polygons when the number of their sides is a prime num-
ber of a particular form. There are likewise some hints
given for forming a table of logarithms, or numbers which
may be employed in algebra to great advantage, as com-
mon logarithms are in arithmetical computations. In this
work many new symbols are introduced, such as 5E for
congruity, that is a= b (to modulus c), when a ± b is
divisible by c, and many others, which cannot be properly
understood without an attentive perusal of the work
itself.

We are informed by M. Gauss that he had another
work in hand, which will carry this interesting subject
much farther. .

ARBOGAST.

M. Arbogast, professor of mathematics at Strasburgh,
published in 1800, in 1 vol, 4to, a work entitled Du Cal-
cul des Derivations, which is appropriated entirely to the
doctrine of series; where their transformation, summation,
interpolation, and reversion, as also various other particu-
lars relating to this intricate branch of analysis, are treated
of in a manner, as new and ingenious, as it is profound
and learned. By the method here introduced many dif-
ficulties are surmounted relating to the higher branches
of astronomy, which the Calcul Differential was inade-
quate to encounter with success. It seems, however, that
some hints of this method was first given by Dr. Waring
in his Meditationes Analytical, and which, perhaps, fur-
nished Arbogast with the germ of his idea. In this
work many new characters are introduced, not to re-
present quantities, but the operations performed upon
them. Thus, let P (a -+- x) be any function of the bino-
mial a ■+• x, we can expand this function in a series pro-
ceeding according to the powers of x, in this form,

F (a + x) = a -+■ bx h X2, h <- x3 &c. Now, we

11 ' 1.2 ■ 1.2.3

know that a = ra, that b is derived from a, or from Fa;
c from b, d from c &c, by the same law ; therefore, if we
conceive the manner in which we derive b from a, in the
same manner we may derive c from b, d from c, &c.
Thus representing by D, the operation that must be per-
formed on Fa in order to obtain b, we shall have b — l> la,
c = d'm, d = D3Fa &c ; so that the above series may

, , , „ ■ lira n'vii .

be expressed thus, r((i + .v) = Fa + — — x -+- -— - r" ■+•

. X1 &C

14,8.3

It is however impossible to give any thing like an ade-
quate idea of the method, or notation, employed in this
profound work, in the narrow limits to which we arc un-
der the necessity of confining our observations, which have
already exceeded the bounds we had at first prescribed
to this article.

Besides the foregoing, there have been many other
authors who have given treatises on algebra, or who have

A L I

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ALL

made improvements on series and other parts of algebra ; part is such, as being taken or repeated a certain number of

as Schonerus, Coignet, Salignat, Laloubere, Hemischius, times, exactly makes up, or is equal to the whole. So

Degraave, Mescher, Henischius, Roberval, the Bernoul- 1 is an aliquot part of 6", or of any other whole number ;

lis, Malbranche, Agnesi, Wells, Dodson, Manfredi, Beg- 2 is also an aliquot part of 6, being contained just 3 times

nault, Bezout, Bossut, Rowning, Maseres, Waring, Lorg- in 6; and 3 is also an aliquot part of 6, bejna contained

na, de la Grange, de la Place, Bertrand, Kuhnius, Hales, just 2 times ; so that all the aliquot parts of 6 are 1, 2 3.
and many other authors. Dr. Waring and the Rev. Mr. All thealiquot parts of any number may be thus found :

Vince, of Cambridge, have both given many improve- Divide the given number by its least divisor; then divide

ments and discoveries in series, and in other branches of the quotient also by its least divisor; and so on, always

analysis. Those of Mr. Vince are chiefly contained in dividing the last quotient by its least divisor, till the quo-

the latter volumes of the Philos. Trans.; where also are tient 1 is obtained ; and all the divisors, thus taken, are

several of Dr. Waring's : but the bulk of this gentleman's the prime aliquot parts of the given number. Then mul-

improvcments are contained in his separate publications, tiply continually together these prime divisors viz every

particularly the Meditationes Algebraicas published in two of them, every three of them, -every four of them, &c;

1770; the Proprietates Algebraicarum Curvarum, in 1772; and the products will be the other or compound aliquot

and the Meditationes Analytical in 1776. parts of the given number. So, if the aliquot parts of 60

For a more enlarged history of algebra, see my Tracts, be required ; first divide it by 2, and the quotient is 30 :

vol.2, tract33. then 30 divided by 2 also gives 15, and 15 divided by 3

Algebraical, something relating to algebra. Thus gives 5, and 5 divided by 5 "gives 1 : so that all the prime

we say algebraical solutions, curves, characters or sym- divisors or aliquot parts are 1,2, 2, 3,5. Then the com-

bols, &c. pound ones, by multiplying every two, are 4, 6, 10, 15;

Algebraical Curve, is a curve in which the general and every three 10, 20, 30. So that all the aliquot parts

relation between the abscisses and ordinates can be ex- of the given number 60, are 1, 2 3, 4 5 6, 10 12 15

pressed by a common algebraical equation. These are 20, 30. — In like manner it will be found that all the ali-

also called [geometrical lines or curves, in contradistinction quot parts of 360, are 1, 2, 3, 4, 5, 6, 8 0 10 12 15

to mechanical or transcendental ones. 18, 20, 24, 30, 06, 40, 45, 60, 72, 180.

ALGEBRAIST, a person skilled in algebra. ALLEN (Thomas), a celebrated mathematician of the

ALGENEB, or Algenib", a fixed star of the second l(uh century. He was born at Uttoxeter in Staffordshire,

magnitude, on the right side of Perseus. in 1542 ; was admitted a scholar of Trinity college, Ox-

ALGOL, or Medusa's Head, a fixed star of the third ford, in 156l ; where he took his deoree of master' of arts

magnitude, in the constellation Perseus.

ALGORAB, a fixed star of the third magnitude, in the
right wing of the constellation Corvus.

ALGORITHM, is similar to logistics, signifying the art
of computing in any particular way, or about some par-

in 1567. In 1570 he quitted his college and fellowship,
and retired to Glocester hall, where he studied very closely,
and became celebrated for his knowledge in antiquity,
philosophy, and mathematics. He received an invitation
from Henry earl of Northumberland, the great friend and

ticular subject ; or the common rules of computing in patron of mathematicians, and he spent some time at the
any art. As the algorithm of numbers, of algebra, of inte- earl's house ; where he became acquainted with thoseable
gers, of fractions, of surds, &c ; meaning the common mathematicians Thomas Harriot, John Dee, Walter War-
rules for performing the operations of arithmetic, or al- ner, and Nathaniel Torporlcy. Robert earl of Leicester,
gebra, or fractions, &c. too, had a great esteem for Allen, and would have con-

ALHAZEN, Allacen, or Abdilazum, was a ferred a bishopric upon him; but his love for solitude and

learned Arabian, who lived in Spain about the year 1 100, retirement made him decline the offer. His great skill in

according to his editor Risner, and Weidler. He wrote the mathematics gave occasion to the ignorant and vulgar

upon astrology; and his work on optics was printed in to look upon him as a magician or conjurer. Allen was

Latin, at Basil, in 1572, under the title of Optics Thesau- very curious and indefatigable in collecting scattered ma-

rus, by Risner. Alhazen was the first who showed the nuscripts relating to history, antiquity, astronomy, philo-

importance of refractions in astronomy, so little known to sophy, and mathematics: which collections have been

the ancients. He is also the first author who has treated quoted by several learned authors, and mentioned as in the

on the twilight, upon which he wrote a work, in which he Bibliotheca Alleniana. He published in Latin the second

likewise speaks of the height of the clouds. and third books of Ptolemy, Concerning the judgement of

ALIDADE, an Arabic name for the label, index, or the Stars, or, as it is usually called, of the quadripartite

ruler, which is moveable about the centre of an astrolabe, construction, with an exposition. He wrote also notes

quadrant, &c, and carrying the sights or telescope, and on many of Lilly's books, and some on John Bale's work,

by which are shown the degrees cut off the limb or arch De scriptoribus Maj. Britannia?. He died at Glocester "

of the instrument. hall in 1632, at 90 years of age.

ALIQUANT part, is that part which will not exactly Mr. Burton, the author of his funeral oration, calls him

measure or divide the whole, without leaving some re- the very soul and sun of all the mathematicians of his age.

mainder. Or the aliquant part is such, as being taken or And Selden mentions him as a person of the most exten-

repeated any number of times, docs not make up the sive learning and consummate judgment, the brightest or-

whole exactly, but is either greater or less than it. Thus nament of the university of Oxford. Camdonlilso says

4 is an aliquant part of 10 ; for 4 twice taken makes 8, he was skilled in most of the best arts and sciences,

which is less than 10; and three times taken makes 12, A. Wood has also transcribed paTt of his character from

which is greater than 10. a manuscript in the library of Trinity college, in these

ALIQUOT part, is such a part of a number as will words: "He studied polite literature with great appli-

exactly divide it without a remainder. Or the aliquot cation; he was strictly tenacious of academic discipline;

AIL

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A L M

always highly esteemed both by foreigners and those of the
university, and by all of the highest stations in the church
of England, and the university of Oxford. He was a sa-
gacious observer, an agreeable companion, &c." See
Wood, vol. i, p. 492.

ALLIGATION, one of the rules in arithmetic, by which
are resolved questions which relate to the compounding
or mixing together of divers simples or ingredients, being
so called from alligare, to tie or connect together, probably
from certain vincula, or crooked ligatures, commonly used
to connect or bind the numbers together.

It is probable that this rule came to us from the Moorish
or Arabic writers, as we find it, with all the other rules
of arithmetic, in Lucas de Burgo, and other early authors
in Europe.

Alligation is of two kinds, medial and alternate.

Alligation medial is the method of finding the rate or
quality of the composition, from having given the rates and
quantities of the simples or ingredients.

The rule of operation is this : multiply each quantity
by its rate, and add all the products together ; then divide
the sura of the products by the sum of the quantities, or
whole compound, and the quotient will be the rate sought.

For example; Suppose it were required to mix together
6 gallons of wine, at 5s. a gallon ; 8 gallons, at 6s. the
gallon; and 4 gallons, at 8s. the gallon; and to find the
worth or value, per gallon, of the whole mixture.

GalU s. products.

Here -6x5= 30
8x6= 48
_4 x 8 = 32
whole comp. 18 110 sum of prod.

Then 18 ) 110 ( 6^. or 6±s. is the rate sought.
108

Alligation alternate is the method of finding the
quantities of ingredients or simples, necessary to form a
compound of a given rate.

The rule of operation is this : 1st, Place the given rates
of the simples in a column, under each other. 2d, Con-
nect or link with a continued line, each rate which is less
than the proposed compound rate, with one or any num-
ber of those that are greater; and every greater rate with
one or any number of the less. 3d, Write the difference
between the given compound rate and that of each simple
opposite the rates with which they are linked. 4th, Then
if only one difference stand opposite any rate, it will be
the quantity belonging to that rate ; but when there are
several, their sum will be the quantity.

For example, Suppose it were required to mix together
gold of various degrees of fineness, viz, of 19, of 21, and
of 23 carats fine, so that the mixture shall be of 20 carats
fine. Here,

Rates Diffs.

f2l)---l

Comp. rate 20 J 19' 1+3

I23) ---1

That is, there must be an equal quantity of 21 and 23
carats fine, and 4 times as much of 19 carats fine.

Various limitations, both of the compound and the
ingredients, may be conceived ; and in such cases, the
differences are to be altered proportionally.

Questions of this sort are however commonly best and
easiest resolved by common algebra, of which they form

Sum of Diffs.
1 of 21 carats fine,
4 of 19 carats fine,
1 of 23 carats fine.

a species of indeterminate problems, as they admit of
many, or an indefinite number of answers.

There is recorded a remarkable instance of a discovery
made by Archimedes, both by alligation and specific gra-
vity at the same time, namely, concerning the crown of
Hiero, king of Syracuse. The king had ordered a crown
to be made of pure gold; but when brought to him, a
suspicion arose that it was mixed with alloy of either
silver or copper, and the king recommended it to Archi-
medes to discover the cheat,' without defacing the crown.
Archimedes, after long thinking on the matter, without
lighting on the means of doing it, being one day in the
bath, and observing how his body raised the water higher,
conceived the idea that different metals of the same weight
would occupy different spaces, and so raise or expel dif-
ferent quantities of water. Upon which he procured two
other masses, each of the same weight with the crown,
the one of pure gold, and the other of alloy; then im-
mersing them all three, separately, in water, and observing
the space each occupied, by the quantity it raised the
water, he thence computed the quantities of gold and alloy
contained in the crown.

ALLIOTH, a star in the tail of the great bear. The
word in Arabic denotes a horse ; and they gave this name
to each of the three stars, in the tail of the great bear, as
they are placed like three horses, thus arranged for the
purpose of drawing the waggon commonly called Charles's
wain, represented by the four stars on the body of the
same constellation.

ALMACANTAll, or Almicantar, from the Arabic
almocantharat, are circles parallel to the horizon, and
conceived to pass through every degree of the meridian ;
serving to show the height of the sun, moon, or stars, &c ;
and are the same as the parallels of altitude.

Almacantae's Staff, was an instrument formerly used
at sea for observing the sun's amplitude at rising or set-
ting, and thence to determine the variation of the com-
pass, &c.

ALMAGEST, the name of a celebrated book composed
by Ptolemy; being a collection of a great number of the
observations and problems of the ancients, relating to
geometry and astronomy, but especially the latter. Being
the first work of this kind which has come down to us,
and containing a catalogue of the fixed stars, with their
places, besides numerous records and observations of
eclipses, the motions of the planets, &c, this work will
ever be held dear and valuable to the cultivators of astro-
nomy.

In the original Greek it is called <ruvrai;i; ^syi<rTrh the
great composition or collection. And to the word iJ.syi<rrrn
megiste, the Arabians joined the particle al, and thence
called it Almughesli, or, as we call it, from them, the Al-
magest.

Ptolemy was born about the year of Christ 6f), and
died in 147: he wrote this work, consisting of 13 books,
at Alexandria in Egypt, where the Arabians found it on
the capture of that kingdom. It was by them translated
out of Greek, into Arabic, by order of the caliph Alma-
mon, about the year S27 ; and first into Latin about 1230,
by favour of the emperor Frederic II. The Greek text
however was not known in Europe till about the begin-
ning of the 15th century, when it was brought from Con-
stantinople, then taken by the Turks, by George, a monk
of Trabezond, who translated it into Latin, which trans-
lation has several times been published.

A L M

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A L P

Riccioli, an Italian Jesuit, also published, in 1651, a
body of Astronomy, which, in imitation of Ptolemy, he
called Almagestum Novum, the New Almagest; being a
large collection of ancient and modern observations and
discoveries, in the science of astronomy.

ALMAMON, caliph of Bagdat, a philosopher and
astronomer, in the beginning of the 9th century, having
ascended the throne in the year 814. He was son of
Harun Al-Rashid, and grandson of Almansor. ' His name
is otherwise written Mamon, Almaon, Almamun, Alamoun,
or Al-Maimon. Having been educated with great care,
and possessing a love for the liberal sciences, he applied
himself to cultivate and encourage them in his own coun-
try. For this purpose he requested the Greek emperors
to supply him with such books on philosophy as they had
among them; and he collected skilful interpreters to trans-
late them into the Arabic language. He also encouraged
his subjects to study them; frequenting the meetings of
the learned, assisting at their exercises and deliberations.
He caused Ptolemy's Almagest to be translated in 827,
by Isaac Ben-honain, and Thabet Ben-korah, according to
Herbelot, but according to others by Sergius, and Alhazen,
the son of Joseph. In his reign, and doubtless by his en-
couragement, an astronomer of Bagdat, named Habash,
composed three sets of astronomical tables.

Almamon himself made manyastronomical observations,
and determined the obliquity of the ecliptic to be then
23° 35' (or 23° 33' in some manuscripts), but Vossius
says 23° 3 1' or 23° 34'. He also caused skilful observers
to procure proper instruments to be made, and to exercise
themselves in astronomical observations; which they did
accordingly at Shemasi in the province of Bagdat, and
upon Mount Casius near Damas.

Under the auspices of Mamon also a degree of the me-
ridian was measured on the plains of Sinjar or Sindgiar
(or according to some Fingar), on the borders of the Red
Sea ; by which the degree was found to contain 56-f
miles, of 4000 coudees each, the coudce being a foot and
a half: but it is not known what foot is here meant, whe-
ther the Roman, the Alexandrian, or some other. Riccioli
makes this measure of the degree amount to 81 ancient
Roman miles, which value answers to 62046 French toises;
a quantity more than the true value of the degree by al-
most one-third.

Finally, Mamon revived the sciences in the East to such
a degree, that many learned men were found, not only
in his own time, but after him, in a country where the
study of the sciences had been long forgotten. This learn-
ed king died near Tarsus in Cilicia, by having eaten too
freely of some dates, on his return from a military expe-
dition, in the year 833.

ALMANAC, a calendar or table, in which are set
down and marked the days and feasts of the year, the
common ecclesiastical notes, the course and phases of the
moon, &c, for each month : and answers to the fasti of
the ancient Romans.

The etymology of the word is much controverted
anjong grammarians. — Some derive it from the Arabic,
viz, from the particle al, and manah, to count. While
Scallger, and others, derive it from the same al, and the
Greek fMvaaoc, the course of tin- months. But Golius
controverts these opinions, ami ascribes the word to an-
other origin, though he still makes it of Arabic extract,
which it more evidently is. He says that, in the Fast, it is
the custom for the people, at the beginning of the year, to

Vol. I.

make presents to their princes; and that, among the pest,
the astrologers present them with their almanacs, or ephe»
merides, for the year ensuing ; whence these came to be
called almanha, that is, new-year's gifts. But this deri-
vation seems rather strained and improbable; for, by tin-
same rule, the gifts or productions of other artists, or
classes of men, might also be palled almanacs. There
are otlier guesses at the. etymology ; and Verstegan yvritcs
the word almonac, and makes it of German original. Our
ancestors, he observes, used to carve the courses of the
moon, for the whole year, upon a square piece of wood,
which they called al-monaght, which is as much as to say,
in old English or Saxon, all-moon-heed.

An instrument of this kind, it seems, of a very ancient
date, is to be seen in St. John's college at Cambridge ;
and that in the midland counties there are still several
remains of them. An engraved figure of one of them,
with a particular description, may be seen in Brady's
Clavis Calendaria, vol. 1, 1812.

Almanacs are of various kinds and composition, some
books, others sheets, &c, some annual, others perpetual.
The essential part is the calendar of months, weeks, and
days; the motions, changes, and phases of the moon;
with the rising and setting of the sun and moon. To these
are commonly added various matters, astronomical, astro-
logical, chronological, meteorological, and even political,
rural, medical, &c; as also eclipses, solar ingresses,'as-
pects,and configurations of the heavenly bodies, lunations,
heliocentric and geocentric motions of the planets, prog-
nostications of the weather, predictions of other events,
the tides, twilight, equation of time, &c.

Till about the 4th century, almanacs bore the marks of
heathenism only ; from thence to the 7th century, they
were a mixture of heathenism and Christianity ; and ever
since they have been altogether Christian : but at all times
astrological and other predictions have been considered
as an essential part, and are so to this day with several of
them, notwithstanding that most people affect to disbelieve
in such predictions.

Nautical Almanac, and Astronomical Ephaneris, is a
kind of national almanac, chiefly for nautical purpose?,
which was begun in the year 176*7 under the direction of
the Board of Longitude, on the recommendation of the
late worthy Astronomer Royal, Dr. Maskclyne, who had
the immediate conducting of it. It is still published by
anticipation for several years before-hand, for the conveni-
ence of ships going out upon long voyages, for which pur-
pose it is highly useful, and was found eminently so in the
course of the late voyages round the world for making
discoveries. Besides most things essential to general use,
that are to be found in other almanacs, it contains many
new and important particulars; more especially, the di-
stances of the moon from the sun and fixed stars, which
are computed for the meridian of the Royal Observatory
of Greenwich, and set down to every three hours of time,
expressly designed for computing the longitude at sea, by
comparing these with the like distances observed there.

For an explanation of the Hindu almanacs, see Philos.
Trans. Abridg. vol. 17, p. 2.50, &c.

ALl'HONSINE Tables, are astronomical tables com-
piled by order of Alphonsus, king of Castile, in the com-
piling of which, it is thought that prince himself assisted.
See Astronomical Tables.

ALPHONSUS the 10th, king of Leon and Castile, who
has been surnamed The Wise, on account of his attachment

ALP

C »06 ]

ALT

to literature, and is now more celebrated for having been
an astronomer than a king. He was bom in 1203 ; suc-
ceeded his father Ferdinand the 3d, in 1252 ; and died id
1284, at the age of 81.

The affairs of the reign of Alphonsus were very extra-
ordinary, and unfortunate for him. But we shall here
onlyconsider him in that part of his character, on account
of which he has a place in this work, namely, as an astro-
nomer and man of letters. He understood astronomy,
philosophy, and history ; and composed books upon the
motions of the heavens, and on the history of Spain,
which are highly commended. " What can be more sur-
prising," says Mariana, " than that a prince, educated in a
camp, and handling arms from his childhood, should
have such a knowledge of the stars, of philosophy, and the
transactions of the -world, as men of leisure can scarcely
acquire in their retirements? There are extant some
books of Alphonsus on the motions of the stars, and the
history of Spain, written with great skill and incredible
care." In his astronomical pursuits he discovered that
the tables of Ptolemy were full of errors; and thence he
conceived, the first of any, the resolution of correcting
them. For this purpose, about the year 1240, and during
the life of his father, he assembled at Toledo the most
skilful astronomers of his time, Christians, Moors, or Jews,
when a plan was formed for constructing new tables.
This task was accomplished about 1252, the first year of
his reign; the tables being drawn up chiefly by the skill
and pains of Rabbi Isaac Hasan, a learned Jew, and the
work called the Alphonsine Tables, in honour of the prince,
who was at vast expenses concerning them. He fixed the
epoch of the tables to the 30th of May 1252, being the
day of his accession to the throne. They were printed
for the first time in 1483, at Venice, by Radtolt, who
excelled in printing at that time; which edition is ex-
tremely rare: there are others of 1492, 1521, 1545, &c.
(Weidler, p. 280.)

We must not omit a memorable saying of Alphonsus,
which has been recorded for its boldness and pretended
impiety ; namely, " that if he had been of God's privy
council when he made the world, he could have advised
him better." Mariana however says only in general, that
Alphonsus was so bold as to blame the works of Provi-
dence, and the construction of our bodies; and he says
that this story concerning him rested only on a vulgar
tradition. The Jesuit's words are curious : " Emanuel,
the uncle of Sanchez (the son of Alphonsus), in his own
name, and in the name of other nobles, deprived Alphon-
sus of his kingdom by a public sentence: which that
prince merited, for daring severely and boldly to censure
the works of Divine Providence, and the construction of
the human body, as tradition says he did. Heaven most
justly punished the folly of his tongue." Though the
silence of such an historian as Mariana, in regard to Pto-
lemy's system, OllgHt to be of some weight, yet we cannot
think it improbable, that if Alphonsus did pass so bold a.
censure on any part of the universe, it was on the celestial
sphere, and meant to glance upon the contrivers and sup-
porters of that system" For, besides that he. studied no-
thing more, it is certain that at that time astronomers ex-
plained the motions of the heavens by intricate and con-
fused hypotheses, which did no honour to God, nor any-
wise answered the idea of an able operator. So that, from
considering the multitude of spheres composing the system
of Ptolemy, and those numerous eccentric cycles and epi-

cycles with which it is embarrassed, if we suppose Al-
phonsus to have said, " That if God had asked his advice
when he made the world, he would have given him better
council," the boldness and impiety of the censure will be
greatly diminished, if not entirely removed, as probably"
meant for a satirical sneer at the intricate devices and
contrivances of mortals.

ALSTED (John-Henry), a German protestant di-
vine, and one of the most indefatigable writers of the 17th
century. He was some time professor of philosophy and
divinity at Herborn, in the county of Nassau: from
thence he went into Transylvania, to be professor at Alba
Julia ; where he continued till his death, which happened
in 16*38, at 50 years of age. He applied himself chiefly
to compose methods, and to reduce the several branches
of arts and sciences into systems. His Encyclopaedia has
been much esteemed, even by Roman catholics : it was
printed at Lyons, and sold very well throughout all France.
Vossius mentions the Encyclopaedia in general, but speaks
of his treatise of Arithmetic more particularly, and allows
the. author to have been a man of great reading and uni-
versal erudition. His Thesaurus Chronologicus is by some
esteemed one of his best works, and has gone through se-
veral editions, though others speak of it with contempt.
In his Triumphus Biblicus, Alsted endeavours to prove,
that the materials and principles of all the arts and
sciences may be found in the Scriptures; but he gained
very few to his opinion. John Himmelius wrote a piece
against his Theologia Polemica, which was one of Alsted's
best performances. It seems he was a millenariah, having
published, in 16Y2, a treatise De Mille Annis, in which
he asserts that the. faithful shall reign with Jesus Christ
upon earth a thousand years; after whieh will be the ge-
neral resurrection, and the last judgment; and he pretend-
ed that this reign would commence in the year lb'94.

ALTERNATE angles, in Geometry,
are the internal angles, a and B, or a and
b, made by a line cutting two parallels,
and lying on opposite sides of the cutting
line. It is the property of these angles
to be always equal to each other, namely
the angle a = the angle b, and the angle
a = the angle b. And the exterior alternate angles are
also equal.

Alternate Ratio or Proportion, is the ratio of the
one antecedent to the other, or of one consequent to the
other, in any proportion, in which the quantities are of
the same kind. So if a : b : : c : D,

then alternately, or by alternation, a : c : : b : d.

ALTERNATION, or Permutation, of quantities or
things, is the varying or changing the order or position of
them.

As suppose two things a and b; these may be placed
either thus, ab or ba, that is two ways, or 1 x 2. ll there
be these things; a, b, c: then the 3d thing c, may be
placed three different ways villi respect to each of the
two positions ab and ba of the other two things; it may
stand either before them, or between them, or after them
both, that is, it may stand either 1st, 2d, or 3d; and
therefore, with three things there will be three times as
many changes as with two, that is 1 x 2 x 3 or 6' changes
with three things. Again, if there he four things a, b, c,
d; then the fourth thing d may be placed in four different
ways will' respect to each of the six positions of the other
three; for it may be set either 1st or 2d or 3d or 4th in

A L T

[ 107 ]

ALT

the order of each position; consequently, from four things angles, or proportional lines, either to compute, or gt o-

there will be four times as many alternations as there are metrically determine, the altitude of the object sought.
from three things ; and therefore 1x2x3x4 = 24 is There are various ways of measuring altitudes, or

the number of changes with four things. And so on, al- depths, by means of different instruments, and by shadows

ways multiplying the last found number of alternations or reflected images, on optical principles. There are

by the next number of things ; or to find the number of also various ways of computing the altitude in numbers,

changes for any number of things, as n, multiply the se- from the measurements taken as above, either by geome-

ries of natural numbers 1, 2, 3, 4, 5, &c, to n, continually
together, and the last product will be the number of alter-
nations sought ; as, 1x2x3x4x5 n is the

number of changes in n things,

trical construction, or trigonometrical calculation, or by
simple numerical computation from the property of pa-
rallel lines, &c.

The instruments mostly used in measuring altitudes,

So if, for example, it were required to find how many are the quadrant, theodolite, geometrical square, line of

changes may be rung on 12 bells; it would be
1x2x3x4x5x6x7x8x9x10x11x12
=479001600, the number of changes. Now supposing
there might be rung 10 changes in one minute, that is
10 x 12 or 120 strokes in a minute, or 2 strokes in each
second of time; then, according to this rate, it would

shadows, &c ; the descriptions of each of which may be
seen under their respective names.

To measure an altitude geometrically, ■when ive can have
access to the base of the object. Suppose the height of the
accessible tower ab be required. First, by means of two
rods, or staves, of different lengths. Plant the longer staff

take upwards of 91 years, without ever ceasing day or perpendicularly at c ; then move the shorter backwards

night, to ring over all these changes on the 12 bells only.
Also, if but two more bells were added, making 14 bells ;
then, at the same rate of ringing, it would require about
16575 years to ring all the changes on the 14 bells but
once over. And if the number of bells were 24, it would
require more than 1 17000000000000000 years to ring all
the different changes upon them '

till by trials you find such a place, d, that the eye placed
at the top of it at e, may see the top of the other, f, and
that of the object b in a straight line : next measure the
distances da or eg and DC or eh, also hf the difference
between the heights of the rods : then, by similar trian-
gles, as eh : eg : : hf: the 4th proportional gb ; to
which add AG or de, and the sum will be the whole al-

E

ALTIMETRY, Altimetria, the art of taking or titude ab sought. Or, as eh : ia : : fh : ba.

measuring altitudes or heights, whether accessible or in-
accessible. Or Altimetria is that part of practical geo-
metry which relates to the theory and practice of mea-
suring both heights and depths, either in perpendicular or
oblique lines.

ALTING (James), was born at Heidelberg in 1(518.
He travelled into England in 1640, where he was ordained
by the learned Dr. Prideaux, bishop of Worcester. He
afterwards succeeded Gomarus in the professorship of
Groninghen. He died in l6~97 ; and recommended the
edition of his works to Menso Alting (author of Notitia
German. Infer. Antiquae) ; but they were published in
5 vols folio, with his life, by Bekker of Amsterdam. They
contain various analytical, exegetical, practical, proble-
matical, and philosophical tracts, which show his great
industry and knowledge.

ALTITUDE, in Geometry, is the height of an object,
or its elevation above that plane to which its base is re-
ferred; so that the altitude of the same object is different
in different places ; thus, two persons taking the height of
a mountain, being on contrary sides of it, will each con-
sider its base as on a level with the place where he stands,
and therefore in many cases they will give different
results.

Altitude of a figure, is the distance of its, vertex
from the base, or the length of a perpendicular let fall
from its vertex to the base. The altitudes of figures are
necessary to be known in computing their areas" or soli-
dities.

Altitude, or Height of any point of a terrestrial ob-
jret, is the perpendicular let fall from that point to the
plane of the horizon.

To measure the Altitude of bodies or objects. If this
cannot be done by stretching a line from top to bottom,
which is the direct and most accurate way, (hen some in-
direct way is used, by measuring some other line or di-
»tance which may serve as a basis, in conjunction with

I T> . C A

Or, with one rod cf only. Plant it at such a place c,
that the eye at the ground, or near it, at l, may see the
tops f and b, in a right line : then, having measured ic,
ia, cf, the 4th proportion to these will be the altitude
ab sought.

Or thus, by means of shadows. Plant a rod ab perpen-
dicularly at a, and measure its shadow ac, as also the
shadow ac of the object ab, at the same instant; then
the 4th proportional to ac, ab, ac will be the altitude
ab sought. For the angles at b and c being equal, the
triangles are similar.

.6

Or thus, by means of optical reflection. Place a vessel of
water, or a mirror or other reflecting surface, horizontal
at c; and go to such a distance, i>, that the eye e may
see the image of the top of the object in the mirror at c:
then, by similar figures, cd : de : : ca : ab the altitude
sought. This depends on a property in optics, namely,
that the angle of incidence is equal to the angle of reflec-
tion, that is, the angle bca is equal to the angle ecd;
and the angles at a and d being right angles, the two tri-
angles are similar: hence the above proportion. See
Optics.

P2

ALT

[ 103 ]

ALT

IB

Then, from the angle c taking the angle d, leaves the angle
cbd ; hence

As sine £. cbd : sine JL. d :
As rad. : sine Z. acb : : cb

cd : cb; and

; ab, the altitude.

And for a balloon, or
cloud, or other moveable
object c, let two observers
at a and b, in the same
plane with c, take at the

Or thus, by the geometrical square. Which consists of
a plain piece of board, a foot square, one of the sides of
which, as gf, is divided into a number of equal parts, and
having a plumb line suspended at the angular point E,
with two sights or a telescope placed on the side de;
and then the whole fixed to a stand at d. At any place,
as at c, fix the stand, and turn the square about the cen-
tre of motion, d, till the eye there see the top of the ob-
ject through the sights or telescope on the side de of the
square, and note the number of divisions cut off the other same time the angles a and ...--'

side by the plumb line eg : then, by similar triangles, as B)and measure the distance ^i i !

ef : fg : : dh : hb ; to which add ah or cd, for the AB between them; then A B n

whole height ab. If the side gf be divided into 100 calculate the altitude cd exactly as in the last example,
equal parts, we have only to multiply the length of dh It must however be by mere accident that a, b, and c,

in feet by the number of parts cut off by the plumb line, are in the same vertical plane; and consequently this
and to cut off two figures from the product for decimals; method, though it be true in theory, is of no use in prac-
and we shall have the height of bh in feet. tice: we will therefore show how the altitude may be ob-

ftained when the observers are not in a plane with the
object.
#i* In this case, there must be three observers, who take

ij3-'': i^Wu *^e angle of elevation at the same instant, being at given

D«K ^&p distances from each other in the same right line, as at a,

jfc_ |_ b, and c, in the annexed figure; now admitting those

C A three angles to be known, it is required to find the altitude

To measure the altitude of an accessible object trigonome- of the balloon.
trically. At any convenient station, c, with a quadrant, Demit the perpendicular dp

theodolite, or other graduated instrument, observe the to the horizontal plane acp,
angle of elevation acb above the horizontal line ac : and draw the other lines as in
and measure the distance AC Then, a being a right the figure. Make /, t', t", re-
angle, it will be, as radius is to the tangent of the angle c, present respectively the nat
so is ac to ab sought.

If ac be not horizontal, but an inclined plane; then
the angle above it must be observed at two stations c and
d in a right line, and the distances ac, cd both mea-
sured. Then, from the angle c take the angle d, and
there remains the angle cbd ; hence in the triangle bcd,
are given the angles and the side DC, to find the side
cb ; and then in the triangle abc, are given the two sides
ca and cb, with the included angle c, to find the third
side ab.

cotan. of the angles observed
at a, B, and c ; also put
ab = a, BC = b, and ac =c;
and call dp=jt. Then ap
== tx, bp = t'x, and cp = t x : hence from the common

principles of geometry we find dv — \Z-jp — -rrj

Ce

)«

Or thus, measure only the distance ac, and the angles
a and c: then, in the triangle abc, are given all the
angles and the side ac, to find the side ab.

To measure the altitude of an inaccessible object, as a
hill, cloud, or other body. This is commonly done, by
observing the angle of its altitude at two stations, and
measuring the distance between them. Thus, for the
height a a of a hill, measure the distance cd at the foot
of it, and observe the quantity of the two angles c and d.

forthealtitudesought. IfZ.A = Z.B, then iip=/,.,_ „,i

if Z.B = /.c, then dp = ^ ." ■,; and if a = b, then

will dp = */— —, — —, =j-

v it" — t- — t"

To find the height of an object, by

knowing the utmost distance at which

its top can be just seen in the horizon.

As, suppose the top n of a tower fii can I

be just seen from E when the distance e f \

is '25 miles, supposing the circumference ^

of the earth to be 25000 miles, or the

radius 3979 miles or 2100yi20 feet. First, as 25000 :

25 : : 360° : 21' 36" equal to the angle c. ; then as radius

: sec. Z. g : : eg : bh, which will be found to be

21009536 feet; from which take eg or gf, anil there

remains 4lS feet, for 111 the height of the tower sought. —

Or rather thus, as 10000000 radius : 1<)8 = sec. io —

radius : : 21009120 = eg : 4l6 = fji, as before.

ALT [109

Or the same may be found easier thus : The horizon
dips nearly 8 inches or | of a foot at the distance of 1
mile, anil according to the square of the distance for
other distances; therefore as 1* or 1 : 252 or 625 : : f :
f of 625 or 4l6 feet, the same as before.

Or conversely, to find the distance r.F, from the known
height Fit : Here | : 1, or as 2 : 3 : : 416 : 625, the
square root of which gives 25 miles, for the distance if.
That is, in general, take the square root of \ of the alti-
tude, in feet, for the distance in miles.

There is a very easy method of taking great terrestrial
altitudes, such as mountains, or the heights to which bal-
loons ascend, &c, by means of the difference between the
heights of the mercury in the barometer observed at the
bottom and top of the same. Which see under the article
Barometer.

Altitude of the Eye, in Perspective, is a right line
let fall from the eye, perpendicular to the geometrical
plane.

Altitude, in Astronomy, is the arch of a vertical circle,
measuring the height of the sun, moon, star, or other ce-
lestial object, above the horizon.

This altitude may be either true or apparent. The ap-
parent altitude, is that which appears by sensible observa-
tions, made at any place on the surface of the earth. And
the true altitude, is that which results by correcting the
apparent, on account of refraction and parallax.

The quantity of the refraction is different at different
altitudes; and the quantity of the parallax is different
according to the distance of the different luminaries: in
the fixed stars, this is too small to be observed ; in the
sun, it is but about Sf seconds; but in the moon it is
about 52 minutes.

Altitudes are observed by a quadrant, or sextant, or
by the shadow of a gnomon or high pole, and by va-
rious other ways, as may be seen in most books of astro-
nomy.

Meridian Altitude, is an arch of
the meridian, intercepted between ■e/'
any point in it and the horizon. So,
if ho be the horizon, and hezo the
meridian; then the arch he, or the
angle hce, is the meridian altitude
of an object in the meridian at the
point e.

Altitude, or elevation, of the Pole, is the angle ocp,
or arch op of the meridian, intercepted between the ho-
rizon and pole p.

This is equal to the latitude of the place ; and it may
be found by observing the meridian altitude of the polar
star, when it is both above and below the pole, and taking
half the sum, when corrected on account of refraction.
Or the same may be found by the declination and meri-
dian altitude of the sun.

Altitude, or elevation, of the equator, is the angle
hce, or arch he of the meridian, between the horizon
and the equator at E ; and it is equal to zp the co-latitude
of the place.

Altit ude of the Tropics, the same as what is otherwise
called the solstitial altitude ol the sun, or his meridian al-
titude when in the solstitial points.

Altitude, or height, of the horizon, or of stars, &c,
seen in it, is the quantity by which it is raised by re-
fraction.

Refraction of Altitude, is an arch of a vertical circle,

U7

] ALT

by which the true altitude of the moon, or a star, or
other object, is increased by means of the refraction; and
is different at different altitudes, being nothing in the
zenith, and greatest at the horizon, where it is about
33'.

Parallax of Altitude, is
an arch of a vertical circle,
by which the true altitude,
observed at the centre of the
earth, exceeds that which is
observed on the surface ; or
the difference between the an-
gles J lm and j ik of altitude there ; and is equal to
the angle I ]) l, formed at the moon or other body, and
subtended by the radius il of the earth.

It is evident that this angle is less, as the luminary is
farther distant from the earth ; .and also less, for any one
luminary, as it is higher above the horizon ; being greatest
there, and nothing in the zenith.

Altitude of the Nonagesimal, is the altitude of the
CjOth degree of the ecliptic, counted upon it from where
it cuts the horizon, or of the middle or highest point of it
which is above the horizon, at any time; and is equal to
the angle made by the ecliptic and horizon where they in-
tersect at that time.

Altitude of the cone of the earth's or moon's shadow,
the height of the shadow of the body, made by the sun,
and measured from the centre of the body. To find it,
say, As the tangent of the angle of the sun's apparent semi-
diameter, is to radius, so is 1 to a 4th proportional, which
will be the height of the shadow, in semidiameters of the
body.

So, the greatest height of the earth's shadow, is 217'8
semidiameters of the earth, when the sun is at his greatest
distance, or his semidiameter subtends an angle of about
15' 47"; and the height of the same is 210 7 semidiame-
ters of the earth, when the sun is nearest the earth, or
when his semidiameter is about 16' 10," — And propor-
tionally between these limits, for the intermediate di-
stances or semidiameters of the sun.

The altitudes of the shadows of the earth and moon, are
nearly as 1 1 to 3, the proportion of their diameters.

Altitude of motion, is a term used by Dr. YVallis, for
the measure of any motion, estimated in the line of direc-
tion of the moving force.

Altitude, in speaking of fluids, is more frequently ex-
pressed by the term depth. The pressure of fluids, in
every direction, is in proportion to their altitude or depth.
And on this principle is constructed the sea-gage, for find-
ing the depth of the water of the ocean. See Gage.

Altitude of the mercury, in the barometer and ther-
mometer, is marked by degrees, or equal divisions, placed
by the side of the tube of those instruments.

The altitude of the barometer, or of the mercury in its
tube, at London, is usually comprised between the limits
of 2S and 31 inches; and the mean height, observed lor
every day in several years, is found nearly 29'S7 inches.

Altitude of the pyramids- in Egypt, was measured so
long since as the time of Thales, which he effected by
means of their shadow, and that of a pole set upright be-
side them, making the altitudes of the pole and pyramid
proportional to the lengths of their shadows, l'lutarch
has given an account of the manner of this operation,
which is one of the first geometrical observations we hav<
an exact account of.

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Altitude, circles of, parallels of, quadrant of, fyc. See
the respective words.

Equal Altitude Instrument, is an instrument used to
observe a celestial object, when it has the same or an
equal altitude, on both sides of the meridian, or before
and after it passes the meridian: an instrument very use-
ful in adjusting clocks, &c, and for comparing equal and
apparent time.

AMALGAM, in Chemistry and the Arts, a mixture or
alloy of mercury with any of the other metals. — For a
description of the amalgam used by electricians, and the
method of preparing it, see Electricity.

AMBIENT, encompassing round about; as the bodies
which are placed round any other body, are called am-
bient bodies, and sometimes circum-ambient bodies; and
the whole mass of the air or atmosphere, because it en-
compasses all things on the face of the earth, is called the
ambient air.

AMBIGENAL Hyperbola,
a name given by Newton, in his
Enumeratio linearum tertii
ordinis, to one of the triple
hyperbolas egf of the second
order, having one of its infinite
legs, as eg, falling within the
angle acd, formed by the
asymptotes ac, CD, and the other leg gf falling without
that angle.

AMBIT, of a figure, in Geometry, is the perimeter, or
line, or sum of the lines, by which the figure is bounded.

AMBLIGON, or Ambligostal, in Geometry, signifies
obtuse-angular, as a triangle which has one of its angles
obtuse, or consisting of more than 90 degrees.

AMICABLE numbers, denote pairs of numbers, of which
each of them is mutually equal to the sum of all the ali-
quot parts of the other. So the first or least pair of ami-
cable numbers are 220 and 284 ; all the aliquot parts of
which, with their sums, are as follow, viz,
of 220, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55,

110, their sum being ... - 284;
of 284, they are 1, 2, 4, 71, 142, and their sum is 220.

The 2d pair of amicable numbers are 6232 and 636S,
which have the same properties.

The 3d pair of amicable numbers are 17296 and 18416,
which have also the same property as above.

And the 4th pair of amicable numbers are 9363584 and
9437056.

The 1st, 3d, and 4th pairs of amicable numbers were
found out by F.Schooten,sect.9 of his Excrcitationes Ma-
thematicae, who I believe first gave the name of amicable
to such numbers, though their properties it seems had be-
fore been treated of by Rudolphus, Descartes, and others.

To find the first pair, Schooten puts 4jt and tyz, or
a"x and a2yz for the two numbers, where a = 2; then
making each of these equal to the sum of the aliquot parts
of the other, gives two equations, from which are found
the values of a; and 2; and consequently, assuming a pro-
per value for y, the two amicable numbers themselves A-x
and 4yz.

In like manner for the other pairs of such numbers; in
which he finds it necessary to assume l6,r and \6yz or
<i,x and ayz for the 3d pair, and 128x- and 128jc or a7x
and (Hyz for the 4th pair.

Schooten then gives this practical rule, from Descartes,
lor finding amicable numbers, viz, Assume the number 0,

or some power of the number 2, such that if unity or 1 be
subtracted from each of these three following quantities,
viz;

from three times the assumed number,
also from six times the assumed number,
and from 18 times the square of the assumed number,
the three remainders may be all primes; then the last
prime number being multiplied by double the assumed
number, the product will be one of the amicable numbers
sought; and the sum of its aliquot parts will be the other.
That is, if«a be put = the number 2, and n some integer
number, such that 3an — 1, and 6an — 1, and lSa'-n — 1
be all three prime numbers; then is (18a2n — 1) x 2an
one of the amicable number ; and the sum of its aliquot
parts is the other.

Or this rule may be otherwise expressed thus. Suppose
a and Bare the pairs of amicable numbers sought; then find
such a power of 2, that is, a = 2", such that 3a — 1 = b,
6a — 1 = c, 18a'2 — 1 = d, the numbers b, c and d may
be primes; then will the numbers sought be a = 2n+} d,
and b = 2n+}bc.

The demonstration of which rule is as follows. Let
/a represent the sum of the divisors of a, andy"B the sum
of the divisors of b, the whole values of a and b not being

comprised, then since 1 ■+■ 21 -+■ 22 •+- 23 . . . -t- 2 =
2n+- 1, we shall have /a = (2n+- 1) . (1 + d) -
2n+1.d = (4a- l)!So2- 2«(lSo2 - 1) = b, and

/b= (2"+Jl) . (1 + 6) . (1 +c) - 2n+\bc = (4a
— I) . 18a2 — 2a(3a — 1) . (6a — 1) = a, as is evi-
dent; because (1 -+- 21 ■+■ 2s . . -1- 2 ) . (d ■+■ 1) will
be the sum of all the divisors of a ; and in the same man-
ner (1 -h 21 + 22 ... •+■ 2n+1) .(i+1) . (c -+- 1) is
the sum of all the divisors of B.

Mr. John Gough has lately investigated some proper-
ties of these numbers, especially of the Cartesian form,
where ax and ayz are amicable numbers, consisting of a
common measure a, multiplied by the primes x,y, z. See
Leybourn's Math. Repos. 11°. 7> »■ s.

AMONTONS (William), an ingenious French expe-
rimental philosopher, was born in Normandy the 31st <>i
August 1663. While at the grammar-school, he by sick-
ness contracted a deafness that almost excluded him
from the conversation of mankind. In this situation he
applied himself to the study of geometry and mechanics;
with which he was so delighted, that it is said he refused to
try any remedy for his disorder, either because he deemed
it incurable, or because it increased his attention to his
studies. Among other objects of his study, were the arts
of drawing, of land-surveying, and of building; and shortly
after he acquired some knowledge of those more sublime
laws by which the universe is regulated. He studied with
great care the nature of barometers and thermometers ;
and wrote his treatise of Observations and Experiments
concerning a new Hour-glass, and concerning Barometers,
Thermometers, and Hygroscopesj as also some pieces in
the' Journal des Savans. In 1087 , be presented n in w hy-
groscope to the Academy of Sciences, which was much
approved. He recommended a methodol conveying in-'
teiligence to a great distance in a short spaceoftime: this
was by making signals from one person to another, placed
at as great distances from each other as they could see the
signals by means of telescopes. When the Royal Academy

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was new regulated in 1699> Amontons was chosen a mem-
ber of it, as an eleve under the third Astronomer; and he
read there his New Theory of Friction, in which he hap-
pily cleared up an important object in mechanics. In
fact he had a particular genius for making experiments :
his notions were just and delicate : and he knew how to
prevent the inconveniences of his new inventions, and had
a wonderful skill in executing them. He died of an in-
flammation in his bowels, the 11th of October 1705, being
only 42 years of age.

The eloge of Amontons may be seen in the volume of
the Memoirs of the Academy of Sciences for the year
1705, Hist. pa. 150. And his pieces contained in the
different volumes of that work, which are pretty numerous,
and on various subjects, as the. air, action of fire, barome-
ters, thermometers, hygrometers, friction, machines, heat,
cold, rarefactions, pumps, &c, may be seen in the volumes
for the years 1696, l6'99, 1702, 1703, 1704, and 1705.

AMPHISCil, or Amphiscians, are the people who
inhabit the torrid zone; which are so called, because they
have their shadow at noon turned sometimes one way and
sometimes another, namely, at one time of the year to-
wards the north, and at the other towards the south.

AMPLITUDE, in Gunnery, the range of a projectile, or
the right line upon the ground subtending the curvilinear
path in which it moves.

Amplitude, in Astronomy, is an arch of the horizon,
intercepted between the true east or west point, and the
centre of the sun or a star at its rising or setting: so that
the amplitude is of two kinds; ortive or eastern, and occi-
duous or western. Each of these amplitudes is also either
northern or southern, according as the point of rising or
setting is in the northern or southern part of the horizon :
and the complement of the amplitude, or the arch of dis-
tance of the point of rising or setting, from the north or
south point of the horizon, is the azimuth.

The amplitude is of use in navigation, to find the varia-
tion of the compass or magnetic needle. And the rule to
find it is this; As the cosine of the latitude is to radius,
so is the sine of the sun's or star's declination, to the
sine of the amplitude. So in the latitude of London, viz,
51° Si', when the sun's declination is 23° 28'; then
cos. 51° 31' the lat. - — 97939907
sin. 23 2S the decl. - ■+■ 9-6001181
sin. 39 47 the ampl. - 9'806l274
That is, the sun then rises or sets 39° 47' from the east or
west point, to the north or south according as the decli-
nation is north or south.

Magnetictil Amplitude, is an arch of the horizon.con-
tained between the sun or star, at the rising or setting, and
the magnctical east or west point of the horizon, pointed
out by the magnetical compass, or the amplitude or azi-
muth compass. And the difference between this magne-
tical amplitude, so observed, and the true amplitude, as
computed in the last article, is the variation of the com-
pass.

So if, for instance, the magnetical amplitude be ob-
served, by the compass, to be 6l° 47', at the time when
it is computed to be - 39 47,
then the difference - - 22 0 is the variation west.

ANABIBAZOtV, a name sometimes given to the dra-
gon's tail, or northern node of the moon.

ANACAMPTICS, or the science of the reflections of
sounds, frequently used in reference to echoes, which are

said to be sounds produced anacamptically, or by reflec-
tion^ And in this sense it was used by the ancients for
that part of optics which is otherwise called Catoptrics.

ANACHRONISM, in Chronology, an error in compu-
tation of time, by which an event is placed earlier than it
really happened. Such is that of Virgil, who makes Dido
to reign at Carthage in the time of ./Eneas, though, fe
reality, she did not arrive in Africa till 300 years after
the taking of Troy.

An error on the other side, by which a fact is placed
later, or lower than it should be, is called a parachronism.
But in common use, this distinction, though proper, is not
attended to; and the word anachronism is used indiffe-
rently for the mistake on both sides.

ANACLASTICS, or Anaclatics, an ancient name
for that part of optics which considers refracted light;
being the same as what is more usually called diop-
trics. See the Compendium of Ambrosius Rhodius, lib. 3.
Optica;, pa. 3S4. & seq.

Anaclastic Curves, a name given by M. de Mairan
to certain apparent curves formed at the bottom of a
vessel full of water, to an eye placed in the air; or the
vault of the heavens, seen by refraction through the at-
mosphere. M. de Mairan determines these curves by a
principle not admitted by all authors ; but Dr. Barrow, at
the end of his Optics, determines the same curves by other
principles.

ANALEMMA, a planisphere, or projection of the
sphere, orthographically made on the plane of the meridian,
by perpendiculars from every point of that plane, the eye
supposed to be at an infinite distance, and in the east or
west point of the horizon. In this projection, the solstitial
colure, and all its parallels, are projected into concentric
circles, equal to the real circles in the sphere; and all
circles whose planes pass through the eye, as the horizon
and its parallels, are projected into right lines equal to
their diameters; but all oblique circles arc projected into
ellipses, having the diameter of the circle for the trans-
verse axis. This instrument, having the furniture drawn
on a plate of wood or brass, with an horizon fitted to it,
is used for resolving many astronomical problems; as the
time of the sun's rising and setting, the length and hour of
the day, &c. It is also useful in dialling, for laying down
the signs of the zodiac, with the lengths of days, and other
matters of furniture, upou dials.

The oldest treatise we have on the analemma, was writ-
ten by Ptolemy, which was printed at Rome 156'2, with a
commentary by F. Commandine. Pappus also treated of
the same. Since that time, many other authors have
treated very well of the analemma; as Aguilonius, Ta-
quet, Dechales, Witty, &c.

ANALOGY, the same as proportion, or equality, or
similitude of ratios. Which see.

ANALYSIS, is, generally, the resolution of any thing
into its component parts, in order to discover it or its com-
position. And in mathematics it is properly the method
of resolving problems, by reducing them to equations.
Analysis may be distinguished into the ancient and the
modern.

The ancient analysis, as described by Pappus, is the
method of proceeding from the thing sought as taken for
granted, through its consequences, to something that»is
really granted or known; in which sense it is the reverse
of synthesis or composition, in which we lay that down

A N A

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first which was the last step of the analysis, and tracing
tht: steps of the analysis back, making that antecedent
here which was consequent there, till we arrive at the
thing sought, which was taken or assumed as granted in
the first step of the analysis. This chiefly respected geo-
metrical inquiries.

The principal authors on the ancient analysis, as re-
counted by Pappus, in the 7th book of his Mathematical
Collections, are Euclid in his Data, Porismata, et de
Locis ad Superficiem; Apollonius de Sectione Rationis,
de Sectione Spatii, de Tactionibus, de Inclinationibus, de
Locis Planis, et de Sectionibus Conicis ; Aristoeus, de
Locis Solidis; and Eratosthenes, de Mediis Proportionali-
bus : from which Pappus gives many examples in the same
book. To these authors we may add Pappus himself.
The same sort of analysis has also been well cultivated
by many of the moderns ; as Fermat, Viviani, Ghetaldus,
Snellius, Huygens, Simson, Stewart, Lawson, Playfair,
Leslie, &c, and more especially Hugo d'Omerique, in his
Analysis Geometrica, in which he has endeavoured to
restore the analysis of the ancients. And on this head,
Dr. Pemberton tells us, " that sir Isaac Newton used to
censure himself for not following the ancients more closely
than he did; and spoke with regret of his mistake, at the
beginning of his mathematical studies, in applying himself
to the works of Descartes, and other' algebraical writers,
before he had considered the Elements of Euclid with that
attention so excellent a writer deserves:, that he highly
approved the laudable attempt of Hugo d'Omerique to re-
store the ancient analysis."

In the application of the ancient analysis in geometri-
cal problems, every thing cannot be brought within strict
rules; nor any invariable directions given, by which we
may succeed in all cases; but some previous preparation
is necessary, a kind of mental contrivance and construc-
tion, to form a connexion between the data and qusesita,
which must be left to every one's judgment to find out;
being various, according to the various nature of the pro-
blems proposed. Right lines must be drawn in particular
directions, or of particular magnitudes; bisecting perhaps
a given angle, or perpendicular to a given line; or perhaps
tangents must be drawn to a given curve, from a given
point; or circles described from a given centre, with a
given radius, or touching given lines, or other circles; or
such-like operations. Whoever is conversant witli the
works of Archimedes, Apollonius, or Pappus, well knows
that they founded their analysis on some such previous ope-
rations; and the great skill of the analyst consists in dis-
covering the most proper affections on which to found his
analysis: for the same problem may often be effected in
many different ways : of which it may be proper to give
here an example or two. Let there be taken, for instance,
this problem, which is the 155th prop, of the 7th book of
Pappus.

From the extremities of the base a b, of a given seg-
ment of a circle, required to draw two lines ac, bc, meet-
ing at. a point c in the circumference, so that they shall
have a given ratio to each other, suppose that of F to o.

The solution of this problem, as given by Pappus, is
thus.

Analysis. <

Suppose the thing done, and that the point C is found:
and let CD be drawn a tangent to the circle at c, and meet-
ing the line ab produced in the point d. Now by the

hypothesis ac : bc : : r : g, and also ac1 : bc* : : da
de, as may be thus proved.

Since dc touches the cir6le, and bc cuts it, the angle
bcd is equal to bac by Euc. iii 32 ; also 'he angle d is
common to both the triangles d c a, d c b ; these are there-
fore similar, and so, by vi 4, da : dc : : dc : db, and
hence da" : dc'2 : : da : db by cor. vi 20. But also, by
vi 4, da : ac : : dc : cb, and by permutation da : dc
: : ac : bc, or da2 : dc2 : : ac2 : bc2; and hence, by
equality, ac- : bc2 : : da : db.

But the ratio of acx to bc'2 is given, because the ratio
of ac to bc is given, and consequently that of da to db
is given. Now since the ratio of da to db is given, there-
fore also, by Data vi, that of da to ab, and hence, by
Data ii, da is given in magnitude.

And here the analysis properly ends. For it having
been shown that da is given, or that a point d may be
found in ab produced, such, that a tangent being drawn
from it to the circumference, the point of contact will
be the point sought; we may now begin the composition,,
or synthetical demonstration ; which must be done by
finding the point d, or laying down the line a d, which, it
was affirmed, was given, in the last step of the analysis.
Synthesis.

Construction. Make as r2 : g2: : ad : »b, (which may
be done, since ab is given, by making it as f2 — g1 : g* : :
ab : db, and then by composition it will be as f2 : g'2 : :
ab : db); then from the point d, thus found, draw a
tangent to the circle, and from the point of contact c
drawing ca and cb, the thing is done.

Demonstration. Since, by the constr. f'2 : GJ : : ad :
db, and also ad : db : : ac2 : bc2, which has beep al-
ready demonstrated in the analysis, and might be here
proved in the same manner. Therefore f'2 : g2 : : AC1 :
bc2, and consequently f : g : : ac : bc 3. E. D.

Here we see an instance of the method of resolution
and composition, as it was practised by the ancients, the
solution here given being that of Pappus himself. But as
the method of referring and reducing every thing to the
Data, and constantly quoting the same, may appear now
to be tedious and troublesome: and indeed it is unne-
cessary to those who have already made themselves ma-
sters of the substance of that valuable book of Euclid,
and have by practice and experience acquired a facility
of reasoning in such matters: 1 shall therefore now show
how we may abate something of the rigour and strict
form of the ancient method of solution, without diminish-
ing any part of its admirable elegance and perspicuity.
And this may be done by means of another solution, of
the many more which might be given, of the same pro-
blem, as follows.

An a lysis.

Let us again suppose that the thing is done, viz, 1C :
BC ; : F : O, and let there, be drawn nil making the angle
ami equal to the angle aci\, and meeting ac produced
in ii. Then, the angle a being also common, the two
triangles abc and abii are equiangular, and therefore,
by vi 4, ac : bc : : ah : Bii, in a given ratio; and, «

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beinc given, therefore bh is given in position and magni-
tude. '

Synthesis.

Construction. Draw eh making the angle abh equal
to that which may be contained in the given segment, and
take ab to bh in the given ratio of f to c. Draw ach,
and bc.

Demonstration. The triangles abc, abh are equiangu-
lar, therefore, vi 4, AC : cb : : ab : bh, which is the
given ratio by construction.

Modern Analysis, consists chiefly of- algebra, arith-
metic of infinites, infinite series, increments, fluxions, &c;
of each of which a particular account may be seen under
their respective articles.

These form a kind of arithmetical and symbolical ana-
lysis, depending partly on modes of arithmetical computa-
tion, partly on rules peculiar to the symbols made use of,
and partly on rules drawn from the nature and species of
the quantities they represent, or from the modes of their
existence or generation.

The modern analysis is a general instrument, by which
the finest inventions, and the greatest improvements, have
been made in mathematics and philosophy, for two cen-
turies past. It furnishes the most perfect examples of the
manner in which the art of reasoning should be employ-
ed ; it gives to the mind a wonderful skill for discovering
things unknown, by means of a small number that are
given ; and by employing short and easy symbols fo_r ex-
pressing ideas, it presents to the understanding things
which otherwise would seem to lie above its sphere. By
this means, geometrical demonstrations may be greatly
abridged ; a long train of arguments, in which the mind
cannot, without the greatest effort of attention, discover
' the connexion of ideas, is converted into visible symbols;
and the various operations which they require, are simply
effected by the combination of those symbols. And,
what is still more extraordinary, by this artifice, a great
number of truths are often expressed in one line only: in-
stead of which, by following the ordinary way of expla-
nation and demonstration, the same truths would occupy
whole pages or volumes. And thus, by the bare contem-
plation of one line of calculation, we may understand in a
short time whole sciences, which otherwise could hardly
be comprehended in several years.

It is true that Newton, who best knew all the advan-
tages of anal} sis in geometry and other sciences, laments,
in several parts of his works, that the study of the an-
cient geometry is abandoned or neglected. And indeed
the method employed by the ancients in their geometrical
writings, is commonly regarded as more rigorous than
that of the modern analysis : and though it be greatly in-
ferior to the latter, in point of dispatch and facility of in-
vention; it is nevertheless highly useful in strengthening
the mind, improving the reasoning faculties, and in ac-
customing the young mathematician to a pure, clear, and
accurate mode of investigation and demonstration, though
by a long and laboured process, which he would with
difficulty have submitted to if his taste had before been
vitiated, as it were, by the more piquant sweets of the
modern analysis. And it is principally on this that the
complaints of Newton are founded, who feared lest by the
too early and frequent use of the modern analysis, the
science of geometry should lose that rigour and purity
which characterise its investigations, and the mind become
debilitated by the facility of our analysis. This great man

Vol. I.

was therefore well founded, in recommending, to a certain
extent, the study of the ancient geometricians : for, their
demonstrations being more difficult, give more exercise to
the mind, accustom it to a closer application, <rive it a
greater scope, and habituate it to patience and resolution,
so necessary for making discoveries. But this is the only
or principal advantage from it; for if we should look no
farther than the method of the ancients, it is probable
that, even with the best genius, we should have made but
few or small discoveries, in comparison with those ob-
tained by means of the modern analysis. And even with
regard to the advantage given to investigations made in
the manner of the ancients, namely of being more rigo-
rous, it may perhaps be doubted whether this pretension
be well founded. For, to instance in those of Newton
himself, though his demonstrations be managed in the
manner of the ancients ; yet at the same time it is evident
that he investigates his theorems by a method different
from that employed in the demonstrations, which are
commonly analytical calculations, disguised by substitu-
ting the name of lines for their algebraical value : and
though it be true that his demonstrations are rigorous, it
is no less so that they would be the same when translated
and delivered in algebraic language ; and what difference
can it make in this respect, whether we call a line a b, or
denote it by the algebraic character a ? Indeed this last
designation has this peculiarity, that when all the lines
are denoted by algebraic characters, many operations can
be performed upon them, without thinking of the lines or
the figure. And this circumstance proves of no small ad-
vantage : the mind is relieved, and spared as much as pos-
sible, that its whole force may be employed in overcoming
the natural difficulty of the problem alone.

Upon the whole therefore, the state of the comparisqn
seems to be this ; That the method of the ancients is fittest
to begin our studies with, to form the mind and to esta-
blish proper habits; and that of the moderns to succeed,
by extending our views be3rond the present limits, and
enabling us to make new discoveries and improvements.

Analysis is divided, with respect to its object, into that
of finites, and that of infinites.

Analysis of finite quantities, is what is otherwise called
algebra, or specious arithmetic.

Analysis of infinites, called also the new analysis, is that
which is concerned in calculating the relations of quan-
tities which are considered as infinite, or infinitely small;
one of its chief branches being the method of fluxions, or
the differential calculus. And the great advantage of the
modern mathematicians over the ancients, arises chiefly
from the use of this modern analysis.

Analysis of powers, is the same as resolving them into
their roots, and is otherwise called Evolution.

Analysis of curve lines, shows their constitution, na-
ture and properties, their points of inflection, station, re-
trogradation, variation, &c.

ANALYST, a person who analyses something, or
makes use of the analytical method. In mathematics it
is a person skilled in algebra, or in the mathematical analy-
sis in general.

Analyst, the title of an ingenious, though sophistical
book, written by the celebrated Dr. Berkeley, against the
doctrine of fluxions.

ANALYTIC, or Analytical, something belonging to,
or partaking of, the nature of analysis ; or performed by
the method of analysis.

Q

ANA

[ H4 ]

ANA

Thus we say analytical demonstration, analytical en-
quiry, analytical table or scheme, analytical method, &c.
The analytical stands opposed to the synthetical, or that
which proceeds by the way of synthesis.

ANALYTICS, the science, or doctrine, and use of
analysis.

ANAMORPHOSIS, in perspective and painting, a
monstrous projection ; or a representation of some image,
either on a plane or curve surface, deformed or distorted;
but which in a certain point of view shall appear regular,
and drawn in just proportion.

To construct an Anamorphosis, or monstrous projection, on
aplane. — Draw the square abcd (fig. 1), of any size at
pleasure, and divide it by crossing lines into a number of
areolae or smaller squares : and then in this square, or re-
ticle, called also the craticular prototype, draw the re-
gular image which is to be distorted.-^Or, about any
image, proposed to be distorted, draw a reticle of small
squares.

_E 6

D

c

T/StoiJ

• 1 :

WW

A

1. B

V

Then draw the line ab (fig. 2) equal to ab, dividing it
into the same number of equal parts, as the side of the
prototype ab; and on its middle point e erect the per-
pendicular ev, as also vs perpendicular to ev, making
ev so much the longer, and vs so much the shorter, as
it is intended the image shall be more distorted. From
each of the points of division draw right lines to the point
v, and draw the right line as. Lastly, through the points
c, e,f,g, &c, draw lines parallel to ab: So shall abed be
the space upon which the monstrous projection is to be
drawn ; which is called the craticular ectype. ,

Then, in every areola, or small trapezium, of the space
abed, draw what appears contained in the corresponding
areola of the original space abcd : so shall there be pro-
duced a deformed image in the space abcd, which yet. will
appear in just proportion to an eye distant from it the
length of ev, and raised above it by a height equal to vs.

It will be amusing to contrive it so, that the deformed
image may not represent a mere chaos, but some certain
figure: thus, a river with soldiers, waggons, and other
objects on the side of it, have been so drawn and distort-
ed, that when viewed by an eye at S, it appeared like the
face of a satyr.

An image may also be distorted mechanically, by per-
forating it through in several places with- a fine pin; then,
placing it against a candle or lamp, obsirve where the rays,

which pass through these small holes, fall on any plane or
curve superficies; for they will give the correspondent
point; of the image deformed, and by means of which the
deformation may be completed.

To draw an Anamorphosis on the convex surface of a cone.
It appears from the construction above, that we have only
to make a craticular ectype on the surface of the cone,
which may appear equal to the craticular prototype, to an
eye placed at a proper height above the vertex of the
cone. Hence,

Let the base, or circumference, abcd, of the cone
(fig. 3) be divided by radii into any number of equal parts;
and let some one radius be likewise divided into equal,
parts; then through each point of division draw concen-
tric circles : so shall the. craticular prototype be formed.

i-; ii

4.

G F T

With double the diameter ab, as a radius; describe the
quadrant efg (fig. 4) so as the arch eg be equal to the
whole periphery; then this quadrant, being plied or bent
round, will form the superficies of a cone,- whose base is
the circle. '

Next divide the arch eg into the same number of equal
parts as the craticular prototype is divided into ; and
draw radii from all the points of division. Produce G p
to i, so that fi be equal to fg; from the centre l, with
the radius if, describe the quadrant fkh; and draw the
right line IE. Then divide the arch kf into the same
number of equal parts as the radius of the craticular pro-
totype is divided into; and from the centre i draw radii
through all the points of division, meeting ef in 1, 2, 3,
&c. Lastly, from the centre p, with the radii Fl, f2,
f3, &c, describe concentric circles. So will the craticu-
lar ectype be formed, whose areolas will appear equal to
each other.

Hence, what is delineated in every areola of the crati-
cular prototype, being transferred into the areolas of the
craticular ectype, the images will be distorted or deform-
ed ; and yet they will appear in just proportion to an eye
elevated above the vertex at a height equal to the height
of the cone itself.

If the chords of the quadrants be drawn in the crati-
cular prototype, and chords of each of the 4th parts in
the craticular ectype, every thing else remaining the same,
there will be obtained the craticular ectype in a qua-
drangular pyramid.

And hence it will be easy to deform an image, in any
other pyramid, whose base is any regular polygon.

Because the illusion is more perfect when the eye, by
the contiguous objects, cahnot estimate the distance of
the parts of the deformed image, it is therefore proper to
view it through a small hole.

Anamorphoses, or monstrous images, may also be made
to appear in their natural shape and just proportions, by
means of mirrors of Certain shapes, from which those

ANA

[ US ]

AND

images are reflected again; and then they are said to be
reformed. For farther particulars, see Wolfius's Catop-
trics and Dioptrics, and some other optical authors.

ANASTROUS signs, in Astronomy, a name given to
the duodecatemoria, or the 12 portions of the ecliptic,
which the signs possessed anciently, but have since de-
serted by the precession of the equinox.

ANAXAGORAS, one of the -most celebrated philoso-
phers among the ancients. He was born at Clazomene in
Ionia, about the 70th Olympiad. He was a disciple of
Anaximenes ; and he gave up his patrimony, to be more
at leisure for the study of philosophy, giving lectures in
that science at Athens. Being persecuted in this place,
and at last banished from it, he opened a school at Lamp-
sacum, where he was greatly honoured during his life, and
still more after his death, statues having been erected to
his memory. It is said he made predictions relative to
the phenomena of nature, as earthquakes, &c, upon which
he wrote some treatises. His principal tenets may be re-
duced to the following : — All things were in the beginning
confusedly mixed together, without order and without
motion. The principle of things is at the same time: one
and multiplex, which had the name of homoemeries, or
similar particles, deprived >of life. But there is beside
this, from all eternity, another principle, an infinite and
incorporeal spirit, who gave motion to these particles; in
virtue of which, such as were homogeneal united, and
such as were heterogeneal separated according to their
different kinds. All things being thus put into motion by
the spirit, and every thing being united to such as are si-
milar, those that had a circular motion produced heavenly
bodies, the lighter particles ascending, while those that
were heavier descended. The rocks of the earth, being
drawn up by the whirling force of the air, took fire, and
became stars, beneath which the sun and moon took their
stations. — It was said he also wrote upon the quadrature
of the circle; the treatise on which, Plutarch says, he
composed during his imprisonment at Athens.

ANAXIMANDER, a very celebrated Greek philoso-
pher, was born at Miletus in the 42d Olympiad ; for, ac-
cording to Apollodorus, he was 64 years of age in the 2d
year of the 58th Olympiad. He was one of the first who
publicly taught philosophy, and wrote upon philosophical
subjects. He was the kinsman, companion, and disciple
ofThales. He wrote also upon the sphere and on geome-
try, &c. And he carried his researches into nature very
far, for the time in which he lived. It is said that he dis-
covered the obliquity of the zodiac; that he first pub-
lished a geographical table ; that he invented the gnomon,
and set up the first sun-dial in an open place at Laceda?-
mon. He taught that infinity of things was the prin-
cipal and universal element ; that this infinite always pre-
served its unity, but that its parts underwent changes;
that all things came from it ; and that all were about to
return to it. By this obscure and indeterminate princi-
ple he probably meant the chaos of other philosophers.
He held that the worlds are infinite ; that the stars
are composed of air and fire, which are carried about in
their spheres, and that these spheres are gods ; and that
the earth is placed in the midst of the universe, as in a
common centre. Farther, that infinite worlds were the
produce of infinity; and that corruption proceeded from
separation.

ANAXIMENES, an eminent Greek philosopher, born
at Miletus, the friend, scholar, and successor of Anaxi-

mander. He diffused some degree of light upon the ob-
scurity of his master's system. He made the first prin-
ciple of things to consist in the air, which he considered
as infinite or immense, and to which he ascribed a per-
petual motion ; that this air was the same as spirit or
God, since the divine, power resided in it, and agitated
it. The stars were as fiery nails in the heavens ; the
sun a flat plate of fire ; the earth an extended flat sur-
face, &c.

ANDERSON (Alexander), one of the brightest or-
naments of the mathematical world, who flourished about
200 years ago. It appears that he was born at Aberdeen
in Scotland, towards the latter part of the l6th century,
and he was professor of mathematics at Paris in the early
part of the 17th, where he published several ingenious
works in geometry and algebra, b6th of his own, and of
his friend Vieta's. He published his

1. " Supplementum Apollonii Redivivi ; sive analysis
problematis hactenus desiderati ad Apollonii Pergasi doc-
trinam 7T£f< vavtrawv, a Marino Ghetaldo Patritio Ragusino
hujusque, non ita pridem restitutam. In qua exhibetur
mechanice aequalitatum tertii gradus sive solidarum, in
quibus magnitudo omnino data, asquatur homogeneas sub
altero tantum coefficiente ignoto. Huic subnexa est va-
riorum problematum practice." Paris, l6l2, in 4to.

2. " AirioAoyia: Pro Zetetico Apolloniani problematis
a se jam pridem edito in supplemento Apollonii Redivivi.
Ad clarissimum et ornatissimum virum Marinum Ghe-
taldum Patritium Ragusinum. In qua ad ea quae obiter
mihi perstrinxit Ghetaldus respondetur, et analytices
clarius detegitur." Paris, 1615, in 4to. 1 >

3. " Francisci Vietae Fontenacensis de Aequationura
Recognitione etEmendationeTractatusduo." Paris, l6l5,
in 4to ; with a Dedication, Preface, and an Appendix,
by Anderson.

4. " Ad Angularium Sectionum Analyticen Theo-
remata KaQoMKoregcc. A Francisco Vieta Fontenacensi,
excogitata, at absque ulla demonstratiolie ad nos trans-
missa, jam tandem demonstrationibus confirmata. Opera
et studio Alexandri Andersoni Scoti." Parisiis, l6l5,
in 4to.

5. " Vindicia? Archimedis. Sive, Elenchus Cyclometrias
nova5, a Philippo Lansbergio nuper edita;. Per Alexan-
drum Andersonum Scotum." Parisiis, 1616, in 4to.

6. " Alexandri Andersoni Scoti Exercitationum Mathe-
maticarum Dicas Prima. Continens, Qua?stionum ali-
quot, quae nobilissimorum turn hujus turn veteris sevi,
Mathematicorum ingenia exercuerc, Enodationem." Pa-
risiis, 1619, in 4to.

All these pieces, of this excellent geometrician, are re-
plete with the finest specimens of pure geometrical exer-
cises that have perhaps ever been produced by any au-
thors, ancient or modern. So that it is to be lamented
that the limits of this work will not admit of giving the
full account of all the particulars contained in them,
which I had prepared indeed expressly for that purpose.
Besides these, I am not aware of any more publications
of our author, though perhaps there may have been others.
Indeed from the last piece, it fully appears that he had at
least written, if not published, another, viz. A treatise on
the mensuration of solids, perhaps with a reference to
gauging; as, in several problems, where he critically ex-
amines the treatise on cask-gauging of Kepler, our author
often refers to his own work on stereometry, which I have
not learned had ever been published.
Q-2

A N D

[ 116 ]

A N E

Alexander was cousin-german to a Mr. David Anderson,
of Finshaugh, a gentleman who also possessed a singular
turn for mathematical and mechanical knowledge. This
mathematical genius appears to have been hereditary
in the family of the Andersons, and from them transmitted
to their descendants of the name of Gregory in the same
country: the daughter of the said David Anderson having
been the mother of the celebrated mathematician James
Gregory, and who herself first instructed her son James
in the elements of the mathematics, on observing in him,
while yet a child, a strong propensity to those sciences.

The time either of the birth or death of our author
Alexander, has not come to my knowledge.

ANDERSON (Robert), an ingenious mathematician,
who flourished in the latter half of the 17th century, and
was much noticed and encouraged in those studies by the
mathematical John Collins, among others, so often men-
tioned in the early labours of the Royal Society. Mr.
Anderson was not in the profession of mathematics, or of
those other branches which it appears he usefully culti-
vated and improved ; but was a silk-weaver by trade, and
must have been a person of some consideration and sub-
stance, as he was able to make, at his own charge, some
thousands of experiments with cannon, for improving the
art of gunnery; which he did in a considerable degree,
as appears by the tracts on that art which he published,
as deduced from those experiments ; viz, The Genuine
Use and Effects of the Gun, in 1674 ; To Hit a Mark, in
1690 ; and, To Cut the Rigging, &c. in 1691, all in 4to.
He published also, Stereometrical Propositions, variously
applicable, but particularly intended for Gauging: in
small 8vo, 166S. This little work is an elaborate treatise
on the measurement, of all kinds of solids that can be
generated by the rotation of circles and the conic sections,
or the ellipse, hyperbola, and parabola; viz, all spheres,
spheroids, conoids, and spindles, &c, with their segments,
zones, ungulas, &c.

ANDROMEDA, in Astronomy, a constellation of the
northern hemisphere, representing the figure of a woman
almost naked, her feet at a distance from each other, and
her arms extended and chained ; being one of the ori-
ginal 48 asterisms, or figures under which the ancients
comprehended the stars, as derived to us from the Greeks,
who probably had them from the Egyptians or Indians,
and who, it is suspected, altered their names, and accom-
panied them with fabulous stories of their own. Accord-
ing to them, Cepheus, the father of Andromeda, was
obliged to give her up to be devoured by a monster, to
preserve his kingdom from the plague ; but that she was
delivered by Perseus, who slew the monster, and espoused
her. And the family were all translated by Minerva to
heaven, the mother being the constellation Cassiopeia.

She is sometimes called, in Latin, Penea, Mulier cate-
nala, Virgo devota, &c. The Arabians, whose religion
did not permit them to draw the figure of the human body
on any occasion whatever, have changed this constellation
into the figure of a sea-calf. Schickard has changed the
name for that of the Scripture name Abigail. And Schiller
has also changed the figure of the constellation, for that
of a sepulchre, and calls it the Holy Sepulchre.

This constellation contains about 27 stars, that are
visible to the naked eye ; of which the principal arc, a
Andromeda's head ; (3 in the girdle, and called Mirach or
Mizar ; y on the south foot, and named Alamak, and
.sometimes Alhamcs.

The number of stars placed in this constellation, by the
catalogue of Ptolemy is 23, by that of Tycho Brahe
also 23, by that of Hevelius 47, and by that of Flam-
steed 66.

ANEMOMETER, an instrument for measuring the
force of the wind.

An instrument of this sort, it seems, was first invented
by Wolfius, in the year 1708, and first published in his
Areometry in l'OJ), also in the Acta Eruditorum, of the
same year ; afterwards in his Mathematical Dictionary,
and in his Elem. Matheseos. He says he tried the good-
ness of it, and observes that the internal structure may be
preserved, so as to measure the force of running water, or
that of men or horses when they draw or pull. The ma-
chine consists of sails, a, b, c, like those of a wind-mill,
against which the wind blows, and by turning them about
raises an arm k with a weight l upon it, to different
angles of elevation, shown by the index m, according to
the force of the wind. (Plate iii. fig. 3.)

' Mr. Leutmann improved on Wolfius's anemometer, by
placing the sails horizontal, instead of vertical, which are
easier then to move, and turn whatever way the wind
blows.

Mr. Benjamin Martin also (Plate iii. fig. 5) improved on
the same. He made the axis like the fusee of a watch,
having a cord winding upon it, with two weights at the
ends, which make always a balance to the force of the
wind on the sails. See his Philos. Britan.

And M. D'Ons-en-Bray invented a new anemometer,
which of itself expresses on paper, not only the several
wiirds that have blown during the space of 24 hours, and
at what hour each began and ended, but also their dif-
ferent velocity. See Mem. Acad. Scienc. an. 1734.

In the Philos. Trans, another anemometer is described,
in which the wind being supposed to blow directly against
a flat side, or board, which moves along the graduated
arch of a quadrant, the number of degrees it advances
shows the comparative force of the wind.

In the same Transactions, for the year 1766, Mr. Alex.
Brice describes a method, successfully practised by him,
of measuring the velocity of the wind, by means of that
of the shadow of clouds passing over a plane upon the
earth.

Also in the same Transactions, for the year 1775, Dr.
Lind gives a description of a very ingenious portable
Wind-Gauge, by which the force of the wind is easily
measured ; and which seems to have been suggested by a
hint of Dr. Hales, in his Description of Ventilators, pa. 12,
and described in the Huetiana, long before Dr. Lind's
time. A brief description of the principal parts of it here
follows. This simple instrument consists of two glass
tubes, ab, CD (Plate iii. fig. 4), which should not be less
than 8 or 9 inches long, the bore of each being about -^
of an inch diameter, and connected together by a small
bent glass tube ab, only of about -^ of an inch diameter,
to check the undulations of the water caused by a sudden
gust of wind. On the upper end of the leg ab is fixed at
right angles to it a thin metal tube, and having its mouth
open to receive the wind blowing horizontally into it.
The two tubes, or rather the two branches of the tube,
arc connected to a steel spindle kl, by slips of brass near
the top and bottom, on the sockets of which at e mu\f,
by the wind blowing in at the orifice at iy the whole in-
strument, turns easily about the spindle, which is fixed
into a block by a screw in its bottom. When the instru-

A N E

[ H7 ]

A N G

ment is used, a quantity of water is poured in, till the
tubes are about half full; then exposing the instrument
to the wind, by blowing in at the orifice f, it forces the
water down lower in the tube ab, and raises it so much
higher in the other tube; then the distance between the
surfaces of the water in the two tubes, estimated by a scale
of inches and parts hi, placed by the sides of the tubes,
will be the height of a column of water whose weight is
equal to the force or momentum of the wind, blowing or
striking against an equal base. And as a cubic foot of
water weighs 1000 ounces, or 62-| pounds, the 12th part
of which is 5^- or 5| pounds nearly, therefore for every
inch the surface of the water is raised, the force of the
wind will be equal to so many times 5j- pounds on a
square foot. Thus, suppose the water stand 3 inches
higher in the one tube, than in the other; then 3 times
5-j, or 15-f- pounds, is equal to the pressure or force of the
wind on the surface of a square foot.

This instrument of Dr. Lind's, measures only the force
or momentum of the wind, but not its velocity. How-
ever, the velocity of the wind may be deduced from its
force so obtained, by help of some experiments performed
by me at the Royal Military Academy, in the years 17S6,
1787, and 1788 ; from which it appears that a plane sur-
face of a square foot suffers a resistance of 12 ounces, or
|- of a pound, from the wind, when blowing with a velo-
city of 20 feet per second ; and that the force is nearly
as the square of the velocity. Hence then, taking the
force of 15| pounds, above found, for the force of the
wind when it sustains 3 inches of water, and taking the
square roots of the forces, it will be, as */■% : \/15y : :
20 : o,lj- the 4th proportional, that is a velocity of Ql-j
feet per second, or 62 miles per hour, is the rate or velo-
city at which the wind blows, when it raises the water 3
inches higher in the one tube than the other. And far-
ther, as the said height is as the force, and the force as
the square of the velocity, we shall have the force and
velocity, corresponding to several heights of the water in
the one tube, above that in the other, as in the following
table.

Table of the corresponding height of water, force of the wind
on a square foot, and its velocity per hour.

In one instance, Dr.
Lind found, that the
forceofthe wind was such
as to be equal to 34-;%
pounds, on a square
foot; and this by pro-
portion, in the follow-
ing table, will be found
to answer to a velocity
of 93 miles per hour.

Height of

Force of

Velocity of
wind per nour.

water.

wind.

Inches.

Pounds.

Wiles.

of

13..

18-0

of

2-6

25-6

1

5-2

36-0

2

104

50-8

3

15-6'

62-0

4

20-8

7C0

5

26-0

80-4

6

31-25

880

7

36-5

95-2

8

41-7

I0r6

9

46-9

1080

10

52-1

113-6'

11

57-3

119-2

12

62-5

1240

Sec also Wind- Gauge.

ANEMOSCOPE, is sometimes used to denote a ma-
chine invented to foretell the changes of the wind, or
weather ; and sometimes for an instrument showing by an
index what the present direction of the wind is. Of this

latter kind, it seems, was that used by the ancients, as
described by Vitruvius ; and we have many of them at
present in large or public buildings, where an index with-
inside a room or hall, points to the name of the quarter
from which the wind blows without; which is simplv ef-
fected by connecting an index to the lower end of the spin-
dle of a weather-cock.

It has been observed that hygroscopes made of cat-out
or such like, prove very good anemoscopes ; seldom failing
by the turning of the index, to foretell the shifting of the
wind. See accounts of two different anemoscopes; one
by Mr. Pickering, vol.43 Philos. Trans, the other by Mr.
B. Martin, vol.2 of his Philos. Britan.

Otto Guericke also gave the title anemoscope to a ma-
chine invented by him to foretell the change of the wea-
ther, as to rain and fair. It consisted of the small wooden
figure of a man, which- rose and fell in a glass tube, as the
atmosphere was more or less heavy. Which was only an
application of the common barometer, as shown by M.
Couriers in the Acta Eruditorum for l6"84.

ANGLE, Angulus, in Geometry, the opening or mutual
inclination of two lines, or two planes, meeting in a point,
called the vertex or angular point. Such as the angle
formed by, or between the two lines ab and Ac, at the
vertex or angular point a. — Also the two lines ab and ac^
are called the legs or the sides of the angle.

A e EC I W &

Angles are sometimes denoted, or named, by the singles
letter placed at the angular point, as the angle a ; and some-
times by three letters, as kac, placing always that of the
vertex in the middle. The former method is used when
only one angle has the same vertex;, and the latter method
it is necessary to use when several angles have the same
vertex, to distinguish them from one another.

The measure of ah angle, by which its quantity or mag-
nitude is expressed, is an arch, as be, described from the
centre a, with any radius at pleasure, and contained be-
tween its legs ab and AC. — Hence angles are compared
and distinguished by the ratio of the arcs which subtend
them, to the whole circumference of the circle; or by the
number of degrees contained in the arc de by which they
are measured, to 36o, the number of degrees in the whole
circumference of the circle. And thus an angle is said to
be of so many degrees as are contained in the arc de, by
which it is measured.

Hence it matters not with what radius the are is de-
scribed, for the arcs de, de, being similar, they have the-
same ratio as their respective radii or circumferences, and
therefore they contain the same number of degrees. —
Hence it follows, that the quantity or magnitude of the
angle remains still the same, though the legs be ever so
much increased or diminished. — And thus, in similar
figures, the like or corresponding angles are equal'.

The taking or measuring of anglesT is an operation of
great use and extent in surveying, navigation, geography,
astronomy, &c. And the instruments chiefly used for
this purpose, are quadrants, sextants, octants, theodolites,
circumfcrentors, &c. Mr. Hadley invented an excellent
instrument for taking the larger kind of angles,, where'

A N G

[ H8 ]

A N G

much accuracy is required, or where the motion of the
object, or any circumstance causing an unsteadiness in
the common instruments, renders the observations diffi-
cult, or uncertain. And Mr. Dollond contrived an in-
strument for measuring small angles: See Hadlei/'s Qua-
drant, Micrometer, and the Philos. Trans. Numbers 420,
425, and vol. 48.

To measure the Quantity of an Angle.

1. On paper. Apply the centre of a protractor to the
vertex a of the angle, so that the radius may coincide
with one leg, as" ab ; then the degree on the arch that is
cut by the other leg ac, will give the measure of the
angle required.

Or thus, by a line of chords. Take off the chord of
60 with a pair of compasses ; and with that radius, from
the centre a, describe an arc, as de. Then take this
arc de between the compasses, and apply the extent to
the scale of chords, which will give the degrees in the
angle as before.

M. De Lagny gave, in several memoirs of the Royal
Academy of Sciences, a new method of measuring angles,
which he called Goniometry. The method consists in mea-
suring with a pair of compasses, the arc which subtends
the proposed angle ; not by applying its extent to a pre-
constructed scale of chords, but in the following manner:
From the angular point as a centre, with a pretty large
radius, describe a circle, producing one leg of the angle
backwards to cut off a semicircle ; then search out what
part of the semicircle the arc is which measures the given
angle in this manner ; viz. take the extent of this arc with
a very fine pair of compasses, and apply it several times
to the arc of the semicircle, to find how often it is con-
tained, with a small part remaining over ; in the same
manner take the extent of this small part, and apply it to
the first arc, to find how often it is contained in it ; and
what remains this 2d time, apply in like manner to the
first remainder; then the 3d remainder apply to the 2d,
and so on, always counting how often the last remainder
is contained in the next foregoing, till nothing remain,
or till the remainder is insensible, and too small to be
measured. Then, beginning at the last, and returning
backwards, make a series of fractions, of which the nu-
merators are always 1, and the denominators are the
number of times each remainder is contained in its next
remainder, with the fractional part more, as derived from
the following remainder ; then the last fraction, thus ob-
tained, will show what part the given angle is of ISO0, or
the semicircle ; and being turned into degrees &c, will be
the measure of the angle, and nearer, it -is asserted, than
it can be obtained by any other means ; whether it be
measuring, or calculating by trigonometrical tables. —
Thus, if it be required to measure the angle gfh: With
a large radius describe the semicircle giii, meeting the
leg fg produced in i ; then take the extent of the arc
on in the compasses, and applying it from a upon the se-
micircle, suppose it contains 4 times to the point 4, and
the part 4i over; take 4i and apply it from H to 1, so
that hg contains 4i once, and 1g over; also apply this re-
mainder to the former 4i, and it is contained 5 times, from
4 to 5, and 5l over; and lastly the remainder 5i is just
twice contained in the former remainder 1G or 12, with-
out any remainder. Here then, the series of quotients, or
numbers of times contained, arc 4, 1, 5, 2; therefore, be-
ginning at the last, the first fraction is •£, or the last re-
mainder is half the preceding one; and the 2d fraction is

-z or -X- • the 3d is — or -H ; and the fourth is — or if ;

5§ , . ]fl 4TL

that is, the arc gh is |i of a semicircle, or the angle
gfh is -if of two right angles, or of 180°, which is equi-
valent to 37 j- degrees, or 37° 8' 34"f.

2. On the ground. Place a surveying instrument with
its centre over the angular point to be measured, turning
the instrument about till 0, the beginning of its arch, fall
in the line or direction of one leg of the angle; then
turn the index about to the direction of the other leg, and
it will cut off from the arch the degrees answering to the
given angle.

To plot or lay down any given angle, either on paper
or on the ground, may be performed in the same manner;
and the method is farther explained under the articles
Plotting and Protracting, also under the names- of
the several instruments.

To bisect a given angle, as suppose the angle lkm. From
the centre k, with any radius, describe the arc isi ; then
with the centres L and m, describe two arcs intersecting
in n ; and draw the line kn, which will bisect the given
angle lkm, dividing it into the two equal angles lkn,
mkn.

To trisect an angle, see Trisection, and Angular
Sections.

• Angles are of various kinds and denominations. With
regard to the form of their legs, they are divided into
rectilinear, curvilinear, and mixed.

Rectilinear, or right-lined Angle, is that whose legs are
both right lines; as the foregoing angle mkl.

Curvilinear Angle, is that whose legs are both of them
curves.

Mixt, or mixtilinear Angle, is that of which one leg is
a right line, and the other a curve.

With regard to their magnitude, angles are again di-
vided into right and oblique, acute and obtuse.

Right Angle, is that which is formed by one line per-
pendicular to another ; or that which is subtended by a
quadrant of a circle. As the angle bac. — Therefore the
measure of a right angle is a quadrant of a circle, or 90°;
and consequently all right angles are equal to each other.

Oblique Angle, is a common name for any angle that
is not a right one; and it is either acute or obtuse.

Acute Angle, is that which is less than a right angle,
or less than 00 degrees; as the angle bad.
. Obtuse Angle, is greater than a right angle, or whose
measure exceeds 90 degrees ; as the angle bae.

With regard to their situation in respect of each other,
angles are distinguished into contiguous, adjacent, vertical,
opposite, and alternate.

Contiguous Angles, are such as have the same vertex,
and one leg common to both. As the angles bad, cad,
which have ad common.

Adjacent Angles, are those of which a leg of the one
produced forms a leg of the other: as the angles GFH and
IFH, which have the legs if and fg in a straight line. —
Hence, adjacent angles are supplements to each other,
making together 180 degrees. And therefore if one of

A N G

[ "9 ]

A N G

these be given, the other will be known by subtracting
the given one from 180 degrees. Which property is use-
ful in .surveying, to find the quantity of an inaccessible
angle; viz, measure its adjacent accessible one, and sub-
tract this from ISO degrees.
H/

Vertical or opposite 'Angles, are such as have their
legs mutually continuations of each other; as the two
angles a and b, or c and d. — The property of these is,
that the vertical or opposite angles are always equal to
each other, viz, /ia-= /.b, an'd Z.c = Ad. And hence
the quantity of an inaccessible angle of a field, &c, may
be found, by measuring its accessible opposite angle.

Alternate Angles, are those made on the opposite
sides of a line cutting two parallel lines; so, the angles
eand/, or, g and h, are alternates. And these are always
equal to each other; viz, the/.e =A/j or ^-g = £-h.

External Angles, are the angles of a figure made
without it, by producing its sides; as the angles i, k, l,m.
All the external angles of any right-lined figure, taken to-
gether, are equal to 4 right angles; and the external an-
gle of a triangle is equal to both the internal opposite
ones taken together; also any external angle of a trape-
zium inscribed in a circle, is equal to the internal oppo-
site angle.

Internal Angles, are the angles within any figure, made
by the sides of it ; as the angles n, o, p, q. — In any right-
lined figure, an internal angle as n, and its adjacent ex-
ternal angle k, together make two right angles, or 180
degrees; and all the internal angles ji, o, p, q, taken to-
gether, make twice as many right angles, wanting 4, as
the figure has sides ; also any two opposite internal an-
gles of a trapezium inscribed in a circle, taken together,
make two right angles, or 180 degrees.

Homologous, or like Angles, are such, angles in two
figures, as retain the same order in both.

Angle at the centre, is that whose vertex is in the cen-
tre; as the angle afc, formed by two radii af, fc, and it
is measured by the arc a DC. — An angle at the centre, as
afc, is always double the angle abc at the circumfe-
rence, standing on the same arc adc ; and all angles at
the centre are equal that stand upon the same or equal
arcs : also all angles at the centre arc proportional to the
arcs they stand upon ; and so also are all angles at the
circumference.

Angle at the circumference, is that whose vertex is
somewhere in the circumference of a circle; as the an-
gle ABC.

Angle in a segment, is that whose legs meet the extre-
mities of the base of the segment, and its vertex is any-
where in its arch ; as the angle b is in the segment abc ;
and is comprehended between two chords ab and bc. —
An angle at the circumference is measured by half the
arc adc on which it stands; and all the angles abc, aec,
in the same segment, are equal to each other.

Angle in a semicircle is an angle at the circumference
standing on a semicircle, or on a diameter. — An angle in
a semicircle, is a right angle ; in' a greater segment, the
angle is less, and in a less segment is greater, than a rioht
angle.

Angle of a segment, is that made by a chord with a
tangent, at the point of contact. So ihk is the angle of
the less segment imh, and ihl, the angle of the greater
segment inh. — And the measure of each of these angles,
is half the alternate or supplemental segment, or equal
to the angle in it; viz, the /.ihk = Z.INH, and the

^.IHL == Z.IMH.

Angle out of the centre, as g, is one whose vertex is
not in the centre of the circle. — And its measure is half
the sum -%a + ^b of the arcs intercepted by its legs when
it is within the circle, or half the difference \a — $b
when it is without.

Angle of a semicircle, is the angle which the diameter
of a circle makes with the circumference. Euclid de-
monstrates that this is less than a right angle, but greater
than any acute angle.

Angle of contact, is that made, by a curve line and a
tangent to it, at the point of contact; as the angle ihk.
It is proved by Euclid, that the angle of contact between
a right line and a circle, is less than any right-lined angle
whatever; though it does not therefore follow that it is
of no magnitude or quantity. This has been the subject
of great disputes among geometricians, in which Peleta-
rius, Clavius, Taquet, Wallis, &c, bore a considerable
share; Peletarius and Wallis contending that it is no an-
gle at all, against Clavius, who rightly asserts that it is
not absolutely nothing in itself, but only of no magnitude
in comparison with a right-lined angle, being a quantity
of a different kind or nature; like as a line in respect to
a surface, or a surface in respect to a solid, &c. And
since his time, it has been proved by sir Isaac Newton,
and others, that angles of contact can be compared to
each other, though not to right-lined angles, and wJiat.

A N G

C 120 ]

A N G

the proportions are which they bear to each other. Thus, Angle, in Trigonometry. See Triangle, Trigono-

the circular angles of contact ihk, ihl, are to each metry, Sinu, Tangent, Sfc.

other in the reciprocal subduplicate ratio of the diameters Angle, in Mechanics. — Angle of Direction, is that

hm, hn. And hence the circular angle of contact may which is comprehended between the lines of direction of

be divided, by describing intermediate circles, into any two conspiring forces.

number of parts, and in any proportion. And if, instead Angle of Elevation, is that which is comprehended

of circles, the curves be parabolas, and the point of con- between the line of direction and any plane on which the

tact H or o the common vertex of their axes; the angles projection is made, whether horizontal or oblique.

of contact will then be reciprocally in the subduplicate An gle of Incidence, is that made by the line of di-

ratio of their parameters. But in such elliptical and hy- rection of an impinging body, at the point of impact. As

perbolical angles of contact, these will be reciprocally in the angle abc.

the subduplicate of the ratio compounded of the ratios Angle of Reflection, is that made by the line of direc-

of the parameters, and the transverse axes. Moreover, if tion of the reflected body, at the point of impact. As the

toq be a common parabola, to the axis op, and tangent angle dbe.

vow, and whose equation is 1 x i=js, on —y*> where Instead of the angles of incidence and reflection bein<*

x is the absciss op, and y the ordinate pq, the parameter estimated from the plane on which the body impinges,

being 1: and if or, os, &c, be other parabolas to the sometimes the complements of these are understood, viz,

same axis, tangent, and parameter, their ordinate y being as estimated from a perpendicular to the reflecting plane;

as the two angles abf and DBF.
F b

K

PR, or PS, &c, and their equations x = y3, x =y\ x = y5,
&c : then the series of angles of contact will be in suc-
cession infinitely greater than each other, viz, the angle
of contact woq infinitely greater than woe, and this in-
finitely greater than wos, and so in infinitum. •

And farther, between the angles of contact of any two
of this kind, may other angles of contact be found ad

infinitum, which shall infinitely exceed each other, and Angle, in OpticsV — Visual or Optic Angle, is the angle
yet the greatest of them be infinitely less than the smallest included between the two rays drawn from the two ex-
right-lined angle. So also x2 =y3, x3 = yi, xi =ys, &c, treme points of an object to the centre of the pupil of the
denote a series of curves, of which every succeeding one eye : as the angle hgi. The apparent magnitude of ob-
makes an angle with its tangent, infinitely greater than the jects, when their distance cannot be compared, is greater
preceding one ; and the least of these, viz, that whose or less, according to the angle under which they appear,
equation is x2 = y3, or the semicubical parabola, is infi- — Objects seen under the same or an equal angle, appear
nitely greater than any circular angle of contact. equal. — But there is an exception to this rule in the case

Angles are again divided into plane, spherical, and of the heavenly bodies, which are really seen under a
solid. less angle in the horizon than when they have attained

Plane Angles, are all those above treated of ; which any sensible degree of elevation (see Refraction);
are defined by the inclination of two lines in a plane, notwithstanding which, it must have been observed by
meeting in a point. every one that they appear larger in the horizon, than in

Spherical Angle, is an angle formed on the surface of any other part of their course ; this becomes still more
a sphere by the intersection of two great circles ; or, it is sensible if we observe the distance of two stars at their
the inclination of the planes of those circles. rising, and again when they are near the zenith, their ap-

The measure of a spherical angle, is the arc of a great parent distance in the former case being considerably
circle of the sphere, intercepted between the two planes greater than in the latter, yet if their angular distance be
which form the angle, and which cuts the said planes at measured by a sextant, it will be found to increase as their
right angles. For their properties, &c, see Sphere, distance from the zenith decreases. (See Apparent Mag-
Spiierical, and Spherical Trigonometry. nitude.)

Solid Angle, is the mutual inclination of more than The least visible angle, or least angle under which a
two planes, or plane angles, meeting in a point, and not body can be seen, according to Dr. Hooke, is one mi-
contained in the same plane; like the angles or corners nute ; but Dr. Jurin shows, that at the time of his debate

with Hevelius on this subject, the latter could probably
discover a single star under so small an angle as 20". But
bodies are visible under smaller angles as they are more
bright or luminous. Dr. Jurin states the grounds of this
controversy, and discusses the question at large, in his
Essay on distinct and indistinct Vision, published in
Smith's Optics, pa. 148, et seq.

Angle of the interval, of two places, is the angle sub-
tended by two lines directed from the eye to those places.
Angle of incidence, or reflection, or refraction, &c. See
the respective words Incidence, Reflection, Re-
fraction, &c.

Angle, in Astronomy. — Angle of Commutation. See
Commutation.

Anclb of elongation, or Angle at the Earth. Sec
Elongation.

of solid bodies. For their measure, properties, &c, see
Solid Angle.

Angles of other less usual kinds and denominations, are
also to be found in some books of Geometry. As,

Horned Angle, angulus cormitus, that which is made by
a right line, whether a tangent or secant, with the circum-
ference of a circle.

Lunular Angle, angulus lunuluris, is that which is
formed by the intersection of two curve lines, the one con-
cave, and the other convex.

Cissoid Angle, angulus cissoides, the inner angle made
by two spherical convex lines intersecting each other.

Sistroid An gle, angulus sistroides, is that which is in
the form of a sistrum.

Pelccoid Angle, angulus pclecoides, is that in the form
of a hatchet.

ANU

[ 121 ]

A N G

Parallactic Angle, or the parallax, is the angle made Angle of the curtin, or of the flank, is the angle giii,

at the centre of a star, or the sun, &c, by two lines drawn, made between the curtin and the flank,

the one to the centre of the earth, and the other to its Angle of the epaule, or shoulder, is the angle agh,

surface. See Parallactic, and Parallax. made by the flank and the face of the bastion.

■ Angle of the position of the sun, of the sun's apparent Angle of the tenaille, or exterior flanking angle, is the

semi-diameter, &c, See the respective words. angle akb, made by the two rasant lines of defence, or

Angle at the sun, is the angle under which the distance the faces of two bastions produced,

of a planet from the ecliptic is seen from the sun. Angle of the counterscarp, is the angle made by the

Angle of the East. See Nonagesimal. two sides of the counterscarp, meeting before the middle

Angle of obliquity, of the ecliptic, or the angle of of the curtin.
inclination of the axis 'of the earth to the axis of the An tus. flanking inward, is the, angle made by the flank-
ecliptic, is now nearly 23° 28'. See Obliquity, and Jng line with the curtin.
Ecliptic Angle forming the flank, is that consisting of one flank

Angle of longitude, is the angle which the circle of a and one demigorge.

star's longitude makes with the meridian, at the pole of
the ecliptic.

Angle of right ascension, is the angle which the circle
of a star's right ascension makes with the meridian at the
pole of the equator.

Angle, in Navigation. Angle of the rhumb, or loxo-
dromic angle. See Rhumb and Loxodromic

Angles, in Fortification, are understood of those form-
ed by the several lines used in fortifying, or making a
place defensible.

These are of two sorts ; real and imaginary. — Real an-

Angle forming the face, is that composed of one flank
and one face.

Angle of the moat, is that made before the curtin,
where it is intersected.

Re-entering, ox re-entrant Angle, is that whose vertex
is turned inwards, towards the place ; as h or l.

Saliant, or sortant Angle, is that turned outwards, ad-
vancing its point towards the field ; as a or g.

Dead Angle, is a re-entering angle, which is not
flanked or defended.

Angle of a wall, in Architecture, is the point or cor-

bv the number of sides in the regular polygon.

A 3

K.

gles are those which actually exist and appear in the ner where the two sides or faces of a wall meet,
works. Such as the flanked angle, the angle of the epaule, ANGUINEAL Hyperbola, a name given by Sir Isaac
angle of the flank, and the re-entering angle of the coun- Newton to four of his curves of the second order, viz,
terscarp. Imaginary, or occult angles, are those which species 33, 3+, 35, 36, expressed by the equation ay2 -+- ey
are only subservient to the construction, and which exist = _ ax3 -+- be" ■+■ ex ■+- d ; being hyperbolas of a ser-
no 'more after the fortification is drawn. Such as the pontine figure. See Curves.

angle of the centre, angle of the polygon, flanking angle, ANGULAR, something relating to, or that hath angles,

saliant angle of the counterscarp, &c. At a distance, angular bodies appear round ; the angles

Angle of, or at, the centre, is the angle formed at the and small inequalities disappearing at a much less distance
centre of the polygon, by two radii drawn from the cen- than the bulk of the body.

tre to two adjacent angles, and subtended by a side of it, ANGULAR Motion, is the motion of a body which

as the angle acb. This is found by dividing 360 degrees moves circularly about a point; or the variation in the

angle described by a line, or radius, connecting a body
with the centre about which it moves. — Thus, a pendu-
lum has an angular motion about its centre of motion ;
and the planets have an angular motion about the sun.—
Two moveable points m and o, of which
the one describes the arc mn, and the
other the arc op, in the same time,
have an equal, or the same angular
motion, though the real motion of the
point o be much greater than that of
the point m, viz, as the arc op is greater than the arc
mn. The angular motions of revolving bodies, as of the
\ / planets about the sun, are reciprocally proportional to

£•'' their periodic times. And they are also, as their real or

Angle of the Polygon, is the angle intercepted between absolute motions directly, and as their radii of motion
two sides of the polygon; as dab, or abe. This is the inversely,
supplement of the angle at the, centre, and is therefore
found by subtracting the angle c from 180 degrees.

' angles into some
or to trisect an

Angular motion is also a kind of compound motion
composed of a circular and a rectilinear motion ; like

Angle of the Triangle, is half the angle of the poly- the wheel of a coach, or other vehicle.

"on; as cab or cba ; and is therefore half the supple- Angular Sections, the dividing ot

ment of the angle c at the centre. ™mb" of eclual PaitsJ as> t0 blsect>

Angle of the Bastion, is the angle fag, made by the angle, &c.

two faces of the bastion. And is otherwise called the
flanked angle.

Diminished Angle, is the angle bag, made by the
meeting of the exterior side of the polygon with the lace
/, a of the bastion.

Vol. I.

Euclid taught how to bisect an angle, viz, in prop. 9,
book 1 ; but his geometry goes not farther. The other
ancients after him endeavoured to trisect an angle; in
which attempt, as Euclid's geometry failed them, they
had recourse to the conic sections, which had been ad-
R

ANN

[ 122 ]

mitted into geometry, by means of which, and a problem
in Apollonius's Inclinations, they effected the trisection;
which was equivalent to solving a cubic equation. Pap-
pus, in the 34th, and following props, of book 4, treats of
the trisection ; and in the 36th et seq. of any section in
general. Many of the moderns have also treated of an-
gular sections ; as, Vieta, Alex. Anderson, Wallis, Briggs,
l'Hopital, Guisne, Emerson, &c. See Section and
Trisection, &c.

ANNUAL, in Astronomy, something that returns every
year, or which terminates with the year.

Ann ual motion of the earth. See Earth.

Annual argument of longitude. See Argument.

Annual epacts. See Epact.

Annual equation of the mean motion of the sun and
moon, and of the moon's apogee and nodes. See Equa-
tion.

ANNUITIES, any periodical incomes, arising from
money lent, or from houses, lands, salaries, pensions, &c;
payable from time to time ; either annually, or at other
intervals of time.

Annuities may be divided into such as are certain, and
such as depend on some contingency, as the continuance
of a life, &c. See Life Annuities.

Annuities are also divided into annuities in possession,
and annuities in reversion; the former meaning such as
have commenced ; and the latter such as will not com-
mence till some particular event has happened, or till
some given period of time has elapsed.

Annuities may be farther considered as payable either
yearly, or half yearly, or quarterly, &c.

The present value of an annuity, is that sum, which,
being improved at interest, will be sufficient to pay the
annuity.

The present value of annuity certain, payable yearly,
is calculated in the following manner. — Let the annuity
be 1/'. and let r denote the amount of \l. for a year, or 1/.
increased by its interest for one year. Then, 1 being the
present value of the sum r, and having to find the present
value of the sum 1, it will be, by proportion thus, r : 1 : :

1 : - the present value of 1/. due a year hence. And for

the same reason - is the present value of a pounds due a

year hence; the present value being always in direct pro-
portion to the yearly payment. So that, in what follows,
we shall only consider the annuity as 1/. ; and whatever
results are thus obtained, being multiplied by a, will cor-
respond to an annuity of a pounds.

Now since- is the present value of 11, due a year

hence, so likewise -3 will be the present value of 1/. due 2

years hence ; for r : 1 : : - : — . So also -j, -^ -r, &c,

will be the present value of 1/. due at the end of 3, 4, 5,

&c, years, respectively ; and in general, — will be the

value of li. to be received after the expiration of n years.

Consequently the sum of all these, or-H--;-t--T-+--i-t-
1 "-. " ' r 1 r1 r

&c, continued to n terms, will be the present value of all

then years annuities. And the value of the perpetuity,

is the sum of the series continued to infinity.

But this scries, it is evident, is a geometrical progres-

sion, the first term and common ratio being each -, and

the number of its termsw; therefore the sum s of all the
terms, or the present value of all the annual payments,

■*" r— 1 r— 1 ,jr» '

When the annuity is a perpetuity, it is plain that the

last quantity — vanishes, and therefore x — also va-

nishes ; consequently the expression becomes barely « =
; that is, is the present value of the perpetuity of

an annuity of 1/. per aim. and consequently will be

the present value of the perpetuity of a pounds per ann.;
hence any annuity divided by the interest of 1/. for one
year, is the value of the perpetuity. So, if the rate of in-
terest be 5 per cent, and yearly paj'ments, then xs*a- a =
20a is the value of the perpetuity. Also J-f- a = 25a is
the value of the perpetuity at 4 per cent. And ^-f^2- a =
33-|a is the value of the perpetuity at 3 per cent, in-
terest. And so on.

If the annuity is not to be entered on immediately, but
after a certain time, as in years; then the present value of
the reversion is equal to the difference between two pre-
sent values, the one of the first term of m years, and
the other for the end of the last term n : that is, equal to

the difference between x —and x

r— 1 r— 1 ?" r— 1 r— 1

1 1 11
— , or = — x ( -).

Annuities certain differ in value, as they are made pay-
able yearly, half-yearly, or quarterly. And by proceeding
as above, using the interest or amount of a half year, or a
quarter, as those for the whole year were used, the fol-
lowing set of theorems will arise ; where r denotes, as be-
fore, the amount of \l. and its interest for a year, and ?i
the number of years, during which, any annuity is to be

paid ; also P denotes the perpetuity , Y denotes

x —the present value of the annuity supposed

payable yearly, 11 the present value of the same when it is
payable half-yearly, and o_ the same when payable quar-
terly ; or universally, m the present value when it is pay-
able every m part of a year.

Theor. 1. y = p — p

Theor. 2. 11 = p — p

Theor. 3. q = p — p

Theor. 4. m = p — r

r + m—\'

Exam. 1. — Let the rate of interest be 4 per cent, and
■the term 5 years; and consequently r = 1,04, n = 5, p
= 25a ; also let m = 12, or the interest payable monthly
in theorem 4 : then the present value of an annuity of a
pounds a year, for 5 years, according as it is supposed
payable yearly, or every half year, or every quarter, or
every month or -jij-th part of a year, will be as follows:
y =3 25 — 25 x -821.028 = 4'451S
11 = 25 - 25 x -S20348 = 4-4,013
q = 25 — 25 x -81.0543 = 4-5114
M = 25 - 25 x -8180.06 = 4-5251
Exam. 2. Supposing the annuity to continue 25

ANN

[ 123 ]

ANN

years, the rate of interest and every thing else being as be-
fore; then the values of the annuities for 25 years will be
y = 25 - 25 x -375118 = 15-6221
h = 25 - 25 x -371527 = 15-7118
q = 25 - 25 x -363709 = 15-7573
m = 25 - 25 x -36S477 = 157881
Exam. 3. And if the term be 50 years, the values
will be

Y = 25 - 25 x -140713 = 21-4822
h = 25 — 25 x -13S032 = 21-5492
q = 25 - 25 x -136685 = 21-5829
m = 25 - 25 x -135775 = 2T6056'

Exam. 4. Also if the terra be 100 years, the values
will be

Y = 25 - 25 x -019800 = 24-5050*
h = 25 - 25 x -019053 = 24'5237
q = 25 - 25 x -018683 = 24'5329
m = 25 - 25 x -018435 = 24-5391

Hence the difference in the value by making periods of
payments smaller, for any given term of years, is the more
as the intervals are smaller, or the periods more frequent.
The same difference is also variable, both as the rate of
interest varies, and also as the whole term of years n
varies ; and, for any given rate of interest, it is evident
that the difference, for any periods m of payments, first
increases from nothing as the term n increases, from 0, to
some certain finite term or value of n, when the difference
d is the greatest or a maximum ; and that afterwards, as
n increases more, that difference will continually decrease
to nothing again, and vanish when n is infinite : also the
term or value of n, for the maximum of the difference,
will be different according to the periods of payment, or
value of m.

Annuities may also be considered as in arrears, or as
forborn, for any number of years ; in which case each
payment is to be considered as a sum put out to interest
for the remainder of the term after the time it becomes
due. And as ]/. due at the end of 1 year, amounts to r
at the end of another year, and to r2 at the end of the 3d
year, and to r3 at the end of the 4th year, and so on ;
therefore, by adding always the last year's annuity, or 1,
to the amounts of all the former years, the sum of all the
annuities and their interests, will be the sum of the fol-
lowing geometrical series, 1 -t- r -)- r2 -t- r3 -t- r4 -

to r°-i, continued till the last term be r"-1, or till the
number of terms be n, the number of years the annuity is
forborn. But the sum of this geometrical progression is

, which therefore is the amount of 11. annuity for-
born for n years. And this quantity being multiplied by
a instead of 1, will produce the amount for the annuity a.
But the amounts of annuities, or their present values,
are easiest found by the two following tables of numbers
for the annuity of 1/. ready computed from the foregoing
principles.

Table 1. — Continued.

The Amount of an Annuity of it. at Comp. Interest.

<<

at 3 per

3-i per

4 per

4^ per

5 per

6 per

6

cent.

cent.

cent.

cent.

cent.

cent.

6-46841

6-55015

6-63298 6-71689

6-80191

6-97532

7

7-65246

7-77941 7-89829

8-01915

8-14201

8-39384

8

8.89234

9-05169

9-21423

9-38001

9-54911

9-89747

9

10-15911

10-36850

10-58280

10-80211

11-02656

11-49132

10

11-46388

11-73139

12-00611

12-28821

12-57789

13-18079

11

12-80780

13-14199

13-48635

13-84118

14-20679

14-97164

12

14-19203

14-60196

15-02581

15-46403

15-91713

16-86994

13

15-61779

16-11303

16-62684

17-15991

17-71298

18-88214

14

17-08632

17-67699

18-29191

18-93211

19-59863

21-01507

15

18-59891

19-29568

20-02359

20-78405

21-57856

23-27597

16

20-15688

20-97103

21-82453

22-71934

23-65749

25-67253

17

21-76159

22-70502

23-69751

24-74171

25-84037

28-21288

18

23-41444

24-49969

25-64541

26-85508

28-13238

30-90565

19

25-11687

26-35718

27-67123

29-06356

30-53900

33-75999

20

26-87037

28-27968

29-77808

31-37142

33-06595

36-78559

.'1

28-67649

30-26947

31-96920

33-78314

35-71925

39-99273

22

30-53678

32-32890

34-24797

36-30338

38-50521

43-39229

23

32-45288

34-46041

36-61789

38-93703

41-43048

46-99583

24

34-42647

36-66653

39-08260

41-68920

44-50200

50-81558

25

36-45926

38-94986

41-64591

44-56521

47-72710

54-86451

26

38-55304

41-31310

44-31174

47-57064

51-11345

59-15638

27

40-70963

43-75906

47-08421

50-71132

54-66913

63-70577

28

42-93092

46-29063

49-96758

53-99333

58-40258

68-52811

29

45-21885

48-91080

52-96629

57-42303

62-32271

73-63980

30

47-57542

51-62268

56-08494

61-00707

66-43885

79-058 19

31

50-00268

54-42947

59-32834

64-75239

70-76079

84-80168

32

52-50276

57-33450

62-70147

68-66625

75-29883

90-88978

33

55-07784

60-34121

66-20953

72-75623

80-06377

97-34316

34

57-73018

63-45315

69-85791

77-03026

85-06696

104-18375

35

60-46208

66-67401

73-65222

81-49662

90-32031

111-43478

36

63-27594

70-00760

77-59831

86-16397

95-83632

119-12087

3/

66-17422

73-45787

81-70225

91-04134

101-62814

127-26812

38

69-15945

77-02889

85-97034

96-13820

107-70955

135-90421

39

72-23423

80-72491

90-40915

101-46442

1 14-09502

145-05846

40

75-40126

84-55028

95-02552

107-03032

120-79977

154-76197

41

78-66330

88-50954

99-82654

112-84669

127-83976

165-04768

42

8202320

92-60737

104-8 1 960

118-92479

135-23175

175-95054

43

85-48389

96-84863

110-01238

125-27640

142-99334

187-50758

44

89-04841

101-23833

115-41288

131-91384

151-14301

199-75803

45

92-71986

105-78167

121-02939

138-84997

159-70016

212-74351

46

96-50146

110-48403

1 26-87057

146-09821

168-68516

226-50812

47

100-39650

115-35097

132-9453S

153-67263

178-11942

241-09861

48

104-40840

120-38826

139-26321

161-58790

188-02539

256-56452

49

108-54065

125-60185

145-83373

169-85936

198-42666

272-95840

SO

112-79687

130-99791

152-66708

178-50303

209-34801

290-33590

51

117-18077

136-58284

159-77377

187-53566

220-8 154C

308-75606

5 2

121-69620

142-36324

167-1647:.

196-97477

232-85617

328-28142

53

126-34708

148-34595

174-8513

206-83863

245-49895

348-97831

54

131-13750

154-5380C

182-8453(

217-14637

258-7739:,

370-91701

Table 1.

The Amount of an Annuity of ll. at Comp. Interest.

7

ri

at 3 per

s\ per

4 per

4i ijer 1 * per

6 per

cent.

rent.

1-00000

cent. j cent.

cent.

1 -00000

1 -00000

1-00000 1-00000

1-00000

:>

2.03000

2-03500

2-04000

2-04500 2-05000

2-06000

a

3-09090

3-10623

3-12160

3-13703 3-15250

3-18360

A

4-18363

4-21494

4-24646

4-27819 4-31013

4-37462

5

5-30914

5-36247

5-41632

5-470711 5-52563

5-63709

The Use of Table I.
To find the Amount of an annuity forborn any number of
years. Take out the amount from the 1st table, for the
proposed years and rate of interest; then multiply it by
the annuity in question ; and the product will be its
amount, for the same number of years, aud rate of in-
terest.— And the converse, to find the rate or time.

Exam. 1. To find how much an annuity of 50/. will
amount tc in 20 years at 3| per cent, compound interest.
— On the line of 20 years, and in the column of 3i per
cent, stands 28-27968, which is the amount of an annuity
of 1/. for the 20 years; and therefore 28-27968 multiplied
by 50, gives 1413984/. or 1413/. 19s. Sd. for the answer.
Exam. 2. In what time will an annuity of 20/. amount
to 1000/. at 4 per cent, compound interest ? — Here the
amount of 1000/. divided by 20/. the annuity, gives 50,
the amount of 1/. annuity for the same time and rate.

R2

ANN

[ '24 ]

A N O

Then, the nearest tabular number in the column of 4 per
cent, is 49 96758, which standing on the line of 28, shows
that 28 years is the answer.

Exam, 3. If it be required to find at what rate of in-
terest an annuity of 20/. will amount to 1000/. forbom
for 28 years. — Here 1000 divided by 20 gives 50 as be-
fore. Then looking along the line of 28 years, for the
nearest to this number 50, we find 49-96758 in the column
of 4 per cent, which is therefore the rate of interest re-
quired^ ■

Table II.
The present Value of an Annuity of 1 /■

t< at a pe

cer

0-97087
1-91347
2-82861
3-71710
4-57971
5-41719
6-23028
7-01969
7-78611
8-53020
9-25262
9-95400
10-63496
11-29607
11-93794
12-56110
13-1)612
13-75351
14-32380
14-87747
15-41502
15-9369-'
16-44361
16-93554
17-41315
17-87684
18-32703
18-76411
19-18845
19-60044
20-00043
20-38877
20-76579
21-13184

A2 P'r
cent.

0-96618
1-89969
2-80164
3-67308
4-51505
5-32855
6-11454
6-87396
7-60769
8-31661
9-00155
9-66333
10-30274
10-92052
11-51741
12-09412
12-6513
13-18968
13-70984
14-21240
14-69797
15-1671

4 per 4j per
cent. cent.

0-96154
1-88610
2-77509
3-62990
4-45182
5-24214
6-00205
6-73274
7-43533
8-11090
8-76048
9-38507
9-98565
10-56312
11-11839
11-65230
12-16567
12-65930
13-13394
13-59033
14-02916
. 14-45112
15-62041 14-85684
16-05837 15-24696
16-48151

0-95694
1-87267
2-74896
3-58753
4-38998
5-15787
5-89270
6-59589
7-26879
7-91272
8-52892
9-11858
9-68285
10-22283
10-739
11-23402

1 6-89035
17-28536
17-66702
18-03577
18-39205
18-73628
19-06887
19-39021
19-70068

15-32208
15-98277

21-48722 20-00066
21-83225 20-29049
22-16724 20-57053
22-49246 20-84109
39 22-80822 21-10250

23-11477
23-41240
23-70"! 36
23-98 1 90
24-25427
24-51871
24-77545
25-0247 1
2667 I
50 1 66
25-7297,0
25-95J23
26- 1 S624
26-37499

54126-57 76

0-95238
1-85941
2-72325
3-54595
4-32948
5-07569
5-78637
6-46321
7-10782
7-72173
8-30641
8-86325
9-39357
9-89864
10-37966
10-83777

11-70719 11-27407

12-15999 11-68959

12-59329 12-08532

00794 12-46221

21-35507

21-59910

21-83483

2-06269

2-28279

22-49545

22-70092

22-89944

23-09 1 24

23-27656

3-45562

i-62862

1-79576

23-95726

24-11330

16-32959
16-66306
16-98371
17-29203
17-58849
17-87355
8-14765
18-41120
18-66461
18-90828
19-14258
19-36786
19-58448
19-79277
19-99305
20-18563
20-37079
20-54884
20-72004
20-88465
21-04294
21-19513
21-34147
21-48218
21-61749
21-74758
21-87267
21-99296

13-40472
13-78442
14-14777
14-49548
14-82821
15-14661
15-45130
15-74287
16-02189
16-28889
16-54439
16-78889
1 7-02286
17-24676

17-46101 16-37419
17-66604! 16-54685

12-8211,5
13-16300
13-48857
13-79864
14-09394
14-37519
4-6430
14-89813
15-14107
15-37245
15-59281
1 5-80268
16-00255
16-19290

17-86224
18-04999
18-22966
18-40158
8-56611
8-72355
18-87421
19-01838
19-15635
19-28837
19-41471
19-53561
19-65130
19-76201

19-8679
19-96933
20-06634
20-15918

16-71129
16-86789
17-01704
17-15909
17-29437
17-42321
17-54591
17-66277
17-77407
17-88007
17-98102
18-07716
18-1687

0-94340
1-83339
2-67301
3-46511
4-21236
4-91732
5-58238
6-20979
6-80169
7-36009
7-88687
8-38384
8-85268
9-29498
9-71225
10-10590
10-47726
0-82760
11-15812
11-46992
11-76408
•04158
12-30338
12-55036
12-78336
13-00317
13-21053
13-40616
13-59072
13-76483
'13-92909
|14-08404
14-23023
14-36814
14-49825
14-62099
14-73678
14-84602
14-94907
15-04630
15-13802
1 5-22454
15-30617
15-38318
15-45583

18-

.9 1

I8-U898
18-41807
1 8-49340
18-5651 5

15-52437
15-58903
1 5-65003
15-70757
15-76186
15-81308
15-86139
15-90697
15-94998

Tii b Use of Table II.
Exam- 1. To find the present value of an annuity of 50/.
which is to continue 20 years, at 3f- per cent. — By the
table, the present value of 1/. for the same rate and time,

is 14-21240; therefore 14*2124 x 50 = 710-62/. or 710/.
12s. 4d. is the present value sought.

Exam. 2. To find the present value of an annuity of
20/. to commence 10 years hence, and then to continue
for 40 years, or to terminate 50 years hence, at 4 per
cent, interest. — In such cases as this, it is plain we have
to find the difference between the present values of two
equal annuities, for the two given times; which therefore
will be effected by subtracting the tabular value of the
one term from that of the other, and multiplying by the
annuity. Thus,

tabular value for 50 years 21 482 18
tabular value for 10 years 8-1 1090
the difference 13-37128
mult, by - - 20

gives - - 2674256
or - - 267l.%s.6d. the answer.

The foregoing observations, rules, and tables, contain
all that is important in the doctrine of annuities certain.
And for further information, reference may be had to
arithmetical writings, particularly Malcolm's Arithmetic,
page 595; Simpson's Algebra, sect. 16; Dodson's Mathe-
matical Repository, page 298, &c ; Jones's Synopsis, ch.
10 ; Philos. Trans, vol. lxvi, page 109.

For what relates to the doctrine of annuities on lives,
see Assurance, Complement, Expectation, Life
Annuities, Reversions, &c

ANNULETS, in Architecture, are small square mem-
bers, in the Doric capital, placed under the quarter
round.

Annulet is also used for a narrow flat moulding, com-
mon to other parts of a column, as well as to the capital ;
and so called, because it encompasses the column around.
In which sense annulet is frequently used for baguette, or
little astragal.

ANNULUS, a species of Voluta. See also Ring-.
ANOMALISTICAL Year, in Astronomy, called also
periodical year, is the space of time in which the earth,
or a planet, passes through its orbit. The anomalistical,
or common year, is somewhat longer than the tropical
year ; by reason of the precession of the equinox.

And the apses of all the planets have a like progressive
motion ; by which it happens, that a longer time is neces-
sary to arrive at the aphelion, which has advanced a little,
than to arrive at the same fixed star. For example, the
tropical revolution of the sun, with respect to the equinox,
is - - - 365^ 5h 4Sm 45s ;

but the sidereal, or return to the

same star, - 365 6 9 101

and the anomalistic revolution is 305 6 15 20,
because the sun's apogee advances each year 65"| with
respect to the equinoxes, and the sun cannot arrive at the
apooee till he has passed over the 6'5"| more than the re-
volution of the year answering to the equinoxes.

To find the anomalistic revolution, say, -\s the whole
secular motion of a planet minus the motion 61 Us aphelion,
is to 100 years or 3155760000 seconds, so is 300°, (o the
duration of the anomalistic revolution.

ANOMAL* lUS, is something irregular, or lhat deviates,
from the ordinary rule and method of other things of the
same kind.

ANOMALY, in Astronomy, is an irregularity in the
motion of a planet, by which it deviates from the aphelion
or apogee ; or it is the angular distance of the planet from

A N O

[ «S5 ]

A N O

the aphelion or apogee ; that is, the angle formed by
the line of the apses, and another line drawn through the
planet.

Kepler distinguishes three kinds of anomaly ; mean,
eccentric, and true.
- Mean or Simple Anomaly, in the ancient astronomy,
is the distance of a planet's mean place from the apogee.
Which Ptolemy calls the angle of the mean motion.

But in the modern astro-
nomy, in which a planet p
is considered as describing
an ellipse apb about the
sun s, placed in one focus,
it is the time in which the
planet moves from its aphe-
lion a, to the mean place or
point of its orbit.

Hence, as the elliptic

area asp is proportional to ""■ ■■"

the time in which the planet describes the arc ap, that
area may represent the mean anomaly. — Or, if pd be
drawn perpendicular to the transverse axis ab, and meet
the circle in d described on the same axis ; then the
mean anomaly may also be represented by the circular
trilineal asd, wnich is always proportional to the elliptic
one asp, as is proved in my Mensuration, pr 8, page 223,
4th edition — Or, drawing sg perpendicular to the radius
DC produced ; then the mean anomaly is also proportional
to sg ■+■ the circular arc ad, as is demonstrated by Keil
in his Lect. Astron. — Hence, taking dh = so, the arc
ah, or angle ach will be the mean anomaly in practice,
as expressed in degrees of a circle, the number of those
degrees being to 360', as the elliptic trilineal area asp, is
to the whole area <<f the ellipse; the degrees of mean ano-
maly, being those in the arc ah, or angle acii.

Eccentric Anomaly, or of the centre, in the modern
astronomy, is the arc ad of the circle adb intercepted
between the aphelion a, and the point D, determined by
the perpendicular dpe to the line of the apses, drawn
through the place p of the planet. Or it is the angle
ACD at the centre of the circle. — Hence the eccentric
anomalyis to the mean anomaly, as ad to ad -t- sG,oras
ad to ah, or as the angle acd to the angle ach.

True or Equated Anomaly, is the angle asp at the
sun, which the planet's distance ap from the aphelion,
appears under; or the angle formed by the radius vector,
or line sp, drawn from the sun to the planet, with the
line of the apses.

The true anomaly being given, it is easy from it to
find the mean anomaly. For the angle asp, which is the
true anomaly, being given, the point p in the ellipse is
given, and thence the proportion of the area asp to the
whole ellipse, or of the mean anomaly to 360 degrees.
And for this purpose, the following easy rules for practice
are deduced from the properties of the ellipse, by M. de
la Caille, in his Elements of Astronomy, and M. de la
Lande, art. 1240 &c of his Astronomy : 1st, as the square
root of SB the perihelion distance, is to the square root of
sa the aphelion distance, so is the tangent of half the true
anomaly asp, to the tangent of half the eccentric ano-
maly acd. 2d, The difference dh or sg between the
eccentric and mean anomaly, is equal to the product of
the eccentricity cs, by the sine of scg the eccentric ano-
maly just found. And in this case, it is proper to express
the eccentricity in seconds of a degree, which will he

found by this proportion, as the mean distance 1 ; the ec-
centricity : : 206264-8 seconds, or 57° 17' 44",S, in the
arch whose length is equal to the radius, to the seconds in
the arc which is equal to the eccentricity cs; which being
multiplied by the sine of the eccentric anomaly, to radius
1, as above, gives the seconds in sg, or in the arc dh,
being the difference between the mean and eccentric ano-
malies. 3d, To find the radius vector sp, or distance of
the planet from the sun, say either, as the sine of the true
anomaly is to the sine of the eccentric anomaly, so is half
the less axis of the orbit, to the radius vector sp; or as
the sine of half the true anomaly is to the sine of half the
eccentric anomaly, so is the square root of the perihelion
distance sb, to the square root of the radius vector, or
planet's distance sp.

But the mean anomaly being given, it is Hot so easy to
find the true anomaly, at least by a direct process. Kepler,
who first proposed this problem, could not find a direct
way of resolving it, and therefore made use of an indirect
one, by the rule of false position, as may be seen page
695 of Kepler's Epitom. Astron. Copernic. See also § 628
Wolfius Elem. Astron. Now the easiest method of per-
forming this operation, would be to work first for the ec-
centric anomaly, viz, assume it nearly, and from it, so as-
sumed, compute what would be its mean anomaly by the
rule above given, and find the difference between this re-
sult and the mean anomaly given ; then assume another
eccentric anomaly, and proceed in the same way with it,
finding another computed mean anomaly, and itsdifference
from the given one; and treating these differences as in the
rule of position for a nearer value of the eccentric ano-
maly: repeating the operation till the result comes out
exact. Then, from the eccentric anomaly, thus found,,
compute the true anomaly by the 1st rule above laid down.

Of this problem, Dr. Wallis first gave the geometrical
solution by means of the protracted cycloid ; and Sir
Isaac Newton did the same at prop. 31, lib. 1 Princip.
But these methods being unfit for the purpose of the prac-
tical astronomer, various series for approximation have
been given, viz, several by Sir Isaac Newton in his Frag-
menta Epistolarum, page 26", as also in the Schol. to the
prop, above-mentioned, which is his best, being not only
proper for the planets, but also for the cornets, whose or-
bits are very eccentric. Dr. Gregory, in his Astron. lib.
3, has also given the solution by a series, as well as M.
Reyneau, in his Analyse Demontree, p. 713, &c. And a
better still for converging is given by Keil in his Pra?lect.
Astron. p. 375: he says, if the arc ah be the mean ano-
maly, calling its sine e, cosine /, the eccentricity g, also
putting z = ge, and a = 1 +fg; then the eccentric ano-
maly ad will be = — x (1 - &c), supposing r =

57'2£)57S decrees ; of which the first term - is sufficient

for all the planets, even for Mars itself, where the error
will not exceed the 200th part of a degree; and in the
orbit of the earth, the error is less than the 10000th part
of a degree.

Dr. Seth Ward, in. his Astronomia Geometrica, takes
the angle afp at the other focus, where the son is not, for
the mean anomaly, and thence gives an elegant solution.
But this method is not sufficiently accurate when the or-
bit is very eccentric, as in that of the planet Mars, as is
shown by Bullialdus, in his defence of the Philolaic.
Astron, against Dr. Ward. However,, when Newton's cor-

ANT

[ 126 ]

ANT

rection is made, as in theschol. above-mentioned, and the
problem resolved according to Ward's hypothesis, Sir
Isaac affirms that, even in the orbit of Mars, there will
scarce ever be an error of more than one second.

ANSjE, Anses, in Astronomy, those seemingly pro-
minent parts of the ring of the planet Saturn, perceived in
its opening, and appearing like handles to the body of the
planet ; from which appearance the name ansae is taken.

ANSER, in Astronomy, a small star, of the 5th or 6th
magnitude, in the inilky-way, between the eagle and swan,
first brought into order by Hevelius.

ANTARCTIC pole, denotes the southern pole, or
southern end of the earth's axis. — The stars near the an-
tarctic pole never appear above our horizon in these nor-
thern latitudes.

ANTARCTIC circle, is a small circle parallel to the
equator, at the distance of 23° 28' from the antarctic or
south pole. — At one time of the year the sun never rises
above the horizon of any part within this circle; and at
other times he never sets.

ANTARES, in Astronomy, the scorpion's heart; a fixed
star of the first magnitude, in the constellation Scorpio.

ANTECANIS, is used by some astronomers, to denote
the constellation otherwise called Canis Minor, or the
star Procyon. It is so called, as preceding, or being
the forerunner of, the Canis Major, and rising a little be-
fore it.

ANTECEDENT, of a ratio, denotes the first of the
two terms of the ratio, or that term which is compared
with the other. Thus, if the ratio be 2 to 3, or a to b;
then 2 or a is the antecedent.

ANTECEDENTAL Method, is a branch of general
geometrical proportion, or universal comparison, and is
derived from an examination of the Antecedents of ratios,
having given consequents, and a given standard of com-
parison, in the various degrees of augmentation and dimi-
nution, which they undergo by composition and decom-
position. This is a method invented by Mr. James Glenie,
and published by him in 1793 ; a method which he says
he always used instead of the fluxional and differential
methods, and which is totally unconnected with the ideas
of motion and time. See the author's treatise above-
mentioned, and also his Doctrine of Universal Compa-
rison, or General Proportion, 1789, upon which it is
founded.

ANTECEDENTIA, a term used by astronomers when
a planet &c moves westward, or contrary to the order of
the signs aries, taurus, &c. When it moves eastward, or
-according to the order of the signs aries, taurus, &c, it is
then said to move in consequentia.

ANTECIANS, or Antoeci, in Geography, the inha-
bitants of the earth which occupy the same semicircle of
the same meridian, but equally distant from the equator,
the one north and the other south ; as Peloponnesus and
the Cape of Good Hope.

These have their noon, or midnight, or any other hour
at the same time ; but their seasons are contrary, being
spring to the one, when it is autumn with the other ; and
summer with the one, when it is winter with the other;
also the length of the day to the one, is equal to the length
of night to the other.

ANTES, in Architecture, are small pilastrcs placed at
the corners of buildings.

ANTICS, in Architecture, figures of men, beasts, &c,
placed as ornaments to buildings.

ANTICUM, in Architecture, a porch before a door;
also that part of a temple, which is called the outer tem-
ple, and lies between the body of the temple and the
portico.

ANTILOGARITHM, the complement of the logarithm
of a sine, tangent, secant, &c, to that of the radius. This
is found by beginning at the left hand, and subtracting
each figure from 9, but the last figure from 10.

ANT1NOUS, in Astronomy, a part of the constella-
tion aquila, or the eagle.

ANTIOCIIIAN Sect, or Academy, a name given to the
fifth academy or branch of academics. It took its name
from being founded byAntiochus, a philosopher contem-
porary with Cicero ; and it succeeded the Philonian aca-
demy. Though Antiochus was really a stoic, and only
nominally an academic.

Antiochian epocha, a method of computing time from
the proclamation of liberty granted to the city of Antioch,
about the time of the battle of Pharsalia.

ANTIPARALLELS, in Geometry, are those lines
which make equal angles with two other lines, but con-
trary ways; that is, calling the former pair the 1st and
2d lines, and the latter pair the 3d and 4th lines, if .the
angle made by the 1st and 3d lines be equal to the angle
made by the 2d and 4th, and contrariwise the angle made
by the 1st and 4th equal to the angle made by the 2d and
3d; then each pair of lines are antiparallels with respect
to the other, viz, the 1st and 2d, and the 3d and 4th.
Thus, if ab and ac be any two lines, and fc and rz be
two others, cutting them so,

that the angle b is equal to the angle e,
and the angle c is equal to the angle d ;
then EC and t>e are antiparallels with respect to ab and
ac; also these latter are antiparallels with regard to the
two former. — See also Subcontra,ry.

It is a property of these lines, that each pair cuts the
other into proportional segments, taking them alternately,

VIZ, AB

: ac :

: ae :

ad :

: de

and fe

: fc :

: fb :

fd :

: db

ANTIPODES, in Geography, are the inhabitants of
two places on the earth which lie diametrically opposite
to each other, or that walk feet to feet; that is, if a line
be continued down from our feet, quite through the cen-
tre of the earth, till it arrive at the surface on the other
side, it will fall on the feet of our Antipodes, and vice

versa. Antipodes are 180 degrees distant from each

other every way, on the surface of the globe ; they have
equal latitudes, the one north and the other south, but
they differ by 180 degrees of longitude: they have there-
fore the same climates or degrees of heat and cold, with
the same seasons and length of days and nights ; but all
of these at contrary times, it being day to the one, when
it is night to the other, summer to the one when it is win-
ter to the other, &c : they have also the same horizon, the
one being as far distant on the one side, as the other on
the opposite, and therefore when the sun, &c, rises to the
one, it sets to the other. The Antipodes to London are a
part a little south of New Zealand.

It has been said that Flato first started the notion of

A P H

[ 127 ]

API

Antipodes, and gave them the name; which is likely
enough, as he conceived that the earth was of a globular
figure. But there have been many disputes on this point,
the fathers of the church having greatly opposed it, espe-
cially Lactantius and Augustine, who pretended to laugh
at it, and were greatly perplexed to think how men and
trees should hang pendulous in the air with their feet up-
permost, as they thought must be the case in the other
hemisphere.

ANT1SCIANS, or ANTISCII, in Geography, are peo-
ple who dwell in the opposite hemispheres of the earth, as
to north and south, and whose shadows at noon fall in con-
trary directions. This term is more general than antoeci,
with which it is often confounded. The Antiscians stand
contradistinguished from Periscians.

ANTCECI, see Antecians.

APERTURE, .in Optics, is the hole next the object-
glass of a telescope or microscope, through which the
light and the image of the object come into the tube, and
are thence conveyed to the eye. Aperture is also under-
stood of that part of the object-glass itself which covers
the former, and which is left pervious to the rays.

A great deal depends on having a just aperture. — To
find it experimentally: apply several circles of dark pa-
per, of various sizes, on the face of the glass, from the
breadth of a straw, to such as leave only a small hole in
the glass ; and with each of these, separately, view some
distinct objects, as the moon, stars, &c ; then that aper-
ture is to be chosen through which they appear the most
distinctly.

Huygens first found the use of apertures to conduce
much to the perfection of telescopes ; and he found by ex-
perience (Dioptr. prop. 56) that the best aperture for an
object-glass, for example of 30 feet, is to be determined
by this proportion, as 30 to 3, so is the square root of 30
times the distance of the focus of any lens, to its proper
aperture : and that the focal distances of the eye-glasses
are proportional to the apertures. And M. Auzout says
he found, by experience, that the apertures of telescopes
ought to be nearly in the subduplicate ratio of their
lengths. It has also been found by experience that ob-
ject-glasses will admit of greater apertures, if the tubes
be blacked within side, and their passage furnished with
wooden rings.

It is to be noted, that the greater or less aperture of an
object-glass, does not increase or diminish the visible area
of the object; all that is effected by this, is the admit-
tance of more or fewer rays, and consequently the ap-
pearance of the object more or less bright. But the
largeness of the aperture or focal distance, causes the ir-
regularity of its refractions. Hence, in viewing Venus
through a telescope, a much less aperture is to be used
than for the moon, or Jupiter, or Saturn, because her
light is so bright and glaring. And this circumstance
somewhat invalidates and disturbs Auzout's proportion,
as is shown by Dr. Hook, Philos. Trans. No. 4.

Apertuee in Hydraulics. See Hydraulics.

APHELION, or Apiielium, in Astronomy, that point
in the orbit of the earth, or a planrt, in which it is at the
greatest distance from the sun. Which is the point a (in
the fig. to the art. Anomaly) or extremity of the trans-
verse axis, of the elliptic orbit, farthest from the focus s,
where the sun is placed; and diametrically opposite to the
perihelion B, or nearer extremity of the same axis. In the

Ptolemak system, or in the supposition that the sun moves
about the earth, the aphelion becomes the apogee.

The times of the aphelia of the primary planets, may
be known by their apparent diameter appearing the small-
est, and also by their moving slowest in a given time: and
the same of the moon's apogee. Calculations and me-
thods of finding them have been given by many astrono-
mers, as Riccioli, Almag. Nov. lib. 7, sect. 2 and 3; Wol-
fius, Elem. Astron. § 659; Dr. Halley, Philos. Trans.
No. 128; Sir 1. Newton, Princip. lib. 3, prop. 14; Dr.
James Gregory, Astron. lib. 3, prop. 14; Keil, Astron.
Lect. ; De la Lande, Memoires de l'Acad. 1755, 1757, 1766,
and in his Astron. liv. 22; also in the writings of MM. Eu-
ler, D'Alembert, Clairaut, &c, on attraction ; Vince's
Astron.; vol. i, p. 133; and Gregory's Astron. p. 220.

The aphelia of the planets are not fixed ; for their mu-
tual actions on one another keep those points of their
orbits in a continual motion, which is greater or less in
the different planets. This motion is made in consequentia,
or according to the order of the signs ; and Sir I. Newton
shows that it is in the sesquiplicate ratio of the distance
of the planet from the sun, that is, as the square root of
the cube of the distance.

The quantities of this motion, as well as the place of
the aphelion for a given time, are variously given by dif-
ferent authors. Kepler states them, for the year 1700, as
in the following table.

Planets. Aphelion".

Mercury Vf 8° 25' 30"

Venus ZZ 3 24 27

Mars tij 0 51 29

Jupiter =2: S 10 40

Saturn £ 28 3 48

The Earth sb S 25 30

By De la Hire they are given as follows, for the same
year 1700.
Planets. Aphelion. Annual Motion.

Mercury
Venus
Mars
Jupiter
Saturn
And Lalande states them as follows, for the year 1750.

Annual Motion.
1' 45"
1 IS
1 7

0 47

1 10

JCP 13°

3'

40"

1'

39

zz 6

56

10

1

20

nj, o

35

25

1

7

=s= 10

17

14

1

34

t 29

14

41

1

22

Planets
Mercury
Venus
Mars
Jupiter
Saturn
The Earth

Aphelion.

S9 130 33'
10 8

5 1

6 10
8 29

13
28

22

53
3S

Secular Motion.

1° 57' 40"

4 10

1 51

1 43

2 23
1 49

0
40
20
20
TO

Planets.
Mercury
Venus
Mars
Jupiter
Saturn
Herschel

And for the year 1800, they are thus :

Aphelion.
140 20'

7

2

11

29

16

50"

1

4
20
11
31

Secular Motion.

• 33'
21
51
34
50
29

45"

0

40

33

7

APHELLAN, the name of a bright star in the constel-
lation Gemini.

APHRODISIUS, in Chronology, denotes the eleventh
month in the Bithynian year, commencing on the 25th
July in ours.

APIAN or Aitian (Peter), called in Gerrr

A P I

C 128 ]

A P O

newitz, a celebrated astronomer and mathematician, was
born at Leisnig or Leipsick in Misnia, 1495, and became
professor of mathematics at Ingolstadt, in 1524, where he
died in the year 1552, at 57 years of age.

Apian wrote treatises on many of the mathematical
sciences, and greatly improved them ; more especially
astronomy and astrology, which in that age were much
the same thing; also geometry, geography, arithmetic,
&c. He particularly enriched astronomy with many in-
struments, and observations of eclipses, comets, &c. His
principal work was the Astronomicum Osarcum, pub-
lished in folio at Ingolstadt in 1540, and which contains a
number of interesting observations, with the descriptions
and divisions of instruments. In this work he predicts
eclipses, and constructs the figures of them in piano. In
the 2d part of the work, or the Meteoroscopium Planum,
he gives the description of the most accurate astronomical
quadrant, and its uses. To it are added observations of
five different comets, viz, in the years 1531, 1532, 1533,
1538, and 1539; where he, first shows that the tails of
comets are always projected in a direction from the sun.
Apian also wrote a treatise on Cosmography, or Geo-
graphical Instruction, with various mathematical instru-
ments. This work Vossius says he published in 1524,
and that Gemma Frisius republished it in 1540. But
Weidler says he wrote it only in 1530, and that Gemma
Frisius published it at Antwerp in 1550 and 1584, with
observations of many eclipses. The truth may be, that
perhaps all these editions were published.

In 1533 he made, at Norimberg, a curious instrument,
which from its figure he called Folium Populi ; which, by
the sun's rays, showed the hour in all parts of the earth,
and even the unequal hours of the Jews.

In 1534 he published his Inscriptiones Orbis*
In 1540, his Instrumentum Sinuum, sive Primi Mo-
bilis, with 100 problems.

Beside these, Apian was the author of many other
works : among which may be mentioned the Ephemerides
from the year 1534 to 1570: Books upon Shadows:
Arithmetical Centilogues : Books upon Arithmetic, with
the Rule of Coss (Algebra) demonstrated : On Gauging :
Almanacs, with astrological directions: A book on Con-
junctions: Ptolemy, with very correct figures, drawn in
a quadrangular form: Ptolemy's works in Greek: Books
of Eclipses : the works of Azoph, a very ancient astro-
loger: the works of Gebre: the Perspective of Vitello: of
Critical Days, and of the Rainbow: a new Astronomical
and Geometrical Radius, with various uses of Sines and
Chords: Universal Astrolabe of Numbers: Maps of the
World, and of particular countries : &c, &c.

Apian left a son, who many years afterwards taught
mathematics at Ingolstadt, and at Tubinga. Tycho has
preserved (Progymn. p. 6'43) his letter to the Landgrave
of Hesse, in which he gives an opinion on the new star in
Cassiopeia, of the year 1572.

One of the comets observed by Apian, viz, that of 1532,
had its elements nearly the same as of one observed 128$
years after, viz, in l66'l, by Hcvelius and other astro-
nomers; and hence Dr. llalley judged that they were the
same comet, and that therefore it might be expected to
appear again in the beginning of the year 1789- , But it
was not found that it returned at this period, though the
astronomers then looked anxiously for it ; and it is
doubtful whether the disappointment might be owing to
its passing unobserved, or to any errors in the observations

of Apian, or to its period being disturbed and greatly al-
tered by the actions of the superior planets, &c.

APIS, musca, the Bee, or Fly, in Astronomy, one of the
southern constellations, containing 4 stars.

APOGEE, Apogccum, in Astronomy, that point in the
orbit of the sun, moon, &c, which is farthest distant from
the earth. It is at the extremity of the line of the apses;
and the point opposite to it is called the perigee, where
the distance from the earth is the least.

The ancient astronomers, considering the earth as the
centre of the system, chiefly regarded the apogee and pe-
rigee : but the moderns, placing the sun in the centre,
change these terms for the aphelion and perihelion. —
The apogee of the sun, is the same thing as the aphelion
of the earth ; and the perigee of the sun is the same as the
perihelion of the earth.

The manner of finding the apogee of the sun or moon,
is shown by Ricciolus, Almag. Nov. lib. 3, cap. 24; by
Wolfius in Elem. Astr. § 6'l S ; by Cassini, De la Hire, and
many others : see also Memoires de 1'Academie, the
Philos. Trans, vol. 5, 47, &c.

The apogee and perigee, or the higher and lower apses,
would be two fixed points, if the moon, or planet, were
only acted on by the single body about which they revolve
in an elliptic orbit. But, by the compound actions of
the other planets, those points become changed, as if the
orbit itself had a motion about its lower focus. The
moon,, in particular, has her regular orbit about the earth
disturbed, and that in an irregular manner, according to
their different positions with regard to the earth.

The quantity of motion in the apogee may be found by
comparing two observations of it made at a great distance,
of time ; converting the difference into minutes, and divi-
ding them by the number of years elapsed between the
two observations; the quotient gives the annual motion
of the apogee. Thus, from an observation made by Hip-
parchus in the year before Christ 140, by which the sun's
apogee was found o° 30' of n ; and another made by Ric-
ciolus, in the year 1646, by which it was found 7° 26' of
S5 ; the annual motion of the apogee is found to be 1' 2".
And the annual motion of the moon's apogee is about
Is I0« 39' 52".

But the moon's apogee moves unequably. When she
is in the syzygy with the sun, it moves forwards; but in
the quadratures, backwards; and these progressions and
regressions are not equable, but it goes forward slower
when the moon is in the quadratures, or perhaps goes re-
trograde ; and when the moon is in the syzygy, it goes for-
ward the fastest of all. — See the articles Aphelion,
Apsis, and Laplace; also Newton's Theory of the
Moon, for more upon this subject, and his Princip. lib. i.
pr. 43, 44, 45.

APOLLODORUS, a celebrated architect, under Tra-
jan and Adrian, was born at Damascus, and flourished
about the year of Christ 100. He had the direction of
the stone bridge which Trajan ordered to be built over the
Danube in the year 104, which was esteemed the most
magnificent of all the works of that emperor. Adrian,
one day as Trajan was discoursing with this architect
on the buildings he hud raised at Rome, would neods
give his judgment, in which he showed that he knew no-
thing of the matter. Apollodorus turned upon him blunt-
ly, and said to him, Go paint Citruls, for you are very ig-
norant of the subject we are talking on. Adrian at this
time boasted of his painting Citruls well. This was the

A P O

r 129 3

A P O

first step towards the ruin of Apollodorus; a slip which
he was so far from attempting to retrieve, that he even
added a new offence, and that too after Adrian was ad-
vanced to the empire, on the following occasion : Adrian
sent to him the plan of a temple of Venus; and though
he asked his opinion, yet to show that he had no need of
him, and that he did not mean to be directed by it, the
temple was already built. Apollodorus wrote his opinion
very freely, and remarked such essential faults in it, as
the emperor could neither deny nor remedy. He showed
that it was neither high enough nor large enough; that
the statues in it were disproportioned to its bulk : for, said
he, if the goddesses should have a mind to rise and 'go out,
they could not do it. This put Adrian into a great pas-
sion, and prompted him to the destruction of Apollodorus.
He banished him at first; then under the pretext of cer-
tain supposed crimes, of which he had him accused, heat
last put him to death.

APOLLONIUS, of Perga, a city in Pamphylia, was a
celebrated geometrician, who flourished in the reign of
Ptolemy Euergetes, about 2+0 years before Christ ; being
about 16*0 years after Euclid, and 49 years later than Ar-
chimedes. He studied a long time in Alexandria under
the disciples of Euclid ; and afterwards he composed se-
veral curious and ingenious geometrical works, of which
only his books of Conic Sections are now extant, and even
these are not perfect. For it appears from the author's de-
dicatory epistle to Eudemus, a geometrician in Pergamus,
that this work consisted of S books; only 7 of which
however have come down to us.

From tlie Collections of Pappus, and the Commentaries
of Eutocius, it appears that Apollonius was the author
of various pieces in geometry, on account of which he
acepjired the title of the Great Geometrician. His Conies
was however the principal of them. Some have thought
that Apollonius appropriated the writings and discoveries
of Archimedes ; Hcraclius, who wrote the life of Archi-
medes, affirms it; though Eutocius endeavours to refute
him. But though if should be allowed a groundless sup-
position, that Archimedes was the first who wrote upon
conies, yet as his treatise on that subject was greatly
esteemed, it is highly probable that Apollonius would
avail himself of the writings of that author, as well as
others who had gone before him; and, upon the whole,
he has the honour of explaining a difficult subject better
than had been done before ; having made several im-
provements, both in Archimedes's problems, and inEuclid.
His work was doubtless the most perfect of the kind
among the ancients, and in some respects even among the
moderns. Before Apollonius, it had been customary, as
we are informed by Eutocius, for the writers on conies
to require three different kinds of cones to cut the diffe-
rent sections from, viz, the parabola from a right-angled
cone, the ellipse from an acute, and the hyperbola from
an obtuse cone; because they always supposed the sec-
tions made by a plane cutting the cones to be perpen-
dicular to the side of them : but Apollonius cut his sec-
tions all from any one cone, by only varying the incli-
nation or position of the cutting plane; an improvement
that has been followed by all other authors since his time,
lint that Archimedes was acquainted with the. same man-
ner of cutting any cone, is sufficiently proved, against
liuto'oiu's, Pappus, and others, by Guide Ubaldus, in the
beginning of his Commentary on the 2d book of Archi-
uedes's F.quiponderantes, published at Pisa in 1588.

Vol.. I.

The first four books of Apollonius's Conies only have
come down to us in their original Greek language ; but
the next three, the 5th, 6th, and 7th, in an Arabic ver-
sion ; and the 8th not at all. These have been com-
mented on, translated, and published by various authors.
Pappus, in his Mathematical Collections, has left some
account of his various works, with notes and lemmas up-
on them, and particularly on the conies. And Eutocius
wrote a regular elaborate commentary on the propositions
of several of the books.

The first four books were badly translated by Joan.
Baptista Memmius. But a better translation of them in
Latin was made by Commandiue, and published at Bo-
nonia in 1566 ; — Vossius mentions an edition of the Co-
nies in l650 ; the 5th, 6th, and 7th books being recovered
by Golius. — Claude Richard, professor of mathematics in
the imperial college of his order at Madrid, in the year
1632, explained, in his public lectures, the first four
books of Apollonius, which were printed at Antwerp in
l6'55, in folio. — The Grand Duke Ferdinand the 2d, and
his brother Prince Leopold de Medicis, employed a pro-
fessor of the Oriental languages at Rome to translate the
5th, 6th, and 7th books into Latin. These were pub-
lished at Florence in l66l, by Borelli, with his own
notes, who also maintains that these books are the Ge-
nuine production of Apollonius, by many strong autho-
rities, against Mydorgius and others, who suspected that
these three books were not the real production of Apol-
lonius.

As to the 8th, some mention is made of it in a work
of Golius's, where he states that it had not been trans-
lated into Arabic ; because it was wanting in the Greek
copies from which the Arabians translated the others.
But the learned Merscnnc, in the preface to Apollonius's
Conies, printed in his Synopsis of the Mathematics, quotes
the Arabic philosopher A ben Nedin for a work of his
about the year 4-00 of Mahomet, in which is part of that
8th book, and who asserts that all the books of Apol-
lonius are extant in his language, and even more than arc
enumerated by Pappus ; and Vossius says he has read the
same; De Scientiis Mathematicis, pa. 55. — A neat edi-
tion of the first four books in Latin was published by
Dr. Barrow, in 4to, at London in l675. — A magnificent
edition of all the 8 books was published in folio, .by
Dr. Halley, at Oxford in 1710; together with the Lem-
mas of Pappus, and the Commentaries of Eutocius.
The first four in Greek and Latin, but the latter four in
Latin only, the 8th book being restored by himself. — The
other writings of Apollonius, mentioned by Pappus, are,

1. The Section of a Ratio, or Proportional Sections,
two books.

2. The Section of a Space, in two books.

3. Determinate Section, in two books.

4. The Tangencics, in two books.

5. The Inclinations, in two books.
6 The Plane Loci, in two books.

The contents of all these are mentioned by Pappus, and
many lemmas are delivered relative to them ; but none,
or very little of these books themselves have descended
down to the moderns. From the account however that
has been given of their contents, many restorations ha\e
been made of these works, by the modern mathematicians,
as follow : viz,

Vie/a, Apollonius Gallus. The Tangcncies. Puiis,
1600, in4to.

S

A P O [ 130

Determinate Section,

]

A P P

Snellius, Apollonius Batavus.
Lugd. 1601, 4to.

Snellius, Sectio Rationis et Spatii. 1607.
Ghetaldus, Apollonius Redivivus. The Inclinations.
Venice, 16c 7, 4to.

GhetnLnu, Supplement to the Apollonius Redivivus.
Tangencies. l607.

Ghetuldus, Apollonius Redivivus, lib. 2. l6l3.
Alex. Anderson, Supplem. Apol. Redivivi. Inclin. Paris,
1.612, 4to

Alex. Anderson, Pro Zetetico Apolloniani problematis a
se jam pridem edito in Supplemento Apollonii Redivivi.
Paris, 16'15, 4to.

Schooten, Loca Plana restituta. Lug. Bat. 1656.
Fermat, Luca Plana, 2 lib. Tolos. l679, folio.
Halley, Apol. de Sectione Rationis libri duoex Arabico
MS. Latine versi duo restituti. Oxon. 1706, 8vo.
Simson, Loca Plana, libri duo. Glasg. 1749, 4to.
Simson, Sectio Determinat. Glasg. 1776, 4to.
Horsley, Apol Inclinat. libri duo. Oxon. 1770, 4to.
Lawson, The Tangencies, in two books. Lond. 1771, 4to.
Lateson, Determinate Section, two books. Lond. 1772,
4to.

Wales, Determinate Section, two books. Lond. 1772,4to.
P. Giannini, Sectio Determinat. Parma, 1773, 4to.
Burrow, The Inclinations. Lond. 1779* 4to.
APONO (Peter de) a learned astronomer and phi-
losopher, was born at Apono near Padua, about the year
1250. He described the Astrolabium Planum, by which
were shown the equations of the celestial houses, for any
hour and minute, and for any part of the world : it was
published at Venice in 1502. He acquired the name of
the Conciliator, on account of a book of his, in which he
reconciles the writings of the ancient philosophers and
physicians ; the book was published at Venice in 1483.
He resided at Padua, where, from his practising medicine,
and his skill in astronomy, he fell under the suspicion of
magic. He died in 1 3 16, at 66 years of age.

APOPHYGE, in Architecture, is a concave part or
ring of a column, lying above or below the flat member;
and it owes its origin to the ring by which the ends of
wooden columns were hooped, to prevent them from split-
ting.

APOTOME, the remainder or difference between two
lines or quantities which are only commensurable in
power. Such is the difference between 1 and v/2, or the
difference between the side of the square and its diagonal.
The term is used by Euclid ; and a pretty full expla-
nation of such quantities is given in the tenth book of his
Elements, where he distinguishes six kinds of apotomes,
ijnd shows how to find them all geometrically.

Apotomf. Prima, is when the greater term is rational,
and the difference of the squares of the two is a square
number : as the difference 3 — a/ 5.

Apo 1 ome Secundtt, is when the less number is rational,
and the square root of the difference1 of the squares of the.
two terms, has to the greater term, a ratio expressible in
numbers ; such is y/ 18 — 4, because the difference of the
squares 18 mid 16 is 2, and a/2 is to a/1% as a/ 1 to a/9
or as 1 t' 3

' Apoi'nj] e Tenia, is when both the terms are irrational,
and, a^ in ihe -.econd, the square root of the difference
of then sq'uan'S, has to the greater term, a rational ratio:
as y/24 — ^/ 1 S ; for the difference of their squares 24 and
IS is 6, and y'O is to ^/24 as ^1 to v'*) or as l t0 »•

Apotome Siuarta, is when the greater term is a rational
number, and the square root of the difference of the squares
of the two terms has not a rational ratio to it : as 4 —
*/5, where the difference of the squares 16 and 3 is 13,
and v/13 has not a ratio in numbers to 4.

Apotome Siuinta, is when the, less term is a rational
number,and the square root of the difference of the squares
of the two has not a rational ratio to the greater : a^
a/6 — 2, where the difference of the squares 6 _and 4 is
2, and ^/2 to a/6 or y^l to a/3 or 1 to a/3 is not a
rational ratio.

Apotome Sexta, is where both terms are irrational, and
the square root of the difference of their squares has not a
rational ratio to the greater ; as a/6 — »/2 ; where the
difference of the squares 6 and 2 is 4, and ^4 to a/6 ,or
2 to a/6, is not a rational ratio.

The doctrine of apotomes, in lines, as delivered by Eu-
clid in the tenth book, is a very curious subject, ind has
always been much admired and cultivated by all mathe-
maticians who have rightly understood this part of the
elements ; so that Peter Ramus has greatly exposed his
judgment by censuring that book. And the first alge-
braical writers in Europe commonly employed a consi-
derable portion of their works on an algebraical exposition
of it, which led them to the doctrine of surd quantities;
as Lucas de Burgo, Cardan, Tartalea, Stifelius, Peletarius,
&c, &c. See also Pappus, lib. 4, prop. 3, and the introduc.
to lib. 7- And Dr Wallis's Algebra, pa. 109.

Apotome, in Music, is the difference between a greater
and less semitone, bein- expressed by the ratio: if 128 to 125.
APPARENT, that which is visible, or evident to the
eye, or the understanding.

Apparent Conjunction of the planets, is when a right
line, supposed to be drawn through their centres, passes
through the eyeof the spectator,and not through thecentre
of the earth. — And, in general, the apparent conjunction
of any objects, is when they appear or are plao d in the
same right line with the eye.

Apparent lt/turle Diameter, Distance, Horizon, Mag-
nitude, Motion, Place, Time, &c. See the respective sub-
stantives, for the quantity and measure of it.

The apparent state of things, is commonly very different
from their real state, either as to distance, figure, magni-
tude, position, &c, &c. Thus,

Apparent Diameter or Magnitude, as for example of
the heavenly bodies, is not the real length of the diameter,
but the angle which they subtend at the eye, or under
which they appear. And hence, the angle, or apparent ex-
tent, diminishing with the increased distance of the object,
a very small object, as ab, may have tin same apparent
diameter as a very large one fg ; and indeed the objects
have all the same apparent diameter, that are contained
in the same angle, feg. And if these are parallel, the real
magnitudes are directly proportional to their distances.

<;

JE. B D

But the apparent magnitude varies, not only by the di-
stance, but also by the position of it. So, if (he object
cd be changi d from the direct position, to the oblique one
cd, its apparent magnitude would then be only the angle
CErfj instead ol the angle ced.

APP

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A I

B

i#,!V

•. c a 15

If tbe eye e be placed between two parallels ab, cd,
these parallels will appear to converge, or come nearer
and nearer to each other, the farther they are continued
out, till at last they will appear to coincide in that point
where the sight terminates, which will happen when the
optic angle bed becomes equal to about one minute of a
degree, the smallest angle under which a common object
" is distinctly visible. — Also the apparent magnitudes of the
same object fg or bd, seen at different distances, that is
the angles i'eg, bed, are in a less ratio than the recipro-
cal ratio of the distances, or the distance increases in a
greater ratio than the angle or apparent magnitude dimi-
nishes. But when the object is very remote, or the optic
angle is very small, as one degree or thereabouts, the angle
then varies nearly as the distance reciprocally.

But though the optic angle be the usual or sensible mea-
sure of the apparent magnitude of an object, yet habit,
and the frequent experience of looking at distant objects,
by which we know that they are larger than they appear,
has so far prevailed on the imagination and judgment, as
to cause this too to have some share in our estimation of
apparent magnitudes ; so that these will be judged to be
more than in the ratio of the optic angles.

The apparent magnitude of the same object, at the same
distance, is different to different persons, and different ani-
mals, and even to the same person, when viewed in dif-
ferent lights ; all which may be occasioned by the different
magnitudes of the eye, causing the optic angle to differ as
that is greater or less: and since, in the same person, the
more light there comes from an object, the less is the pu-
pil of the eye, looking at that object ; therefore the optic
angle will also be less, and consequently the apparent mag-
nitude of the object. Every one must have experienced
the truth of this, by looking at another person in a room,
and afterwards abroad in the sunshine, when he always
appears smaller than in a room where the light is less. So
also, objects elevated in the air, having more light coming
from them than when they are upon the ground, or near
it, may appear less in the former case than in the latter;
thu* the ball of the cross on the top of St. Paul's church,
which is 6* feet in diameter, appears less than an object of
the same diameter seen at the same distance below, near
the ground. And this is, by some, supposed to be the
reason why the sun and moon appear so much larger when
seen in the horizon, where their beams are weak, than
when they are more elevated, and their light stronger.

This apparent variation in the magnitude of the hea-
venly bodies, which is mentioned under the article optic
angle, is a very singular phenomenon, which has engaged
the attention of many eminent philosophers; and various
attempts have accordingly been made to explain the cause
from which it arises; the most satisfactory of which ap-
pears to be the following.

In judging of the magnitude of a body, we do not es-
timate it from the angle only under which we perceive it,
but also from its distance, and this distance is estimated
by a comparison with that of other objects. Now there
is nothing between us and the moon, when she is near the
zenith, from which to draw that comparison, and there-

fore, deceived by the absence of intermediate bodies, we
involuntarily conclude that she^s nearer to us than when
we observe her rising or setting in the horizon, beyond
hills, valleys, forests, and other objects, which we know to
be at a considerable distance; and hence it happens, that
seeing this, and the other heavenly bodies, always under
the same angle, and conceiving them to be sometimes
near and sometimes more remote, our senses are de*-
ceived, and we attribute to them, alternately, a greafer
and less magnitude.

That this is the cause of the phenomenon is demon-
strable from the following circumstance, that the illusion
ceases as soon as we lose srght of other objects. To prove
this, it will be sufficient to observe any of the heavenly
bodies through a tube, or through a small hole pierced in
a card; in either case, they will appear no larger in the.
horizon than in the zenith, because the smallness of the
aperture prevents us from seeing any other object than
that on which the observation is made, and consequently
all means of comparison are destroyed.

Again, if the eye be placed in a rare medium, and view
an object through a denser, as glass or water, having plane
surfaces; the object will appear larger than it really is:
and contrariwise, smaller. And hence it is that fishes,
and other objects, seen in the water, by an eye in the air,
always appear larger than they are. — In like manner, an
object will appear larger when viewed through a globe
of glass or water, or any transparent convex spherical
segment; and, on the contrary, it will appear smaller

when viewed through a concave of glass or water. See

Magnitude, Apparent; also Robins's Tracts, vol. 2,
p. 225, &c.

Apparent Distance, is that distance which we jud^e
an object to be from us, when seen afar off. This is com-
monly very different from the true distance ; because we
are apt to think that all very remote objects, whose parts
cannot well be distinguished, and which have no other vi-
sible objects near them, are at the same distance from us ;
though perhaps they may be thousands or millions of
miles off; as in the cases of the sun and moon and stars..
The apparent distances of objects are also greatly altered
by the refraction of the medium through which they are
seen.

Apparent Figure, is the figure or shape which aw ob-
ject is seen under when viewed at a distance ; and is often
very different from the true figure. For a straight line,
viewed at a distance, may appear but as a point; a sur-
face, as a line; and a solid, as a surface. Also these mav
appear of different magnitudes, and the surface and solid
of different figures, according to their situation with re-
spect to the eye : thus, the arc of a circle may appear a
straight line; a square, a trapezium, or even a triangle;
a circle, an ellipsis; angular magnitudes, round ; and a
sphere, a circle. Also all objects have a tendency to
roundness and smoothness, or appear less angular, as their
distance is greater': for, as the distance is increased, the
smaller angles and asperities first disappear, by subtend-
ing a less angle than one minute; after these, the next
larger disappear, for the same reason ; and so on conti-
nually, as the distance is more and more increased ; the
object seeming still more and more round and smooth.
So, a triangle, or square, at a great distance, shows only
as a round speck; and' the edge of the moon appears
round and regular to the eye, notwithstanding the hills
and valleys oil her surface. And hence it is also, that
S2

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near objects, as a range of lamps, or such like, seen at
a great distance, appear to be contiguous, and to form one
uniform continued magnitude, by the intervals between
them disappearing, from the smallness of the angles sub-
tended by them.

Apparent Motion, is either that motion which we
perceive in a distant body that moves, the eye at the same
time being either in motion or at rest; or that motion
which an object at rest seems to have, while the eye itself
only is in motion.

The motions of bodies at a great distance, though
really moving equally, or passing over equal spaces in
equal times, may appear to be very unequal and irregular
to the eye, which can only judge of them by the muta-
tion of the angle at the eye. And motions, to be equally
visible, or appear equal, must be'directly proportional to
the distances of the objects moving. Again, very swift
motions, as those of the luminaries, may not appear to be
any motions at all, but like that of the hour hand of a
clock, on account of the great distance of the objects :
and this will always happen, when the space actually
passed over in one second of time, is less than about the
14000th part of its distance from the eye ; for the hour
hand of a clock, and the stars about the earth, move at
the rate of 15 seconds of a degree in one second of time,
which is only the 13751 part of the. radius or distance
from the eye. On the other hand, it is possible for the
motion of a body to be so swift, as not to appear any mo-
tion at all; as when through the whole space it describes
there constantly appears a continued surface o'r solid, as
it were generated by the motion of the object, as when
any thing is whirled very swiftly round, describing a ring,
&c.

Also, the more oblique the eye is to the line which a
distant body moves in, the more will the apparent motion
differ from the true one. So, if a body revolve with an
equable motion in the circumference of the circle abcd

&c, and the eye be at e in the »'. . ^ ™
plane of the circle; as the body
moves from a to b and c, it seems

to move slower and slower along

the line alk, till when the body

arrives at c, it appears at rest at k ;

then while it really moves from c by

D to F, it appears to move quicker

and quicker from k by l to a,

where its motion is quickest of all ;

after this it appears to move slower

and slower from a to n, while the

body moves from v to it : there be-
coming stationary again, it appears

to return from n to a in the straight

line, while its real motion is from

h by i to a in the circle. And thus it appears to move

in the line kn, by a motion continually varying between

the least, or nothing, at the extremes k and N, and the

greatest of all at the middle point a. Or, if the motion

be referred to the concave side of the circle, instead of the

line km, the appearances will be the same.

If an eye move directly forwards Q. P

from e to o, &c; any remote ob-
ject at rest at p, will appear to move

the contrary way, or from r to q,

with the same velocity. But if the

object P move the same way, and e ' o

t^E

with the same velocity as the eye ; it will seem to stand
still. If the object have a less velocity than the eye, it
will appear to move back towards Q with the difference of
the velocities ; and if it move faster than the eye, it will ap-
pear to move forwards from q, with the ;arae difference of
the velocities. And so likewise when the object p moves
contrary to the motion of the eye, it appears to move
backwards with the sum of the motions of the two. And.
the truth of all this is experienced by persons sitting in a
boat moving on a river, or in a wheel-carriage when run-
ning fast, and viewing houses or trees, &c, on the sbore or
side of the road, or other boats or wheel- carriages also i»
motion.

Apparent Place of an object, in Optics, is that in
which it appears, when seen in or through glass, water, or
other refracting mediums; which is commonly different
from the true place. So, if an object be seen in or
through glass, or water, either plane or concave, it will
appear nearer to the eye than its true place ; but when
seen through a convex glass, it appears more remote.

Apparent Place of the Image of an object, in Catoptrics,
is that where the image of an object made by the reflection
of a speculum appears to be; and the optical writers, from
Euclid downwards, give it as a general rule, that this is
where the reflected rays meet the perpendicular to the
speculum drawn from the object: so that if the speculum
be a plane, the apparent place of the image will be at the
same distance behind the speculum, as the eye is before
it ; if convex, it will appear behind the glass nearer to the
same; but if concave, it will appear before the speculum.
And yet in some cases there are exceptions to this rule, as
is shown by Kepler in his Paralipomena in Vitellionem,
prop. 18. See also Wollius Catoptr. § 51, 1SS, 233, 234.

Apparent Place of a Planet, &c, in Astronomy, is
that point in the surface of the sphere of the fixed stars,
where the centre of the luminary appears from the sur-
face of the exirth.

APPARITION, in Astronomy, denotes a star's or other
luminary's becoming visible, which before was hid. So,
the heliacal rising, is rather an apparition than a proper
rising.

APPEARANCE, in Perspective, is the representation
or projection of a figure, body, or the like object, on the
perspective plane.— The appearance of an objective right
line, is always a right line. See Perspective. — Having
given the appearance of an Opake body, and of a lumi-
nary, to find the appearance of the shadow ; see Shadow.
Appearances, in Astronomy, &c, are more usually
called phenomena and phases. — In Optics, the term di-
rect appearance is used for the view or sight of any object
by direct rays; without either refraction or reflection.

APPLICATE, Applicata, Ordinate Appi.icate, in
Geometry, is a right line drawn to a curve, and bisected
by its diameter; This is otherwise called an Ordinate j
which see.

Applicate Number. See Concrete.
Application, the act of applying one thing to ano-
ther, by approaching or bringing them nearer together. So
a longer space is measured by the continual application
of a less; as a foot or yard by an inch, && And motion
is determined by a successive application of any thing to
different parts of space.

Application is sometimes used, both in arithmetic
and geometry, for the rule or operation of division, or
what is similar to it in geometry. Thus, 20 applied to,

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A PP

or divided by 4, gives 5. And a rectangle ab, applied to it; and, making no difference between the known and un-

r • 4, ,., ... . "l> .i r known quantities, consider how they depend upon, or are

a line c, gives the 4th proportional — , or another line , ,M \, ^, ■ J , ■■ i iT »

r ' c' related to, one another; that it may be perceived what

which, with the given line c, will contain another rectan- quantities, if they are assumed, will, by proceeding syn-

gle which shall be equal to the former rectangle ab. And thetically, give the rest, and that in the simplest manner.

this is the sense in which Euclid uses the term, lib. 6, And in this comparison, the geometrical figure is to be

pr. 28. , feigned and constructed at random, as if all the parts were

Application, in Geometry, is also used fof the act or actually known or given; and any other lines drawn that
supposition of putting or placing one figure upon another, may appear to conduce to the easier and simpler solution
to find whifiher they be equal or unequal ; which seems to of the problem. Having considered the method of com-
be the primary way in which the mind first acquires both putation, and drawn out the scheme, names are then to
the idea and proof of equality. And in this way Euclid, be given to the quantities entering into the computation,
and other geometricians, demonstrate some of the first or that is, to some few of them, both known and unknown ;
leading properties in geometry. Thus, if two triangles from which the rest may most naturally and simply be
have two sides in the one triangle equal to two sides in derived or expressed, by means of the geometrical proper-
the other, and also the angle included by the same sides ties of figures, till an equation be obtained, by which the
equal to each other; then are the two triangles equal in value of the unknown quantity may be derived by the
all respects: for by conceiving the one triangle placed on, ordinary methods of reduction of equations, when only
or applied to the other, it is proved that they coincide or one unknown quantity is in the notation ; or till as many
exactly agree in all their parts. And the same happens equations are obtained as there are unknown letters in
if, of two triangles, one side and the two adjacent angles the same.

of the one triangle, are equal, respectively, to one side For example, suppose it were

and the two corresponding angles of the other. Thus also required to inscribe a square in a

it may be proved that the diameter of a circle divides it given triangle. Let abc be the

into two equal parts, as also that the diagonal of a square given triangle; and suppose defg

or parallelogram bisects or divides it into two equal parts, to be the required square; also

Application of one science to another, as of algebra draw the perpendicular bp of the

to geometry, is said of the use made of the principles and triangle, which will be given, as

properties of the one, for augmenting. and perfecting the well asall the sides of it. Then, con- A. D P G- ' ' C

other. Indeed all arts and sciences mutually receive aid sideringthat the triangles bac, bef are similar, it will be

from each other. But the application here meant, is of proper to make the noiation as follows, viz, making the

a more express and limited nature; as will appear by base AC = b, the perpendicular bp = p, and the side of

what follows. the square de or ef = x. Hence then bq = bp — ed

Application of Algebra or of Analysis to Geometry. = p _ x ■ consequently, by the proportionality of the

The first and principal applications of algebra, were to parts of those two similar triangles, viz, bp : AC : : bq :

arithmetical questions and computations, as being the first EF) hisp:b::p — x:x; then, multiply extremes and

and most useful science in all the concerns of human life, means &c, there arises px == bp — bx, or bx -+- px=; bp,

Afterwards algebra was applied to ceometry and all the , bp ' . , ' L \ ,.. ' . '.., " .' .

■ „u • * " tl v »• flu and x =— , the side of the square sought; that is, a

other sciences in their turn. 1 he application of algebra h+p' ,

to geometry, is of two kinds; that which regards the plane fourth proportional to the base and perpendicular, and

or common geometry, and that which respects the higher the sum of the two, taking this sum for the first term, or

geometry, or the nature and properties of curve lines. ac ■+- bp : bp : : ac : ef.

The first of these, or the application of algebra to com- Evitmp. 2. In a right-angled triangle, having given the

mon geometry, is concerned in the algebraical solution of perimeter, or sum of all the sides, and the perpendicular

geometrical problems, and finding out theorems in geo- let fall from the right angle on the hypothenuse, to deter-

metrical figures, by means of algebraical investigations mine the sides of the triangle.

or demonstrations. This kind of application has been Let abc represent the tri-

madc from the time of the earliest writers on algebra, as angle, and bd the given perpen-

Diophantus, the Indians and Arabians, Lucas de Burgo, dicular. Also put bd = p, ab

Cardan, Tartalea, &c, &c, down to the present times. = x, ec -=3/, ac = z, and the

Some of the best precepts and exercises of this kind of sum of the sides = s. Now then

application, are. to be met with in Newton's Universal we have three unknown quanti-

Arithmetic, and in Simpson's Algebra and Select Exer
ciscs. Geometrical problems are commonly resolved more
directly and easily by algebra than by the geometrical
analysis, especially by young beginners; but then the
:/Mhcsis, or construction and demonstration, is most ele-
gant as deduced from the latter method. Now it com-
monly happens, that the algebraical solution succeeds best
in such problems as respect the sides and other lines in
geometrical figures ; while, on the contrary, those pro-
blems in which angles arc concerned, are best effected by
the geometrical analysis. Newton gives these, among
many other remarks on this branch. Having any problem

ties, and therefore we must

establish three equations before it will be possible to pro-
ceed in the solution, which are as follows;

x- -4- y — z: Euclid prop. 47, book I.

xy = pz. Euclid prop. 8, book 4.

x -t- y h- z .— .s. ■ By the question,

From the first and second equation we find .r! -+- Ixy -t-

y = z2 -1- Izp, or x -+- y = s/ (z1 ■+■ 2pz) ; but x •+■ y — s

— ;, by the third equation; therefore, l/'{z* -+-'2pz) =s—z,

or by squaring both sides and transposing (2p + 2s) z =

?, hence s = t^ — . Having thus found the value of z,
'ip-

proposed: compare together the quantities concerned in we may proceed to find x and y, and consider s as

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C 13* ]

A P P

known quantity, only substituting in the final expressions
its value determined as above. This may be done by
means of any two of the original equations, but the first
and second are the most proper for this purpose; from
them we determine at once sr4-f- 2xy -t- y2 = z* ■+■ 2ps, or
x -hy = ^(z2 + 2pz) ; x2 — 2xy ■+■ y2 = z* — 2/>z,or x -y
= </{zl- 2p2); hence x = iK/(z2-t- 2pz) +i>/{^- 2PZ)>
and y = -IVO2 •+- 2pz) — i\Z(.z2 _ ^P3) > and now re'
establishing the value of 2 as determined above, the sides
of the triangle will be as follows:

ab = vtc—) + — ] + V[(— )

];

*&<&?&

vcc— )- — 1;

The other branch of the application of algebra to geo-
metry, was introduced by Descartes, in his Geometry,
which is the new or higher geometry, and relates to pro-
perties of curve lines. In this branch, the nature of the
curve is expressed or denoted by an algebraic equation,
which is thus derived : A line is conceived to be drawn,
as the diameter or some other principal line about the
curve; and on any indefinite points of this, other lines are
erected perpendicularly, which are called ordinates, while
the parts of the first line cut off by them, are called ab-
scisses. Then, calling any absciss x, and its correspond-
ing ordinate y, by means of the known nature, or relations
of the other lines in the curve, an equation is derived, in-
volving x and y, with other given quantities in it. Hence,
as x and y are common to every point in the primary line,
that equation, so derived, will belong to every position or
value of the absciss and ordinate, and so is properly con-
sidered as expressing the nature of the curve in all points
of it; and is commonly called the equation of the curve.
In this way it is found that any curve line has a pecu-
liar form of equation belonging to it, and which is dif-
ferent from that of every other curve, either as to the
number of the terms, the powers of the unknown letters x
and y, or the signs or coefficients of
the terms of the equation. Thus, if
the curve line hk be a circle, of
which hi is part of the diameter, and
IK a perpendicular ordinate : then
putting hi = x, ik =-y, and/) = the
diameter, the equation of the circle will be px — x2 = y2.
But if hk be an ellipse, an hyperbola, or parabola, the
equation of the curve will be different, and for all the four
curves, will be respectively as follows, viz,
for the circle - - px — x2=y2,
for the ellipse - - «r_?r2- ,fi

px —

px ■+■

And,

for the hyperbola

for the parabola - px - - = y2,
where t is the transverse axis, and ]> its parameter.
in like manner for other curves. Sec Absciss.

This way of expressing the nature of curve lines, by al-
gebraic equations, has given occasion to the greatest im-
provement and extension of the geometry of curve lines;
for thus, all the properties of algebraic equations, and
their roots, are transferred and added to curves, whose
abscisses and ordinates have similar properties. Indeed
the benefit of this sort of application is mutual and reci-
procal, the known properties of equations being transferred

ab

b?-

2
a

ab

to the curves that they represent; and, on the contrary,
the known properties of curves are transferred to their re-
presentative equations. See Curves.

Application of Geometry to Algebra. Besides the
use and application of the higher geometry, for obtaining
the nature and roots of equations, and for finding the
values of those roots by the geometrical construction of
curve lines, even common geometry may be made subser-
vient to the purposes of algebra. Thus, to take a very
plain and simple instance, if it were required to square
the binomial a + i; by forming a square, as in the an-
nexed figure, whose side is equal to a ■+■ b, or the two lines
or parts added together denoted by the letters a and b ;
and then drawing two lines parallel to the sides, from the
points where the two parts join, it will be immediately
perceived that the whole square of the
compound quantity a ■+■ b, is equal to
the squares of both the parts, together
with two rectangles under the two
parts, or a2 and b2 and "air, that is the
square of a -t- b is equal to a2 ■+- 61 •+■
2ab, as derived from a geometrical a b

figure or construction. And in this very manner it was,
that the Arabians, and the early F,uropean writers on al-
gebra, derived and demonstrated the common rule for re-
solving compound quadratic equations. And thus also,
in a similar way, it was, that Tartaleaand Cardan derived
and demonstrated all the rules for the solution of cubic
equations, using cubes and parallelopipedons instead of
squares and rectangles. And many other instances might
be given of the use and application of geometry in algebra.
Application of Algebra and Geometry to Mechanics.
This is founded on the same principles as the application
of algebra to geometry. It consists principally in repre-
senting by equations the curves described by bodies in
motion, by determining the equation between the spaces
which the bodies describe, when actuated by any forces,
and the times employed in describing them, &c. A fa-
miliar instance also of the application of geometry to me-
chanics, maybe seen at thearticle Acceleration, where
the perpendiculars of triangles represent the times, the
bases the velocities, and the areas the spaces described by
bodies in motion; a method first given by Galileo. In
short, as velocities, times, forces, spaces, &c, may be re-
presented by lines and geometrical figures; and as these
again may be treated algebraically 1 it is evident how the
principles and properties, of both algebra and geometry,
may be applied to mechanics, and indeed to all the other
branches of mixed mathematics.

Application of Mechanics to Geometry. This consists
chiefly in the use that is sometimes made of the centre of
oravity of figures, for determining the contents of solids
described by those figures. This is called the ccntrobaric
method.

Application of Geometry and Astronomy to Geogra-
phy. This consists chiefly in three articles. 1st, In de-
termining the figure of the" globe we inhabit, by means of
geometrical and astronomical operations. 2d, In deter-
mining the positions of places, by observations of latitudes
and longitudes. 3d, In determining, by geometrical ope-
rations, the position of such places as arc not far distant
from one another.

Geometry and Astronomy are also of great use in Na-
vigation.

A P P

[ 13S ]

A P P

Application of Geometry and Algebra to Physics or
Natural Philosophy. This application we owe to Newton,
whose philosophy may therefore be called the geometri-
cal or majhematical philosophy; and on this application
are founded all the physico-mathematical sciences. Here
a single observation or experiment will often produce a
whole science : so, when we know, as we do by ex-
perience, that the rays of light, in reflecting, make the
angle of incidence equal to the angle of reflection ; we
thence deduce the whole science of catoptrics : for, that

connected by parallels or lines of communication.— The
besieged commonly make counter-approaches, to inter-
rupt and defeat the approaches of the besiegers.

The ancients made their approaches towards the place
besieged, much after the same manner as the moderns.
Folard shows, that they had their trenches, their paral-
lels, saps, &c; which, though usually thought of modern
invention, it appears, have been practised long before, by
the Greeks, Romans, Asiatics, &c.

APPROXIMATION, a continual approach, still nearer

experiment once admitted, catoptrics becomes a science and nearer, to a root or any quantity sought. — Methods

purely geometrical, since it is reduced to the comparison of continual approximation for the square roots and cube

of angles and lines given in position. And the same in roots of numbers, have been employed by algebraists and

many other sciences. arithmeticians, from Lucas de Burgo down to the present

Application of one thins; to another, in general, is time. And the later writers have given various approxi-

employed to denote the use that is made of the former, to mations, not only for the roots of higher powers, or all

understand or to perfect the latter. Thus, the applica- simple equations, but for the roots of all sorts of com-

tion of the cycloid to pendulums, means the use made of pound equations whatever: especially Newton, Wallis,

the cycloidal curve for improving the doctrine and use of Raphson, Halley, De Lagny, Lagrange, &c, &c; all of

pendulums. them forming a kind of infinite series, either expressed or

APPLY. This term is used two different ways, in geo- understood, converging nearer and nearer to the quantity

nietry. sought, according to the nature of the process.

1st, It signifies to transfer or place a given line, either It is evident that if a number proposed be not a true

in a circle or some other figure, so that the extremities of square, that no exact square root of it can be found, ex-

the line shall be in the perimeter of the figure. plicable by rational numbers, whether integers or frac-

2d, It is also used to express division in geometry, or tions: therefore, in such cases, we must be content with

to find one dimension of a rectangle, when the area and
the other dimension are given. As the area ab applied to

the line c, is — .

APPROACH, the curve of equable approach. This

approximations, or coming continually nearer and nearer
to the truth. In like manner, for the cubic and other
roots, when the proposed quantities are not exact cubes,
or other powers.

The most easy and general mode of approximation, for

was a problem first proposed by Leibnitz, namely, to find any power above the square, is perhaps by the rule of
a curve, down which a body descending by the force of Double Position, or, what is sometimes called, the Me-
oravity, shall make equal approaches to the horizon in thod of Trial-and-error; which sec under its own name.

qual portions of time. It has been found by Bernoulli
and others, that the curve is the second cubical parabola,
placed with its vertex uppermost, and which the descend-
in" bou\ must enter with a certain determinate velocity.

-Varignon rendered the question general for any law of

And among all the rules for finding the roots of pure
powers, of which there are many, I believe the best is that
which was discovered by myself, and given in the first vo-
lume of my Mathematical Tracts, (printed in J 786) in
point of ease, both of execution and for remembering

gravity, by which a body may approach towards a given it. Which is this: if n denote any nurnber, out of

point by equal spaces in equal times. And Maupertuis
also resolved the problem in the case of a body descend-
ing in a medium which resists as the square of the velo-
city. See Hist, del' Acad, des Sciences for l6"99and 1730.
Method of Approaches, a name given by Dr. Wallis,
in his Algebra, to a method of resolving certain problems

/hich is to be extracted the root whose index is denoted
by r, and if n be the nearest root first taken; then will

n be the required root of n very

(r + i) N + (r

(r— I) N + (r -t- ljrar

nearly ; or as r — I times the given number added to
r •+• 1 times the nearest power, is to r h- 1 times the given
relating to square numbers, &c. This is done by first number added to r — 1 times the nearest power, so is the
assigning certain limits to the quantities required, and assumed root n, to the required root, very nearly. Then
then approaching nearer and nearer, till a coincidence is this last value of the root, so found, if one still nearer is
obtained. — In this sense, the method of Trial-and-error, wanted, is to be used for n in the same theorem, to repeat

or double rule of False Position, may be considered as a
method of approaches.

APPROACHES, in Fortification, the several works
made by the besiegers, lor advancing or getting nearer to

a fortress or pbice besieged. Such as the trenches, mines, cube, or to find the cube root of the number 2
saps, lodgrrfents; batteries, galleries, cpaulments, &c. -

Approaches, or Lines of Approach, are particularly
used for trenches dug in the ground, and the earth thrown
up on tin side towards the place besieged; under the de-
fence or shelter of which, the besiegers may approach
without lusj, to raise batteries, and plant guns, &c, to
batter u. — The lines of approach are commonly carried
oil, in ii aigzag bum, parallel to the opposite faces of the
besieged work, or nearly so, that they may not be en-
filaded by the guns from the enemy. And they are also

the operation with it. And so on, repeating the operation
as often as necessary. Which theorem includes all the
rational formula; of Halley and De Lugny.

For example, suppose it were required to double the

Here
r = 3 ; consequently r -+- 1=4, and r — I = 2; and

therefore the general theorem becomes — -. x n or

2N 4- 4n

' N + x n for the cube root of n ; or as s + 2»3 : n3

-h n3 : : n : the root sought nearly. Now, in this case,
N = 2, and therefore the nearest root n is 1, and its cube
n3 = 1 also : hence n + 1ni = 2 + 2=4, and 2n -+- n'
= 4+1 = 5; therefore, as 4 : 5 : : 1 : 4 or 14- = 1-25 tho

A P P [ 136 ] A P P

and the terras will be thus;

x2 = 64 -+- l6e -+- :5

— 5x = — 40 — 5c

- 31 = - 31
the sum is — 7" ■+■ 1 1 ~ -+- z° = 0,

gures. And by taking f#§ For a new value of n, and re- Then, rejecting f, it is 1 Is — J — 0, and i
peating the process again, a great many more figures may &c, or = '6' nearly. ■

quently r3 = *-££ ;

hence n

and 2n

therefore as 378

1-259921, which

->- ~n £ ■*■ -54 SS 1

* »3 = 4 + W = \V. „,
381, or as 120 : 127 : : £• : r« T
the cube root of 2, true in all the fi-

7 = -6363,

be found.

Of the Roots of Equations by Approximation.— Ste-
■vinus and Vieta gave methods for finding values, always
nearer and nearer, of the roots of equations. And Ough-
tred and others pursued and improved the same. These
however were very tedious and imperfect, and required a
different process for every degree of equations. But New-
ton introduced, not only general methods for expressing
radical quantities by approximating infinite series, but
also for the roots of all sorts of compound equations what-
ever, which are both easy and expeditious, and which will
be more particularly described under their respective arti-
cles. His method of approximating, is in substance this:
First take a value of the root as near as may be, by trials,
either greater or less ; then assuming another letter to de-
note the unknown difference between this and the true va-
lue, substitute into the equation the sum or, difference of
the approximate root and this assumed letter, instead of
the unknown letter or root of the equation, which will
produce a rfgw equation having only the assumed small
difference for its root or unknown letter; and find, from
this equation, a near value of this small assumed quantity.
Assume then another letter for the small difference be-
tween this last value and the true one, and substitute the
sum or difference of them into the last equation, by which
will arise a third equation, involving the second assumed
quantity; whose value is found as before. Proceeding
thus as far as we please, all the near values, connected to-
gether by their proper signs, will form a series approach-
ing still nearer and nearer to the true root of the first or
proposed equation. The approximate values of the seve-
ral small assumed differences, may be found in different
ways: Newton's method is this: As the quantity sought
is small, its higher powers decrease more and more, and
therefore neglecting them will not lead to any great error.
He therefore neglects all the terms having in them the 2d
and higher powers, leaving only the 1st power and the ab-
solute known term; from which simple equation he al-
ways finds the value of the assumed unknown letter nearly,
in a very simple and easy manner. Halley's method of
doing the same thing, was to neglect all the terms above
the squ;ue or 2d power, and then to find the root of the
remaining quadratic equation ; which would indeed be a
nearer value of the assumed letter than Newton's was, but
then it is much more troublesome to perform. — Raphson
has another way, which is a little varied from that of
Newton's again, which is this: Having found a near value
of the first assumed small quantity or difference, by this
he corrects the first approximation to the root of the pro-
fjpsed equation; and then, assuming another letter for the
next, or smaller difference, he introduces it into the origi-
nal equation in the same way as before. And thus he
proceeds, from one correction to another, employing al-
ways the fust proposed equation to find them, instead of
the successive iu w equations used by Newton.

For example, let it be required to find the. root of the
< q nation x! — j.r = 31, or x- — 5x — 3 1 = 0 : — lleie the
rootx, it is evident, is nearly = 8; for x therefore take
S -+- z, and substitute S ■+ c for x iu the given equation

Assume now z = •& -t- y: the*
z- = -36" -f- V2y -f-jr
11; = 6'6 + Uy

the sum — -04 ■+• \2'2y ■+- y1 = 0,

— where y = = -00327S nearlv. .

j i ■>■■_>

Hence then, collecting all the assumed difference*, with
their signs, it is found that x = 8 -t- z ■+■ y = S -t- -6 ■+■
•003278 = 8-60327S the root of the equation required,
by Newton's method.

The same by Rqpksoa's '•■'■ay,
' First x = S -i- z ;

then x2 = 6'4 -+- 1 6z -+- ;-

— 5x = — 40 — 5s
-31 = - 31

the sum — 7 -+- lis -t-c" = t) :
hence z = -t\ = '6 nearly, a-s before,
and x = 8 +se S'6 nearly.
.Assume it x = S'6 ■+- y;

then x- = 73-96 '■+- 17/23/ ■*? y

— 5x = — 43 — by

— 31 = -31

the sum — -04 ■*■ I2'2y •+- y- = 0 ;

hence y = — = "00327 S nearly,

and x = S'6 ■+- y = S*60327S, as before.
And in the same manner Newton performs the approxi-
mation for the roots of literal equations, that is, equa-
tions having literal coefficients ; so the root of this equation
y1 -+- axy -tr a'y — x3 — 2«3 = 0, is
X xr I31ar 50o.i '
y = a-

&c.

64« 612a" lfi:jS4(i

See also a memoir on this method by the Marquis de
Courtivron, in the Memoires de l'Academie for 1744.

Other Methods of Approximation. Besides the fore-
going general methods, other particular ways of .approxi-
mating, for various purposes, have been given by many
different persons. — As lor example, methods of approxi-
mating, by series, to the roots nf cubic equations belong-
ing to the irreducible case, by Nicole in the same Me-
nioires de l'Acatleniie, by M. Clairaut in his Algebra, and
by myself in the Philos. Trans, for 17S0. See also seve-
ral parts of Simpson's works, or my Tracts vol. f, and my
Course of Mathematics, vol. 1 ; or the treatise on nume-
rical equations by Lagrange. Also the methods of infi-
nite series by \V;illis, Newton, Gregory, Mefcator, &C,
may be considered as approximations, in quadratures, and
other branches of the mathematics, many instances of
which may be seen in N\'ul lis,'s Algebra, and other books :
■ — Likewise the method of exhaustions of the ancients, by
which Archimedes and others have approximated to the
quadrature and rectification of the circle, ecc, which was
performed by continually bisecting the sides of polygons,
both inscribed in a circle and circumscribed about it; by
which means the stun ol thesides of I he like polygons ap-
proach continually liearej antl nearer together, and the
circumference of the circle is nearly a mean between the
two sums. See alsti EQUATIONS.

APS

[ 137 ]

AQ V

APPULSE, in Astronomy, means the near approach
of two luminaries to each other, so as to be seen, for in-
stance, within the same telescope.

The appulses of the planets to the fixed stars have al-
ways been very useful to astronomers, as serving to fix
and determine the places of the former. The ancients,
wanting an easy method of comparing the planets with the
ecliptic, which is not visible, had scarce any other way
of fixing their situations, but by observing their track
among the fixed stars, and marking their appulses to some
of those visible points. See Hist. Acad. Scienc. for 1710,
pa. 417- And Philos. Trans. No. 369, where Dr. Halley
has given a method for determining the places of the pla-
nets, by observing their near appulses to the fixed stars.
See also Philos. Trans. No. 76, pa. 36l, and Mem. Acad.
Scienc. for 1708, where Flamsteed and De la Hire have
given observations of the moon's appulses to the Pleiades.
See also Flamsteed's Historia Coelestis, where a multitude
of observations of appulses, or small distances, of the
moon and planets, from the fixed stars, are recorded. And
Dr. Halley has published a map or planisphere of the
starry zodiac, in which are accurately laid down all the
stars to which the moon's appulse has ever been observed
in any part of the world. See Philos. Trans. No. 369 ;
or my Abridg. vol. 6, pa. 530.

APRIL, the 4th month of the year according to the
common computation, and the 2d from the vernal equi-
nox.— The word is derived from Aprilis, of aperio, I open;
because the earth, in this month, begins to open her bo-
som for the production of vegetables. — In this month the
sun travels through part of the signs Aries and Taurus.

APRON, in Gunnery, a piece of thin or sheet lead,
used to cover the vent or touch-hole of a cannon.

APSES, or Apsides, in Astronomy, are the two points
in the orbits of planets, where they are at their greatest
and least distance from the sun or the earth. The point
at the greatest distance being called the higher apsis, and
that at the nearest distance the lower apsis. Also the higher
apsis is more particularly called the aphelion, or the apo-
gee ; and the lower apsis, the perihelion, or the perigee.
The diameter which joins these two points, is called the
line of the apses or of the apsides ; it passes through the
centre of the orbit of the planet, and the centre of the sun
or the earth ; and in the modern astronomy this line makes
the longer or transverse axis of the elliptical orbit of the
planet. In this line is counted the excentricity of the
orbit ; being the distance between the centre of the orbit
and the focus, where is placed the sun or the earth.

The foregoing definitions suppose the lines of the great-
est and least distances to lie in the same straight line ;
which is not always precisely the case ; as they arc some-
times out of a right line, making an angle greater or less
than ISO degrees, and the difference from 180 degrees is
called the motion of the line of the apses: when the an-
gle is less than 180 degrees, the motion of the apses is said
to be contrary to the order of the signs ; on the other
hand, when the angle exceeds 180 degrees, the motion is
according to the order of the signs.

Different means have been employed to determine the
motion of the apses. Dr. Keil explains, in his Astronomy,
the method used by the ancients, who supposed the orbits
of the planets to be perfectly circular, and the sun out
of the centre. But since it has been discovered that they
describe elliptic orbits, various other methods have been
devised for determining it. Dr. Halley has given one,
Vol. I.

which supposes to be known only the time of the planet's
revolution, or periodic time. Seth Ward has also given a
determination from three different observations of a planet,
in any three places of its orbit: but his method being
founded on an hypothesis not strictly true, Euler has given ■
one much more exact in vol. 7 of the Petersburg Com-
mentaries. See various ways explained in the Astronomy
of Keil and Monnier.

Newton has also given, in the Principia, an excellent
method for determining the same motion, on the supposi-
tion that the orbits of the planets differ but little from
circles, which is the case nearly. That great philosopher
shows, that if the sun be immoveable, and all the planets
gravitate towards him in the inverse ratio of the squares
of their distances, then the apses will be fixed, or their
motion nothing ; that is, the lines of greatest and least
distance will form one right line, and the apses will be di-
rectly opposite, orat 180 degrees distance from each other.
But, because of the mutual tendency of the planets to-
wards each other, their gravitation towards the sun is not
precisely in that ratio ; and hence it happens, that the
apses are not always exactly in a right line with the sun.
And Newton has given \a very elegant method of deter-
mining their motion, on the supposition that we know
the force which is thereby added to the gravitation of the
planet towards the sun, and that this additional force is
always in that direction. Dr. Chas. Walmesley has a trear
tise on theTheoriedu Mouvamens des Apsides, in 8vo.

APSIS. See Apses.

APUS or Apous, Avis Indica, in Astronomy, a constel-
lation of the southern hemisphere, situated near the south
pole, between the triangulum australe and the chameleon,
and supposed to represent the bird of paradise. Also
supposed to be one of the birds named Apodes, as having
no feet.

The number of stars contained in this constellation are,
11 in the British Catalogue, in Bayer's" Maps 12, and a
still greater number in La Caille's Catalogue; the princi-
pal star being but cf the 4th or 5th order of magnitude.
See Ccelum Australe Stelliferum, and the Memoires de
l'Acad. for 1752, pa. 569.

AQUARIUS, in Astronomy, one of the celestial con-
stellations, being the eleventh sign in the zodiac, reckon-
ing from Aries, and is marked by the character zx, repre-
senting part of a stream of water, issuing from the vessel
of Aquarius, or the water-pourer. This sign also gives
name to the eleventh part of the ecliptic, through which the
sun moves in part of the months of January and February.

The poets feign that Aquarius was Ganymede, whom
Jupiter ravished under the shape of an eagle, and carried
away into Heaven to serve as a cup-bearer, instead of
Hebe and Vulcan ; whence the name. Others hold, that
the sign was thus called, because that when it appears in
the horizon, the weather commonly proves rainy.

The stars in the constellation Aquarius are, in Ptole-
my's Catalogue 45, in Tycho's 41, in Hevelius's 47, and
in Flamsteed's 108. See the article Constellation ;
also Catalogue.

AQUEDUCT, or Aquteduct, that is to say, ductus
aqua;, a conduit of water, is a construction of stone «r
timber built on uneven ground, to preserve the level of
water, and convey it, by a canal, from one place to ano-
ther.— Some aqueducts are under ground, being conducted
through hills, &c ; and others are raised above ground, and
supported on arches, to conduct the water ovlil valleys, &c.

A II A

[ 133 ]

ARC

The Romans were very magnificent in their aqueducts ;
having some that extended a hundred miles, or more.
Frontvnus, a man of consular dignity, who had the direc-
tion of the aqueducts under the emperor Nerva, speaks
of nine that emptied themselves through 13,594 pipes, of
an inch diameter. And it is observed by Vigenere, that in
the space of 24 hours, Rome received from these aque-
ducts not less than 500,000 hogsheads of water. The chief
aqueducts now in being, are these : 1st, that of the Aqua
Virginia, repaired by pope Paul iv; 2d, the Aqua Felice,
constructed by pope Sixtus v, and is called from the name
lie assumed before he was exalted to the papal throne;
3d, the Aqua Paulina, repaired by pope Paul v, in the
year l6ll; and 4'tbly, the aqueduct built, by Louis xiv,
near Maintenon, to convey the river Bure to Versailles,
which is perhaps the largest in the world ; being 7000 fa-
thoms long, elevated 2560 fathoms in height, and contain-
ing 242 arcades. See Philos. Trans, for 1685, No. 171;
or my Abridg. vol. 3, pa. l6'7 and 231.

AQUEOUS Humour, or the watry humour of the
eye, is the first or outermost, and the rarest of the three
humours of the eye. It is transparent and colourless,
like water ; and it fills up the space that lies between the
cornea tunica, and the crystalline humour.

AQUILA, the Eagle, or the Vulture as it is sometimes
called, is a constellation of the northern hemisphere, usu-
ally joined with Antinous. It is one of the 48 old con-
stellations, according to the division of which Hipparchus
made his Catalogue of the Fixed Stars, and which are
described by Ptolemy. The number of stars in Aquila,
and those near it, now in the later-formed constellation
Antinous, amount to 15 in Ptolemy's Catalogue, to 19 in
Tycho's, to 42 in that of Hevelius, and to 71 in Flam-
steed's. But in Aquila alone, Tycho counts only 12 stars,
and Hevelius 23 ; the principal star being Lucida Aquila,
and is between the 1st and 2d magnitude. The Greeks,
as usual, relate various fables of this constellation, to make
the science appear as of their own invention.

ARA, the Altar, one of the 4S old constellations, men-
tioned by the ancient astronomers, and is situated in the
southern hemisphere; containing only 7 stars in Ptolemy's
Catalogue, and 9 in that of Flamsteed; none of which
exceed the 4th magnitude.

ARATUS, celebrated for his Greek poem entitled $a<-
veasva, the Phenomena, flourished about the 127th Olym-
piad, or near 300 years before Christ, while Ptolemy Phi-
ladelphia reigned in Egypt. Being educated under Dio-
nysius Heracleotes, a Stoic philosopher, he espoused the
principles of that sect, and became physician to Antigo-
nus Gonatus, the son of Demetrius Poliorcetes, king of
Macedon. The Phenomena of Aratus gives him a title to
the character of an astronomer, as well as a poet. In
this work he describes the nature and motion of the stars,
and shows their various dispositions and relations; he de-
scribes the figures of the constellations, their situations in
the sphere, the origin of the names which they bear in
Greece and in Egypt, the fables which have given rise to
them, the rising and setting of the stars, and he indicates
the manner of knowing the constellations by their respec-
tive situations.

The poem of Aratus has been commented on and trans-
lated by many authors: of whom, among the ancients,
were Cicero, Germanicus Ca;sar, and Festus Avienus, who
made Latin translations of it; apart of the former of
which is still extant. Aratus must have been much es-

teemed by the ancients, since we find so great a number
of scholiasts and commentators on him ; among whom are
Aristarchus of Samos, the Arjstylli the geometricians,
Apollonius, the Evseneti, Crates, Numenius the gramma-
rian, Pyrrhus ot Magnesia, Thales, Zeno, and many others,
as may be seen in Vossius, p. 156; Suidas ascribes seve-
ral other works to Aratus. Virgil, in his Georgics, has
translated or imitated many passages from this author:
Ovid speaks of him with admiration, as well as many
others of the poets. And St. Paul has quoted a passage
from him ; which is in his speech to the Athenians (Acts
xvii. 28.) where he tells them that some of their own poets
have said, "For we are also his offspring," these words
being the beginning of the 5th, line of the Phenomena of
Aratus.

His modern editors are as follow : Henry Stephens pub-
lished his poem at Paris in 1566, in his collection of the
poets, in folio. Grotius published an edition of the Phe-
nomena at Leyden in quarto, l600, in Greek and Latin,
with the fragments of Cicero's version, and the translations
of Germanicus and Avienus ; all which the editor has il-
lustrated with curious notes. Also a neat and correct
edition of Aratus was published at Oxford, 1672, in 8vo.
with the Scholia.

AR.EOMETER, see Areometer.

ARC, or Arch ; which see.

ARCADE, in Architecture, denotes an opening in the
wall of a building formed by an arch.

ARC-BOUTANT, is a kind of arched buttress, formed
of a fiat arch, or part of an arch, abutting against the feet,
or sides of another arch or vault, to support them and
prevent them from bursting or giving way.

ARCH, Arc, Arcus, in Geometry, a part of any curve
line ; as, of a circle, or ellipsis, or the like.

It is by means of circular arcs, or arches, that all an-
gles are measured; the arc being described from the an-
gular point as a centre. For this purpose, every circle is
supposed to be divided into 360 degrees, or equal parts;
and an arch, or the angle it subtends and measures, is es-
timated according to the number of those degrees it con-
tains: thus, an arc, or angle, is said to be of 30 or SO, or
100 degrees. Of late years the French mathematicians
have divided the circle into 400 degrees, or the quadrant
into 100, called the centigrade division. Circular arcs
are also of great use in finding of fluents.

Concentric Arcs, are such as have the same centre.

Equal Arcs, are such arcs, of the same circle, or of
equal circles, as contain the same number of degrees.
These have also equal chords, sines, tangents, &c.

Similar Arcs, of unequal circles, &c, are such as con-
tain the same number of degrees, or
that are the like part or parts of their q\
respective whole circles. Hence, in
concentric circles, any two radii cut
off, or intercept, similar arcs mn and
op. — Similar arcs are proportional to
the radii lm, lo, or to the whole cir-
cumferences.— Similar arcs of other
like curves, are also like parts of the
wholes, or determined by like parts similarly posited.

Of the Length of Circular A RCS. The lengths of circular
arcs, as found and expressed in various ways, may be seen
in my large Treatise on Mensuration, pa. 88 et seq. 4th edi-
tion: some Of which are as follow. The radius of a circle
being 1 ; and of any arc a, if the tangent be t, the sine s,

ARC

[ 133 ]-

ARC

tie cosine c, and the versed sine v: then the arc a will be
truly expressed by several series, as follow, viz, the arc
a = t — 1 t* -+- 4*s — -fi7 + IP &c.

V

1.3 s
2.4.5 S

1.3 zr 1.3.5

a = 4 .-r

a = s -+•

0 t ° e

&c.

C = a/2v X (1 -t-
3-14159&C

&C.

2.3 2 2.4.5 23 2.4.6.

d = "01745329 &c x d; where d denotes

the number of degrees in the given arc. Also a = — - —

nearly ; where c is the chord of the arc, and c the chord
of half the arc ; whatever the radius is.

To find the length of the arc of any curve. Put x = the
absciss, y = the ordinate, of the arc z, of any curve what-
ever. Put z = ■v/C*2 ■+■ yz) 5 then, by means of the equa-
tion of the curve, find the value of x in terms of y, or of j
in terms of x, and substitute that value instead of it in the
above expression z = \Z(x"~ -t-y1); hence, taking the flu-
ents, they will give the length of the arc z, in the terms of
x or y.

Arc, in Astronomy. Of this, there are various kinds.
Thus, the latitude, elevation of the pole, and the declina-
tion, are measured by an arc of the meridian; and the lon-
gitude, by an arc of a parallel circle, &c.

Diurnal Aitc of the sun, is part of a circle parallel to
the equator, described by the sun in his course from his ri-
sing to the setting. And his Nocturnal Arc is of the same
kind ; excepting that it is described from setting to rising.

Arc of Progression, or Direction, is an arch of the eclip-
tic, which a planet seems to pass over, when its motion is
direct, or according to the order of the signs.

Arc of Relrogradation, is an arch of the ecliptic, de-
scribed while a planet is retrograde, or moves contrary to
the order of the signs.

Arc between the Centres, in eclipses, is an arc passing
from the centre of the earth's shadow, perpendicular to
the moon's orbit, meeting her centre at the middle of an
eclipse. — If the aggregate of this arc and the apparent semi-
diameter of the moon, be equal to the semi-diameter of the
shadow, the eclipse will be total for an instant, or without
any duration; and if that sum be less than the radius of
the shadow, the eclipse will be total, with some duration;
but if greater, the eclipse will be only partial.

Arc of Vision, is that which measures the sun's depth
below the horizon, when a star, hefore hid by his rays, be-
gins to appear again. — The quantity of this arc is not al-
ways the same, but varies with the latitude, declination,
right ascension, or descension, and distance, of any planet
or star. Ricciol. Almag. vol. 1, pa. 42. However, the fol-
lowing numbers will serve nearly for the stars and planets.
Table exhibiting the Arc of Vision of the Planets and
Fixed Stars.

PLANETS.

FIXED STARS.

Magnitude.

Mercury - v - - 10° 0'

1

-

-

12°

Venus - - - - 50

2

-

.

13

Mars - - - - 11 30

3

-

.

14

Jupiter - - - 10-0

4

.

.

15

Saturn - - - 11 0

5

-

.

\6

6

-

-

17

Arch, in Architecture, is a concave structure, raised
or turned upon a mould, called the centering, in form of
the arch of a curve, and serving as the inward support of

some superstructure. Sir Henry Wotton says, An arch is
nothing but a narrow or contracted vault; and a vault is
a dilated arch.

Arches are used in large intercolumniations of spacious
buildings; in porticoes, both within and without temples;
in public halls, as ceilings, the courts of palaces, cloisters,
theatres, and amphitheatres. They are also used to cover
the cellars in the foundations of houses, and powder maga-
zines; also as buttresses and counter-forts, to, support
large walls laid deep in the earth ; for triumphal arches,
gates, windows, &c ; and, above all, for the foundations
of bridges and aqueducts. And they are supported by
piers, butments, imposts, &c.

Arches are of several kinds, and are commonly deno-
minated from the figure or curve of them; as circula'r, el-
liptical, cycloidal, catenarian, &c, according as their curve
is in the form of a circle, ellipse, cvcloid, catenary, &c.

There are also other denominations of circular arches,
according to the different parts of a circle, or manner of
placing them. Thus,

Semicircular Arches, which are those that make an
eNact semicircle, having their centre in the middle of the
span or chord of the arch ; called also by the French
builders, perfect arches, and arches en plein centre. The
arches of Westminster Bridge are semicircular.

Scheme Arches, or skene, are those which are less than
semicircles, and are consequently flatter arches, contain-
ing 120, or 90, or 60, degrees, &c. They are also called
imperfect and diminished arches.

Arches of the third and fourth point, or Gothic arches;
or, as the Italians call them, diterzo and quarto acuto, be-
cause they always meet in an acute angle at top. These
consist of two excentric circular arcs, meeting in an angle
above, and are drawn from the division of the chord into
three or four more parts at pleasure. Of this kind are
many of the arches in churches and other old Gothic
buildings.

Elliptical Arches, usually consist of semi-ellipses.
These were formerly much used instead of mantle-trees in
chimneys; and are still employed, from their bold and
beautiful appearance, for many purposes, and particularly
for the arches of a bridge, like that at Black-Friars, both
for their strength, beauty, convenience, and cheapness.

Straight Arches, are those which have their upper and
under edges parallel straight lines, instead of curves. These
are chiefly used over doors and windows; and have their
ends and joints usually pointing towards one common
centre.

Aucii is particularly used for the space between the
two piers of a bridge, intended for the passage of the
water, boats, &c.

Arch of equilibration, is that which is in equilibrium
in all its parts, having no tendency to break in one part
more than in another, and which is therefore safer and
stronger than any other figure. Every particular figure
of the cxtrados, or upper side of the wall above an arch,
requires a peculiar curve for the under side of the arch
itself, to form an arch of equilibration, so that the incum-
bent pressure on every part may be proportional to the
strength or resistance there. When the arch is equally
thick throughout, a case that can hardly ever happen,
then the catenarian curve is the arch of equilibration; but
in no other case: and therefore it is a great mistake in
some authors to suppose that this curve is the best figure
for arches in all cases; when iti reality it is commonly the
T 2

K—

A R C

[ 140 ]

ARC

worst. This subject is fully treated in my Collection of supported as in the last article, and pointing all to the
Tracts, tract 1, on the Principles of Bridges, pr. xi, exam. 2, same centre; none of them can sink, till the supporters or
where the proper intrados is investigated for every extra- butments give way, because they want room in that situa-
dos, so as to form an arch of equilibration in all cases tion to descend perpendicularly. But this is a weak struc-
wliatever. It there appears that, when the upper side of ture; because the supporters are subject to too much im-
wall is a straight horizontal line, as in the annexed figure, pulsion, especially where the line is long ; for which reason

the form of straight arches is seldom used, excepting over
doors and windows, where the line is short, and the side
walls strong. In order to fortify the work, therefore, we
must change not only the figure of the materials, but also
their position.— 4th, If the materials be shaped wedge-like,
and be disposed in form of an arch, and pointing to some,
centre ; in this case, neither the pieces of the said arch
can sink downwards, for want of room to descend perpen-
dicularly; nor can the supporters or butments suffer much
violence, as in the preceding flat form: for the convexity
will always make the incumbent rather rest upon the
supporters, than thrust or push them outwards. His rea-
soning, however, afterwards, on the effect of circular and
other arches, is not accurate, as he attends only to the
side pressure, without considering the effect of different
vertical pressures.

The chief properties of arches of different curves may
be seen in the 2d sect, of my Tract on the Principles of
Bridges, above quoted. It there appears that none, except
the mechanical curve of the arch of equilibration, can ad-
mit of a horizontal line at top : that this arch is of a form
both graceful and convenient, as it may be made higher
or lower at pleasure, with the same span or opening : that
all other arches require extrados that are curved, more or
less, either upwards or downwards : of these, the elliptical
arch approaches the nearest to that of equilibration, for
equality of strength and convenience ; and it is also the
best form for most bridges, as it can be made of any
height to the same span, its hanches being at the same
time sufficiently elevated above the water, even when it is
very flat at top : elliptical arches also look bolder and
lighter, are more uniformly strong, and much cheaper than
most others, as they require less materials and labour. Of
the other curves, the cycloidal arch is next in quality to
the elliptical one, for those properties, and lastly the circle.
As to the others, the parabola, hyperbola, and catenary,
they are quite inadmissible in bridges that consist of se-
veral arches; but may, in some cases, be employed for a
bridge of one single arch, which may be intended to rise
very high, as in such cases they are not much loaded at
the hanches.

See also Bridge, and Magazine.
Av.cn Mural. See Mural Arch.
ARCHER, or Sagittarius, one of the constellations of the
See my Tracts, vol. i, art. 1. northern hemisphere, and one of the twelve signs of the

The doctrine and use of arches are neatly delivered by zodiac, placed between the Scorpion and Capricorn. See
sir Henry VVotton, though he is not always mathematically Sagittarius.

accurate in the principles. He says ; 1st, All matter, un- ARCHIMEDES, one of the most celebrated Anathema*
less impeded, tends to the centre of the earth in a perpen- ticians among the ancients, who flourished about 28J)years
dicular line. 2d, All solid materials, as bricks, stones, before Christ, being about 1 10 years later than Euclid.
&c, in their ordinary rectangular form, if laid in numbers, He was born at Syracuse in Sicily, and was related to
one by the side of another, in a level row, and their ex- Hiero, according to Plutarch, the then king of that city,
treme ones sustained between two supporters; those in by his father. In his youth, he devoted himself to the
the middle will necessarily sink, even by their own gravity, study of geometry ; and in his maturer years, he travelled
much more if forced down by any superincumbent weight. intoEgypt, whither the Greeks usually resorted in the pur-
To make them stand, therefore, either their figure or their suit of science. After un absence of several years, which
position must be altered. — 3d, Stones, or other materials, he spent in the society of Conon and other eminent men,
being figured cuneatim, or wedge-like, broader above than and during which time he gave very promising indications
below, and laid in a level row, with their two extremes of his future fame, he returned to his own country; and

the equation of the curve is thus expressed,
l0g. of ^^-aJ)

KI Or PC = y = h X ■ rrrf jr ;

log. of 2—

where h = aq, w = ci, a = dk, c = kq.

Or ci = w = ; n = the number whose log. is

2K d

cy+A,andA = log. of a, and c =- x log. of ^ — — .

And hence, when a, h,'c, are any given numbers, a ta-
ble is formed for the corresponding values of iv and y, by
which the curve is constructed for any particular occa-
sion. Thus supposing a or dk = 6, h or aq = 50, and c
or kq = 46 ; then the corresponding values of ki and ic,
or horizontal and vertical lines, will be as in this table.
Table for constructing the Curve of Equilibration.

Value of

Value of

Value of

Value of

Value of

Value of

KI

IC.

KI.

IC.

KI.

IC.

O

6-000

21

10381

36

21-774

2

6-035

22

J 0-858

37

22-948

4

6-144

23

11-368

38

24-190

6

6-324

24

11-911

39

25-505

8

6-580

25

12'489

40

26-894

10

6-914

26

13'106

41

28-364

12

7-330

27

13-761

42

29919

13

7-571

28

14-457

43

31-563

14

7-834

29

15'19S

44

33-299

15

8-120

30

15980

45

35-135

16

8-430

31

16-811

46

37-075

! u

8-766

32

17-693

47

39-126

' 18

9-168

33

18-627

48

41-293

19

9-517

34

19-617

49

43-581

1 20

9-934

35

20-665

50

46-000

ARC

L «1 ]

ARC

then continued his studies with the greatest assiduity.
Such indeed were the ardour and intenseness of his appli-
cation to mathematical science, that he prosecuted his
studies to the neglect both of food and sleep, and improved
the minutest circumstance that occurred into an occasion
of making very important and useful discoveries. The
mathematical genius of Archimedes set him with such di-
stinguished excellence in the view of the world, as render-
ed him both the honour of his own age, and the admira-
tion of posterity. He was indeed the prince of the ancient
mathematicians, being to them what Newton is to the mo-
derns, to whom in his genius and character he bears a
very near resemblance. He was frequently lost in a kind
of reverie, so as to appear hardly sensible ; he would study
for days and nights together, neglecting his food ; and Plu-
tarch tells us that he used to be carried to the baths by
force. Many particulars of his life, and works, mathe-
matical and mechanical, are recorded by several of the
ancients, as Polybius, Livy, Plutarch, Pappus, &c. He
was equally skilled in all the sciences, astronomy, geome-
try, mechanics, hydrostatics, optics, &c, in all of which
he excelled, and made many and great discoveries.

Archimedes, it is said, made a sphere of glass, of a most
surprising contrivance and workmanship, exhibiting the
motions of the heavenly bodies in a very pleasing manner.
Claudiai) has an epigram on this invention, which has
been thus translated :

When in a glass's narrow space confin'd,
Jove saw the fabric of th' almighty mind,
He smil'd, and said, Can mortals' art alone,
Our heavenly labours mimic with their own ?
The Syracusan's brittle work contains
Th' eternal law. that through all nature reigns.
Fram'd by his art, see stars unnumber'd burn,
And in their courses rolling orbs return :
His sun thiuugh various signs describes the year;
And every month his mimic moons appear.
Our lival's laws his little planets bind,
And rule their motions with a human mind.
Salmoneus could our thunder imitate,
But Archimedes can a world create.
Many wonderful stories are told of his discoveries, and
of his very powerful and curious machines, &c. Hiero
once admiring them, Archimedes replied, these effects are
nothing, " But give me," said he, " some other place to fix a
machine on, and I will move the earth." He fell upon a
curious device for discovering the deceit which had been
practised by a workman, employed by the king to make a
golden crown. Hiero, hating a mind to make an offering
to the gods of a golden crown, agreed for one of great
value, and weighed out the gold to the artificer. After
some time he brought the crown home of the full weight ;
but it was afterwards discovered or suspected that a part
of the gold had been stolen, and the like weight of silver
substituted in its stead. Hiero, being angry at this im-
position, desired Archimedes to take it into consideration,
how such a fraud might be certainly discovered. While
engaged in the solution of this difficulty, he happened to go
into the bath ; where observing that a quantity of water
overflowed, equal to the bulk of his body, it presently oc-
curred to him, that Hiero's question might be answered by
3 like method : upon which he leaped out, and ran home-
ward, crying out Et>fi;xa! gSfijxa ! I have found it! 1 have
found it ! He then made two masses, each of the same
weight as the crown, one of gold and the other of silver :

this done, he rilled a vessel to the brim with water, and
put the silver mass into it, upon which a quantity of water
overflowed equal to the bulk of the mass; then taking the
mass of silver out he filled up the vessel again, measuring
the water exactly, which he put in; this showed him what
measure of water answered to a certain quantity of silver.
He then tried the gold in like manner, and found that it
caused a less quantity of water to overflow, the gold being
less in bulk than the silver, though of the same weight.
He then filled the vessel a third time, and putting in the
crown itself, he found that it caused more water to over-
flow than the golden mass of the same weight, but less than
the silver one; so that, finding its bulk between the two
masses of gold and silver, and that in certain known pro-
portions, he hence computed the real quantities of gold and
silver in the crown, and so manifestly discovered the fraud.

Archimedes also contrived many machines for useful and
beneficial purposes; among these, engines' for launching
large ships ; screw pumps, for exhausting the water out of
ships, marshes, or overflowed lands, as Egypt, &c, which
they would do from any depth.

But he became most famous by his curious contrivances,
by which the city of Syracuse was so long defended, when
besieged by the Roman consul Marcellus ; showering up-
on the enemy sometimes long darts, and stones of vast
weight, and in great quantities ; at other times lifting their
ships up into the air, that had come near the walls, and
dashing them to pieces by letting them fall down again;
nor could they find their safety in removing out of the
reach of his cranes and levers, for there he contrived to fire
them with the rays of the sun reflected from burning glasses.

However, notwithstanding all his art, Syracuse was at
length taken by storm, or by corrupting the guards. "What
gave Marcellus the greatest concern," says Plutarch, "was
the unhappy fate of Archimedes, who was at that time in
his museum ; and his mind, as well as his eyes, so fixed
and intent upon some geometrical figures, that he neither
heard the noise and hurry of the Romans, nor perceived
the city to be taken. In this depth of study and con-
templation, a soldier came suddenly upon him, and com-
manded him to follow him to Marcellus; which he refu-
sing to do, till he had finished his problem, the soldier, in
a rage, drew his sword, and ran him through." Livy says
he was slain by a soldier, not knowing who he was, while
he was drawing schemes in the dust: that Marcellus was
grieved at his death, and took care of his funeral ; and
made his name a protection and honour to those who could
claim a relationship to him. His death it seems happened
about the 142d or 143d Olympiad, or 210 years before the
birth of Christ. According to Tzetzes he lived 75 years.

When Cicero was quesfor for Sicily, he discovered the
tomb of Archimedes, all overgrown with bushes and bram-
bles ; which he caused to be cleared, and the place set in
order. There was a sphere and cylinder cut upon it, with
an inscription, but the latter part of the verses quite worn
out.

Many of the works of this great man are still extant,
though the greatest part of them are lost. The pieces re-
maining are as follow : 1. Two books on the Sphere and
Cylinder.— 2. The Dimension of the Circle, or proportion
between the diameter and the circumference.— 3. Of Spi-
ral lines. — 4. Of Conoids and Spheroids. — 5. Of Equipon-
derants, or Centres of Gravity. — 6. The Quadrature of the
Parabola. — 7. Of Bodies floating on Fluids. — 8. Lemma-
ta.— 9. Of the Number of the Sands.

«W

ARC

[ 142 ]

ARC

Araoncr the works of Archimedes which are lost, may He determined also the relation between the circle and the

be reckoned the descriptions of the following inventions, ellipse, with that of their similar parts. And it is proba-

which may be gathered from himself and other ancient ble that he also attempted the hyperbola ; bat it is not to

authors. 1. His account of the method which he employ- be expected that he met with any success, since approxi-

ed to discover the mixture of gold and silver in the crown, mations to its area are all that can be given by the various

mentioned by Vitruvius, — 2. His description of the Coch- methods that have since been invented,

leon, or engine to draw water out of places where it is Besides these figures, he determined the measures of the

stagnated, still in use under the name of Archimedes's spiral, described by a point moving uniformly along a right

Screw. Athenaeus, speaking of the prodigious ship built line, the line at the. same time revolving with a uniform

by the order of Hiero, says, that Archimedes invented the angular motion ; determining the proportion of its area to

cochleon, by means of which the hold, notwithstanding its that of the circumscribed circle, as also the proportion of

depth, could be drained by one man. And Diodorus Si- their sectors.

cuius says, that he contrived this machine to drain Egypt, Throughout the whole works of this great man we every
and that by a wonderful mechanism it would exhaust the where perceive the deepest design, and the finest invention,
water from any depth. — 3. The Helix, by means of which, He seems to have been, with Euclid, exceedingly careful
Athenseus informs us, he launched Hieio's great ship. — of admitting into his demonstrations nothing but princi-
4. The Trispaston, which, according to Tzetzes and Ori- pies perfectly geometrical and unexceptionable : and
basius, could draw the most stupendous weights. — 5. The though his' most general method of demonstrating the re-
machines, which, according to Polybius, Livy, and Plu- lations of curved figures to straight ones, be by inscribing
tarch, he used in thedefence of Syracuse against IMarcellus, polygons in them; yet to determine those relations, he
consisting of Tormenta,Balistae,Catapults, Sagittarii,Scor- does not increase the number, and diminish the magni-
pions, Cranes, &c. — 6. His Burning Glasses, with which tude, of the sides of the polygon ad infinitum, but from
he set fire to the Roman galleys. — 7. His Pneumatic this plain fundamental principle, allowed in Euclid's Ele-
and Hydrostatic engines, concerning which subjects he ments, (viz, that any quantity may be so often multiplied,
wrote some books, according to Tzetzes, Pappus, and Ter- or added to itself, as that the result shall exceed any

tullian. — 8. His Sphere, which exhibited the celestial mo-
tions. And probably many others.

A whole volume might be written on the curious me-
thods and inventions of Archimedes, that appear in his
mathematical writings now extant only. He was the first
who squared a curvilineal space ; unless Hypocrates must
be excepted, on account of his lunes. In his time the co-

proposed finite quantity of the same kind,) he proves
that to deny his figures to have the proposed relations,
would involve an absurdity. And when he demonstrated
many geometrical properties, particularly in the parabo-
la, by means of certain progressions of numbers, whose
terms are similar to the inscribed figures ; this was still
done without considering such series as continued ad in-

nic sections were admitted into geometry, and he applied Jinitum, and then collecting or summing up the terms of

himself closely to the measuring of them, as well as other such infinite series.

figures. Accordingly, he determined the relations of There have been various editions of the existing wri-

spheres, spheroids, and conoids, to cylinders and cones; tings of Archimedes. The whole of these works, together

and the relations of parabolas to rectilineal planes, whose with the Commentary of Eutocius, were found in their

quadratures had long before been determined by Euclid, original Greek language, on the taking of Constantinople,

He has left us also his attempts upon the circle : he proved whence they were brought into Italy ; and here they were

that a circle is equal to a right-angled triangle, whose found by that excellent mathematician John Muller,

base is equal to the circumference, and its altitude equal otherwise called Regiomontanus, who brought them into

to the radius ; and consequently, that its area is equal to
the rectangle of half the diameter and half the circum-
ference; thus reducing the quadrature of the circle to the
determination of the ratio between the diameter and cir-
cumference; which determination however has never yet

Germany : where they were, with that Commentary,
published long afterwards, viz, in 1544, at Basil, being
most beautifully printed in folio, both in Greek and Latin,
by Hervagius, under the care of Thomas Gechauff Vena-
torius. — A Latin translation was published at Paris 1557,

been done. Being disappointed of the exact quadrature by Pascalius Hamellius. — Another edition of the whole,

of the circle, for want of the rectification of its circumfe- in Greek and Latin, was published at Paris 1615, in folio,

rence, which all his methods would not effect, he proceed- by David Rivaltus, illustrated with new demonstrations

ed to assign a useful approximation to it : this he effected and commentaries : a life of the author is prefixed ; and

by the numeral calculation of the perimeters of the in- at the end of the volume is added some account, by way

scribed and circumscribed polygons; from which calcu- of restoration, of our author's other works, which have

lation it appears, that the perimeter of the circumscribed been lost; viz, The Crown of Hiero; the Cochleon or

regular polygon of 192 sides, is to the diameter-, in a less Water Screw ; the Helicon, a kind of endless screw ; the

ratio than that of 3f or 3f § to 1 ; and that the perimeter Trispaston, consisting of a combination of wheels and

of the inscribed polygon of £)6 sides, is to the diameter, in axles; the Machines employed in the defence of Syracuse;

a greater ratio than that of 3f£ to 1 ; consequently, that the Burning Speculum ; the Machines moved by Air and

the ratio of the circumference to the diameter, lies be- Water; and the Material Sphere. — In lG75, Dr. Isaac

tween these two ratios. Now the first ratio, of 3-f to 1, Barrow published a neat edition of the works, in Latin,

reduced to whole numbers, gives that of 22 to 7, for at London, in 4to ; illustrated, and succinctly demon-

3y : 1 : : 22 : 7; which therefore is nearly the ratio of the strated in a new method. — But the most complete of any,

circumference to the diameter. From this ratio between is the magnificent edition, in folio, lately printed at the

the circumference and the diameter, Archimedes com- Clarendon press, Oxford, 17£)2. This edition was pre-

putcd the approximate area of the circle, and he found pared ready for the press by the learned Joseph Torclli,

, that it is to the square of the diameter, as 11 is to 14. of Verona, with his preface, and in that state presented to

ARC

[ 143 ]

ARC

llie University of Oxford. The Latin translation is a new Fortification, is the art of strengthening and fortify-in"

one. We find here also Eutocius's Commentary on some places, to secure them from the insults or attacks of ene-

ofthe-pieces, and notes on the whole. An account of the mies, and the violence of arms ; by erecting forts castles

life and writings of Torelli is prefixed, by Clemens Sibi- and other fortresses, with ramparts, bastions &c. The

liati. And at the end a large appendix is added, in two authors who have chiefly excelled in this art are Coehorn

parts; the first being a Commentary on Archimedes's Pagan, Vauban, Scheiter, Blondel, and Montalembert.

paper on Bodies that float on Fluids, by the Rtv. Abram Naval Architecture, or Ship-build in", is the art of

Robertson of Christ-Church College, who had the care of constructing ships, galleys, and other vessels proper to float

this edition; and the latter is a large collection of various on the water.

readings in the manuscript works of Archimedes, found ARCHITRAVE, is that part of a column which bears

in the library of the late king of France, and of anotherat immediately on the'capital. It, is the lowest member of

Florence, as collated with theBasiledition above mentioned, the entablature, and is supposed to represent the principal

There are al-o extant other editions of certain parts of beam in timber buildings, in which it is sometimes called

the works of Archimedes. Thus, Commandine published, the Reason-piece, or Master-piece. Also, in, chimneys it

in 4to, at Bologna- 1 565, the two books concerning Bodies' is called the Mantle-piece; and the Hyperthyron over

that float on Fluids, with a Commentary. Commandine the jaumbs of doors, or lintels of windows,

published also a translation of the Arenarius. And Borelli Architrave Cornice. See Corkice.

published, in folio, at Florence, l66l, Archimedes's Liber Architrave doors, are those which have an archi-

Assumptorum, translated into Latin from an Arabic ma- trave on the jaumbs, and over the door; on the cap piece

nuscript copy. This is accompanied with the like trans- if straight; or on the arch, if the top be curved,

lation, from the Arabic, of the 5th, 6th, and 7th books of Architrave windows, of timber, are usually an o»ee

Apollonius's Conies. Mr. G. Anderson published (in Svo. raised out of the solid timber, with a list over it: though

Lond. 1784) an English translation of the Arenarius of sometimes the "mouldings are struck, and laid on; and

Archimedes, with learned and ingenious notes and illus- sometimes they are cut in brick.

trations. Dr. Wallis published a translation of the Are- AKCHIVAULT or Archivolt, the contour of an

narius. Lastly, the translation of Torelli's work into the arch ; a band or frame adorned with mouldings, running

French language, by F. Peyrard, in 2 vols, 8vo, at Paris, over the faces of the voussoirs or arch-stones, and bearin

in 18 — , and 1808. And there may be other editions be-
side the above; but these are all that I have seen, or know of.

Archimedes's Screw. See Screw of Archimedes.

Archimedes's Burning-glass. See Bvrk mo -glass.

ARCHITECT, a person skilled in architecture, or the
art of building; who forms plans and designs for edifices,
conducts the work, and directs the various artificers em-
ployed in it.

The most celebrated architects are, Vitruvius, Palladio,

upon the imposts.

ARCHYTAS, of Tarentum, a celebrated mathemati-
cian, cosmographer, and Pythagorean philosopher, whom
Horace calls

Maris ac Terra;, numeroque carentis Arenaj,

Mensorem.
He flourished about 400 years before Christ ; and was the
master of Plato, Eudoxus, and Philolaus. He gave a me-
thod of finding two mean proportionals between two given

Scamozzi, Serlio, Vignola, Barbaro, Cataneo, Alberti, lines, and thence the duplication of the cube, by means

Viola, Inigo Jones, De Lorme, Perrault, S. Lc Clerc, Sir of the conic sections. His skill in mechanics was such,

Christopher Wren, the Earl of Burlington, &c. that he was said to be the inventor of the crane and the

ARCHITECTURE, Architectura, the art of planning screw; and he made a wooden pigeon that could fly about,

and building or erecting any edifice, so as properly to an- when it was once set off, but it could not rise again of it-

swer the end proposed, for solidity, conveniency, and self, after it rested. He wrote several works, though none

beauty; whether houses, temples, churches, bridges, halls, of them are now extant, particularly a treatise rfsg ) tb

theatres, &c, &c. — Architecture is divided into civil, mili- Havro;, De Universo, cited by Simplicius in Aristot.

tary, and naval or marine

Civii Architecture, is the art of designing and erect-
ing edifices of every kind for the uses of civil life in every
capacity ; -as churches, palaces, private houses, &c ; and
it has been divided into five orders or manners of buildino,
under the names of the Tuscan, Doric, Ionic, Corinthian,
and Composite,

Categ. It is said he invented the ten categories. He ac-
quired great reputation both in his legislative and mili-
tary capacity; having commanded an army seven times
without ever being defeated. He was at last shipwrecked,
and drowned in the Adriatic sea.

ARCTIC Circle, is a lesser circle of the sphere, parallel
to the equator, an'd passing through the north pole of the

There were many authors on architecture among the ecliptic, or distant from the north or arctic pole, by a

Greeks and Romans, before Vitruvius ; but he is the first quantity equal to the obliquity of the ecliptic, which was

whose work is entire and extant, lie lived in the reigns formerly estimated at23Q 30', but its mean quantity is now

of Julius Ca?sar and Augustus, and composed a complete 23° 28' nearly. This, and its opposite, the antarctic circle,

system of architecture, in ten books, which he dedicated are also called the polar circles, where the longest day and

to this prince. The principal authors on architecture longest night are 24 hours, and within all the space of these

since Vitruvius, are Philander, Barbaras, Salmasius, Bal- circles, at one time of the year, the sun never sets, and at

dus, Alberti, Gauricus, Dernoniosius, Perrault, De the opposite season he never rises for some days, more or

l'Orme, Rivius, Wotton, Serlio, Palladio, Strada, Vignola,
Scamozzi, Dieussart,Catanei, Freard, De Cam bray, Blon-
del, Goldman, Sturmy, Wolfius, De Rosi, Uesgodetz,
Baratteri, Mayer, Gulielumus, Ware, &c, &c. See also
Architect.

less according as the place is nearer the pole.

Arctic Pole, the north pole of the world, and so called
from aproj, ursa, the bear, from its proximity to the
constellation of that name.

ARCTURUS, a fixed star of the first magnitude, bc-

Mililary Architecture, otherwise more usually called tween the thighs, of the constellation Bootes. So called

«.^

ARE

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A R G

from af-tro;, bear, and ci^d, tail ; as being near the bear's be contained in the immense sphere of the fixed stars,

tail. This star is twice mentioned in the book of Job, viz
ix. 9, and xxxviii. 32, by the name Aish, if the translation
be right; and by many of the ancients under its Greek
name Arctium. The-Greeks gave the fabulous history of
this star, or constellation, to this purport : Calisto, who
was afterwards, in form of the great bear, raised up into a
constellation, they tell us, brought forth a son to Jupiter,
whom they called Areas. That Lyacon, when Jupiter
afterwards came to visit him, cut the boy in pieces, and
served him up at table. Jupiter, in revenge, as well as
by way of punishment, called down lightning to consume
the palace, and turned the monarch into a wolf. The
limbs of the boy were gathered up, to which the god gave
life again, and he was taken and educated by some of the
people. His mother, who was all this time a bear in the
woods, fell in his way : he chased her, ignorant of the
fact, and, to avoid him, she threw herself into the temple
of Jupiter : he followed her thither to destroy her ; and
this being death by the laws of the country, Jupiter took
them both up into heaven, to prevent the punishment,
making her the constellation of the great bear, and con-
verting the youth into this single star behind her.

Dr. Kornsby, theSavilian Professor of Astronomy, con-
cludes that Arcturus is the nearest star to our system visi-
ble in the northern hemisphere, because the variation of
its place, in consequence of a proper motion of its own, is
more remarkable than that of any other of the stars ; and
by comparing a variety of observations respecting both the
quantity and direction of the motion of this star, he in-
fers, that the obliquity of the ecliptic decreases at the rate
of oS" in 100 years ; a quantity which nearly corresponds
to the mean of the computations framed by Euler, La-
lande, and Laplace, on the principles of attraction.
Philos. Trans, vol. 63.

AREA, in general, denotes any plain surface to walk
upon ; and derived from arere, to be dry.

Area, in Architecture, denotes the space or site of
ground on which an edifice stands. It is also used for
inner courts, and such like portions of ground.

Area, in Geometry, denotes the
superficial content of any figure. The
areas of figures are estimated in squares
and parts of squares. Thus, suppose
a rectangle efgh have its length eh
equal to 4 inches, or feet, or yards,
&c, and its breadth ef equal to 3 ; its
area will then be 3 times 4, or 12
squares, each side of which is respec-
tively one inch, or foot, or yard, &c. The areas of other
particular figures may be seen under their respective names.

The areas of all similar figures, are in the duplicate
ratio, or as the squares of their like sides, or of any like
linear dimensions. — Also the law by which the planets
move round the sun, is regulated by the areas described
by a line connecting the sun and planet ; that is, the time
in which the planet describes, or passes over, any arc of
its elliptic orbit, is proportional to the elliptic area de-
scribed in that time by the said line, or the sector con-
tained by the said arc and two radii drawn from its extre-
mities to the focus in which the sun is placed.

Area, in Optics. See Field.

ARENARIUS, the name of a book of Archimedes, in
which he demonstrated, that not only the sands of the
earth, but even a greater quantity of particles than could

I

1

V

"

I

might be expressed by numbers, in a way by him invented
and described. This notation proceeds by certain geo-
metrical progressions; and in denoting and producing
certain very distant terms of the progression, he here first
of any one makes use of a property similar to that of lo-
garithms, viz, adding the indices of the terms to find the
index of their product. See Archimedes.

AREOMETER, Arsometrum, an instrument to mea-
sure the density or gravity of fluids.

The areometer, or water-poise, is commonly made of
glass; consisting of a round hollow ball, which termi-
nates in a long slender neck, hermetically sealed at top ;
having first as much running mercury put into it, as will
serve to balance or keep it swimming in an erect position.
The stem, ,or neck, is divided into degrees or parts, which
arc numbered, to show, by the depth of its descent into
any liquor, the lightness or density of it : for that fluid is
heaviest in which it sinks least, and lightest in which it
sinks deepest.

Another instrument of this kind is described by Hom-
berg of Paris, in the Memoirs of the Acad of Sciences for
the year 1699 ; also in the Philos.Trans. N3 262, where a
table of numbers is given, expressing the density of various
fluids, as determined by this instrument both in summer
and winter. By this table it appears that the density, or
specific gravity of quicksilver and distilled water, in the
two seasons, were as follow, viz,

in summer as 13"6l to 1,

in winter as - 13'53 to 1;

and the medium of these two is as 13'57 to 1.

See the Philos. Trans, vol. 36, or my Abridg. vol.7, for
the description and use of another new areometer. See
also Hydrometer.

AREOMETRY, the science of measuring the gravity
and density of fluids. See the Philos. Trans, vol. 68, for an
essay on areometry, &c. Also my Abridgement, vol. 14.

AREOSTYLE, in Architecture, a sort of intercolum-
nation in which the columns were placed at a great dis-
tance from one another.

ARGETENAR, a star of the fourth-magnitude, in the
flexure of the constellation Eridanus.

ARGO NAVIS, or the ship, is a constellation of fixed
stars, in the southern hemisphere, being one of the 48 old
constellations. The number of stars in this constellation,
arc, in Ptolemy's catalogue 45, in Tycho Brahe's 11, in
Flamsteed's 64.

The Greeks tell us, that this was the noted ship in
which the Argonauts performed that celebrated expedi-
tion, which has been so famous in all their history.

ARGUMENT, in Astronomy, is an arch given, by
which another arch is foundinsomc proportion toit. Hence,

Argument of Inclination, or Argument of Latitude,
of any planet, is an arc of a planet's orbit, intercepted
between the .ascending node, and the place of the planet
from the sun, numbered according to the succession of the
signs.

Menstrual Argument of Latitude, is the distance of
the moon's true place from the sun's true place. — By this
is found the quantity of the real obscuration in eclipses,
or how many digits are darkened in any place.

Annual Argument of the ?>ioon's apogee, or simply,
Annual Argument, is the distance of the sun's place from
the place of the moon's apogee ; that is, the arc of the
ecliptic comprised between those two places.

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ARIES, or the Ram, in Astronomy, one of the con- moon: this was translated into Latin, and commented on

stellations of the northern hemisphere, being the first of by Commandine, who first published it with Pappus's Ex-

the old twelve signs of the zodiac, and marked V in imi- planations, in 1572. Dr. Wallis afterwards published it

tation of a ram's head. It gives name to a twelfth part of in Greek, with Commandine's Latin version, in 1088, and

the ecliptic, which the sun enters commonly about the which he inserted again in the 3d volume of his Mathe-

20th of March. — The stars of this constellation in Ptole-
my's catalogue are 18, in Tycho Brahe's 21,' in Hevelius's
27> and in Flamsteed's 66: but they are mostly very
small, only one being of the 2d magnitude, two of the 3d
magnitude, and all the rest smaller.

The fabulous account of this constellation, as given by
the Greeks, is to this effect. That Nephele gave Phryxus,
her son, a ram, which bore a golden fleece, as a guard
against the greatest dangers. Juno, the step-mother both
of him and Helle, laid designs against their lives; but

matical Works, printed in folio at Oxford, 1659. The
piece was animadverted on by Mr. Foster, in his Mis-
cellanies. There is another piece which has gone under
the name of Aristarchus, on the Mundane System, its
parts, and motions, published in Latin by Roberval, and
by Mersenne, in his Mathematical Synopsis. But this
piece is censured, by Menagius (in Diog. Laert.), and Des-
cartes, in his Epistles, as a fictitious piece of Roberval's,
and not the genuine work of Aristarchus.

ARISTOTELIAN, something that relates to the phi

Phryxus, remembering the admonition of his mother, took losopher Aristotle. Thus we say, an Aristotelian dogma,
his sister with him, and getting upon the back of the ram, the Aristotelian school, &c.

they were carried to the sea. The ram plunged in, and
the youth was carried over; but Helle dropping off, was
drowned, and so gave name to the Hellespont. When he
arrived in Colchis, iEeta the king received him kindlv ;

Aristotelian Philosophy, the philosophy taught by
Aristotle, and maintained by his followers. It is other-
wise called the Peripatetic Philosophy, from their prac-
tice of teaching while they were walking. — The principles

and, sacrificing the ram to Jupiter, dedicated the fleece to of Aristotle's philosophy, the learned agree, are chiefly laid

the god ; which was afterwards carried off by Jason. The down in the four books de Coelo. Instead of the more an-

animal itself, they say, Jupiter snatched up into the hea- cient systems, he introduced Matter, Form, and Privation,

vens, and made of it the constellation Aries. They have as the principles of all things ; but it does not seem that

other fables also to account for its origin. But it is most he derived much benefit from them in natural philosophy,

probable that the inventors of this sign, placed it there as And his doctrines are, for the most part, so obscurely ex-

the father of those animals which are brought forth about pressed, that it has not yet been satisfactorily ascertained,

the time the sun approaches to that part of the heavens, what were his sentiments on some of the most important

and so marking the beginning of spring. subjects. He attempted to refute the Pythagorean doc-

Aries also denotes a battering ram; being a military trine concerning the two-fold motion of the earth; and

engine with an iron head, much used by the ancients, to pretended to demonstrate, that the matter of the heavens

batter and beat down the walls of places besieged. See is ungenerated, incorruptible, and not subject to any al-

• Ram, and Battering Ram. teration : and he supposed that the stars were carried

ARISTARCHUS, a celebrated Greek philosopher and round the earth in solid orbs.

astronomer, was a native of the city of Samos; but of
what date is not exactly known ; it must have been how-
ever before the time of Archimedes, as some parts of his
writings and opinions are cited by that author, viz, in his
Arcnarius; he probably flourished about 420 years be-

ARISTOTELIANS, a sect of philosophers, so called,
from their leader Aristotle, and are otherwise called Peri-
patetics.— The Aristotelians and their dogmata prevailed
for a long while, in the schools- and universities; even in
spite of all the efforts of the Cartesians, Newtonians, and

fore Christ. He held the opinion of Pythagoras as to the other corpuscularians. But the systems of the latter have
system of the world, but whether before or after him, is un- at length gained the ascendency ; and the Newtonian phi-
certain, teaching that the sun and stars were fixed in the losophy in particular is now very generally received.

heavens, and that the earth moved in a circle about the
sun, at the same time that it revolved about its own centre
or axis. He taught also, that the annual orbit'of the earth,
compared with the distance of the fixed stars, is but as a
point. On this head Archimedes says, " Aristarchus, the
Samian, confuting the notions of astrologers, laid down

ARISTOTLE, a Grecian philosopher, the son ofNico-
machus, physician to Amyntas king of Macedonia, .was
born 384 years before Christ, at Stagira, a town of Mace-
donia, or, as others say, of Thrace ; from which he is also
culled the Stagirite. Not succeeding in the profession of
arms, to which it seems he first applied himself, he turned

certain positions, from which it follows, that the world is his views to philosophy, and at 17 years of age entered

much larger than is generally imagined ; for he lays it himself a disciple of Plato, and attended in the academy

down, that the fixed stars and the sun are immoveable; till the death of that philosopher. Aristotle then retired

and that the earth is carried round the sun in the circum- to Atarna, where the prince Hermias gave him his dauoh-

ference of a circle. On which account, though he might ter to wife. Repairing afterwards to the court of kin<r

not suffer persecution and imprisonment like Galileo, yet Philip, he became preceptor to his son, Alexander the

he did not escape censure for his supposed impiety; for it Great, whose education he attended for the space of 8

is said Cleanthus was of opinion, that Greece ought to years; and by the magnificent encouragement of this

have tried Aristarchus for irreligion, for endeavouring to prince he was afterwards enabled to procure all sorts of

preserve the regular appearance of the heavenly bodies, by animals, from the inspection of which to write their his-

^upposing that the heavens themselves stood sill; but tory. On his quitting Macedon, he settled at Athens

that the earth revolved in an oblique circle, and at the where he established his school, having the Lyceum

same time turned round its own axis. assigned him, by the magistrates, for the place of his in-

Afistarchus invented a peculiar kind of sun-dials, men- struction or disputation ; where he became the head and
tioned by Vitruvius. There is extant of his works only a founder of the sect called after his name, as also Peri pa-
treatise on the magnitude and distance of the sun and tetics, from the circumstance of his giving instructions

Vol. I. U

A R 1

t "6 ]

A R I

while walking. But being here accused of impiety by
Eurymedon, priest of Ceres, and fearing the fate of So-
crates, he retired to Chalcis, where he died at 63 years of
age, and 322 years before Christ. Some say tliat he poison-
ed himself, others that he died of a colic, and others
again pretend that he threw himself into the sea, for grief
that he could not discover the cause of the flux and re-
flux of the waters. Laertius, in his life of Aristotle, esti-
mates his books at the number of 4000; of which how-
ever scarce 20 have come down to us : these may be com-
prised under five heads ; the first, relating to poetry and
rhetoric; the second, to logics; the third, to ethics and
politics ; the fourth, to physics; and the fifth, to meta-
physics. In the schools, Aristotle has been called the
philosopher, and the prince of philosophers. Indeed, such
was the veneration paid to him, that his opinion was al-
lowed to stand on a level with reason itself: nor was any
appeal from it admitted, the parties, in every dispute, be-
ing obliged to show, that their conclusions were no less
conformable to the doctrine of Aristotle than to truth.

Aristotle's Mechanics were published in Greek and
Latin, at Paris, in 4to, 1599; by Henry Monanthol, phy-
sician, and royal professor of the Mathematical arts, with
a large commentary in Latin. This is a very short tract,
and consists chiefly in explaining the reasons or causes of
certain motions, and machines, and natural effects ; some
being true, and others erroneous. In explaining many
physical phenomena, he takes occasion to advert to the
properties of the simple mechanical powers, the lever, the
balance, the statera, the wheel-and-axle, the wedge, the
screw, and the pulley ; of which, most or all, he ob-
serves, may be referred to the lever, by means of which
most mechanical effects are produced. — The form of the
work, is by way of answering certain queries, and explain-
ing natural effects, and seeming contraries or paradoxes :
as,

1. Of a circular line which, without any breadth, has
both convexity and concavity.

2. How two things, the one moving the other, may
move contrary ways at the same time ; as when one wheel
turns another, by means of teeth on their circumferences,
while the one turns from right to left, the other turns
from left to right.

3. In a balance, because a greater line revolves circu-
larly quicker than a less, therefore the larger balances are
more exact than the smaller.

4. That two lines moved parallelwise, in different di-
uctions, with velocities having always the same ratio,
have their intersection always in the diameter or diagonal
of the parallelogram having its sides in the directions of
the motions. Or, one line moved parallelwise, while a
point moves along the line, with amotion in a constant
iatio to that of the line; the point moves always in the
diagonal of the said parallelogram. And this is the same
as the celebrated theorem of the parallelogram of motions,
<>r of forces. But if the motions be not always in the same
ratio, then the intersections will not be in the diagonal of
a parallelogram.

5. On the variable motions, to cause the intersections
to describe the periphery of a circle.

<i. On the affections and properties of scales or balances.

7. On the cause of the power of the lever. — That the
weight and power arc reciprocally proportional to the arms
t.t the lever.

8. Hence the reason why the oars arc the most effica-

cious of those rowers who are seated in the middle of the
ship.

9. On the cause of the power of the rudder, in turning
the ship.

10. On the cause of the power of the sail-yard in a ship.
As the yard is placed higher up, on the mast, so much
quicker will the ship be moved, with the same wind and
the same sail ; because the lever (or distance up the mast}
is so much the longer.

11. On shifting the sails and rudder when turning the
corners and windings of a river.

12. Why round or circular figures are more moveable.

13. Why the greatest circles are the most moveable.

14. Why balista without weight are more easily moved.

15. Why a heavy or loaded carriage is easier than a
light or running one.

16. Why a stone is thrown farther from a sling, than
when thrown by the hand.

17. On the wheel-and-axle, and tooth-and-pinion work.

18. On the breaking of wood. Why a stick is more ea-
sily broken over the knee, when the extremities are held
by the hands, than when held nearer the middle.

19. On the roundness of pebbles, or why those about
the sea-shore are round ; viz, from the continual rolling
by the motion of the waters.

20. Why wood is weaker as it is longer.

21. The cause of the power of the wedge, is chiefly ow-
ing to percussion.

22. Of the pulley or little wheel.

23. Why an ax by striking divides; but by pressing
not : because in the former way it acts as a thin wedge.

24. Of the Statera, or Steelyard.

25. Of the Dentiduco, or Toothdrawer.

26. Of the Nutcracker.

27. Why, in a Rhombus, both the extreme points do
not describe or pass over equal right lines. That is, ei-
ther extreme point moving along an adjacent side, while
that side is carried by an equal parallel motion, then the
point describes the adjacent diagonal of the rhombus.
But the diagonal connecting the acute angles of the rhom-
bus, being greater than the other which connect the ob-
tuse angular points; therefore the motion of the acute an-
gular point, is so much greater than that of the other.

28. Why a greater circle revolves equally with a less,
when posited about the same centre.

29- Of Beds.— Why the sides of a bed are made in a
double ratio; the one being 6 feet, or little more; the
other only 3. And why the cordings are not extended dia-
gonalwise. Or that they be made in such magnitude as to
correspond with the bodies. For thus the sides will be
double, as 4 cubits in length, and 2 in breadth.

30. Why it is more difficult to support a long stick at
the extremity of the arm, than in the middle of it.

31. Why it is more difficult to support, at the extremity
of the arm, a very long weight, than a short one.

32. Why a tolleno for drawing water, acts better as it
works with a heavy weight at the end of the cross bar.

33. Why, when two men carry a weight on a pole be-
tween them, they are not equally pressed when it is not in
the middle between them.

34. Why those who rise up from a seat, put themselves
into a particular position; viz, putting their feet back-
wards, and bending their body forward.

35. Why a moving body is more easily moved, than a
quiescent one

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36. Why a lath or rod projected loses of its motion —
This, and the former cause, Aristotle seems Hot to under-
stand.

37. Why any thing proceeds not on in the direction of
its motion, but relaxes, when impelled in the contrary di-
rection. As, when its weight causing it to descend down-
wards, is greater than the impulsive force can impel it up-
wards.

3S. Why any things, either small or large, when pro-
jected, do not go far.

39. Why alight body, floating in a vortex of water, goes
into the middle of the vortex.

ARITHMETIC, is the art andscienceof numbers; or,
that part of mathematics which considers their powers and
properties, and teaches how to compute or calculate tru-
ly, with ease and expedition. It is by some authors also
defined the science of discrete quantity. Arithmetic con-
sists chiefly in the four principal rules or operations of
Addition, Subtraction, Multiplication, and Division; to
which may perhaps be added involution and evolution, or
raising of powers and extraction of roots. But besides
these, for the facilitating and expediting of computations,
mercantile, astronomical, &c, many other useful rules
have been contrived, which are applications of the former,
such as, the rules of proportion, progression, alligation,
false position, fellowship, interest, barter, rebate, equa-
tion of payments, reduction, tare and tret, &c ; besides
the doctrine of the curious and abstract properties of
numbers.

Very little is known of the origin and invention of arith-
metic. In fact it must have commenced with mankind,
or as soon as they began to hold any sort of commerce to-
gether; and must have undergone continual improvements,
as occasion was given by the extension of commerce, and
by the discovery and cultivation of other sciences. It is
therefore very probable that the art has been greatly in-
debted to the Phoenicians or Tyrians; and indeed Proclus,
in his commentary on the first book of Euclid, says, that
the Phoenicians, by reason of their traffic and commerce,
were accounted the first inventors of Arithmetic. From
Asia the art passed into Egypt, whither it was carried by
Abraham, according to the opinion of Josephus. Here it
was greatly cultivated and improved; insomuch that a
considerable part of the Egyptian philosophy and theology
seems to have turned altogether upon numbers. Hence
those wonders related by them about unity, trinity, with
the numbers 4, 7, 9, &c. In effect, Kircher, in his Oedip.
jEgypt. shows, that the Egyptians explained every thing
by numbers; Pythagoras himself affirming, that the nature
of numbers pervades the whole universe; and that the
knowledge of numbers is the knowledge of the Deity.

From Egypt arithmetic was transmitted to the Greeks,
by means of Pythagoras and other travellers ; among whom
it was greatly cultivated and improved, as appears by the
writings of Euclid, Archimedes, and others: with these
improvements it passed to the Romans, and from them it
has descended to us.

The nature of the arithmetic however that is now in
use, is very different from that above alluded to; this art
having undergone a total alteration by the introduction of
the Arabic or Indian notation, about 8 or 9 hundred years
since, into Europe: so that nothing now remains of use
from the Greeks, but the theory and abstract properties of
numbers, which have no dependence on the peculiar na-
ture of any particular scale or mode of notation. That

used by the Hebrews, Greeks, and Romans, was chiefly by
means of the letters of their alphabets. The Greeks, par-
ticularly, had two different methods ; the first of these was
much the same with the Roman notation, which is suffici-
ently well known, being still in common use with us, to
denote dates, chapters and sections of books, &c. After-
wards they had a better method, in which the first nine let-
ters of their alphabet represented the first numbers, from
one to nine, and the next nine letters represented any
number of tens, from one to nine, that is, 10, 20, 30, &c,
to 90. Any number of hundreds they expressed by other
letters, supplying what they wanted with some other marks
or characters : and in this order they went on, using the
same letters again, with some different marks, to express
thousands, tens of thousands, hundreds of thousands, &c :
in which it is evident that they approached very near to
the more perfect decuple scale of progression used by the
Arabians, and who acknowledge that they had received it
from the Indians. Archimedes also invented another pe-
culiar scale and notation of his own, which he employed
in his Arenarius, to compute the number of the sands.
Ptolemy is supposed to have invented the sexagesimal di-
vision of numbers, with its peculiar notation and opera-
tions : a mode of computation still used in astronomy, &c,
for the subdivisions of the degrees of circles. Those no-
tations however were ill adapted to the practical opera-
tions of arithmetic : and hence it is that the art advanced
but very little in this part ; for, setting aside Euclid, who
has given many plain and useful properties of numbers in
his Elements, and Archimedes, in his Arenarius, they
mostly' consist in dry and tedious distinctions and divisions
of numbers ; as appears from the treatises of Nicomachus,
supposed to be written in the 3d century of Rome, and
published at Paris in 1538; as also that of Boethius, writ-
ten at Rome in the 6th century. A compendium of the
ancient arithmetic, written in Greek, by Psellus, in the
9th century, was published in Latin by Xylander, in 155,6.
A similar work was written soon after in Greek by Jodo-
cus Willichius ; and a more ample work of the same kind
was written by Jordanus, in the year 1200, and published
with a comment by Faber Stapulensis in 1480.

Since the introduction of the Indian notation into Eu-
rope, about the 10th century, arithmetic has greatly
changed its form, the whole algorithm, or practical ope-
rations with numbers, being quite altered, as the notation
required ; and the authors of arithmetic have gradually
become more and more numerous. This method was
brought into Spain by the Moors or Saracens; whither the
learned men from all parts of Europe repaired, to learn
the arts and sciences of them. This, Dr. Wallis proves,
began about the year 1000; particularly that oneGerbert,
a monk, afterwards pope Sylvester the 2nd, who died in
the year 1003, brought this art from Spain into France,
long before the date of his death : and that it was known
in Britain before the year 1 150, where it was brought into
common use before 1250, as appears by the treatise of
arithmetic of Johannes de Sacro Bosco, or Halifax, who
died about 1256. Since that time, the principal writers
on this art have been, Barlaam, Lucas de Burgo, Tonstall,
Aventinus, Purbach, Cardanj Schcubelius, Tartalia, Fa-
ber, Stifelius, Regius, Recorde, Ramus, Maurolycus, He-
mischius, Peletarius, Stevinus, Xylander, Kersey, Snellius,
Tacquet, Clavius, Metius, Gemma Frisius, Buteo, Ursinus,
Romanus, Nnpier,Ceulcn, Wingatc, Kepler, Briggs, Ulacq,
Oughtrcd, Crugcr, Van Schooten, Wallis, Dec, Newton,
U2

A R I

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ARI

Morland, Moore, Jeake, Ward, Hatton, Malcolm, &c, &c;
the particular inventions or excellencies of whom, will be
noticed under the articles of the several species or kinds of
arithmetic here following, which may be included under
these heads, viz, Theoretical, Practical, Instrumental, Lo-
garithmical, Numerous, Specious, Universal, Common or
Decadal, Fractional, Radical or of Surds, Decimal, Duo-
decimal, Sexagesimal, Dynamical or Binary, Tetractycal,
Political, &c.

Theoretical Arithmetic, is the science of the proper-
ties, relations, &c, of numbers, abstractedly considered ;
with the reasons and demonstrations of the several rules.
Such is that contained in the 7th, 8th, and 9th books of
Euclid's Elements; the Logistics of Barlaam the monk,
published in Latin by J. Chambers, in 1600; the Summa
Arithmetica of Lucas de Burgo, printed 1494, who gives
the several divisions of numbers from Nicomachus, and
their properties from Euclid, with the algorithm, both in
integers, fractions, extraction of roots, &c ; Malcolm's
New System of Arithmetic, Theoretical and Practical, in
1730, in which the subject is very completely treated, in
all its branches, &c.

Practical Arithmetic, is the art or practice of num-
bering or computing ; that is, from certain numbers given,
to find others which shall have any proposed relation to
the former. As, having the two numbers 4 and 6 given ;
to find their sum, which is 10; or their difference, which
is 2 ; or their product, 24; or their quotient, 1-J; or a third
proportional to them, which is 9; &c. — Lucas de Burgo's
works contain the whole practice of arithmetic, then used,
as well as the theory. Tunstall gave a neat practical trea-
tise of Arithmetic in 1526; as did Stifelius, in 1544, both
on the practical and other parts. Tartalea gave an entire
body of practical arithmetic, which was printed at Venice
in 1556, consisting of two parts; the former, the applica-
tion of arithmetic to civil uses; the latter, the grounds of
algebra. And most of the authors in the list before enu-
merated, joined the practice of arithmetic with the theory.

Binary or Dyadic Arithmetic, is that in which only
two figures are used, viz, 1 and 0. See Binary.— Leibnitz
and De Lagny both invented an arithmetic of this sort,
about the same time: and Dangicourt, in the Miscel. Be-
rol. gives a specimen of the use of it in arithmetical pro-
gressions; where he shows, that the laws of progression
may be more easily discovered by it, than by any other
method where more characters are used.

Common or Vulgar Arithmetic, is that which is con-
cerning integers and vulgar fractions.

Decimal or Decadal Arithmetic, is that which is per-
formed by a series of ten characters or figures, the pro-
gression being ten-fold, or from 1 to 10's, 100's, &c; which
includes both integers and decimal fractions, in the com-
mon scale of numbers; and the characters used are the
ten Arabic or Indian figures 0, 1,2, 3, 4, 5, 6, 7, 8, 9.
This method of arithmetic was not known to the Greeks
and Romans; but was borrowed from the Moors while
they possessed a great part of Spain, and who acknowledge
that it came to them from the Indians. It is probable
that this method took its origin from the ten fingers of the
hands, which were used in computations before arithmetic
was brought into an art. The Eastern missionaries assure
us, that to this day the Indians are very expert at compu-
ting on their fingers, without any use of pen and ink. And
it is asserted, that the Peruvians, who perform all compu-
tations by the different arrangements of grains of maize,

outdo any European, both for certainty and dispatch, with
all his rules.

Duodecimal Arithmetic, is that which proceeds from
12 to 12, or by a continual subdivision according to 12.
This is greatly used by most artificers, in squaring or cal-
culating the quantity of their work; as Bricklayers, Car-
penters, Painters, Tilers, &c.

Fractional Arithmetic, or of fractions, is that which
treats of fractions, both vulgar and decimal.

Harmonical Arithmetic, is so much of the doctrine of
numbers, as relates to making the comparisons, reduc-
tions, &c, of musical intervals.

Arithmetic of Infinites, is the method of summing up
a series of numbers, of which the number of terms is infi-
nite. This method was first invented by Dr. VVallis, as
appears by his treatise on that subject ; where he shows
its uses in geometry, in finding the areas of superficies, the
contents of solids, &c. But the method of fluxions, which
is a kind of universal arithmetic of infinites, performs all
these more easily ; as well as a great many other things,
which the former cannot reach.

Instrumental Arithmetic/Is that in which the common
rules are performed by instruments, or some sort of tangi-
ble or palpable substance. Su,ch are the methods of com-
puting by the ten fingers and the grains of maize, by the
East-Indians and Peruvians, above-mentioned ; by the
Abacus or Shwanpau of the Chinese; the several sorts of
scales and sliding rules; Napier's bones or rods ; the arith-
metical machine of Pascal, and others; Sir Samuel Mor-
land's instrument, described in 16'66"; that of Leibnitz, do-
scribed in the Miscell. Berol.; that of Polenus, published
in the Venetian Miscellany, 1709; and that of Dr. Saun-
derson, of Cambridge, described in the introduction to his
Algebra.

Integral Arithmetic, or of integers, \<, that which re-
spects integers, or whole numbers.

Literal Arithmetic, or Algebra, is that which is per-
formed by letters, which represent any numbers indefinitely.

Logarithmical Arithmetic is performed by the tables
of logarithms. These were invented by baron Napier;
and the best treatise on the subject, is Brigg's Arithmetica
Logarithmica, l6"24.

Logistical Arithmetic. See Logistical.

Numerous or Numeral Arithmetic, .is that which
teaches the calculus of numbers, or of abstract quantities;
and is performed by the common numeral or Arabic cha-
racters.

Political Arithmetic, is the application of arithmetic
to political subjects; such as, the strength and revenues of
nations, the number of people, births, burials, &c. See
Political Arithmetic. To this head may also be referred
the doctrine of Chances, Gaming, &c.

Arithmetic of Radicals, Rutioiuds, and Irrationals.
See Radical, &c.

Arithmetic of Sines. See Sines.

Sexageshnal or Sexagenary Arithmetic, is that which
proceeds by sixties; or the doctrine of sexagesimal frac-
tions : a method which, it is supposed, was invented by-
Ptolemy, in the 2d century; at least they were used by
him. In this notation, the integral numbers from 1 to 59
were expressed in the common way, by the alphabetical
letters': then sixty was called a Sexagena Prima, and mark-
ed with a dash to the character l, thus i'; twice sixty, or
120, thus ll'; and so on to 59 times (>0, or 3540, which is
Lix'. Again, 60 times 6'0, or 3000, was called Scxagena

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A R I

Secunda, and marked with two dashes, thus i"; twice 3600,
thus n"; and ten times 3600, thus x"; &c. And in this
way tha notation was continued to any length. But when
a number less than sixty was to be joined with any of the
sexagesimal integers, their proper expression was annexed
without the dash: thus 4 times 00 and 25, isiv'xxv;
the sum of twice 60 and 10 times 3600 and 15, is x"u'xv.
So near did the inventor of this method approach to the
Arabic notation : instead of the sexagesimal progression,
he had only to substitute decimal ; and to make the signs
of numbers, from 1 to 9, simple characters, and to intro-
duce another character, which should signify nothing by
itself, but serving only to fill up places. — The Sexagenas
Jntegrorum were soon laid aside, in ordinary calculations,
after the introduction of the Arabic notation ; but the
sexagesimal fractions continued till the invention of deci-
mals, and indeed are still used in the subdivisions of the
degrees of circular arcs and angles; except by the French,
who now divide their circles into 400°; and were it not
for the difficulty in overcoming the prejudices for old esta-
blished customs, this method would doubtless be generally
introduced, as it is much better in practice than that in
common use.

Sam. Reyher has invented a kind of sexagenal rods, in
imitation of Napier's bones, by means of which the sexa-
gesimal arithmetic is easily performed.

A good treatise on the Greek or Sexagenary Arith-
metic has lately (1808) been given by M. Delambre, in
F. Peyrard's French translation of the works of Archi-
medes ; and a very good account is given of the same in
the Edinburgh Review, vol. 18, pa. 185.

Specious Arithmetic, is that which gives the calculus
of quantities as designed by the letters of the alphabet : a
method which was more generally introduced into alge-
bra, by Vieta; being the same as literal arithmetic, or al-
gebra.— Dr. Wallis has joined the numeral with the lite-
ral calculus; by which means he has demonstrated the
rules for fractions, proportions, extraction of roots, &c ;
of which a compendium is given by himself, under the ti-
tle of Elementa Arithmetics, in the year l6'98.

Tabular Arithmetic, is that in which the operations
of multiplication, division, (kc,"are performed by means of
tables calculated for that purpose : such as those of Her-
wart, in l6l0; and my tables of powers and products, pub-
lished by order of the Commissioners of Longitude, in 1781.

Tetractic Arithmetic, is that in which only the four
characters 0, 1,2, 3, are used. A treatise of this kind of
arithmetic is extant, by Erhard or Echard Weigel. But
both this and binary arithmetic are little better than cu-
riosities, especially with regard to practice ; as all num-
bers are much more compendiously and conveniently ex-
pressed by the common decuple scale.

Vulgar, or Common Arithmetic, is that which relates
to integers and vulgar fractions.

Universal Arithmetic, is the name given by Newton
to the science of algebra ; of which he left at Cambridge
an excellent treatise, being the text-book drawn up for the
use of his lectures, while he was professor of mathematics
in that university.

ARITHMETICAL, something relating to, or after the
manner of arithmetic.

Arithmetical Complement, of a logarithm, is what the
logarithm wants of 10*00000 &c ; and the easiest way to
find it is, beginning at the left hand, to subtract every fi-
gure from 9, and the last from 10. So, the arithmetical

complement of 8-2501396 is 17498604.— It is commonly
used in trigonometrical calculations, when the first term
of a proportion is not radius ; in that case, adding all to-
gether, the logarithms of the 3d, 2d, and arithmetical com-
plement of the 1st term.

Arithmetical Instruments, or Machines, are instru-
ments for performing arithmetical computations; such as
Napier's bones, scales, sliding rules, Pascal's machine, &c.

Arithmetical Mean, or Medium, is the middle term
of three1 quantities in arithmetical progression ; and is al-
ways equal to half the sum of the extremes. So, an arith-
metical mean between 3 and 7, is 5 ; and between a and
b, is \a -+- f b.

Also, an arithmetical mean among any number of quan-
tities, is often found by adding the quantities all together,
and then dividing the sum by the number of the terms.
So, of the six numbers 1, 3, 4, 6, 7, 9, the sum is 30,
which divided by 6, gives 5, for the mean or medium a-
mong them all.

Arithmetical Progression^ a series of three or more
quantities that have all the same common difference : as
3, 5,7, &c, which have the common difference 2; and a,
a -+- d, a ■+■ Id &c, which have all the same difference d.

In an arithmetical progression, the chief properties are '
these : 1st, The sum of any two terms, is equal to the sum
of every other two that are taken at equal distances from
the two former, and equal to double the middle term when
there is one equally distant between those two: so, in the
series 0, 1, 2, 3, 4, 5, 6, &c, 0-t-6=l-+-5=2+4 =
twice 3 or 6. — 2d, The sum of all the terms of any arith-
metical progression, is equal to the sum of as many terms
of which each is the arithmetical mean between the ex-
tremes; or equal to half the sum of the extremes multi-
plied by the number of terms: so, the sum of these ten

terms 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, is ^ii x 10, or 0 * 5,

which is 45: and the reason of this will appear by invert-
ing the terms, setting them under the former terms, and
adding each two together, which will make double the
same series;

thus 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
inverted 9, 8, 7, 6, 5, 4, 3, 2, 1, 0,
sums 9, 9, 9, 9, 9, 9, 9, 9, 9, 9;
where the double series being the same number of 9's, or
sum of the extremes, the single series must be the half of
that sum. — 3d, The last, or any term, of such a series, is
equal to the first term, with the product added of- the com-
mon difference multiplied by 1 less than the number of
terms, when the series ascends or increases ; or the same
product subtracted when the series descends or decreases:
so, of the series 1, 2, 3, 4, &c, whose common difference
is 1, the 50th term is I -+- 1 x 49, or 1 ■+■ 49, that is 50;
and of the series 50, 49, 48, &c, the 50th term is 50 —
1 x 49, or 50 — 49, which is 1. Also, if

a denote the least term,

s the greatest term,

d the common difference,

n the number of the terms,
and s the sum of them all ;
then the principal properties are expressed by these equa-
tions, viz, 2 =? a -+- d(n — 1),

a =z — d(n — 1),

s = (a -+- z)|»,

s = [z - id(n - 1)] n,

s = \a -1- \d(n — 1)] n.

ARM

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ARM

Also, when the first term a, is 0 or nothing; the theorems
become 2 = d (n — 1), and s = \zn.

Arithmetical Proportion, is when the difference be-
tween two terms, is equal to the difference between two
other terms. So, the four terms, 2, 4, 10, 12, are in a-
rithmetical proportion, because the difference between 2
and 4, which is 2, is equal to the difference between 10
and 12. — The principal property, besides the above, and
which indeed depends upon it, is this, that the sum of the
first and last, is equal to the sum of the two means : so 2
-+■ 12, or the sum of 2 and 12, is equal to 4 + 10, or the
sum of 4 and 10, which is 14.

Arithmetical Ratio, is the same as the difference
of any two terms : so, the arithmetical ratio of the series
2, 4, 6", 8, is 2 ; and the arithmetical ratio of a and b, is
a — b.

Arithmetical Scales, a name given by M. Buffon, in
the Memoirs of the Acad, for 1741, to different progres-
sions of numbers, according to which, arithmetical com-
putations might be made. It has already been remarked
above, that our common decuple scale of numbers was
probably derived from the number of fingers on the two
hands, by means of which the earliest and most natural
mode of computation was performed; and that other scales
of numbers, formed in a similar way, but of a different
number of characters, have been devised ; such as the bi-
nary and tetractic scales of arithmetic. In the memoir
above cited, Buffon gives a short and simple method to
find, at once, the manner of writing down a number given
in any scale of numbers whatever ; with remarks on dif-
ferent scales. The general effect of any number of cha-
racters, different from ten, is, that by a smaller number
of characters, any given number would require more
places of figures to express or denote it by, but then arith-
metical calculations, by multiplication and division, would
be easier, as the small numbers 2, 3, 4, &c, are easier to
use than the larger 7, 8, 9 ; and by employing more than
ten characters, though any given number would be ex-
pressed by fewer of them, yet the calculations in arith-
metic would be more difficult, as by the larger numbers
11, 12, 13, &c. It is therefore concluded, on the whole,
that the ordinary decuple scale is a good convenient me-
dium among them all, the numbers expressed being tole-
rably short and compendious, and no single character re-
presenting too large a number. The same might also be
said, and perhaps more, of a duodecimal scale, by twelve
characters, which would express all numbers in a more
compendious way than the decuple one, and yet no single
character would represent a number too large to compute
by; as is confirmed by the now common practice of ex-
tending the multiplication table, in school books, to 12
numbers or dimensions, each way, instead of 10; and
every person is taught, with sufficient ease, to multiply
and divide by 11 and 12, as easily as by 8 or 9 or 10.
Another convenience might be added, namely, that the
number 12 admitting of moresubmultiplcs than the num-
ber 10, there would be fewer expressions of interminate
fractions in that way than in decimals. So that on all
accounts, it is very probable that the duodecimal would
be the best of any scale of numbers whatever.

Aritii m etica l Triangle. SceTuiANGLE Arithmetical.

ARMED. A magnet or loadstone is said to be armed,
when it is capped, cased, or set in iron or steel ; to make
it take up a greater weight ; and also readily to distin-
guish its poles.

It is surprising, that a little iron fastened to the poles
of a magnet, should so greatly improve its power, as to
make it even 150 times stronger, or more, than it is na-
turally, or when unarmed. The effect however, it seems,
is not uniform ; but that some magnets, by arming, gain
much more, and others much less, than one would ex-
pect; and that some magnets even lose some of their effi-
cacy by arming. In general however, the thickness of
the iron armour ought to be nearly proportioned to the
natural strength of the magnet ; giving thick irons to a
strong magnet, and to the weaker ones thinner : so that
a magnet may easily be over-loaded.

The usual armour of a load-stone, in form of a right-
angled parallelopipedon, consists of two thin pieces of iron
or steel, of a square figure, and of a thickness propor-
tioned to the goodness of the stone ; the proper thickness
being found by trials; always filing it thinner and thin-
ner, till the effect be found to be the greatest possible. —
The armour of a spherical load-stone, consists of two steel
shells, fastened together by a joint, and covering a good
part of the convexity of the stone. This also is to be filed
away, till the effect is found to be the greatest.

Kircher, in his book De Magnete, says, that the best
way to arm a load-stone, is to drill a hole through the
stone, from pole to pole, in which is to be placed a steel
rod of a moderate length : this rod, he asserts, will take
up more weight at the end, than the stone itself when
armed in the common way. And Gasscndus and Cabsus
prescribe the same method of arming. But Muschen-
broek found, by repeated trials, that the usual armour, al-
ready mentioned, is preferable to Kircher's; and he gives
the following directions for preparing it. When, by means
of steel filings and a small needle, the poles of a magnet
have been discovered, he directs that the adjacent parts
should be rubbed or ground into parallel planes, without
shortening the polar axis ; and the magnet may be after-
wards shaped into the figure of a cube or parallelopipedon,
or any other figure that may be more convenient. Plates
of the softest iron are then prepared, of the same length
and breadth with the whole polar sides of the magnet :
the thickness of which plates, so as that they may admit
and convey the greatest quantity of the magnetic virtue,
is to be previously determined by experiment, in a man-
ner which he prescribes for the purpose. A thicker piece
of iron is to be annexed at right angles to these plates,
which is called pes armatura?, the foot or base of the ar-
mour: then the plates, nicely smoothed and polished, arc
to be firmly attached to each of the polar sides, while the
thicker part or base is brought into close contact with the
lower part of the magnet. In this way, he says, almost
all the magnetic virtue issuing from the poles, enters into
the armour, is directed to the base, and condensed by
means of its roundness, so as to sustain the greatest weight
of iron. Phys. Exper. and Geom. Dissert. 1729, pa. 131.

ARMILLARY Sphere, a name given to the artificial
sphere, composed of a number of circles of metal, wood,
or paper, which represent the several circles of the system
of the world, put together in their natural order. It serves
to assist the imagination to conceive the disposition of the
heavens, and the motion of the celestial bodies.

This sphere is represented at Plate 2, Fig. 0", where P
and Q represent the poles of the world, ad the equator,
el the ecliptic and zodiac, tagd the meridian, or the
solstitial colure, t the earth, fg the tropic of cancer, ax
the tropic of Capricorn, mn the arctic circle, ov the ant-

A S C

[ W

aTctic, n and o the poles of the ecliptic, and es the
horizon.

The Armillary sphere constructed by Dr. Long, in Pem-
broke-hall, Cambridge, was 18 feet in diameter; and will
contain more than 30 persons sitting within it, to view, as
from a centre, the representation of the celestial spheres.
The lower part of the sphere, which is not visible to Eng-
land, is cut off; and the whole apparatus is so contrived,
that it may be turned round with as little labour as is em-
ployed to wind up a common jack. See also Mr. Fergu-
son's sphere, in his Lectures, p. 194.

Armillary Trigonometer,za instrument first contrived
by Mr. Mungo Murray, and improved by Mr. Ferguson,
consisting of five semicircles ; viz, meridian, vertical cir-
cle, horizon, hour circle, and equator ; so adapted to each
other by joints and hinges, and so divided and graduated,
as to serve for expeditiously resolving many problems in
astronomy, dialling, and spherical trigonometry. The
drawing, description, and method of using it, may be seen
in Ferguson's Tracts, pa. 80, &c.

ARTILLERY, the heavy equipage of war; compre-
hending all sorts of large fire-arms, with their appurte-
nances ; as cannon, mortars, howitzers, carronades, balls,
shells, petards, musquets, carbines, &c; being what is
otherwise called Ordnance. The term is also applied to
the larger instruments of war used by the ancients, as the
catapult, balista, battering-ram, &c.

The term Artillery, or Royal Artillery, is also applied
to the persons employed in that service ; as also to the art
or science itself; and formerly it was used for what, is
otherwise called Pyrotechnia, or the art of fire-works, with
the apparatus and instruments belonging to the same.

There have been many authors on the subject of artil-
lery; the principal of which are, Bucherius, Braunius,
Tartalea, Collado, Sardi, Ul'ano, Hanzelet, Digges, Mo-
retti, Simienowitz, Mieth, d'Avelour, Manesson, Mallet,
St. Julien ; and the later authors, of still more conse-
quence, are Belidor, St. Remy, le Blond, Valiere, Moro-
gue, Puget, Coudray, Robins, Muller, Eukr, Antoni, Tig-
nola, Scheele; to which may be added the extensive and
accurate experiments published in the 2d and 3d volumes
of my Tracts, and in the Philos. Trans, for 177S.

Park of Artillery, is that place in a camp which is
set apart for the artillery, or large fire-arms.

Traile or Train of Artillery, a number of pieces of
ordnance, mounted on carriages, with all their furniture
fit for marching. To this commonly belong mortars, can-
non, balls, shells, &c— There arc trains of artillery in
most of the royal magazines; as in the Tower, at Ports-
mouth, Plymouth, &c, but, above all, at Woolwich, from
whence the ships commonly receive their ordnance, and
where they are all completely proved before they are re-
ceived into the public service.

The officers and men of the artillery were formerly called
also the Train of Artillery, but are now called the Royal
Regiment of Artillery; consisting at present of ten bat-
talions, and several troops of Horse or Cavalry Artillery.

ASCENDING, in Astronomy, a term used to denote
any star, or degree, or other point of the heavens, rising
above the horizon.

Ascending Latitude, is the latitude of a planet when
going towards the north.

Ascending Node, is that point of a planet's orbit where
it crosses the ecliptic, in proceeding northward. It is
otherwise called the Northern Node, and is denoted by

1 ]
this character

A S C

representing a node, or knot, with the
larger part upwards ; and the same character reversed is
used to denote the opposite, or Descending Node Q .

Ascending Signs, are such as are upon their ascent,
or rise, from the nadir or lowest point of the heavens, to-
wards the zenith, or highest point.

ASCENSION-Day, otherwise called Holy Tliursday, is
a festival of the church, held 10 days before Whit-sunday,
in memory of our Saviour's Ascension.

ASCENSION, in Astronomy, is either Right or Oblique.

Right Ascension of the sun, or of a star, is that de-
gree of the equinoctial, accounted from the beginning of
Aries, which rises with them, in a right sphere. — Or, Right
Ascension, is that point of the equinoctial, counted as be-
fore, which comes to the meridian with the sun or star, or
other point of the heavens. And the reason of thus re-
ferring it to the meridian, is, because this is always at right
angles to the equinoctial ; whereas the horizon is so only
in a right or direct sphere. — The right ascension stands
opposed to the right descension ; and is similar to the lon-
gitude of places on the earth. All the fixed stars, &c,
which have the same right ascension, that is, which are at
the same distance from the first point of Aries, or, which
comes to the same thing, which are in the same meridian,
rise at the same time in a right sphere, that is, to the peo-
ple who live at the equator. And if they be not in the
same meridian, the difference between their times of rising,
or of coming to the meridian of any place, is the precise
difference of their right ascension.* — But, in an oblique
sphere, where the horizon cuts all the meridians obliquely,
different points of the same meridian never rise or set to-
gether: so that two or more stars on the same meridian,
or having the same right ascension, never rise or set at the
same time in an oblique sphere; and the more oblique
the sphere is, the greater is the interval of time between
them.

To find the right ascension of the sun, stars, &c, by
trigonometry, say, As radius is to the cosine of the sun's
greatest declination, or obliquity of the ecliptic, so is the
tangent of the sun's or star's longitude, to the tangent of
the right ascension.

Right Ascension of the Mid-heaven, often used by as-
tronomers, especially in calculating eclipses by means of
the nonagesimal degree, is the right ascension of that point
of the equator which is in the meridian; and it is equal to
the sum of the sun's right ascension and the horary angle
or true time reduced to degrees, or to the sum of the
mean longitude and mean time.

Oblique Ascen sion, is an arch of the equator intercept-
ed between the first point of Aries, and that point of the
equator which rises together with the star, &c, in an ob-
lique sphere.— The oblique ascension is counted from
west to east ; and is greater or less, according to the vari-
ous obliquity of the sphere. — To find the oblique ascen-
sion of the sun, see Ascensional and Globe.

The Arch of Oblique Ascension, is ah arch of the horizon
intercepted between the beginning of Aries, and the point
of the equator which rises with a star or planet, in an ob-
lique sphere; and it varies with the latitude of the place.

Refraction of Ascension and Descension. See Re-
fraction.

ASCENSIONAL Difference, is the difference be-
tween the right and oblique ascension of the same point
on the surface of the sphere. Or it is the time the sun
rises or sets before or after 6 o'clock.

A S C

[ 152 ]

ASP

To find the ascensional difference, having given the
sun's declination and the latitude of the place, say, As ra-
dius is to the tangent^of the latitude, so is the tangent of
the sun's declination, to the sine of the ascensional differ-
ence sought. The sun's ascensional difference converted
into time, shows how much he rises before, or sets after,
6 o'clock. When the sun has north declination, the right
ascension is greater than the oblique ; but the contrary,
when the sun has south declination ; and the difference, in
either case, is the ascensional difference.

ASCENT, the motion of a body from below tending
upwards ; or the continual recess of a body from the earth,
or from some other centre of force. And it is opposed to
descent, or motion downwards.

An acclivity is also sometimes called an ascent, as a
declivity is called a descent.

The Peripatetics attributed the spontaneous ascent of
bodies to a principle of levity, inherent in them. But the
moderns have demonstrated that there is no such thing as
spontaneous levity; and they show, that whatever ascends,
does so by virtue of some external impulse or extrusion.
Thus it is that smoke, and other rare bodies, ascend in the
atmosphere; and oil, light woods, &c, in, water; not by
any inherent principle of levity, but superior gravity, or
.pressure of the medium in which they ascend and float.

The ascent of light bodies in heavy mediums, is produ-
ced after the same manner as the ascent of the lighter scale
of a balance. It is not that such scale has an internal
principle, by which it immediately tends upwards ; but it
is impelled upwards by the preponderancy of the other
scale ; the excess of the weight in the one having the same
effect, by augmenting its impetus downwards, as so much
real levity in the other: because the tendencies mutually
oppose each other, and action and reaction are always
equal. — See this farther illustrated under the articles
Fluid, and Specific Gravity.

Ascent of Bodies on Inclined Planes. See the doctrine
and laws of them under Inclined Plane.

Ascent of Fluids, is particularly understood of their
rising above their own level, between the surfaces of nearly
contiguous bodies, or in slender capillary glass tubes, or
in vessels filled with sand, ashes, or the like porous sub-
stances. Which is an effect that takes place as well in va-
cuo, as in the open air, and in bent, as well as straight
tubes. Indeed some fluids ascend swifter than others, as
spirit of wine, and oil of turpentine; and some rise after
a different manner from others. The phenomenon, with its
causes, &c, in the instance of capillary tubes, will be
treated more atlarge under Capillary Tube.

As to planes: Two smooth polished plates of glass,
metal, stone, or other matter, being placed almost conti-
guous, have the effect of several capillary tubes, and the
fluid rises in them accordingly: the same may be said of
a vessel filled with sand, &c; the various small interstices
of which form, as it were, a kind of capillary tubes. So
that the same principle accounts for the appearance in
tliem all. And to the same cause may be ascribed the
ascent of the sap in vegetables. And on this subject Sir
Isaac Newton says, " If a large pipe of glass be filled with
sifted ashes, well pressed together, and one end dipped into
stagnant water, the fluid will ascend slowly in the ashes,
so as in the space of a week or fortnight, to reach the height
of 30 or 40 inches above the stagnant water. This ascent
is wholly owing to the action of those particles of the ashes
which are on the surface of the elevated water ; those

within the water attracting as much downwards as up-
wards: it follows, that the action of such particles is very
strong ; though being less dense and close than those of
glass, their action is not equal to that of glass, which keeps
quicksilver suspended to the height of 60 or 70 inches,
and therefore acts with a force which would keep water
suspended to the height of above 60 feet. By the same
principle, a sponge sucks in water, and the glands in the
bodies of animals, according to their several natures and
dispositions, imbibe various juices from the blood." Op-
tics, pa. 367.

Again, if a drop of water, oil, or other fluid, be dropped
upon a glass plane, perpendicular to the horizon^ so as to
stand without breaking, or running off; and another plane
touching it at one end, be gradually inclined towards the
former, till it touch the drop ; then will the drop break
and move along towards the touching end of the planes ;
and it will move the faster in proportion as it proceeds far-
ther, because the distance between the planes is constantly
diminishing. And after the same manner, the drop may
be brought to any part of the planes, either upward or
downward, or sideways, by altering the angle of incli-
nation.

Lastly, if the same planes be so placed, as that two of
their sides meet, and form a small angle, the other two
being only kept apart by the interposition of some thin
body; and thus immerged in a fluid, tinged with some co-
lour to render it visible ; the fluid will ascend between the
planes, and that the highest where the planes are nearest;
so as to form a curve line which is found to be a true hy-
perbola, of which one of the asymptotes is the line of the
fluid, the other being a line drawn along the touching sides.

And the physical cause of all these phenomena, is the
same power of attraction.

Ascent of Vapour. See Cloud and Vapour.

Ascent, in Astronomy, &c. See Ascension.

Ascii, are those inhabitants of the globe, who, at cer-
tain times of the year, have no shadow. Such are the in-
habitants of the torrid zone, who twice a year having the
sun at noon in their zenith, have then no shadow.

ASELLI, two fixed stars of the fourth magnitude, in
the constellation Cancer.

ASH-Wednesday, the first day of Lent, supposed to
have been so called from a custom in the church, of
sprinkling ashes that day on the heads of penitents then
admitted to penance.

ASPECT, is the situation of the stars and planets in re-
spect of each other. Or, in Astrology, it denotes a certain
configuration and mutual relation between the planets,
arising from their situations in the zodiac, by which it is
supposed that their powers are mutually either increased
or diminished, as they happen to agree or disagree in their
active or passive qualities. Though such configurations
may be varied and combined a thousand ways, yet only a
few of them are considered. Hence, Wolfius more accu-
rately defines aspect to be, the meeting of luminous rays
emitted from two planets to the earth, cither posited in
the same right line, or including an angle which is an ali-
quot part, or some number of aliquot parts, of four right
angles, or of 300 degrees.

The doctrine of aspects was introduced by the astrolo-
gers, as the foundation of their predictions. And hence
Kepler defines aspect to be, an angle formed b) the rays
of two planets meeting on the earth, capable of exciting
some natural power or influence.

ASS

[ 153 ]

ASS

The ancients reckoned five aspects, viz. conjunction, sex-
tile, quartile, trine, and opposition.

Conjunction is denoted by this character 6 , and is when
the planets are in the same sign and degree, or have the
same longitude.

Sextile is denoted by #, and is when the planets are
distant by the 6th part of a circle, or 2 signs, or 60 de-
grees. .

Quartile is denoted by D , and is when the planets are
distant $ of the circle, or 90 degrees, or 3 signs.

Trine is denoted by A, and is when the planets are di-
stant by | of the circle, or 4 signs, or 120 degrees.

Opposition is denoted by g , and is when the planets are
in opposite points of the circle, or differ by J the circle,
or 6 signs, or ISO degrees of longitude.

Or their characters and distances are as in this following
tablet.

NAME.

CHARACTER.

DISTANCE.

Conjunction

• 6

00= 0s

Sextile - - -

*

60 = 2

Quartile - - -

D

00 = 3

Trine - - -

A

120 = 4

Opposition - -

8

180 = 6

These intervals are reckoned according tothe longitudes
of the planets ; so that the aspects are the same, whether
the planet be in the ecliptic or out of it.

ASPERITY, signifies the inequality or roughness of the
surface of any body; by which some parts of it are more
prominent than the rest, so as to hinder the. hand, &c, from
passing over it with ease and freedom ; and thus producing
what is called friction. — Asperity, or roughness, stands
qpposed to smoothness, evenness, &c.

According to the relations of Vermausen, the blind man
so celebrated for distinguishing colours by the touch, it
seems that every colour has its peculiar degree and kind of
asperity. He makes black the roughest, as it is the dark-
est of colours ; but the others are not smoother in propor-
tion as they are lighter; that is, the roughest do not al-
ways reflect the least light : for, according to him, yellow
is two degrees rougher than blue, and as much smoother
than green. See Colours.

ASSIGNABLE Magnitude, is used for any finite mag-
nitude that can be expressed or denoted. And,

Assignable Ratio, for any expressible ratio.

ASSUMPTION, a feast celebrated in the Romish
church, in honour of the miraculous ascent of the Holy
Virgin, as they describe it, body and soul, into heaven. It
is kept on the 15th of August.

ASSURANCE on Lives, a compact by which security
is granted for the payment of a certain sum of money on
the expiration of the life on which the policy is given, in
consideration of such a previous payment made to the as-
surer as is accounted a sufficient compensation for the loss
and hazard to which he exposes himself:

The sum at which this compensation should be valued,
depends principally on these two circumstances, viz. 1st,
On the rate of interest given for the use of money; and
2d| On the probability of the duration of the life assured,
and .the values of annuities. For, 1st, If the interest of
money be high, the value of the insurance will be pro-
portionally low, and the contrary; because the higher the
rate of interest, the less will be the present value which
amounts to a certain proposed sum in any given time.
Also, if the probability of the duration of life be high, the
value of the assurance will again be proportionally low,

Vol. I.

and the contrary; because the longer the time is, the less
will be the principal which will amount to "any assigned
sum. Thus, if it be required to know the premium or
present value, to be given for 100 pounds to be received
at the end of any time, as suppose 10 years; then, if the
interest of money be at the rate of 5 per cent, the answer,
or present premium, would be 6ll 7s lOd; but at 4 per
cent, it would be 6?l lis Id; and at 3 per cent, it would
amount to 74/ &s id. Again, suppose it were required to
assure 100/ on a life, for any time, for instance 1 year;
that is, let 100/ be supposed to be payable a year hence,
provided $, life of a given age fails in that time : here it is
evident that, whatever be the rate of interest, the less the
probability of the life failing within the year, the kss the
risk is, and the less the premium ought to be. In effect,
the rate of interest being 5 per cent, if it were sure that
the life would fail in that year, the value of the assurance
would be the same as the present value of 100/. payable
at the end of the year, which is 95/ 4s Qd.

But, if it be an equal chance whether the life does or
does not fail in the year, in which case the probability of
failing is ■§; then the- value of the assurance will be but
half the former value, or 47/ 12s 4|d. Or if the odds
against its failing be as 2 to 1, that is, if one person out of
every 3 die at the age of the proposed life, the probability
of dying being only j, the value of the assurance will be
y of the first value, or 31/ 14s lid. And if the odds be
19 to 1, or one person die out of 20, of that age, the pro-
bability of dying will be ^, and the value of the assu-
rance will also be ^ of the first value, or 4/ 1 5s 3d nearly.
Lastly, if only one person die out of 50 at the given age,
the probability of dying will be 3?^, and the value of the
assurance will be accordingly only T^ of the. first sum, or
1/ 18* Id: the interest of money being all along consi-
dered as after the rate of 5 per cent. — Now, according to
Dr. Halley's table of observations, one person dies out of
3, at the age of 87 ; one in 20 at the age of 64 ; and one
in 50 at the age of 39 : It follows, therefore, that the va-
lue of the assurance of 100/. for one year, on a life aged
87, is 31/ 14s lid; on a life aged 64, it is 4/ 15s 3d; and
on a life aged 39, it is 1/ 18s Id: reckoning interest at 5
per cent. But if interest were rated at 3 per cent, these
values would be 32/ 7s 3d, and 4/ 17s Id, and 1/ 18s lOd.
When a life is assured for any number of years ; the
premium or value may be paid, either in one single pre-
sent payment; in consequence of which the sum assured
will become payable without any farther compensation,
whenever, within the given term, the life shall happen to
drop: or the value may be paid in annual payments, to
be continued till the failure of the life, should that happen
within the term; or, if not, till the determination of the
term. And the determination of the value of assurance, in
all cases, is to be made out from the rules for computing
annuities on lives; the principal writers on which a.reHallcy,
De Moivre, Simpson, Smart, Kersseboom, De Parcieux,
Price, Morgan, Maseres, and Baily. See also Life An-
nuties, Reversion, &c.

Assurances may be made either on single lives ; as above
explained; or they may be made on any number of joint
lives, or on the longest of any lives ; that is, an assurer
may bind himself to pay any sums at the extinction of
any joint lives, or the longest of any lives, or at the ex-
tinction of any one or two of any number of lives. There
are further assurances on survivorships ; by which is meant
an obligation, for the value received, to pay a given sum

ASS

[ 154 ]

A S S

or annuity, provided a' given life shall survive any other
given life or lives. For which see Survivorship.

The principal offices for making these insurances, in
England, are the London and the Royal Exchange Assu-
rance Offices ; the Amicahle Society, incorporated for a per-
petual Assurance Office ; the Society for equitable Assu-
rances on Lives and Survivorships; the Westminster So-
ciety for granting Annuities and insuring Money on Lives;
the Phcenix fire-office; the Westminster fire-office; the
Life-assurance Society, for widows and female relations ; the
British fire-office; the Pelican life-insurance; the Laud-
able Society, for widows; the Imperial Insurance Company;
the Globe; the Albion; the Rock ; the Provident Institu-
tion; the Philanthropic; the Hope; the County fire-office ;
the Eagle; the London Life Association; and the Atlas.

Most of these offices, besides various extensive compa-
nies in the country, make also insurances on houses, goods,
ships, &c, from fire and damages, &c. These articles of
assurances are commonly divided into 3 classes : first, com-
mon assurances, which are made at 2s per cent, per annum ;
2d, hazardous assurances, at 3s; and 3dly, doubly hazard-
ous, at 5* per cent, per annum : and the mode of classifi-
cation may be learned from the proposals of any of the
companies. Besides these rates, a duty of It 6d on every
100/ was imposed in 1782, which was increased in 1797
to 2s per cent, and in ] 804 to 2s 6d per cent. From the
produce of this duty, an estimate has been formed of the
total amount of property assured from fire in Great Bri-
tain, which appears to have been at several periods nearly
as follows :

In 1785 =£125 millions.

1789
1793
1797
1801
1806

142
167
184
223
260

As to the assurances on lives, the premium or rates of
the different companies are nearly alike ; as indeed they
must be if founded on correct principles, and equally ad-
vantageous to the public; and the following tables of the
rates are nearly the, same in all.

A Table of Premiums for assuring the Sum of one hundred

Pounds upon the Life o

" any healthy Person, from the

Age of Eight to Sixty-seven.

Age.

For 1 Year.

For 7 Years at an
annual Payment of

For the whole Life
at an annual Pay-
ment of

1 s d

I S d

/ s d

8 to 14

0 17 9

1 1 5

1 17 7

15

0 17 11

1 2 11

1 18 7

10

0 19 2

1 4 7

1 19 8

17

1 1 2

I 6 1

2 0 8

18

1 3 3

1 7 5

2 1 8

19

1 5 0

1 S 0

2 2 S

20

1 7 3

1 9 5

2 3 7

21

1 S 10

1 10 1

2 4 6

22

1 9 3

1 10 6

2 5 4

23

1 9 8

111 0

2 0 3

24

1 10 2

111 6

2 7 1

25

1 10 7

1 12 1

2 8 1

26

111 1

1 12 7

2 9 1

■27

111 7

1 13 2

2 10 1

28

1 12 1

113 9

2 11 1

29

1 12 8

1 14 4

2 12 3

30

113 3

1 14 II

2 1 3 5

31

1 13 9

1 15 7

2 14 7

The

Table Continu

ed.

Age.

For 1 Year.

For 7
annua

Years at an]
Payment of

For th
at an

e who

eLife
lPay-
f

s

d

/

s

d

/

s

d

32

14

4

1

16

3

2

15

9

33

15

0

1

16

10

2

17

1

34

15

8

1

17

8

2

18

5

35

16

4

1

18

10

2

19

10

36

17

0

1

19

7

3

1

4

37

17

9

2

0

8

3

2

10

38

IS

6

2

I

9

3

4

6

39

19

3

2

■ 2

11

3

6

2

40

2

0

8

2

4

1

3

7

11

41

2

2

0

2

5

4

3

9

9

42

2

3

6

2

6

6

3

11

8

43

2

4

6

2

7

9

3

13

8

44

2

5

6

2

9

2

3

15

9

45

2

6

S

2

10

10

3

17

11

46

2

7

10

■2

12

6

4

0

2

47

2

9

0

2

14

4

4

2

7

48

2

10

3

2

16

4

4

5

I

49

2

12

3

2

18

6

4

7

10

50

2

15

1

3

0

8

4

10

s

51

2

17

4

3

2

8

4

13

6

52

2

19

I

3

4

9

4

16

5

53

3

1

0

3

7

0

4

19

7

54

3

3

0

3

9

5

5

2

10

55

3

5

0

3

12

0

5

6

4

56

3

7

3

3

14

8

5

10

1

51

3

9

8

3

17

6

5

14

0

58

3

12

3

4

0

6

5

18

2

59

3

15

1

4

3

8

6

2

8

60

3

18

1

4

7

1

6

7

4

61

4

1

5

4

10

1 1

0

12

4

62

4

3

11

4

15

0

6

17

9

63

4

7

8

4

19

8

7

3

7

64

4

10

9

5

4

10

7

9'

10

65

4

15

2

5

10

10

7

16

9

66

5

0

1

5

17

7.

8

4

1

67

5

5

6

6

5

2

8

12

1

A Table of annual Premiums payable during the joint

Continuance of two lives for assuring one hundred

Pounds, or an equivalent Annuity on the Contingency

of one Life's surviving the other.

Ages.

[Annuity . equivalent to

Life to be
assured.

Life against
which the
Assurance is
to be made.

Premium.

] 00/, to be paid from the
Death of the Life assured,
during the Remainder ol
the oilier Life.

/ .< d

/ * ,1

10

10

1 S 6

5 1 1- 6

20

1 9 1

6 14 10

30

1 8 3

7 14 11

40

1 7 3

9 5 6

50

1 6 11

11 13 0

60

1 6 0

15 13 5

70

1 4 11

23 1 3 0

80

1 3 4

40 1 0 S

20

10

1 16 6

5 (ill

20

1 17 0

6 4 1

30

1 15 9

7 0 0

40

114 8

8 4 11

50

113 6

10 1 9

60

112 1

13 0 7

70

1 10 6

IS 12 S

SO

1 8 3

30 u o

ASS
The Table continued.

[ 1M-]

AST

Life to be
assured.

30

Life against
which the
Assurance is
to he made.

00

07

10
20
30
40
50
60
70
80
10
20
30
4-0
50
60
70
80
10
20
30

40

50

60

70

80

10

20

30

40

50

60

70

80

10

20

30

40

50

60

70

80

4

2

0

18

16

13

19

19

18

15

12

9

5

1

0
1

0
17

13
7
1

15

16

18

.16

14

10
2
9

17
1
2
0

18

15
8

10

5
0
6
9

11

10
7
9
2

10
2

11

10
4

11

10

1 1

10
I

10

10,

7

6

0

9

1

3

0

7

4
10
II

0

9
10

7

6

8

8

9

Annuity equivalent
1 00/, to be paid from the
Death of the Life assured,
during the Remainder of
the other Life.

I

5

5

8

6

2

9

6

19

6

8

3

8

10

0

6

13

0

0

18

12

10

30

9

3

5

3

6

5

19

9

6

16

8

8

1

0

9

16

6

12

14

3

18

5

6

29

19

10

5

1

4

5

16

2

6

12

2

7

16

9

9

12

8

12

6

8

17

11

5

28

12

6

4

19

3

5

12

10

6

7

7

7

10

10

12
17
27
4
5
6
7
9

12
17
27

A Table ol

animal Premiums pay

able during the Conti-

nuance

of two join

Lives for

assuring one hundred

Pounds,

to be paid n

hen either

of the Lives shj

11 drop.

Age.; Age

( s d

Atro.

Arc.; / a

d

Age.

Age.

I s d

10

10

'j

17 1

15

30

3 17

6

20

.55

6

10 2

15

3

1 1

35

4 3

1

60

7

10 2

20

3

5 7

10

4 10

4

67*

9

13 9

25

3

9 3

45

4 19

5

25

25

4

0 10

30

3

13 9

50

5 1 1

3

30

4

5 0

35

3

1') 0

55

6 6

1

35

4

10 3

40

4

1, 10

60

7 6

0

40

4

17 4

45

4

15 1 1

67

9 9

5

45

5

6 2

50

5

7 10

20

20

3 13

11

50

5

17 10

55

6

2 8

25

3 17

5

55

6

12 6

60

7

2 9

30

4 1

9

60

7

12 5

67

9

6 3

35

4 7

3

67

9

15 9

l'5

15

3

5 0

40

4 14

6

30

30

4

8 11

20

3

9 6

45

5 3

i.

35

4

14 1

2 3

3

13 1

50

5 15

4

40

5

0 I I

The Table continued.

Age.

Age.

I s d

Age.

Age.

/ s

(/

Age.

Age.

l s d

30

45

5

9

6

35

67

10 1

2

45

67

10

I 1 J

50

6

1

0

.40

40

5 11

9

50

50

'7

7 S

551

6

15

5

45

5 19

9

55

8

0 3

60

7

15

0

50

6 10

8

60

8

18 2

67

9

18

1

55

7 4

5

67

10

18 10

35

35

4

19

0

60

8 3

4

55

55

8

12 2

40

5

5

6

67

10 5

6

60

9

9 0

45

5

13

10

45

45

6 7

4

67

11

8 5

50

6

5

0

50

6 17

9

60

60

10

4 9

55

6

19

2

55

7 11

0

07

12

2 1

60

7

18

0

00

8 9

6

67

67

13

15 8

From the above specimen of premiums, the reader will
easily judge of the proportional premium for any interme-
diate age.

ASTE11ISM, the same with constellation, or a collec-
tion of many stars, which are usually represented on globes
by some particular image or figure, to distinguish the stars
which compose this constellation from those of others.

ASTRiEA, a name given by some to the sign Virgo, by
others called Erigone, and sometimes Isis. The poets feign
that Justice quitted heaven to reside on earth, in the. golden
age; but, growing weary of the iniquities of mankind, she
left the earth, and returned to heaven, placing herself in
that part of the zodiac called Virgo, where she became a
constellation of stars, and from her orb still looks down on
the ways of men. Ovid. Metam. lib. i. ver. 149.

ASTRAGAL, in Architecture, a small round moulding,
which encompasses the top of the fust or shaft of a column,
like a ring or bracelet. The shaft always terminates at top
with an astragal, and at bottom with a fillet, which in this
place is called ozia.

Astragal, in Gunnery, is a kind of ring or moulding
on a piece of ordnance, at about half a foot distance from
the muzzle or mouth ; serving as an ornament to the gun,
as the former does to a column.

ASTRAL, something belonging to or depending on the
•stars.

Astral Year, or Sidereal Year, See Year.
ASTRO DICTICUM, an astronomical instrument in-
vented by M. Weighel, by means of which many persons
shall be able to view the same star at the same time.

ASTROLABE, from aa-njg, star, and Xayi.Za.vui, I take;
alluding to its use in taking, or observing, the stars. The
Arabians call it in their tongue astharlub; a word formed
by corruption from the common Greek name.

This name was originally used for a system or assemblage
of the several circles of the sphere, in their proper order
and situation with respect to each other. And the ancient
instruments were much the same as our armillary spheres.
The first and most celebrated of this kind, was that of
Hipparchus, which he made at Alexandria, the capital of
Egypt, and lodged in a secure place, where it served for
divers astronomical operations. Ptolemy made the same
use of it: but as the instrument had several inconveni-
ences, he contrived to change its figure, though perfectly
natural, and agreeable to the doctrine of the sphere; and
to reduce the whole astrolabe to a plain surface, to which
he gave the name of the Planisphere. Hence,

Astrolabe is used among the moderns for a Plani-
sphere, or a stereographic projection of the circles of the
sphere on the plane of one of the great circles; which is
usually either the plane of the equinoctial, the eye being
then placed in the pole of the world ; or that of the meri-
X 2

AST

[ 156 ]

AST

dian, the eye being supposed in the point of intersection of " That the heavens are one great volume or book, wherein %
the equinoctial and horizon ; or on that of the horizon. God has written the history of the world ; and in which
The Astrolabe has been treated at large by Stoffler, Gem- every man may read his own fortune, and the transactions
ma Frisius,- Clavius, and our poet Chaucer, who wrote a of his time. The art," say they, " had its rise from the same
Ion" and learned tract upon it, and indeed on the whole people as astronomy itself: while the ancient Assyrians,
practice of astronomy, as the art stood in his day. For a whose serene unclouded sky favoured their celestial obser-
farther account of the nature and kinds of it, see the arti- vations, were intent on tracing the paths and periods of the
cle Planisphere. heavenly bodies, they discovered a constant, settled rela-

Astrolabe, or Sea Astrolabe, more particularly de- tion of analogy, between them and things below; and
notes an instrument chiefly used for taking altitudes at sea; hence were led to conclude these to be the Parcse, the De-
~as the altitude of the pole, the sun, or the stars. stinies so much talked of, which preside at our births, and

The common Astrolabe, represented Plate 2, fig. 7, con^ dispose of our future fate."
sists of a large brass ring, about 15 inches in diameter, " The laws therefore of this relation being ascertained
whose limb, or a convenient part of it, is divided into de- by a series of observations, and the share each planet has
grees and minutes. It is fitted with a moveable label or therein; by knowing the precise time of any person's na-
index, which turns upon the centre, and carries two sights; tivity, they were enabled, from their knowledge in astro-
and having a small ring, at a, to hang it by in time of ob- nomy, to erect a scheme or horoscope of the situation of
servation. the planets, at that point of time; and hence, by consi-

To make use of the Astrolabe in taking altitudes; sus- deririg their degrees of power and influence, and how each
pend it by the ring a, and turn it to the sun, &c, so as that was either strengthened or tempered by some other, to com-
the rays may pass freely through both the sights r and G; pute what must be the result thereof.''
then will the label cut or point out the altitude on the di- Judicial Astrology, it is commonly said, was invented in
vided limb. There a-re many other uses of the Astrolabe; Chalda;a, and from thence transmitted to the Egyptians,
of which Clavius, Henrion, and others have written very Greeks, and Romans; though some insist that it was of
largely. Egyptian origin, and ascribe the invention to Cham. But

The Astrolabe, though now come into disuse, is by ma- it is to the Arabs that we owe it. At Rome the people
ny .esteemed equal to any other instrument for taking the were so infatuated with it, that the astrologers, or, as they
altitude at sea; especially between the tropics, where the were then called, the mathematicians, maintained their
sun comes near the zenith. - ground in spite of all the edicts of the emperors to expel

ASTROLOGICAL, something relating to Astrology. them out of the city. ■ See Genethliaci.
ASTROLOGY, the pretended art of foretelling future Among the Indians, the bramins, who introduced and
events, from the positions, aspects, and influences of the practised this art in the East, have hereby made themselves
heavenly, bodies. The word is compounded of ounftfg, star, the arbiters of good and evil hours, which gives them great
and Aoyoj, discourse; whence, in the literal sense of the authority: they are consulted as oracles; and they have
term, astrology should signify no more than the doctrine taken care always to sell their answers at good rates,
or science of the stars: this indeed was its original accep- The same superstition has prevailed in more modern ages
tation, and constituted the ancient astrology ; which con- and nations. The French historians remark, that in the
sisted formerly of both the branches now called astronomy time of queen Catharine de Medicis astrology was so much
and astrology, under the name of the latter only; and for in vogue, that the most inconsiderable thing was not to be
the sake of making judiciary predictions it was, that astro- done without consulting the stars. And in the reigns of
nomical observations, properly so called, were chiefly made king Henry in and iv of France, the predictions of astro-
by the ancients. And though the two branches be now logers were the common theme of the court conversation,
perfectly separated, and that of astrology almost univer- And this predominant humour in that court was well ral-
sally rejected by men of real learning, this has but lately lied by Barclay, in his Argenis, lib. 2, on occasion of an
been the case, as their union subsisted, in some degree, astrologer, who had undertaken to instruct king Henry in
from Ptolemy till Kepler, who had a strong bias towards the event of a war which was then threatened by the tac-
tile ancient astrology. tion of the Guises.

Astrology may be divided into two branches, natural History of Astrology. — Nothing could be more vain, ri-
and judiciary. , diculous, and absurd, than the pretended science of astro-

To Natuual Astrology belongs the predicting of logy; yet nothing, perhaps, could possibly have led men
natural effects; such as the changes of weather, winds, to a more diligent inquiry info, and finally to a knowledge
storms, hurricanes, thunder, floods, earthquakes, &c. But of, the laws that regulate the motions of the heavenly bo-
this art properly belongs to Physiology, or Natural Philo- dies. Thus it has been said, tint astrology was the daugh-
sophy ; and is only to be deduced, a posteriori, from phe- ter of ignorance, and the mother of astronomy; and cer-
nomena and observations. And for this kind of astrology tainly, if it was not the offspring of passive ignorance, it
it is that Mr. Boyle makes an apology in his History of
the Air. Its foundation and merits may be gathered from
what is said under the articles Air, Atmosphere, and
What her.

Judicial, or Judiciary Astrology, which is what is com-
monly and properly called astrology, is that which pro-

must be acknowledged to have derived its origin from the
natural vanity and credulity of the human mind, aided
perhaps by the coincidence of some accidental circum-
stances, such as an eclipse, the appearance of a comet, or
other celestial phenomenon, at the time of any remarkable
calamity, the death of a prince, or the defeat of an army ;

fesses to foretel moral events, or suchas have a dependence which being once observed, nothing more was necessary,

on the free will and agency of man ; as if they were pro- in this infant stale of the reasoning faculty of man, to lead

duccd or directed by the stars. him to a belief that all great events were connected with

The professors of this kind of astrology maintain, the motion of the heavenly bodies, that the death of princes-

AST

[ m ]

AST

was always announced by some celestial phenomenon; and
finally, that every man had his destiny written in the hea-
vens, in which might be read all the physical and political
events of his life.

Whether or not, astronomical observations were made,
before they were thought necessary to astrological predic-
tions, we have now no means of'determiningj but at all
events, we know that many were made with no other view,
and in consequence of which the knowledge of astronomy
was advanced, if it did not derive its immediate origin from
astrology. It is also probable, that to the same frivolous
motive, we are indebted for the preservation of so many of
the learned works of Greece, which have been handed
down to us. For, as a celebrated author justly observes,
" What could be the charms of truth, purely geometrical,
to a people sunk for so many ages in the deepest abyss of
ignorance and superstition :" but that arithmetic, geome-
try and even optics, such as it was, were the necessary"
steps to astronomy, and this again was the preliminary to
every astrological prediction? In this respect astrology,
if not the mother of astronomy, was at least its nurse, and
the preserver of many of the other sciences: for which
reason its name may still be preserved from that oblivion
which, considered as a science, it so justly merits. At what
time, or in what country this pretended science first ori-
ginated, are alike unknown ; or whether it may not have
been invented, neglected, and revived, in diflerent coun-
tries, at different periods, according as the sciences were
in their infancy, plenitude, or decline. Indeed the latter
seems the most probable, as there were diflerent modes of
drawing horoscopes in different nations; which seems to
indicate that they had not all obtained it from one and
the same original. However this may be, we find its traces
in the ancient records and manuscripts of every nation,
where those records are to be found ; in India, China, Chal-
dea, Greece, Egypt, Arabia, and Europe. In short, in
every country where the sciences have been introduced,
the footsteps of astrology are likewise to be discovered.

In India the pretended knowledge of astrology, or the
art of divination, remained entirely with the bramins,
who were too cunning to promulgate its mysteries, being
aware that it would be the most probable means of detect-
ing the imposition, and their interest was too much in-
volved in the deception, for them not to be anxious to keep
the secret to themselves: it was not only in their hands a
source of emolument and riches, but likewise a conveni-
ent instrument for securing the reverence of the people.
They not only professed a knowledge of judicial astrology,
but also as it regarded meteorology; publishing yearly an
almanac or panjamgam, in which every week, and hour,
both of day and night, were set aside for performing cer-
tain avocations in agriculture and other concerns of life :
they annually presented to the prince a prediction of events
for the following year; and such as were found to deceive
three times were condemned to silence, but the others who
were fortunate in their predictions, enjoyed every distinc-
tion that was due to such profound knowledge. The Chi-
nese had, and still continue to have, their astrologers; and
many records, both of judicial, and natural or meteorolo-
gical astrology of very great antiquity, still remain. The
following is a remarkable instance of the estimation in
which aerology was held in China, and of the brutal bar-
barity and ignorance of their cmpcrorTsin-chi-hoang, who
in the year 246, after having re-united under one head all
the little states, into which the empire had been divided

by intestine broils, and presuming tb&t the sword was suf-
ficient to secure what it had already acquired, he ordered
every book to be burnt, except those relating to agricul-
ture, medicine, and astrology ; three sciences which, it thus
appears, he thought equally serviceable to the nation; while
all those relating to history and pure astronomy were de-
stroyed in the barbarous conflagration. But of all the pro-
fessors of this doctrine, none appear to have been so cele-
brated as the Chaldeans, even the name of Chaldean and
Astrologer became as it were synonymous ; and according
to their records it has been practised among them for thou-
sands of ages : even taking their years for days, which, ac-
cording to some modern authors, is what they are supposed
to represent, they amount to more than 2000 years before
the birth of Christ. Zoroaster is said to have been the
founder of astrology among the Chaldeans ; he was follow-
ed by Hostanes, from whom the Greeks and Egyptians are
supposed to have received this doctrine. Belezes, another
noted Chaldean astrologer, became famous for having pre-
dicted to Arbases the victory that he afterwards gained
over Sardanapalus, a prediction by no means difficult,
when we consider the tyranny and debauchery of the lat-
ter, which caused him to be abhorred and despised, botli
by his troops and his nation. But thus it frequently hap-
pened, that the prediction of an event, which required only
a little foresight and knowledge of mankind to know nearly
to a certainty the result, was the means of raising men to
celebrity, and they were generally too cunning to unde-
ceive the wondering multitude. As to the Greeks, it does
not appear that they were so much infatuated with the be-
lief of judicial astrology, as the nations we have already
mentioned; but to what is called meteorological astrolo-
gy, they paid particular attention ; various predictions ofc
this kind having been given in the Greek calendars by Eu-
doxus, Callippus, Aratus, Hipparchus, and other authors
of eminence. It is along time after this, that we find some
account of the destiny of particular men pretended to be
predicted from an inspection of the stars. We must not
however omit to mention, that there are two works on the
subject of judicial astrology under the name of Ptolemy;,
but it appears there is some reason to suspect that they
arc not his works, and for the honour of the author of the
Almagest, it is to be hoped they are not.

Next to the Chaldeans, the Egyptians were certainly
the most celebrated astrologers; their belief in this doc-
trine being carried to the highest degree of enthusiasm r
all that we suppose to depend upon our will, or on our
determination, they believed to be connected with the mo-
tion of the stars, and that our destinies are written in the-
heavens, wherein may be read our thoughts, inclinations,
constitutions, and every event of our lives.

Till about the commencement of the Christian rera,ora
little before, astrology was circumscribed to the Eastrand;
to Egypt ; but about this time it invaded the Roman em-
pire, and Rome itself was inundated with the professors of
this imaginary science. It, was in vain that the senate
issued decrees against them, under the name of Chaldeans,,
astrologers, or mathematicians, they having arrogated to
themselves the latter appellation; thus degrading one of the
noblest sciences, by associating it with their ridiculous doc-
trine: but notwithstanding these decrees, they still main-
tained themselves, through the support which they expe-
rienced from individuals, and some even gained great ce-
lebrity for their pretended knowledge and ability, as Ni-
gidius I'tgilus Trassylus, the astrologer to Tiberius; Ta-

AST

[ 158 ]

AST

runtius Firmanus, the friend both of Cicero and Varro ;
and particularly the poet Manilius, whose poem entitled
Astronomicon sire Matheseos, lib. viii, published in 1551,
with some Greek and Arabian works of the same kind, is
a sort of course of judicial astrology, having scarcely any
thing in il relating to pure astronomy. Nigidius became
celebrated for predicting to Ocfavius, father of Augustus,
that the infant would be one day master of Rome. Tras-
syluswas the constant attendant of Tiberius; and all those
persons whom the latter had reason, or thought that he had
reason to fear, he caused their horoscope to be drawn, and
if, as it frequently happened, their stars announced them
to be ambitious, or that they had any pretensions to the
empire, they were put to death : such was the effect of
this most abominable, and disgusting doctrine; and who
knows how many may thus have innocently suffered death,
from such predictions, through the tyranny, ignorance,
and superstition of princes, and the rulers of nations? Un-
der these circumstances, astrologers could not fail of hav-
ing a complete ascendency over the people; as it must
have been very dangerous to offend those, who could make
the stars say what they pleased, to gain the favour of their
prince, or to gratify their own private revenge. It should
be observed that this art was not always limited to pre-
dicting events from certain periods of time being given,
but they performed also the converse of the' proposition ;
that is, by knowing the events, they pretended to discover
the time ; this indeed was an easier problem, at least they
were not so likely to be belied by the event, as frequently
happened in other cases. Thus Varro having demand-
ed of Firmanus the time of the birth of Romulus, from the
events recorded of his life, the astrologer discovered that
he was conceived the first year of the second Olympiad,
about 3 o'clock of the 23d day of the Egyptian month
Cha?ac, that is, the 23d of December, and that he was
born under the influence of the sun, the 21st of Septem-
ber following; and moreover, that Rome was founded by
him on the 9th of April, when the moon was in the sign
libra: hence also he determined the fate of the city; for
cities and towns had their horoscopes as well as men ; such
was the pitch to which this most ridiculous science was
carried.

The Arabs are another people who made a considerable
figure in the history of judicial astrology. All the know-
ledge they had of astronomy was made entirely subservient
to that doctrine, and their history is filled with traits on
this subject; a circumstance which rjroves their great pre-
dilection for this pretended science.

The number of Arabian authors, who have written on
astrology, would form a long catalogue; but we shall only
mention some of the most celebrated. Such were Mesalah
and Alchindi, both Jews; and Ali-ben-rodoam, who was
so much admired by Cardan ; Alcabitilis, whoseTwork was
published in 1502 in quarto, and afterwards in folio in
1 503 and 1512; Alfarabius; Alboacen; Aliaben Ragel ;
Albubator; Almansor; and a great number of others,
worthy of the obscurity in .which they tire placed at this
day. Jacob Alchindi the Jew enjoyed great celebrity un-
der the reign of Almanson; but the Arabs, jealous of the
favour which he experienced, accused him of magic, which
led tn a violent dispute between him and a mosulam doctor,
till at last they defied each other's art. This being put to
the test, these two learned professors each drew a circle
about himself, to which they attributed a magical power;
and thus posited, the doctor, by way of challenge, wrote

two words on a piece of paper which he sealed up and de-
livered to the calif, and defied Alchindi to divine what they
were : this was no easy task for Jacob ; but still he cun-
ningly applied himself to his books and instruments; and
then, with all the importance of self-consequence, answer-
ed, that the first of these words was a plant, and the other
an animal ; the calif having opened the paper, he, and all
present, were struck with astonishment to find the words
ussa, mousa. This lucky guess gave Jacob such a decided
advantage over his confounded adversary, that Albuma-
zar, one. of the doctor's pupils, was so much hurt at the
disgrace of his master, that hiding a dagger under his
cloak, he was going to kill Alchindi; when the latter, no
doubt apprised of his design, called out with a stern voice,
" You come to slay me ; quit this design and the poniard
that you carry, and I will teach you astronomy." Astonished
at these words he let fall the dagger, and became one of
Alchindi's most zealous disciples, and afterwards one of
the most celebrated astrologers. He wrote a treatise on
the conjunction of the planets, in order to read, as it were,
the destiny of the earth ; to find the time of its origin, and
the end of the world. He predicted the fates of the dif-
ferent religions; the Christian religion was to subsist for
1460 years, that of Mahomet 544 years; but all these
predictions, like most others of a similar nature, have been
contradicted by facts.

In 1179, the Arabian astrologers predicted for the year
1186, a dreadful inundation, and such terrible tempests,
that would overthrow every thing in their course, in con-
sequence of a conjunction of five planets, and a great
eclipse of the sun. During seven years alarm was spread
in every quarter; but at length the time arrived, and no-
thing extraordinary having been observed, it gave decided-
ly the lie both to the prediction and the art; but notwith-
standing this, the people still continued to pay the same
reverence to the science and its professors as formerly,
nothing being sufficient to open the eyes of the infatuated
multitude.

The Europeans having received their first notions of
astronomy from the Arabians, it was to be expected that
they would partake also of their superstitious belief in as-
trology. Indeed in all those obscure ages which preceded
the regeneration of the arts, none seemed to doubt of the
influence of the stars on the destiny of man, and most of
those who studied or cultivated astronom}', had no other
object in view, than its application to astrological pre-
dictions.

But what must be our opinion of the church of Rome,
when we find one of its cardinals so much infatuated with
his belief in this doctrine, that he actually drew the horo-
scope of Jesus Christ, from a pretended knowledge that he
professed to have of the moment of his birth ; and this be-
ing what may be called predicting after the event, he failed
not to find written in the heavens all the circumstances of
his life and death. And the manner in which he endea-
voured to defend himself from the charge ol impiety al-
leged against him was equally as ridiculous as the asser-
tiop itself.

Among the most celebrated astrologers of Europe, we
find many, whose works on other subjects have done ho-
nour 10 themselves and to their country, and it is to be
wished that they had confined their labours only to these
subjects. Such were Hegiomontaiius, Camerarius, Car-
dan, Argoli, Mori 11, Tycho Brahe, and his patron and
friend count Rantzau; to these we may add, Xjuide

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Botanus de Forli, who was the author of six treatises on
Astrology; Stoffler, who published for many years an ephe-
meris containing astrological predictions. In his almanac
for 1499, ne announced for the year 1514 a terrible in-
undation throughout Europe, on account of the conjunc-
tion of the superior planets in the sign pisces : but this, as
also a similar one by Virdungus, was, when the time arri-
ved, most completely belied by the events, for these years
were rather more dry than usual. These ephemerids were
similar to our present Moore's almanac, which to the dis-
grace of the country still carries on this species of decep-
tion, and which, notwithstanding its many and palpable
contradictions, is at this day regarded as the oracle of wis-
dom among the greater part of our peasantry.

But perhaps none of those above cited, except Tycho
Brahe, was so enthusiastically devoted to astrology as Car-
dan. He even, in imitation of the Roman cardinal, at-
tempted from the configuration of the heavens to draw
the horoscope of Christ; and many other of his notions
were equally absurd and ridiculous. He firmly believed
in the influence of the stars on the fates and destinies of
kingdoms. Thus Butler says :

" Cardan believ'd great states depend

Upon the tip o' th' bear's tail's end ;

That as she whisk'd it t'wards the sun,

Strow'd mighty empires up and down."
He drew his own horoscope ; and it has been said that
he even starved himself, in order that the time of his death
should not give the lie to his doctrine; but this is not
true, for he died before the time that his stars had predict-
ed for that event. Junctinus is another who has written
largely on this subject ; he was astrologer to Catharine de
Medicis, at the same time that he was almoner to the duke
of Anjou; and a column was erected to him in the Halle
au Ble; but he died at last in indigent circumstances at
Lyons; au event which he did not probably foresee not-
withstanding his astrological skill. To these we may far-
ther aud the famous John Dee, who acted in the capacity
of astrologer to queen Elizabeth, by whom he was consult-
ed as to the time propitious for her coronation ; and his
friend and companion Kelly, another of the heroes that
Butler has introduced in that character,
" Kelly did all his fates upon,

The devil's looking-glass, a stone."
From which we may conclude, thai he dealt a little in
the magical way, beside his more common occupation of
astrology. But count Rantzau, and his ally TychoBrahe,
were pel haps the most zealous defenders of astrology ; the
latter having written largely in defence of the truth of this
science, the tendency of which was to show its connexion
with the christian religion, and the impiety in doubting of
the truth of its doctrine. Even Kepler was in some mea-
sure addicted to astrology, as appears from several of his
letters to his friends, where he is seriously discussing the
best method of drawing an horoscope: he was not how-
ever so violent and during. as Stoffler, and Cardan; making
hardy and dangerous predictions, which dishonoured both
the art and artist.

Morin, who was born in 1583, was the last astrologer of
any note, and whose attempts to defend the doctrine of as-
trology may be considered as the last efforts of this expi-
ring art; he was an outrageous opponent to the Copemican
system, with respect to the motion of the earth, and is
said to have laboured 30 yeuis at liis Astrologia G allied,
and predicted the death pf his adversary the learned Gas-

sendi, a number of times ; but the latter seemed lo rise
from under his frequent afflictions, in order to give the lie
to Morin's doctrine. He even foretold the death of Louis
xiii, who never was in better health than at the time an-
nounced by Morin for his death : finally, his works and
his predictions being generally contradicted by the events,
they were despised and ridiculed by all his contemporaries
who had any reputation in the sciences, and his favourite
doctrine seemed to lose ground, as the truth of the modern
system of astronomy became more generally acknow-
ledged. That a nian of this description, an astrological bi-
got, should so strongly combat the idea of the motion of the
ear.th, is by no means surprising ; because the old system
of astronomy was much more favourable to the doctrine
of astrology than the modern. Indeed it was to this sys-
tem having finally prevailed over the obstacles that igno-
rance and superstition had raised against it, that was the
principal cause of plunging astrology into that contempt
it so justly merited. For as soon as it was known that
the earth was not the centre.of the universe; that it was
only one of the little satellites of the sun ; that instead
of the sun and stars turning about the earth, it turned
on its own axis to receive the light and heat of that lu-
minary ; and finally, that the fixed stars are placed at an
immense distance, that even startles the imagination; —
how could it for a moment be admitted, that all this vast
machine was subordinate to this little globe of earth, and
be designed only to preside over the destinies of its inha-
bitants !

Morin died in 1656, since which time astrology has been
declining; and at this day there is not a man of science in
Europe who is not fully aware of the absurdity of this pre- '
tended doctrine.

But in the East, astrology still preserves its credit ; and
astronomy is only cultivated there, so far as it is thought
necessary for astrological purposes; for the Turks can no
more be persuaded that the Europeans study astronomy
without believing in astrology, than imagine that ws can
examine the inscriptions and ruins of antiquity, without
searching for treasure.

It may be thought that too much has been said on this
subject, that even its name ought not to be introduced into-
books of science ; and certainly, were it not for the benefit
that astronomy has derived from it, and the respectability
of many authors who, unfortunately for their memoirs,,
have been its admirers aud supporters,it would long before
this have been lost in total oblivion, and even its name
would not have been remembered by philosophers. But
those circumstances cannot fail of giving it some interest
when we are contemplating the progress of human know-
ledge, and when we reflect, that without this superstitious
stimulus, we might still have remained ignorant of those
grand and sublime laws, that regulate the motion of the
heavenly, bodies, and preserve unimpaired, the universal
harmony of our planetary system.

ASTROMETEOROLOOIA, the art of foretelling the
weather, and its changes, from the aspects and configura-
tions of the moon and planets: a species of astrology distin-
guished by some under the denomination of meteorological
astiolosy.

ASTRONOMICAL, something relating to Astronomy.

Astronomical Calendar, Characters, Column, Horizon r
Hours, Month, Quadrant, Ring-Dial, Sector, Tables, Te-
lescope, Time, Year. Sjee the several substantives.

Astronomical Observations. Of these there are re-

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cords, or mention, in almost all ages. It is said that the
Chinese have observations for a course of many thousand
years. But of these, as well as those of the Indians, we
have yet had but little benefit. But the observations of
most of the other ancients, as Babylonians, Greeks, &c.
among which those of Hipparchus make a principal figure,
are carefully preserved by Ptolemy, in his Almagest.

About the year 8S0, Albategni, a Saracen, applied him-
self to the making of observations ; in which he was fol-
lowed by others of the same nation, as well as Persians
and Tartars j among whom were Nassir-Eddin-Ettusi,
Arzachel, who also constructed a table of sines, and Ulug
Beigh. In 1457 Regiomontanus undertook the province
at Nuremberg; and his disciples, J. Werner and Ber.Wal-
ther, continued the same from 1475 to 1504. Their ob-
servations were published togetherin 1544. — In 1509, Co-
pernicus, and after him the landgrave of Hesse, with his as-
sistants Rothman and Byrge, observed ; and after them
Tycho Brahe, assisted by the celebrated Kepler, from 1582
to l601. — All the foregoing observations, together with
Tycho's apparatus of instruments, are contained in the
Historia Coelestis, published in 1672, by order of the em-
peror Ferdinand. — In 1651, was published at Bononia, by
Ricciolus, Almagestum Novum, being a complete body of
ancient and modern observations, which he so named after
the work of the same nature by Ptolemy. — Soon after, He-
velius, with a magnificent and well-contrived apparatus of
instruments, described in his Machina Coelestis, began a
course of observations. It has been objected to him, that
he only used plain sights, and could never be brought to
take the advantage of telescopic ones ; which occasioned
Dr. Hooke to write animadversions on Hcvelius's instru-
ments, printed in 1674, in which he too rashly despises
them, on account of their inaccuracy: but Dr. Halley,
who at the instance of the Royal Society went over to
Dantzick in the year l679j to inspect his instruments', ap-
proved of their justness, as well as of the observations made
with them. See Sights.— Our two countrymen Jer.
Horrox and Will. Crabtree, are celebrated for their ob-
servations from the year 1635 to lfj45, who first observed
the transit of Venus over the sun in the year 1630. — They
were followed by Flamsteed, Cassini the father, and son,
Halley, de la Hire, Roemer, and Kirchius. — We have also
now had published the accurate observations of Dr. Brad-
ley and of Dr. Maskelyne ; as also those of the French
and other observatories, with the observations of many in-
genious private astronomers, which are to be found in the
Transactions and Memoirs of the various Philosophical
Societies. — There have been also observations of many
other eminent astronomers; as, Galileo, Huygens, and our
countryman Harriot, whose very interesting observations
have lately .been brought to light by the earl of Egrcmont,
and count Bruhl, though they have not been thought fit to
be published. Other publications of celestial observations,
are those of Cassini, LaCaiIlc,Monnier, Lalande, Herschel,
&c. — See farther under Celestial Observations, Cata-
logue, Observatory, &c.

Astronomical Place of a star or planet, is its longi-
tude, or place in the ecliptic, reckoned from the beginning
of aries, in consequential or according to the order of the
sign?.

ASTRONOMICALS, a name used by some writers for
sexagesimal fractions ; on account of their use in astrono-
mical calculations.

ASTRONOMICUS Radius. See Radius.

ASTRONOMY, the doctrine of the heavens, and their
phenomena. Astronomy is properly a mixed mathematical
science, by which we become acquainted with the celes-
tial bodies, their motions, periods, eclipses, magnitudes,
distances, and other phenomena. Some, however, under-
stand the term astronomy in a more extensive sense, as
comprising in it the theory of the universe, with the pri-
mary laws of nature : in which sense it seems to be rather
a branch of physics than of mathematics.

History of Astronomy. — The invention of astronomy
lias been variously given, and ascribed to several per-
sons, several nations, and several ages. Indeed it is pro-
bable that mankind never existed without some know-
ledge of astronomy amongst them. For, besides the mo-
tives of mere curiosity, which are sufficient of themselves
to have excited men to a contemplation of the glorious
and varying celestial canopy, it is obvious that some parts
of the science answer such essential purposes to mankind,
as to make the cultivation of it a matter of indispensible
necessity. — Accordingly we find traces of it in various
nations, and records of its cultivation by the earliest
inhabitants. Josephus informs us, that the sons of Seth
employed themselves in the study of astronomy, and that
they engraved their observations on two pillars, one of
brick and the other of stone, in order that they might be
preserved from the destruction which Adam had foretold
should come upon the earth. He also relates that "Abra-
ham gave lectures in astronomy and arithmetic to the
Egyptians," " which," he says, " they knew nothing of, till
Abraham brought them from Chaldea to Egypt, and from
thence they passed to the Greeks." Berosus also observes,
" that Abraham was a great and just man, and famous Wr
his celestial observations." The same author ascribes to
the antediluvians a knowledge of the astronomical cycle
of 600 years, which may be supposed to mean the period
in which the sun and moon came again into the same situa-
tion with regard to the nodes, apogee of the moon, &c.
" This period," says Cassini, " of which we find no intima-
tion in any monument of any other nation, is the most ac-
curate of the kind that ever was invented : for it brings out
the solar year more exactly than that of Hipparchus and
Ptolemy ; and the lunar month within about one second
of what is determined by modern astronomers." If the an-
tediluvians had such a period of 600 years, they must have
known the motions of the sun and moon more exactly than
their descendants knew them some ages after the flood.

The accounts however which we have of these matters,
anterior to the flood, do not carry with them the degree
of credibility that would justify a more particular mention
of them ; nor would a more copious detail conduct us to
any certain conclusion of the age or country where the
science of astronomy first originated, or by whom it was
first methodised and extended. Whatever credibility these
uncertain testimonies of the ancients may possess ; certain
it is that most authors are agreed that the commencement
of the science originated either in Chaldea, Egypt, or In-
dia; all of which nations pretended to a very high anti-
quity, and equally claimed the honour of producing the
first cultivators of this science. The Chaldeans in parti-
cular boasted of their temple, and enormous tower of Be-
lus, which has been thought by some to have been an as-
tronomical observatory ; and no less of their famous as-
tronomer and philosopher Zoroaster, whom they placed
500 years before the destruction of Troy. While the
Egyptians, with similar ostentation, vaunted of their Col-

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leges of priests, which were the depots of every species of
knowledge ; and of the monument of Osymandyas, in
which tradition asserts, there was a golden circle of 365
cubits in circumference, and one cubit thick, divided into
365 equal parts according to the days of. the year, and
containing the heliacal risings and settings of the stars for
each day, &c.

It is indeed evident that both Chaldea and Egypt were
countries very proper for astronomical observations, on
account of the extended flatness of the country, and the
purity and serenity of the air. The tower of Belus, or of
Babel, being of a great height, was probably an astrono-
mical observatory ; and the lofty pyramids of Egypt, what-
ever they were originally designed for, might perhaps an-
swer the same purpose; at least, they show the skill of
this people in practical astronomy, as they are all placed
with their four fronts exactly facing the cardinal points
of the compass. For it is hardly possible, that they should
have placed them in that striking position, and with such
a degree of exactness, without some particular reason;
and this being the case, it will follow, that they were ac-
quainted with the correct method of drawing a meridian
line, which is a matter of more difficulty than is generally
supposed: it being well known that Tycho Brah6, the
most able astronomer of his time, committed an error of
several minutes in tracing that of the observatory of Ura-
nisburg.

The Chaldeans certainly began to make observations
soon after the confusion of languages, as appears from the
observations found there on the taking of Babylon by
Alexander; and probably much earlier. It hence appears
that they had determined, with tolerable exactness, the
length both of a periodical and synodical month. They
had also discovered, that the motion of the moon was not
uniform; and they even attempted to assign those parts of
the orbit in which the motion is quicker or slower. We
are also assured by Ptolemy that they were not unac-
quainted with the motion of the moon's apogee and nodes,
the latter of which they supposed made a complete revo-
lution in 6585| days, or a little more than 18 years, and
that it contained 223 complete lunations, which period is
called the Chaldean Saros. From Hipparchus, the same
author also gives us several observations of lunar eclipses
made at Babylon above 720 years before Christ. And
Aristotle informs us, that they had many occultations of
the planets and fixed stars by the moon; a circumstance
which led them to conceive that eclipses of the sun were
to be attributed to the same cause. They had also no in-
considerable share in arranging the stars into constella-
tions. Nor had even those eccentric bodies the comets
escaped their observation : for both Diodorus Siculus and
Appollinus Myndicus, Seneca informs us, accounted these
to be permanent bodies, having stated revolutions as well
as the planets, but in much more extensive orbits: al-
though others of them were of opinion, that the comets
were only meteors raised very high in the air, which,
blazing for a while, disappear when the matter of which
they consist is consumed or dispersed. The art of dialling
was also practised among them long before the Greeks'
were acquainted with that science. Herodotus says, that
the Greeks were made acquainted with the use of the Pole
and Gnomon through the medium of the Babylonians ;
the latter of which seems to have been the most ancient
instrument made use of in astronomy.

The Egyptians, it appears from various circumstances,
Vol. 1.

laid claim to nearly the same critical pretensions in the
knowledge of astronomy as the Chaldeans. ' Herodotus
ascribes their knowledge in the science to Sesostris; pro-
bably not the same whom Newton makes contemporary
with Solomon, as they were acquainted with astronomy
at least many hundred years before that Eera. We learn,
from the testimony of some ancient authors, many parti-
culars relative to the state of their knowledge in this sci-
ence; such as, that they believed the figure of the earth
was spherical; that the moon was eclipsed by passing
through the earth's shadow, though it does not certainly
appear that they had any knowledge of the true system
of the universe; that they attempted to measure the mag-
nitude of the earth and sun, though their methods of as-
certaining the latter were very erroneous; and that they
even pretended to foretel the appearance of comets.

Diodorus Siculus also informs us that the Egyptians
discovered that the planets had sometimes a direct, and
sometimes a retrograde motion, and at others were sta-
tionary: that they made the sun move in a circle inclined
to the equator, and in a contrary direction to the diurnal
motion. The idea of dedicating the seven days of the
week to the planets, is also ascribed to the Egyptians.
We are also further informed by Mr. E. Barnard, that
these people discovered that the stars had an annual mo-
tion of 50" 9"' 45"" in a year (Phil. Trans. No. 15§) : and
by Macrobius that the Egyptians made the planets revolve
about the sun in the same order as is ascribed to them by
us: but it does not appear at what time the planets were
first discovered to be of a different nature from, the fixed
stars nor by whom. They were also acquainted, as is
generally supposed, with several other branches of this
science, as indeed were the Chaldeans; which will appear
from the first notions they entertained of the system of
the world, and the agreement which has been found among
several ancient measures of the circumference of the earth.

Many other nations, it is true, have claimed the honour
of the original cultivation of this science ; but among the
various observations of the ancients which have been
transmitted to us, none appear of greater antiquity than
those of the Chinese. The most remarkable of these is a
conjunction of live of the planets which, according to
their annals, is said to have taken place in the reign of
their emperor Tcheun-hiu, about 2500 years before Christ.
They also mention an eclipse of the sun, which happened
in the constellation Scorpio about the year 2150 of the
same aera ; and which is said to have proved fatal to two
Chinese astronomers, of the names of Ho and Hi, who
were condemned to death by the emperor Tchong-kang,
on account of their omitting, through negligence and in-
toxication, to announce the precise time at which it ar-
rived. And from these data, apparently well attested, se-
veral eminent astronomers have endeavoured to discover
whether these events could have possibly happened about
the time here mentioned ; but the subject is attended with
too many difficulties to afford any satisfactory result.

Most of the accounts which we have of the Chinese
astronomers have come from the Jesuit missionaries. 1 hey
inform us, that the Chinese were originally taught the sci-
ence of astronomy by their first emperor Fo-hi, who is
supposed by some to be the same with Noah. He also
constructed astronomical tables, and gave the figures of <>he
heavenly bodies. In the reign of the emperor Hpang-ti,
about 20.97 years before Christ, Yuchi likewise observed
the pole-star and the constellations circumscribing it. He
Y

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also constructed a sphere with several fixed and moveable
circles ; and was the author of an instrument by winch
he could fir.d the four cardinal points, without any refer-
ence to the heavens ; which was probably the same as a
compass. Nearly about this time the cycle of 60 years
was also established, and it is asserted by F. Gaubril,
that for more than a century A. C. the Chinese had dis-
covered the obliquity of the ecliptic, with the theory of
eclipses : and that they were, long before that, acquainted
with the true length of the solar year, the method of ob-
serving meridian altitudes of the sun by the shadow of a
anomon, and of deducing from thence his declination, and
the height of the. pole. The same missionary also says,
that the Chinese have yet remaining some books of astro-
nomy, which were written about 200 years before Christ;
from which it appears, that the Chinese had known the
daily motion of the sun and moon, and the times of the
revolutions of the planet", many years before that period.

Du Halde likewise informs us, that Tcheou-cong, the
most skilful astronomer that ever China produced, lived
more than a thousand years before Christ ; that he passed
whole nights in observing the celestial bodies, and arrang-
ing them into constellations, &c. At present, however,
the state of astronomy is but very low in that country, al-
though it be cultivated at Peking, by public authority, in
like manner as in most of the capital cities of Europe.

The inhabitants of Japan, of Siam, and of the Mogul's
empire, have also been acquainted with astronomy from
time immemorial ; and the celebrated observatory at Be-
nares, is a monument both of the ingenuity of the people,
and of their skill in that science.

According to Porphyry, astronomy must have been of
very ancient standing in the east. He informs us that,
when Babylon was taken by Alexander, there were brought
from thence celestial observations for the space of Jg03
years ; which therefore must have commenced within 115
years after the flood, or within 15 years after the building
of Babel. — Epigenes, according to Pliny, affirmed that the
Babylonians had observations of 720 years engraven on
bricks. — Again, Achilles Tatius ascribes the invention of
astronomy to the Egyptians ; and adds, that their know-
ledge of that science was engraven on pillars, and by that
means transmitted to posterity.

M. Bailly, in his elaborate history of ancient and mo-
dern astronomy, endeavours to trace the origin of this sci-
ence among the Chaldeans, Egyptians, Persians, Indians
and Chinese, to a very early period. And thence he main-
tains, that it was cultivated in Egypt and Chaldea 2800
years before Christ; in Persia, 3209; in India, 3101; and
in China, 29^2 years before that a;ra. He also appre-
hends, that astronomy had been studied even long before
these distant periods, and that we are only to date its re-
vival from thence.

In investigating the antiquity and progress of astronomy
among the Indians, M. Bailly examines and compares four
different sets of astronomical tables of the Indian philoso-
phers, namely that of the Siamese, explained by M. Cas-
sini in lfi8<); that brought from India by M. le Gentil of
the Academy of Sciences; and two other manuscript ta-
bles, found among the papers of the late M. de Lisle ; all
ol whicli he found tn agree very accurately, and referring
to the meridian of Benares, above mentioned. It appears
that the fundamental epoch of the Indian astronomy, is a
conjunction of the sun and moon, which took place at the
amazing distance of 3102 years before Chr'st : and JM.

Baillj' informs us that, by our most accurate astronomical
tables, such a conjunction did really happen at that time.
He further observes that, at present, the Indians calculate
eclipses by the mean motions of the sun and moon ob-
served 5000 years since ; and that their accuracy, with re-
gard to the solar motion, far exceeds that of the best Gre-
cian astronomers. They had also settled the lunar mo-
tions by computing the space through which that lumi-
nary had passed in 1,600,984 days, or a little more than
4383 years. M. Bailly also informs us, that they make
use of the cycle of 19 years, the same as that ascribed by
the Greeks to Meton; that their theory of the planets is
much better than Ptolemy's, as they do not suppose the
earth in the centre of the celestial motions, and believe
that Venus and Mercury move round the sun ; and that
their astronomy agrees with the most modern discoveries
as to the decrease of the obliquity of the ecliptic, the ac-
celeration of the motion of the equinoctial points, &c.

In the 2d vol. of the transactions of the Royal Society
of Edinburgh, is also a learned and ingenious dissertation
on the astronomy of the Brahmins of India, by Mr. Pro-
fessor Playfair ; in which the great accuracy and high an-
tiquity of the science, among them, is reduced to the
greatest probability. It hence appears, that their tables
and rules of computation have peculiar reference to an
epoch, and to observations, 3 or 4 thousand years before
Christ; and many other instances are there adduced, of
their critical knowledge in the other mathematical sci-
ences, employed in their precepts and calculations.

Astronomy, it seems too, was not unknown to the Ame-
ricans ; though in their division of time, they made use
only of the solar, and not of the lunar motions. And that
the Mexicans, in particular, had a strange predilection
for the number 13, by means of which they regulated al-
most every thing : their shortest periods consisted of 13
days; their cycle of 13 months, each containing 20 days;
and their century of 4 periods, of 13 years each: and this
excessive veneration for the number J 3, arose, according
to Siguenza, from its being the number of their greater
gods. And it is very remarkable, that the Abbe Clavigero
asserts it as a fact, that, having discovered the excess of a
few hours in the solar above the lunar year, they made use
of intercalary days, to bring them to an equality, as esta-
blished by Julius Caesar in the Roman Calendar; but with
this difference, that, instead of one day every 4 years, they
interposed 13 days every 52 years, which products the
same effect.

This science however fell into great decay with the
Egyptians, and in the time of the emperor Augustus, it
was entirely extinct among them.

From Chaldea and Egypt the science of astronomy
passed into Phenicia, which this people applied to the
purposes of navigation, steering their course by the north
polar star ; and hence they became masters of the sea,
and of almost all the commerce in the world.

The Greeks, it is probable, derived their astronomical
knowledge chiefly from the Egyptians and Phoenicians, by
means of several of their countrymen who visited these
nations, for the purpose of learning the different sciences.
Newton supposes that most of the constellations were in-
vented about the time of the Argonautic expedition; but
it is more probable that tliey were, at least the greater
part of them, of a much older date, and derived from
other nations, though clothed in fables of their own in-
vention or application. Several of the constellations are

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mentioned by Hesiod and Homer, the two most ancient
writers among the Greeks, and who lived about 870 years
before Christ. Their knowledge in this science, however,
was greatly improved by Thales the Milesian, and other
Greeks, who travelled into Egypt, and brought from
thence the chief principles of the science. Thales was
born about 640 years before Christ ; and he, first of all
among the Greeks, observed the stars, the solstices, the
eclipses of the sun and moon, and predicted the same.
And the same was farther cultivated and extended by his
successors Anaximander, Anaximanes, and Anaxagoras ;
but most especially by Pythagoras, who was born 577 years
before Christ, and having resided for several years in
Egypt, &c, brought from thence the learning of these peo-
ple, taught the same in Greece and Italy, and founded the
sect of the Pythagoreans. He taught that the sun was in
the centre of the universe; that the earth was round, and
people had antipodes ; also that the moon reflected the
rays of the sun, and was inhabited like the earth ; that
comets were a kind of wandering stars, disappearing in
the further parts of their orbits ; that the while colour of
the milky-way was owing to the united brightness of a
great multitude of small stars; and he supposed that the
distances of the moon and planets from the earth, were in
certain harmonic proportions to one another.

Philolaus, a Pythagorean, who flourished about 450
years before Christ, asserted the annual motion of the
earth about the sun. He was persecuted for propagating
this opinion, and obliged to fly ; and it is a circumstance
worthy of remark, that Galileo lost his liberty for main-
taining the same. Not long after, the diurnal motion of
the earth on her own axis, was taught by Hicetas, a Syra-
cusan. About the same time flourished at Athens, Meton
and Euctemon, where they observed the summer solstice
432 years before Christ, and observed the risings and set-
tings of the stars, and what seasons they answered to.
Meton also invented the cycle of 1<) years, which still
bears his name.

The next astronomer was Eudoxus the Cnidian, and
scholar of Plato, who lived about 370 years before Christ,
and was accounted one of the most skilful astronomers
and geometricians of antiquity, being esteemed the in-
ventor of many of the propositions in Euclid's Elements,
and having introduced geometry into the science of astro-
nomy. He travelled into Asia, Africa, Sicily, and Italy,
for improvements in astronomy; and we are informed by
Pliny, that he determined the annual year to contain 365
days 6 hours, and also the periodical times of the planets,
anil made other important observations and discoveries.

Calippus flourished soon after Eudoxus, and his celestial
sphere is mentioned by Aristotle ; but he is better known
by his invented period of 76", containing 4 corrected Me-
tunic periods, and which commenced at the summer sol-
stice in the year 330 before Christ. About his time the
knowledge of the Pythagorean system was carried into
Italy, Gaul, and Egypt, by certain colonies of Greeks.

However, the introduction of astronomy into Greece is
represented by Vitiuvius in a manner somewhat different.
He maintains, that Berosus, a Babylonian, brought it im-
mediately from Babylon itself, and opened an astronomi-
cal school in the isle of Cos. And Pliny says, that in con-
sideration of his wonderful predictions, the Athenians
erected him a statue in the gymnasium, with agilded tongue.
But if this Berosus be the same with the author of the
Chaldaichistories, he must have lived before Alexander.

After the death of this conqueror, the sciences flourished
chiefly in Egypt, under the auspices of Ptolemy Phila-
delphus and his successors. He founded a school there,
which continued to be the grand seminary of learning, till
the invasion of the Saracens in the year of Christ 650.
From the founding of that school, the science of astro-
nomy advanced considerably. Aristarchus, about 270
years before Christ, strenuously asserted the Pythagorean
system, and gave a method of determining the sun's di-
stance by the dichotomy of the moon. — Eratosthenes, who
was born at Cyrene in the year 271 before Christ, mea-
sured the circumference of the earth by means of a gno-
mon ; and being invited to Alexandria, from Athens, by
Ptolemy Euergetes, and made keeper of the royal library
there, he set up for that prince those armillary spheres,
which Hipparchus and Ptolemy the astronomer afterwards
employed so successfully in observing the heavens. He
also determined the distance between the tropics to be -j-}
of the whole meridian circle, which makes the obliquity
of the ecliptic in his time to be 23° 51y . — The celebrated
Archimedes likewise cultivated astronomy, as well as geo-
metry and mechanics : he determined the distances of the
planets from one another, and constructed a kind of pla-
netarium or orrery, to represent the phenomena and mo-
tions of the heavenly bodies.

The next ancient astronomer that deserves our most
grateful acknowledgements, for the number of his correct
observations, as well as the important conclusions he de-
rived from them, is Hipparchus. This celebrated astrono-
mer was the first who cultivated all the branches of this
science ; his first labours were directed to the obliquity
of the ecliptic, observed by Eratosthenes, which he found
to be perfectly correct ; and this was afterwards confirm-
ed by Ptolemy. He next attempted to determine the
length of the tropical year; a thing which had not been
done by his predecessors, as he conceived, with that de-
gree of accuracy as the importance of it deserved : this
he did by observing the interval of the return of the sun
to the same tropic or equinox : he imagined that if he
could get two corresponding observations, after a great
number of revolutions, the error would be proportionally
diminished ; which was the method employed by future
astronomers, in determining the mean motions of all the
planets ; a discovery of very high importance in this sci-
ence. He compared the observation of a solstice by Ari-
starchus, with one made by himself, at the interval- of 145
years, and found that this happened half a day sooner than
it ought to have done, if the year consisted of 365^ days,
as the Greeks believed before him. Thus then he deter-
mined the tropical year to be 36'5d 5h 55m 12*, being
nearly 44 minutes too great. He constructed the first so-
lar tables, of which any mention is made in ihc history
of astronomy ; and discovered the excentricity of the
sun's orbit : he also considered the motion of the moon,
and attempted to estimate the exact time of her revolu-
tion, by a comparison of ancient eclipses ; he likewise dis-
covered the excentricity of her orbit, as indeed he did
that of all the planets : he found that the moon's motion
was not uniform, that she moved slower in the apogee of
her orbit than in the perigee, and that there was a mo-
tion of anticipation of her nodes : he collected the ac-
counts of the Egyptians and Chaldeans relative to eclip-
ses ; and calculated all that were to happen for 600 years
to come. It is also to this celebrated astronomer that \vc
are indebted for the discovery of the principle of the eqtia-
Y2

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'ion of time, and the equally important one of the preces-
sion of the equinoxes : he found that the stars changed
their declinations, but not their latitudes, and on which
account he referred them to the ecliptic. He also knew
that an eclipse of the sun varied on different parts of the
earth ; and thus discovered that the moon had a parallax;
and this suggested to him a method of determining the
sun's distance from the earth more accurately than any
of his predecessors had done; he also enriched geography
with many discoveries, and was the first writer that we
are acquainted with on spherical trigonometry; but his
chief work is a catalogue which he made of the fixed stars,
to the number of 1022, with their longitudes, latitudes,
and apparent magnitudes ; which, with most of his other
observations, are preserved by Ptolemy in his Almagest.

After this period it may be remarked, that little pro-
gress was made in this science from the time of the last-
mentioned astronomer to that of Ptolemy, who was born
at Ptolemais in Egypt, in the first century of Christianity,
and who made the greatest part of his observations at the
celebrated school of Alexandria in that country. Profit-
ing by those of Hipparchus, and other ancient astrono-
mers, he formed a system of his own, which, though er-
roneous, was followed for many ages by all nations. He
compiled a great work, called the Almagest, a name given
it by the Arabs: but which was originally entitled MeycL-
Xtj ivvrafrc, or Great Construction. From its containing
the observations and collections of Hipparchus, Timocha-
ris, and others his predecessors in astronomy, it will ever
be valuable to the professors of that science. Thus did
the .Alexandrian school continue to be enriched, and to
gain celebrity in science, for more than five centuries;
when (as has been observed) an event the most calami-
tous, and the most disgraceful to its perpetrators, laid in
ruins its valuable library, and consumed in one hour the
labours of all the most celebrated astronomers of that
period. But it deserves to be remarked, that the very
people who destroyed those admirable treasures of learn-
ing and invention, became afterwards the protectors of
science and literature, and cultivated astronomy with the
greatest ardour.

Geography likewise is much indebted to Ptolemy for
his meritorious services in collecting all the determina-
tions of the latitudesand longitudes of places then known,
and for his fundamental principle of the methods of pro-
jections for constructing geographical charts; which was
scarcely known before his time. The Almagest was pre-
served from the lamentable conflagration of the Alexan-
drine library by the Saracens, and translated out of Greek
into Arabic in the year S27, and from thence into Latin
in 1230. The Greek original was not known in Europe
till the beginning of the 15th century, when it was brought
from Constantinople, then taken by the Turks, by George,
a monk of Trabczond, by whom it was translated into
Latin ; and various other editions have since appeared.

During the long period from the year 800 till the be-
ginning of the 14th century, the western parts of Europe
were involved in the utmost ignorance and barbarity,
while the Arabians, profiting by the books they had pre-
served from the wreck of the Alexandrine library, culti-
vated and improved all the sciences, and particularly that
of astronomy, in which they had many able professors and
authors. The caliph Al Mansur first introduced a taste
lor the sciences into his empire. His grandson Al Ma-
mas, who ascended the throne in S 14, was a great cn-

courager and improver of the sciences, and especially of
astronomy. Having constructed proper instruments, he
made many observations; determined the obliquity of the
ecliptic to be 23° 35'; and under his auspices a degree of
the circle of the earth was measured a second time in the
plain of Singar, on the border of the Red Sea. About the
same time Alferganus wrote elements of astronomy ; and
the science was from hence greatly cultivated by the Ara-
bians, but principally by Albategnius, who flourished about
the year 880, and who greatly reformed astronomy, by
comparing his own observations with those of Ptolemy :
hence he computed the motion of the sun's apogee from
Ptolemy's time to his own ; settled the precession of the
equinoxes at one degree in 70 years; and fixed the obli-
quity of the ecliptic at 23° 35'. The tables which he
composed, for the meridian of Aracta, were long esteemed
by the Arabians. After his time, though the Saracens had
many eminent astronomers, several centuries elapsed with-
out producing any very valuable observations, excepting
those of some eclipses observed by Ebn Younis, astrono-
mer to the caliph of Egypt, by means of which the quan-
tity of the moon's acceleration since that time may be de-
termined.

Other eminent Arabian astronomers, were, Arzachcl a
Moor of Spain, who observed the obliquity of the eclip-
tic: he also improved trigonometry by constructing ta-
bles of sines, instead of chords of arches, dividing the dia-
meter into 300 equal parts. And Alhazen, his contem-
porary, who wrote upon the twilight, the height of the
clouds, the phenomenon of the horizontal moon, and who
first showed the importance of the theory of refractions in
astronom}'.

Ulug Beg, grandson of the celebrated Tartar prince Ta-
merlane, was a great proficient in practical astronomy ;
he had very large instruments, particularly a quadraivt of
about ISO feet high, with which he made good observa-
tions. From these he determined the latitude of Samer-
cand, his capital, to be 39° 37' 23"; and composed astro-
nomical tables for the meridian, of the san.e so exact, that
they differ very little from those constructed afterwards
by Tycho Brahe ; but his principal work was his Cata-
logue of the fixed stars, made also from his own observa-
tions in the year 1437.

During this period, almost all Europe was immersed in
gross ignorance. But the settlement of the Moors in Spain
introduced the sciences into Europe; from which time
they have continued to improve, and to be communicated
from one people to another, to the present time, when as-
tronomy, and all the sciences, have arrived at a very emi-
nent degree of perfection. The emperor Frederick n,
about 1230, first began to encourage learning; restoring
some decayed universities, and founding a new one in
Vienna: he also caused the works of Aristotle, and Pto-
lemy's Almagest, to be translated into Latin; and from
the translation of this work we may date the revival of
astronomy in Europe. Two years after this, John de Sa-
cro Bosco, that is, of Halifax, likewise compiled, from
Ptolemy, Albategnius, Alferganus, and other Arabic as-
tronomers, his wurk De Sp/taru, which was held in the
greatest estimation lor 300 years after, and was honoured
with commentaries by Cltivius and other learned men. [rj
1240, Alphonso, king of Castile, not only cultivated as-
tronomy himself, but greatly encouraged others; and by
the assistance of several learned men he corrected (he ta-
bles of Ptolemy, and composed those which were denomi-

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natcd from him the Alphonsine Tables. About the same
time also, Roger Bacon, an English monk, wrote several
tracts relative to astronomy, particularly of the lunar as-
pects, the solar rays, and the places of the fixed stars.
And, about the year 1270, Vitello, a Polandef, composed
a treatise on optics, in which he showed the use of re-
fractions in astronomy. Also our poet Chaucer, who was
born in 1328, wrote a large and learned work on the As-
trolabie, as he calls it, and indeed on practical astronomy
in general, detailing all the precepts very particularly
and orderly.

Little other improvement was made in astronomy till
the time of Purbach, who was born in 1423. He com-
posed new tables of sines for every 10 minutes, making
the radius 60, with four ciphers annexed. He construct-
ed spheres and globes, and wrote several astronomical
tracts; as, a commentary on Ptolemy's Almagest; some
treatises on Arithmetic and Dialling, with tables for vari-
ous climates; new tables of the fixed stars reduced to the
middle of that century ; and he corrected the tables of
the planets, making new equations to them where the Al-
phonsine tables were erroneous. In his solar tables, he
placed the sun's apogee in the beginning of Cancer; but
retained the obliquity of the ecliptic 23° 334-', as deter-
mined by the latest observations. He also observed some
eclipses, made new tables for computing them, and had '
just finished a theory of the planets, when he died, in
1462, being only 39 years of age.

Purbach was succeeded in his astronomical and mathe-
matical labours by his pupil and friend, John Muller, com-
monly called Regirtmontanus, from Monteregio, or Ko-
ningsberg) a town of Franconia, where he was born, lie
completed the epitome of Ptolemy's Almagest, which Pur-
bach had begun; and after the death of his friend, was in-
vited to Home, where he also made many astronomical ob-
servations. Being returned to Nuremberg in 1471, by the
encouragement of a wealthy citizen named Bernard Wal-
ther, he made several instruments for astronomical obser-
vations, among which was an armillary astrolabe, like that
used in Alexandria by Hipparchus and Ptolemy, with
which he made many observations, using also a good clock,
which was then but a late invention. He also made cphe-
merides for 30 years to come, showing the lunations,
eclipses, &c ; and, the art of printing having then been
lately invented; he printed the works of many of the most
celebrated ancient astronomers. He wrote the Theory of
the Planets and Comets, and a treatise on triangles, still in
repute { >r several good theorems; computing the table of
sines for every single minute, to the radius 1000000, and
introducing the use of tangents also into trigonometry. Af-
ter his deaih, which happened at Rome in 1476, being only
40 years of a;>e, Walther collected his papers, and conti-
nued the astronomical observations till his own death also.
The obs rvations of both were collected by order of the se-
nate of Nuremberg, and published there in 1544 by John
Schoner: they were also afterwards published in 1618 by
Sir-llius, at the end of the observations made by the Land-
grave of Hesse ; and lastly with those of Tycho Brahe in

Walther was succeeded, as astronomer at Nuremberg,
by John Werner, a clergyman. He observed the motion
of the comet in 1500; and wrote several tracts on geome-
try, astronomy, and geography in a masterly manner; the
jn6st remarkable of which, are those concerning the mo-

tion of the Sth sphere, or of the fixed stars: in this tract,
by comparing his own observations, made in 1514, with
those of Ptolemy, Alphonsus, and others; he showed that,
the motion of the fixed stars, since called the precession
of the equinoxes, is 1° 10' in 100 years. Pie made also
the first star of Aries 26° distant from the equinoctial
point, and the obliquity af the ecliptic only 23° 28'. He
also constructed a planetarium, representing the celestial
motions according to the Ptolemaic hypothesis ; and pub-
lished a translation of Ptolemy's Geography, with a com-
mentary, in which he first proposed the method of finding
the longitude at sea by observing the moon's distance from
the fixed stars; now so successfully practised for that
purpose. Werner died in 1528, at 60 years of age.

The next astronomer of eminence after the celebrated
characters just mentioned, was Nicholas Copernicus, who
was born at Thorn, in Prussia, January 19th, 1472., He
is peculiarly deserving of notice, for having restored the
old Pythagorean system, of the world, which had conti-
nued to be universally neglected from the time of Ptole-
my, and which is now so Universally received throughout
all the civilised countries of Europe. After much time
spent in meditation on the various systems, and an atten-
tive examination of all the hypotheses of his predecessors,
he found that a circular motion of the planets about the
earth was insufficient to solve the phenomena ; and he.
could not admit the revolution of these bodies about an
imaginary centre, which the doctrine of epicycles incul-
cated. He therefore sought for a happier expedient, and
was soon led to the establishment of his own system, by
the opinions which he found had been suggested by other
astronomers, lie learnt that Philolaus had placed the
sun in the centre, and that Nicetas had given the earth a
rotation about its axis; "the previous knowledge of which
was, doubtless, of the greatest utility in facilitating his
laudable enquiries. The notion of placing the most glo-
rious luminary in the heavens in the centre, could not but
strike him with delight, it being so natural an idea ; but,
to bring it forward as the basis of a system, supported by
such new and demonstrative arguments in its favour, was
a circumstance of much greater moment. Speaking of
its merits, he says, " By long observations, I discovered
that if the motions of the planets be compared with that
of the earth, and be, estimated according to the times in
which they perform their revolutions, not only their seve-
ral appearances will follow from this hypothesis, but it
will so connect the order of the planets, their orbits, mag-
nitudes, and distances, and even the apparent motion of
the fixed stars, that it will be impossible to remove one of
these bodies out of its place without disordering the rest,
and even the whole frame of the universe." Having formed
the rudiments of his system, he applied himself with the
greatest assiduity to the improvement of it, by making
numerous observations, and comparing them with those
of his predecessors. By these he constructed new tables,
which, with the true system of. the universe, were given
in his great work of astronomy, entitled Astronomia In-
staurata. But, in consequence of the dread of persecu-
tion from religious bigots, and violent opposition from
others under tiie denomination of astronomers, the pub-
lication of this valuable performance was protracted, and
the priming of it delayed till the year 1543, when Schouer
and Osiander undeitook the care of it, and the work came
out under the title of Revolutioncs Orbium Ccelestium,

AST [ 166 ] AST

a copy of which Copernicus received a few hours before chair of Cassiopeia ; which induced him, like Hipparchus

his dissolution, which happened on the 23d of. May, 1543, on a similar occasion, to make a new catalogue of the

in the 71st year of his age. stars; which he did to the number of 777, and adapted

After the death of Copernicus, the science and prac- their places to the year 1600. In the year 1576, by fa-

tice of astronomy were greatly improved by many other vour of the king of Denmark, he built his new observa-

persons, as Schoner, Nonius, Appian, Gemma Frisius, tory, called Uraniburg, on the small island Huenna, op-

Rothman, Byrgius, the Landgrave of Hesse, &c. — Scho- positc to Copenhagen, and which he very amply furnished

ner reformed and explained the calendar, improved the with many large instruments, some of them so divided as

methods of makine celestial observations, and published a to show single minutes, and in others the arch might be

treatise on cosmography ; but he died four years after Co- read off to 10 seconds. One quadrant was divided ac-

pernicui. Nonius, or Nunez, wrote several works on ma- cording to the method invented by Nonius, that is, by 47

thematics, astronomy and navigation, and invented some concentric circles; but most of them were divided by dia-

useful and more accurate instruments than formerly; one gonals, a method of division invented by a Mr. Richard

of these was the astronomical quadrant, on which he di- Chanceler, an Englishman. Tycho employed his time at

vided the degrees into minutes by a number of concentric
circles; the first of which was divided into 90 equal
parts or degrees, the second into 89, the third into S8, and
so on, to 46; so that, the index of the quadrant always
falling upon or near one of the divisions, the minutes

Uraniburg to the best advantage, till the death of the
king; when, falling into discredit, he was obliged to re-
move to Holstein ; and he afterwards found means of in-
troducing himself to (he Emperor Rodolph, with whom
he continued at Prague till the time of his death in 1601.

would be known by an easy computation. Nunez wrote It is well known that Tycho was the inventor of a system

also a tract on the Twilight; and a neat treatise on Alge- of astronomy, a kind of semi-Ptolemaic, which he vainly

bra; besides other pieces. — The chief work of Appian, endeavoured to establish instead of the Copernican or true

the Cesarean Astronomy, was published at Ingoldstat in system. His works, however, which are very numerous,

1540 ; in which he shows, how to observe the places of show that he was a man of great abilities; and his disco-

the stars and planets by the astrolabe ; to resolve astrono- veries, together with those of Purbach and Regiomonta-

mical problems by certain instruments; to predict eclip- nus, were collected and published together in l621, by

ses, and to describe the figures of them; and the method Longomontanus, the favourite disciple of Tycho.
of dividing and using an astronomical quadrant: at the While Tycho resided at Prague with the emperor, he

end are added observations of 5 comets, one of which has prevailed on Kepler, who had formerly been his disciple

been supposed the same with that observed by Hevelius, and assistant at Holstein, and from his conduct and abi-

and if so, it ought to have returned again in the year lities had rendered himself dear to him, to leave the uni-

1789; — but it was not observed then. Gemma Frisius versity of Glatz, and to come to him, which he did with

wrote a commentary on Appian's Cosmography, accom- his family and library in 1600: but Tycho dying in 1601,

panied with many observations of eclipses : he also in- Kepler enjoyed during his life the title of mathematician

vented the astronomical ring, and several other instru- to the Emperor, who ordered him to finish the tables of

ments, useful in taking observations at sea; and was also Tycho Brahe, which he did accordingly, and published

the first who recommended a time-keeper for determining them in l627 under the title of Rodolphine. He died

the longitude at sea. — Rheticus gave up his professorship about the year 1630 at Ratisbon, where he was soliciting

of mathematics at Wittemberg, that he might attend the the arrears of his pension. From his own observations,

astronomical lectures of Copernicus; and, for improving and those of Tycho, Kepler discovered several of the true

astronomical calculations, he began a very extensive work, laws of nature, by which the motions of the celestial bo

being a table of sines, tangents, and secants, to a very
large radius, and to every 10 seconds, or A of a minute ;
which was completed by his pupil Valentine Otho, and
published in 1594.

dies are regulated. He discovered that all the planets
revolved about the sun, not in circular, but in elliptical or-
bits, having the sun in one of the foci of the ellipse; that
their motions were not equable, but varying, quicker or

About the year 156l, William iv, Landgrave of Hesse slower, as they were near to the sun or farther from him;
Cassel, applied himself to the study of astronomy, having but that this motion was so regulated, that the areas de-
furnished himself with the best instruments that could scribed by the variable line drawn from the planet to the
then be made: with these he made a great number of ob- sun, are equal in equal times, and always proportional to
servations, which were published by Snellius in l6l8, and the limes of describing them. He also discovered, by tri

which were preferred by Hevelius to those of Tycho Brahe
From these observations he formed a catalogue of 400
stars, with their latitudes and longitudes, adapted to the
beginning of the year 1593.

The next astronomer of note was Tycho Brahe, a
noble Dane, who, from his assiduous attention to this
science, became one of the greatest observers that ever
existed: he. began his observations about the same time
with the Landgrave of Hesse, above mentioned, and ob-
served the great conjunction of Jupiter and Saturn; but
finding the usual instruments very inaccurate, he con-

als, that the cubes of the distances of the planets from
the sun, were in the same proportion as the squares of
their periodical times of revolution. By observations also
on comets, he concluded that they moved in very excen-
tric orbits, and crossing those of the planets in all direc-
tions, but which he could not then determine. These dis-
coveries, and many others which are to be found in his
writings, but of which no traces arc discoverable in the
annals of antiquity, will lor ever immortalize his fame, and
render his history interesting to the most distant ages.
In Kepler's time there were many other good proficients

Mrucled many others, much larger and exacter, with in astronomy ; as Edward Wright, Baron Napier, John
which he applied himself diligently to observe the celes- Bayer, &c. Wright made several good meridional obser-
tial phenomena. In 1571 he discovered anew star in the vations of the sun, with a quadrant of 6 feeUadnw, iuthe

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years 1594, 159-5, and 1596; from which he greatly im-
proved the theory nf the sun's motion, and computed more
accurately his declination than any person had done be-
fore. In 1599 he published also an excellent work en-
titled " Certain Errors in Navigation discovered and de-
tected," containing a method which has commonly, though
erroneously, been ascribed to Mercator. — To Napier we
owe some excellent theorems and improvements in sphe-
rics, besides the ever-memorable invention of logarithms,
one of the most useful ever made in the art of numbering,
and of the greatest use in all the other mathematical sci-
ences.— Bayer, a German, published his Uranomelriu, be-
ing a complete celestial atlas, or the figures of all the con-
stellations visible in Europe, with the stars marked on them,
and the stars also accompanied by names, or theletters of
the Greek alphabet; a contrivance by which the stars may
easily be referred to with distinctness and prescision.

The .next astronomer, whose labours deserve to be
particularly recorded, is Galileo, who was born at
Florence in the year 1564'. He discovered the laws by
which bodies falling freely are accelerated, and meditated
deeply on their motions in general. From the valuable dis-
coveries he made in this branch of physico-mechanical
sciences, Newton and Huygens were afterwards enabled to
derive the most complete and satisfactory theories of the
planetary motions. Bein<» informed of the invention of
the telescope, an ever-memorable instrument, and to
which we are indebted for the most brilliant discoveries,
and all the accuracy to which the science of astronomy
has attained, he, by duly considering the principles of re-
fraction, constructed one himself of far superior power, by
which he discovered the inequalities of the moon's sur-
face, the satellites of Jupiter, and the ring of Saturn; also
spots on the sun, by which he determined the revolution
of that luminary about ils axis: he also found that Venus
exhibited the same phases as the moon ; and discovered
that nebulas and the galaxy were full of small stars. He
was also a warm defender of the Copernican system,
for which he was cruelly persecuted by a congregation of
cardinals, who made him abjure, upon his knees, in the
most solemn manner, a truth which nature and his own
conviction had shown to be infallible. He was afterwards
imprisoned in the inquisition at Florence, where he was
born; and died in the year 1642, leaving behind him many
works which testify his great skill in the sciences.

The history of astronomy, about this period, also offers
us many names of very respectable observers; among
whom may be noticed Mercator, Mam-blycus, Homelius,
and Stevin, abroad; and in England, Thomas and Leo-
nard Digges, John Dee, Robert Flood, Harriot, &c ; the
latter of whom, who had hitherto only been distinguished
as an algebraist, made nearly the same discoveries as
Galileo, and at as early a period, if not more so, as appears
by his papers not yet printed, in the possession of the
Earl of Egremont. After whom, Mr. Horrox, a young
astronomer of great talents, made considerable, discoveries
and improvements in this science. In 1633 he discovered
that the planet Venus would pass over the sun's disc on the
24th of November I639, an event which he announced
only to his friend Mr. Crabtrec; and these two were the
only persons in the world that observed this transit, which
was also the first time it had ever been seen by human
eyes. Mr. Horrox made also many other useful obser-
vations, and had even formed a new theory of the moon,
taken notice of by Newton; but his early death, in the

beginning of the year 1640, put a stop to his useful and
valuable labours.

The next discoveries to Kepler's and Galileo's, were
those of Huygens. This celebrated genius was the first
who applied pendulums to clocks, a discovery as brilliant
and advantageous as ever was presented to astronomy.
He was also the first who found that the singular appear-
ances of Saturn are produced by a ring by which the pla-
net is surrounded; and from his great assiduity in observ-
ing it, he was led to the discovery of one of its satellites.
To this memorable man, also, we are indebted for some
of the most useful theorems on centrifugal forces; and,
had he had an idea of combining them with his enquiries
into the developement of curves and the laws of Kepler,
Newton had certainly not been the first to announce to the
world the universal law of gravitation; but these are the
things in which discoveries generally consist.

Next flourished Hevelius, Burgomaster of Dantzic, who
furnished an excellent observatory in his own house, where
he observed the spots and phases of the moon, from which
observations he compiled his Selcnogruphia ; anil published
an account of his apparatus in his work entitled Machina
Cmkstis, a book now very scarce, as most of the copies
were accidentally burnt, with the whole house and appa-
ratus, in 1079. Hevelius died in 168S, aged 76.

About this time the celebrated Dr. Hooke invented
other instruments with telescopic sights, and censured those
of Hevelius, which occasioned a sharp dispute between
them ; to settle which, the celebrated Dr. Halley was- sent
over to Hevelius to examine his instruments. The two as-
tronomers made several observations together, very much
to their satisfaction, and among them was one of an occul-
tation of Jupiter by the moon, when they determined the
diameter of the latter to be 30' 33".

Before the middle of the 17th century the construction
of telescopes had been greatly improved, particularly by
Huygens and Fontana. The former constructed one of
123 feet, with which he long observed the moon and pla-
nets, and discovered that Saturn was encompassed with a
ring. With telescopes, also, of 200 and 300 feet focus,
Cassini saw five satellites of Saturn, with his zones or
belts, and likewise the shadows of Jupiter's satellites pass-
ing over his body. In 1 666 Azout applied a micrometer
to telescopes, to measure the diameters of the planets and
other small distances in the heavens ; but an instrument of
this kind had been invented before, by Mr. Gascoigne,
though it was but little known abroad.

To obviate the difficulties of the great lengths of re-
fracting telescopes and the aberration of the rays, it is
said that Mersene first started the idea of making telescopes
of reflectors, instead of lenses, in a letter to Descartes; and
in 1663 James Gregory of Aberdeen showed how such a
telescope might be constructed. After some time spent
also by Newton, on the construction of both sorts of
telescopes, he discovered the great inconvenience which
arises to refractors from the different rcfrangibility of the.
rays of light, and for which he could not then find a re-
medy; and therefore, pursuing the other kind, in the year
l6'72 he presented to the Royal Society two reflectors,
which were constructed with spherical speculums, as he
could not procureother figures. The inconveniences, how-
ever, arising from the different refrangibility of the raysof
light, have since been fully obviated by the ingenious Mr.
Dollond. Towards the latter part of the 1 7th, and be-
ginning of the 18th century,practical astronomy, it seems,

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rather languished. But at the same time the speculative
part was carried to the highest perfection by the immor-
tal Newton in his Principia, and by the Astronomy of
David Gregory.

Soon after this, however, great improvements of astro-
nomical instruments began to take place, particularly in
Britain. Mr. Graham, a celebrated mechanic and watch-
maker, not only improved clocks and watch work, but
also carried the accuracy of astronomical instruments -to
a surprising degree. He constructed the old 8 feet mural
arch at the Royal Observatory, Greenwich, and a small
equatorial sector for making observations out of the meri-
dian; but he is chiefly to be noticed for contriving the
zenith sector of 24 feet radius, and afterwards one of 12i
feet, with which Dr. Bradley discovered the aberration of
the fixed stars. The reflecting telescope of Gregory and
Newton was also greatly improved by Mr. Hadley, who
presented a very powerful instrument of that kind to the
Royal Society in 1719- The same gentleman has also
immortalized his memory by the invention of the reflect-
ing quadrant or sector, now called by his name, which he
presented to the society in 173), and which is now used so
universally at sea, especially where nice observations, are
required. It appears, however, that an instrument simi-
lar to this in its principles had been invented by Newton;
and a description with a drawing of it given by him to Dr.
Ilalley, when he was preparing for his voyage in 1701, to
discover the variation of the needle. It has also been as-
serted, that a Mr. Godfrey, of Philadelphia in America,
made the same discovery, and the first instrument of this
kind. About the middle of this century, the construct-
ing and dividing of large astronomical instruments were
carried to great perfection by Mr. John Bird ; and re-
flecting telescopes were not less improved by Mr. Short,
who also first executed the divided object-glass micro-
meter, which had been proposed and described by M.
Louville and others. Mr. Dollond also brought refract-
ing telescopes to the greatest perfection, by means of his
acromatic glasses; and, lately, the discoveries of Herschel
are owing to the amazing powers of reflectors of his own
construction.

Thus the astronomical improvements in the ISth cen-
tury have been chiefly owing to the foregoing inventions
and improvements in the instruments, and to the esta-
blishment of regular observatories in England, France,
and other parts of Europe. Roemer, a celebrated Danish
astronomer, first made use of a meridional telescope ; and,
by observing the eclipses of Jupiter's satellites, he first
discovered the progressive motion of light, concerning
which he read a dissertation before the Academy of Sci-
ences in Paris in 1675. — Mr. Flamsteed was appointed the
first Astronomer Royal at Greenwich in 16*75. He ob-
served,' for 44 years, all the celestial phenomena, of the
sun, moon, planets, and fixed stars, of all which he gave
an improved theory and tables, viz, a catalogue of 3000
stars with their places, to the year 1689; also new solar
tables, and a theory of the moon according to Horrox;
likewise, in Sir Jonas Moore's System of Mathematics, ho
gave a curious tract on the doctrine of the sphere, show-
ing how, geometrically, to construct eclipses of the sun
and moon, as well as occultations of the fixed stars by
the moon. And it was on. his tables that Hallcy's tables
and Newton's theory of the moon were also constructed.
Cassini, also, the firit French Astronomer Royal, very

much distinguished himself, making many observations on
the sun, moon, planets, and comets, and greatly improved
the elements of their motions. He also erected the gno-
mon, and drew the celebrated meridian line in the church
of Petronia at Bologna.

In 1719 Mr. Flamsteed was succeeded by Dr. Halley,
as Astronomer Royal at Greenwich. The Doctor had
been sent, at the early age of 21, to the island t>f St. He-
lena, to observe the southern stars, and make a catalogue
of them, which was published in 1679- In 1705 he pub-
lished his Synopsis Astronomicc Cometica, in which he ven-
tured to predict the return of a comet in 1758 or 1750.
He was the first who discovered the acceleration of the
moon, and he gave a very ingenious method for finding
her parallax by three observed phases of a solar eclipse.
He also published, in the Philosophical Transactions,
many learned papers, and among them some that were
concerning the use that might be made of the next transit
of Venus in determining the distance of the sun from the
earth. He composed tables of the sun, moon, and all the
planets, which are still in great repute; with which he
compared the observations he made of the moon at Green-
wich, amounting to near 1500, and noted the differences.
He recommended the method of determining the longi-
tude by tffc moon's distances from the sun and certain
fixed stars; a method which had before been noticed, and
which has since been carried into execution, more parti-
cularly at the instance of the late Astronomer Royal, Dr.
Maskelyne.

About this time a dispute arose concerning the figure
of the earth. Sir Isaac Newton had determined, from a
consideration of the laws of gravity, and the diurnal mo-
tion of the earth, that the figure of it was an oblate sphe-
roid, or flatted at the poles; but Cassini had determined,
from the measures of Picart, that the figure was an oblong
spheroid, or lengthened at the poles. To settle this dis-
pute, it was resolved, under Lewis xv, to measure 2 de-
grees of the meridian ; one near the equator, and the other
as near the pole as possible. For this purpose, the Royal
Academy of Sciences sent to Lapland Mess. Maupertuis,
Clairault, Camus, and Le Monier, &c; and, on the south-
ern expedition were sent Mess. Godin, Condamino, and
Bouguer, to whom the king of Spain joined Don George
Juan and Don Antonio de Ulloa. These set out in 1735,
and returned at different times in 1744, 1745, and 1746";
but the former party, who set out only in 1736', returned
theyear following, having both fulfilled their commissions.
Picart's measure was also revised by Cassini and De la
Caille, which, after his errors .were correcte'l, was found
to agree very well with the other two; and the result of
the whole served to confirm the determination of the
figure before laid down by Newton. On the southern ex-
pedition, it was found that the attraction of the great
mountains of Peru had a sensible effect on the plumb-line
of one of their largest instruments, deflecting it 7 or 8 se-
conds from the true perpendicular.

On the death of Dr. Halley, in 1742, he was succeeded
by Dr. Bradley as Astronomer Royal at Greenwich. The
accuracy of his observations enabled him to detect tin1
smaller inequalities in the motions of the planets and fixed
stars. The consequence, of this accuracy was, the disco-
very of the aberration of light, the nutation of the earth's
axis, and a much greater degree of perfection in the lunar
tables. He also observed the places, and computed the

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elements of the comets which appeared in the years 1723,
1730, 17*3, and 1757. He made new and accurate tables
of the motions of Jupiter's satellites; and from a multitude
of observations of the luminaries, he constructed a most
accurate table of refractions, with a general rule for com-
puting them. Also, with a very large transit instrument,
and a new mural quadrant of S feet radius, constructed
by Mr. Bird in 1750, he made an immense number of
observations, for settling the places of all the stars in the
British catalogue, together witb near 1500 places of the
moon, the greater part of which he compared with Mayer's
tables. Dr. Bradley died in 1762.

In the year 1760 all the learned .Societies in Europe
made preparations for observing the transit of Venus over
the sun, which had been predicted by Dr. Halley more
than 80 years before, with the use that might be made of
it in determining the sun's parallax, and the distances of
the planets from the sun. The same exertions were also
repeated, to observe the transit in 1769, by sending ob-
servers to different parts of the world, for the more con-
venience in observing. From the whole, Mr. Short com-
puted that the sun's parallax was nearly 8-f- seconds,
and consequently the distance of the sun from the earth
about 24114- of the earth's diameters, or 96 millions of
miles.

Dr. Bradley was succeeded, in 1762, in his office of Astro-
nomer Royal, by Mr. Bliss, Savilian professor of astrono-
my ; who being in a declining state of health, did not long
enjoy it. But, dying in 1765, was succeeded by Dr. Maske-
lyne, the late Astronomer Royal, who discharged the duties
of that office with the greatest honour to himself, and be-
nefit to the science. In January 1761 this gentleman was
sent by the Royal Society, at a very early age, to the
Island of St. Helena, to observe the transit of Venus over
the' sun, and the parallax of the star Sirius. The first
of these objects partly failed, by clouds preventing the
sight of the 2d internal contact; and the 2d also, owing
to Mr. Short having suspended the plumb-line by a loop
hom the neck of the central pin. However, our astrono-
mer indemnified himself by many other valuable observa-
tions : Thus, at St. Helena, he observed the tides; the
horary parallaxes of the moon; and the going of a clock,
to find, by comparison with its previous motion which had
been observed in England, the difference of gravity at the
two places : also, in going out and returning, he practised
the method of finding the longitude by the lunar distances
taken with a Hadley's quadrant, making out rules for the
use of seamen, and taught the method to the officers on
board the ship ; which he afterwards explained in a letter
to the secretary of the Royal Society, inserted in the
Philos. Trans, for the year 176?, and still more fully af-
terwards, in the British Mariner's Guide, which he pub-
lished in the year 1763. He returned from St. Helena in
the spring of 176'2, after a stay there of 10 months ; and
in September 1763 sailed for the island of Barbadoes, to
.settle the longitude of the place, and to compare Mr.
Harrison's watch with the time there, when this gentleman
should bring it out: another object was also to try Mr.
Irwin's marine chair, which he did in his way out. While
at Barbadoes, he also made, many other observations; and
among them, several relating to the moon's horary paral-
laxes, not yet published. Returning to England in the
latter part of the year 1764, he was appointed in 176*5 to
succeed Mr. Bliss as Astronomer Royal, and immediately
recommended to the Board ol Longitude the lunar method
Vol. I.

of finding the longitude, and proposed to them the project
of a Nautical Almanac, to be calculated and published to
facilitate that method : this they agreed to, and the first
volume was published for 176"7, and which continued till
his death under his direction, to the great benefit of navi-
gation and universal commerce.

A multitude of other useful writings by this gentleman
are inserted iu the volumes of the Philos. Trans.; and par-
ticularly, in consequence of a proposal made by him to
the Royal Society, the noble project was formed of mea-
suring very accurately the effect of some mountain on the
plumb-line, in deflecting it from the perpendicular; and the
mountain of Schihallien, in Scotland, having been found
the most' convenient in this island for the purpose, at the
request of the Society he went into Scotland to conduct
the business, which he performed in the most accurate
manner; showing that the sum of the deflections on the
two opposite sides was about 11|. seconds of a degree;
and proving, to the satisfaction of the world, the universal
attraction of all matter: from the data resulting from
which measures I have computed the mean density of the
whole matter in the earth, and have found it to be nearly
5 times that of common water. Besides many learned
and valuable papers in the Philosophical Transactions,
the most assiduous exertions in the duties of the observa-
tor}', as abundantly appears by the curious and volumi-
nous observations which he has given to the public, the
world is particularly obliged to his endeavours with the
Board of Longitude, for the publication of the Nautical
Ephemeris, and the method of observing the longitude, by
the distances of the moon and stars, now adopted by all
nations, and by which the practice of navigation has been
brought to the greatest perfection. Dr. Maskelyne died
the 9th of February 1811, and has been succeeded in his
office by Mr. John Pond.

The discoveries of Dr. Herschel form a new jera in
astronomy. He first, in 1781, began with observations on
the periodical star in ColloCeti, and a new method of
measuring the lunar mountains, none of which he made
more than half a mile in height : and, having constructed
telescopes vastly more powerful than any former ones, he
proceeded to other observations, Concerning which he has
had several papers printed in the Philosophical Trans-
actions; as, On the rotation of the planets round their
axes; On the parallax of the fixed stars ; Catalogues of
double, triple, &c stars ; On the proper motion of the sun
and solar system ; On the remarkable appearances of the
solar regions of the planet Mars, &c, &c; and, above all,
his discovery of a new primary planet, on the 13th of
March 17S1 , which he called the Georgian planet; though
at first named the planet Herschel by the French and fo-
reign astronoineis, but lately, and more generally, Uranus;
by which, and its six satellites, he has greatly enlarged
the bounds of the solar system, this new planet being more
than twice as far from the sun as the planet Saturn.

M. Piazzi, astronomer royal at Palermo, on Jan. 1, 1801,
the first day of the 19th century, discovered another pla-
net, moving in an intermediate orbit between Mars and Ju-
piter. It had long been conjectured by Maclaurin, Bax-
ter, Lambert, Bode, Zach. and others, that there was a
primary planet within this space; and the discovery has
answered to some of these conjectures in a very ex-
traordinary manner. This planet is now distinguished
by the name Ceres ; though it would perhaps be better
to call this, and all the new planets, by the names of their

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respective discoverers. Another new and small planet was
discovered by Dr. Olbers of Bremen, on March 28, 1802 :
it is called Pallas, and is distant from Mars nearly the
same as Ceres, their orbits indeed intersecting each other.
— Juno is also another very small planet, in the same in-
terval, discovered by M. Harding, at Lilienthal, on Sept. 1,
1804 ; and is a little farther distant than Ceres and Pal-
las. A foarth of these *mall planetary bodies was disco-
vered by Dr. Olbers the 29th of March, 1807, which is
denoted by the name of Vesta. And indeed it is highly
probable that, as astronomical instruments are improved,
new discoveries of this kind will be made, and our system
be enlarged by the discovery of planets making their ex-
cursions even beyond the orbit of the planet Uranus, or
Herschel, as well as some within the orbit of Mercury.

In the labours of scientific men on physical astronomy,
the most distinguished names after Newton, are Maclau-
rin, Clairaut, Simpson, Euler, Walmesley, Dalembert, Fri-
si, Lagrange, Landen, and Laplace. Simpson, in his tracts,
directed his attention to the theory of the moon. He
showed that the effects of such forces as are proportional
to the cosine of an arc r, are explicable by means of the
cosines of that arc and of its multiples ; and thus deter-
mined a very important point : for, since it hence appears,
that no terms enter into the equation of the moon's orbit,
but whatby a regular increase and decrease do, after a
certain time, return to their former values ; it is therefore
evident that the mean motion, and the greatest quantities
of the several equations, receive no change from gravity.
Frisi, in his Cosmographia, applied a similar mode of rea-
sonipg to the variations in the obliquity of the ecliptic;
which he thus showed to be confined within certain limits.
Laplace was conducted farther: in the general movement
of a system of bodies, such as is actually exemplified in
nature, every thing is in motion; not only every body,
but the plane or position of every orbit. The mutual ac-
tions of the planets change the positions of the planes in
which they revolve; being perpetually made, to depart by
a small quantity, from side to side, each from that plane
in which it would go on continually, if their mutual ac-

the first who deduced, from his solution of the problem, a
complete set of lunar tables, far more accurate than any
former ones. Their accuracy, however, was exceeded by
another set produced by Tobias Mayer of Gottingen, and
founded on a comparison of Euler's solution with correct
observations. The problem of finding the longitude at sea,
which was now understood to depend so much on the ex-
actness with which the moon's place could, be computed,
gave much additional value to these researches, and esta-
blished a very close connexion between the conclusions of
theory and the art of navigation. Mayer's tables were re-
warded by the Board of Longitude in England; and Eu-
ler's, at the suggestion of Turgot, by the Board of Longi-
tude in France.

TShus the lunar theory was brought to a h"igh degree of
accuracy ; the tables constructed by means of it giving the
moon's place true to £ a minute. Yet there remained still
one inequality in the moon's motion, for which the prin-
ciples of gravitation seemed to afford no account; this was
what is known by the name of the moon's acceleration.
Various attempts were made to explain that phenomenon
by Halley, and others : it was at last satisfactorily eluci-
dated by Laplace, who thus gave the finishing touch to the
theory of the moon, nearly a century after it had been pro-
pounded in the first edition of the Prjncipia. See Acce-
leration of the Moon.

That branch of the theory of disturbing forces which re-
lates to the action of the primary planets on one another,
was successfully cultivated between 1740 and 1780. In
the course of these researches, the change in the obliquity
of the ecliptic came first to be perfectly recognized, and
ascribed to the action of the planets, on the earth. Euler
proved, independently of Frisi, that the change in this obli-
quity is periodical ; that it is not a constant diminution,
but a slow and small oscillation, never exceeding 2° alto-
gether on both sides of a mean quantity, by which it al-
ternately increases and diminishes in the course of periods,
which are not all of the same length, but by which, in the
course of ages, a compensation ultimately takes place.

Lagrange, struck with the circumstance that the calcu-

tions were to cease. The calculus of Laplace, in the first " lus had never given any inequalities but such as were pe-

book of his Mecanique Celeste, shows that the inclinations
of these orbits, in the planetary system, are stable, or that
the planes of the orbits oscillate a little, to and fro, on each
side of a fixed and immoveable plane This plane is shown
to be one, on which, if every one of the bodies of the sy-
stem be projected by a perpendicular let fall upon it, and
if the mass of each body be multiplied into the area de-
scribed in a giv^n time by its projection on the said plane',
the sum of all these products shall be a maximum.

The celebrated problem of the three bodies, engaged in
succession the attention of many eminent philosophers.
The problem is this : Having given the masses of three bo-
dies, projected from three points given in position, with
velocities give i.i their quantity and direction; and sup-
posing the bodies to gravitate to one another, with forces
thai are as their masses directly, and the squares of their
distances inverselj ; to find the lines described by those
bodies, and their position at any given instant. Or, the
problem may be rendered universal, by assuming any num-
ber of bodies above three. To resolve the problem gene-
rally, according to either of these enunciations, exceeds
the powers even of the, most refined analysis : but, under
he Condi ti . presented by nature, much has been done
by Clairaul Dalembert, Kuler, and others. Clairaut was

riodical, applied himself to the investigation of a general
question, from which he found, by a method pecuriar to
himself, and independent of any approximation, that the
inequalities produced by the mutual action of the planets,
must in effect be all periodical ; that such changes are con-
fined within narrow limits; that none of the planets ever
has been, nor ever can be a comet, moving in a very ex-
centric orbit; but that the planetary system oscillates as
it were round a medium state, from which it never deviates
far : that amid all the changes which arise from the mu-
tual actions of the planets, two things remain perpetually
the same, viz, the length of the greater a> i.-- of the ellipse
described by the planet, and its periodical time round the
sun; or, which is the same thing, the mean distance of
each planet from the sun, and its mean motion, both re-
main constant. The plnne of the. orbit varies, the species
of the ellipse and its excentricity change, but ncyer, by any
means whatever, the greater axis of the ellipse, or the time
of the entire revolution of the planet. The discovery of
this great principle, which may be considered as the bul
wark that secures the Stability of our system, and excludes
all nccess to confusion and disorder, must render the name
of Lagrange for ever memorable in science, and ever n -
vercd by those who delight in the contemplation of what-

AST [ 171 ] AST

ever is excellent and sublime. After Newton's discovery by virtue of a force varying inversely as the squares of the
of the elliptic orbits of the planets from gravitation, La- distances, and directly as the masses

grange's discovery of their periodical inequalities, is doubt-
less the noblest truth in physical astronomy; and, in re-
spect of the doctrine of final causes, it may be regarded
as the greatest of all.

Nearly allied to this truth, is the following theorem, re-
sulting from one of Laplace's investigations : If the mass
of each planet be multiplied into the square of the excen-

The sun agrees with the fixed stars in the property of
emitting light continually, and in retaining constantly its
relative situation, with very little variation : it is probable
also that these bodies have many other properties in com-
mon. The sun is therefore considered as a fixed star com-
paratively near us: and from the same analogy it is in-
ferred that the stars are possessed of gravitation, and of the

tricity of its orbit, and this product into the square root of other general properties of matter ; they are supposed to

the axis of the same orbit, the sum of all these quantities, emit heat as well as light ; and it has with reason been

when added together, will remain for ever the same: this conjectured, that they serve to cherish the inhabitants of a

sum is of a constant magnitude, which the mutual action multitude of planetary bodies revolving about them,
of the planets cannot change. Hence, no one of the ex- The sun, like many other stars, has probably a progres-

centricities can ever increase to a greater magnitude : for, sive motion, which, from a comparison of the apparent mo-

as the mass of each planet is given, as well as the axis of tions of a great number of the stars, is supposed to be di-

its orbit, the square of the excentricity in each is multi- rected towards the constellation Hercules. It is beyond

plied into a given coefficient, and the sum of all the pro- all question that many of the stars have motions peculiar

ducts so formed is incapable of change. Hence, therefore, to themselves, and it is not certain that any of them are

we have an ''her general property, indicating the stability without such motions : it is therefore highly probable that

of our system within narrow limits. Yet it does not fol- the sun may have such motion ; and indeed Dr. Herschel

low that this permanency is necessary and unavoidable, has confirmed this conjecture by arguments that are al-

under every possible constitution of the planetary orbits : most demonstrative.

for, if the planets did not all move the same way; if their Besides this progressive motion, the sun is subject to a
orbits were not all nearly circular ; if their excentricities small change of place, dependent on the situations of the
were not small » or if the law of planetary deflection planetary bodies; which was long inferred from theory
were different; the permanency of the preceding quantity only, but which has been actually demonstrated by mo-
could not obtain. Such permanency depends on con- dern observations. Supposing all the planets to be in con-
ditions which are not necessary in themselves, and is junction, or nearly in the same direction from the sun, the
therefore an argument of design in the construction of the common centre of inertia of the system is then at the di-
universe. stance of about a diameter of the sun from his centre, or
The subject of the tides had its true theory sketched by half a diameter without his surface: and since the centre
Newton, but was not completed till long after his time, of inertia of the whole system must be undisturbed by any
The state of neither mechanical nor mathematical science reciprocal actions of the bodies composing it, the sun must
was such in his time as would enable any one to determine describe an irregular orbit round this centre, his greatest

the motions of a fluid, acted on by three gravitations, and
having besides a rotatory motion. The dissertations of
D. Bernoulli, Euler, and Maclaurin, which shared the
prize with that of P. Cavalleri on the principle of vortices,
were excellent: but the first man who felt himself in pos-
session of all the principles required in this arduous inves-
tigation, was Laplace, who, in the years 1775, 1779> and
179Q, communicated to the Academy of Sciences a series

distance from it being equal to his own diameter.

The sun revolves on his axis in 25 days 10 hours, with
respect to the fixed stars; this axis is directed towards a
point about half way between the pole-star and Lyra,
the plane of the rotation being inclined a little more than
7 degrees to the plane of the ecliptic. The direction of
this motion is from west to east; and all the rotations of
the different bodies which compose the solar system, as far

of memoirs on this subject, which he has united and ex- as they have been ascertained, are in the same direction
tended in the 4th book of the Mecanique Celeste. See The time, and direction of the sun's rotation is ascer-

Tides; as also Edinb. Review, N°22. — The precession of tained by the change in the situation of the spots, which

the equinoxes is another interesting subject, to which the
attention of Frisi, Walmesley, Simpson, Dalembc it, Landen,
Milner, M. Young, and Robertson, have been successively
directed. — See the articles, Comets, Eclipses, Gravi-
tation, Moon, Planets, Satellites, Sun, Solar
. System, Tides, and the several planets Mercury, Ve-

n us, &C, tiC.

On the Solar System.
The most conspicuous of all the celestial bodies is the
sun, that magnificent luminary occupying the centre of the
system whicS comprehends our earth, with a variety of

are often visible on his disc; and which some astronomers
suppose to be elevations, but others, apparently with bet-
ter reason, to be-excavations or deficiencies in the lumi-
nous matter covering the sun's surface. These poirits arc
often observed to appear and disappear, and they are in the
mean time liable to great variations; but they are gene-
rally found about the middle parts of the sun's surface.
Lalande imagined that they were parts of the solid body
of the sun, which by some agitations of the luminous ocean,
with which he conceived the sun to be surrounded, are left
nearly or entirely bare. Dr. Wilson and Dr. Hnschel are

Jthenprimary and secondary planets, and a still greater disposed to consider this ocean as consistingjather of a

number of comets. — The sun is a body nearly spherical, flame, than of a liquid substance ; and Dr. Herschel attri-

whose diameter is about 883,250 English miles, nearly 4 butes the spots to the emission of an aeriform fluid, not

times the distance of the. moon from the earth. Situated yet in combustion, which displaces the general luminous

nearly at the centre of the system of the planets, it exerts atmosphere, and which is afterward to serve as fuel for

on th'em all a remarkable influence: it heats them, it en- supporting the process: hence he supposes the appearance

lightens them, and enchains them in their elliptical orbits, of copious spots to be indicative of the approach of warm

A S'T

[ 172 ]

AST

seasons cm the surface nf the earth, and he has attempted
to maintain this opinion by historical evidence. Whatever
be the cause of these spots, by their often continuing du-
ring the whole time of the rotation, they have given oc-
casion, by observations, to determine the duration of that
period, as above mentioned.

The planetary system comprises at least 30 bodies,
without including comets. Among these 30 bodies, the
sun is the only one which is really phosphoric, or that
shines with a lustre which is, properly speaking, its own.
All the others are opake ; that is, they intercept the lumi-
nous fluid, and are only visible by the reflected light.
1 1 of these bear the name of primary planets ; the other
IS are known under that of satellites or secondary planets.
The planets perform their revolutions about the sun in
elliptical curves, differing but little from circles, and of
which the centre of the sun (or rather the common centre
of inertia of the whole system) occupies one of the foci.
Commencing with that which is nearest the sun, they have
the following disposition: Mercury, Venus, Terra or the
Earth, Mars, Ceres or Piazzi, Pallas or Others1, Juno or
Harding, Vesta or Olbers-, Jupiter, Saturn,'and Uranus or
Herschel. See plate 31.

Each of the planetary orbits is in a plane which passes
through the sun's centre. The earth's orbit is named the
ecliptic; which is conceived as indefinitely prolonged on
all sides ; and astronomers observe the situations of the
planes of the other orbits by referring them to this.

All the planets move in their orbits from west to east,
perform, their revolutions each in equal times, and in an
orbit always at the same mean distance from the sun. The
velocities in their orbits are variable, being quicker in the
parts nearer the sun, and proportionally slower when far-
ther off, in such sort that equal areas are described by the
radius vector, or line connecting the sun and the planet in
its motion.

Mercury and Venus are nearer the sun than the earth is ;
on which account they are called inferior or interior
planets. Those which are farther from the sun than the
earth, are called superior or exterior planets.

The inferior planets can never be in opposition to the
sun (see Opposition) ; but they will be found twice in
conjunction with that luminary during the course of their
sidereal revolution : first, when they come between the
sun and the earth, making the inferior conjunction; and
again, when the sun is between the planet and the earth,
making the superior conjunction. See Conjunction.

The inferior planets present different phases, when they
are viewed with telescopes: these appearances are more
perceptibk for Venus than for Mercury, and they depend
on the proper motion of those planets. If this motion be
combined with that of the earth in its orbit, which is ef-
fected more slowly by reason of its greater distance from
the sun, new appearances will arise, such as the direct mo-
tion of those planets in the inferior conjunction, and their
retrograde motion in the superior conjunction. See Re-
trograde.

The orbits of the superior planets are above or include
that of the earth; at the same time that the velocity of
The earth is greater than those of the superior planets.
Hence it results that the earth, in its motion, passes be-
tween these planets and the sun; which causes them to
appear in opposition to that luminary. In the opposition
(Ivy have a motion apparently retrogrades but it is direct

in the conjunction, as that of Venus and Mercury is in their
superior conjunctions.

Some of the planets, as the Earth, Jupiter, Saturn, and
Uranus, have satellites or moons, which revolve round thosv
primaries, as these do about the sun. Thus, the earth ha-
1 moon or satellite, Jupiter 4, Saturn 7, and Uranus 6. See
Moon and Satellites.

With respejet to these satellites or secondary planets, tile
following remarkable circumstance takes place. Our moon,
with the 4 satellites of Jupiter, and.l of Saturn's, are found
by observation to turn about an axis, in the same time as
they respectively revolve a bout their primaries. And though
it has not yet been ascertained from observations whether
the same be true for the other satellites. of Saturn, and
those of Uranus; yet, from the uniformity which obviously
pervades the system, it is concluded that the same is true
for all the secondaries.

The planet Saturn is encompassed with a thin flat ring,
or, as it is now found, with two rings, lying one within the
other edgewise towards the planet, and detached from it,
having their planes passing through the centre of Saturn.
If a circular annulus be cut out of a card, and divided into
two parts by a concentric circle, leaving the inner breadth
about 3 times that of the outer ; and a ball be then put
within, of such a size, that the space between the ball and
the annulus may be a little larger than the breadth of the
annulus, a representation will thus be obtained of Saturn
and his two rings. That side next the sun is bright, like
the body of the planet. The rings revolve in their own
plane; and, not being of a regular figure, their centre of
inertia is at a small distance from the centre of Saturn. M.
Laplace computed the time of their rotation to be 10h 33nl
36s ; nearly agreeing with the time found by Dr. Hers-
chel from observation.

If the mean distances, either of the planets or of their
satellites, be compared with the duration of their sidereal
revolutions, it will be easy to trace the beautiful relation
discovered by Kepler; namely, that while several bodies
turn about the same point, the squares of the periodical
times are respectively as the cubes of their mean distances
from that point: and, by combining this law with the theory
of Huygens, viz, that when the squares of the periodical
times, of several bodies circulating about the same point,
are respectively as the cubes of the distances from that
point, the central forces which animate, them, are in the
inverse ratio of the squares of the same distances; it will be
easy to discover the law of gravitation ; and, as it were, to
unveil the. mechanism of the planetary system.

The following tables exhibit a brief view of the solar sys-
tem, according to the latest and correct account.
I. Duration of the Sidereal Revolutions of the Planets, or of
their Periodic Times.

Mercury 87*969255 days.

Venus 224-700817

The Earth ------ 365-256384

Mars -------- 686-979579

Ceres, or Piazzi - - - - 1680-000000

Pallas, or Olbers1 - - - - 1681000000

Juno, or Harding - - - - 2007'50000O

Ditto, by Burckhardt - - - 1582-000000
Vesta, or Olbers*

Jupiter ■ 4332-602208

Saturn -10759077213

Uranus, or Herschel - - - 30689-000000

AST

C i

If. Greater Semiaxes of the Planetary Orbits, or their Mean
Distances from the Sun, that of the Earth (which is about
<)5 millions of miles) being denoted by unity.

Mercury 0-387100

Venus 0723332

The Earth ------- 1-000000

Mars -,--.- ■». • - •■- - - 1-523693
Ceres --------- 2776/55

Pallas 2-77690S

Juno 287673 L

Ditto, by Burckhardt - - - - 2657000

Jupiter 5.202778

Saturn 9-538785

Uranus 19- 1.83475

III. Relations of Excentricity to the Greater Semiaxis 1.

Mercury

- (for 17

50)

- -

0-205513

Venus

ditto

- -

0-006885

The Earth

ditto

- -

0-016814

Mars

ditto

- -

0-093088

Ceres

- (for 1&05)

f '

0-790000

Pallas

ditto

- -

0-246300

Juno

ditto

- -

0-250000

Jupiter

- (for 1750)

- -

0-048077

Saturn

ditto

-. ■ -

0056223

Uranus

ditto

- -

0-046683

IV. Inclination of Orbits

to the

Ecliptic.

Mercury

- (for 1750)

- -

- 7-0000

Venus

ditto

- -

- 3-3930

The Earth

ditto

- -

- 00000

Mars

ditto

- -

- 1-8501

Ceres

- (for 1805)

- -

- 10-6167

Pallas

ditto

-. .

- 337500

Juno

ditto

- -

- 21-0000

Jupiter

- (for 1750)

- -

- 1-3172

Saturn

ditto

- -

- 2-49S6

Uranus

ditto

- -

- 0-7706

V. Diameters of the Planets, that of the Earth (7914 Eng-

lish

miles) being assumed as Unity.

Mercury -

. _ 7

Jupiter

llf

Venus - -

- - 12.

Saturn -

- - - 10,%

The Earth

- - 1

His

Ring

.- - - 23i

Mars - -

" " ?

Uranus -

- - - 3f

Ceres - -

1 00

The

Moon - f-

Pallas - -

" " 7TT

The

Sun

- - - 112T9r

Juno - -

100

" " rrz

VI. Rotations of the Planets.

<1 h m
Mercury ----- unknown

Venus 0 23 20

The Earth 0 23 56T\

The Moon 27 7 43,%

Mars
Jupiter
Saturn
Uranus
The Sun

0 24 40
0 9 56
0 10 17
unknown
25 10 0

The masses of the planets have been found from princi-
ples as follow : The forces that urge two bodies, moving
circularly, are in a ratio composed of the masses, the dis-
tances from the centre, and the inverse square of the pe-
riodic times (see Central Forces): whence it results,
that the gravity of one of the satellites towards its planet,
is to that of the earth towards the sun, as the mean dis-
tance of the satellite from the centre of its planet, divided
by the square of its periodic time, is to the mean distance

73 ] AST

of the earth from the sun, divided by the square of its pe-
riodic time : or, expressing these gravitating tendencies by
G,g, the mean distances by r, r, the periodic times by t, r,

it is g : g : : ^ : -. But, denoting by m the mass of the

sun, and by m, that of the planet about which the satellite

revolves, by the nature of gravitation it is g : s : : - • — •

.1 r R r M m , r3 r3

therefore— : ^ : : — : -; consequently— : -= : : m : m.

TV R" r * J T t-

By applying this result to the planets which have satel-
lites, it is easy to find the value of their masses ; for, we
know the radii of the orbits of the satellites, as well as the
length of their sidereal revolutions, or their periodic times.
Therefore, taking the cubes of the radii of these orbits,
and dividing them successively by the squares of the perio-
dic times, the quotients will give the values of the masses
of the bodies, about which the satellites revolve.

As to the planets which have not satellites, Laplace has
deduced the values of the masses of Venus and Mars, from
the secular diminution of the obliquity of the ecliptic, and
the acceleration of the moon's mean motion. The mass of
Mercury was inferred from its volume, supposing the den-
sities of that planet and the earth reciprocally as their
mean distances from the sun. Mecanique Celeste, torn. 3,
p. 64. Exposition du Systeme du Monde, ed. 2, p. 193.
Thus was deduced the following table.

VII. Masses of the Planets, that of the Sun being taken for
Unity.

Mercury -»---.

J 2025810

Venus -------

3831U0

The Earth - ----- - — !—

329620

Mars — - —

1846082

Jupiter -------

r 1067

Saturn ------- -J —

3359

Uranus -------

19504

The Moon ------

22578970

The densities of bodies being in the direct ratio of their
masses, and theinverse ratio of the volumes ; and in bodies
nearly spherical, the volurnes being as the cubes of their
radii ; it results that the densities are then as the masses
divided by the cubes of the radii. By this process the fol-
lowing numbers are found for the mean densities of the
planets, that of the sun being assumed as unity ; viz,

to the sun 1 . to water I .

Mercury - - - - 10-1743 - - - 12-'912

Venus 5-0446 - - - 6-402

The-Earth - - - - 3'9393 - - - 5000

Mars 2-6734 - - ■ - 3393

Jupiter - - - - 08601 - - - 1-092
Saturn - - - - 0-4951 - - - 0-628
Uranus - - - - 0-1376 - - - 0'174
The Moon - - - 2-4656 - - - 3-129
The Sun - - - - 1-0000 - - - l-26'9
Where the first column of numbers denotes the propor-
tion of the densities of the planets to the sun considered as

unity. Then multiplying these by 1-2692 = , we

J J ° y • y 3-9-192

A S Y

C 174 ]

A S Y

obtain the last column of numbers, denoting the densities
of the planets in respect of that of water 1. See my Tracts,
vol. 2, p. 64, 66. •

Besides the bodies which revolve completely round the
sun, within the limits of our observation, there are others,
of which we only conclude from analogy that they perform
such revolutions. These are the comets ; they generally
appear Attended by a nebulous light, either surrounding
them as a coma, or stretched out to a considerable length
as a tail ; and they sometimes seem to consist of such light
only. Their orbits are so excentric, that in their remoter
situations the comets are no longer visible to us, though at
other times they approach much nearer to the sun than
any of the planets ; for the comet of 1680, when at its
perihelion, was at the distance of only -| of the sun's dia-
meter from his surface. Their tails are often of great ex-
tent, appearing as a faint light, directed always nearly op-
posite to the sun. It is quite uncertain of what substance
they consist ; and it is difficult to determine which of the
conjectures concerning them is the least improbable. For
further particulars, see Comet.

Lists and historical accounts of the principal writings
and authors on astronomy are contained in Weidler's His-
tory of Astronomy, which is brought down to the year
1/37. There is also Bailly's excellent History of astro-
nomy, ancient and modern. For this purpose, consult
also the following authors, viz, Adam, Vossius, Bayle,
Chauffepie, Costard, Perraut, the chronological table of
Riccioli, and that of Sherburn, at the end of his edition of
Manilius ; also the first volume of Lalande's astronomy.
The more modern, and popular books on elementary astro-
nomy, are very numerous, and well known : as those of
Ferguson, Long, Emerson, Vince, Leadbetter, Brent, Keil,
-Whiston, Wing, Street, Gregory, &e, &c.

ASTROSCOPE, a kind of astronomical instrument,
composed of two cones, on the surface of which are de-
lineated the constellations, with their stars, by means of
which Jhese may easily be known in the heavens. The
astroscope was invented by William Shukhard, professor
of mathematics at Tubingen, on which he published a
treatise in 1698.

ASTROSCOPIA, the art of observing and examining
the stars, by means of telescopes; and other instruments, to
discover their nature and properties.

ASTROTHEMATA, the placesor positions of the stars,
in an astrological scheme of the heavens.

ASTROTHESIA, is used by some for a constellation or
collection of stars in the heavens.

ASTRUM, or Astron, a constellation, or assemblage
of stars; in which sense it is distinguished from Aster,
which denotes a single star. Some apply the term, in a
more particular sense, to Sirius the Great Dog, or rather
to the large bright star in his mouth.

ASYMMETR-Y,the want of proportion, otherwise call-
ed Incommensurability, or the relation of two quantities
which have no common measure; as between 1 and ^/2,
or the side and diagonal of a square.

ASYMPTOTIC, is properly a, right line, which approaches
continually nearer and neater to some curve, whose asym-
ptote it is said to be, in such a manner, that when they are
both indefinitely produced, they are nearer together than
by any assignable finite distance; or it may be considered
as a tangent to the curve, when conceived to be produced
to an infinite distance. Two curves are also said to be
asymptotical, when (hey thus continually approach inde-

finitely to a coincidence: thus, two parabolas, placed with
their axes in the same right line, are asymptotes to each
other.

Of lines of the second kind, or curves of tke first kind,
viz. the conic sections, only the hyperbola has asymptotes,
which are two in number. All curves of the second kind
have at least one asymptote; but they may have three.
Those of the third kind may have 4 asymptotes, those of
the fourth kind 5 ; and so on for every order of curves
whatever; viz, that they have as many asymptotes as the
equations expressing them have dimensions, and no more.
The conchoid, cissoid, and logarithmic curve, though not
reputed geometrical curves, have each one asymptote.
And the branch or leg of a curve that has an asymptote,
is said to be of the hyperbolic kind.

The nature of asymptotes will be easily conceived from
the instance of that of the conchoid. Thus, if abc &c
be part of a conchoid, and
the line mn be so drawn,
that the parts fb,gc,hd,
ie, &c, of right lines,
drawn from the pole p, P

to the curve bcd, be equal to each other; then will the
line mn be the aymptote of the curve: because the per-
pendicular cc is shorter than FB, and v>d than cc, &c; so
that the two lines continually approach; yet the points se
&c can never coincide.

Asymptotes of the Hyperbola are thus described.
Suppose a right line de drawn to touch the curve in any
point a, and equal to the conjugate de of the diameter
acb drawn to that point a, viz, ad or ae equal to the
semiconjugate cd or ce ; then the two lines cdf, ceh,
drawn from the centre c, through the points D and e, are
the two asymptotes of the curve.

.N

fi/fc^>-i

The parts of any right line, lying between the curve of
the common hyperbola and its asymptotes, are equal to one
another on both sides, that is, gG =ha. In like manner,
in hyperbolas of the second kind, if there be drawn any
right line cutting both the curve and its three asymptotes
in three points, the sum of the two parts of that right line
extended in the same direction from any two of the asymp-
totes to two points of the curve, is equal to the third part
which extends in the contrary direction from the third
asymptote to the third point of the curve.

If ad k be an hyperbola of any kind, whose nature, with
regard to the curve and asymptote, is expressed by this
general equation xmi/n = a x am+", where x is = cf, and
y = fg drawn any where parallel to the other asymptote
en; and the parallelogram CFGI be completed: Then
in — n is to n, as this parallelogram cfgi is to the hyper-
bolic space fc.k, contained under the determinate line re.
with the asymptote fk and the curve c. k, both indefinitely
continued towards k. So that, if m be greater than ft, the
said asymptotic space is finite and quadrable : but when
m — n, as in the common or conic hyperbola, then m — u

A T L

[ J75 ]

A T M

= 0, the ratio of that space to the said parallelogram, is
as n to 0 : that is, the hyperbolic space is infinitely great,
in respect of the finite parallelogram : and when m is less
than n, then, m — n being negative, the asymptotic space
is to the determinate parallelogram, as a positive number
is to a negative one, and is what Dr. Wallis calls more
than infinite.

Asymptote of the Logarithmic Curve. If lmn
be the logarithmic curve, qon an asymptote, lq and MP
ordinates, mo a tangent, and po the subtangent, which in
this curve is a constant quantity. Then the indeterminate
space lmnq is equal to lq x po, the rectangle under the
ordinate lq and the constant subtangent po ; and the
solid generated by the rotation of that curve space about
the asymptote nq, is equal to half the cylinder, whose al-
titude is the said constant subtangent po, and the radius
of its base is lq.

Asymptotes, by some mathematicians, are distinguish-
ed into various orders. The asymptote is said to be of the.
first order, when it coincides with the bas.e of the curvili-
near figure : of the 2d order, when it is a right line pa-
rallel to the base : of the 3d order, when it is a* right line
oblique to the base : of the 4-th order, when it is the com-
mon parabola, having its axis perpendicular to the base :
and, in general, of the n -t- 2 order, when it is a parabola
whose ordinate is always as the n power of the base. The
asymptote is oblique, to the base, when the ratio of the
first fluxion of the ordinate to the fluxion of the base, ap-
proaches to an assignable ratio, as its limit; but it is pa-
rallel to the base, or coincides with it, when this limit is
not assignable.

The doctrine and determination of the asymptotes of
curves, isacurious part of the higher geometry. Fontenelle
has given several theorems relating to thi-. subject, in his
Geometric de I' Infini. See also Stirling's Lima: Tertii Or-
dinis, prop.vi, where the subject of asymptotes is learnedly
treated ; and Cramer's Introduction d I' analyse des lii>nes
courbes, art. 147 fy SCl]- f°r an excellent theory of asym-
ptotes of geometrical curves and their branches. Likewise
Maclaurin's Algebra and his Fluxions, book i, chap. 10,
where he has carefully avoided the modern paradoxes con-
cerning infinites and infinitesimals. But the easiest way of
determining asymptotes, it seems, is by considering them as
tangents to the curves at an infinite distance from the he-
ginningof the absciss; that is, when theabscissxisinfinitein
the equation of the curve, and in the proportion of x toy,
or in that of the subtangent to the ordinate.

The areas bounded by curves and their asymptotes,
though indefinitely extended, have sometimes limits to
which they may approach indefinitely near: and this hap-
pens in hyperbolas of all kinds, except the first or Apollo-*
nian, and in the logarithmic curve; as was observed above.
But in the common hyperbola, and many other curves,
the asymptotical area has no such limit, but is infinitely
great.-^-Solids, too, generated by hyperbolic areas, re-
volving about their asymptotes, have sometimes their li-
mits; and sometimes they may be produced till they ex-
ceed any given solid. — Also the surface of such solid, when
supposed to be infinitely produced, is either finite or infi-
nite, according as the area of the generating figure is finite
or infinite.

ATLANTIDES, a name given to the Pleiades, or seven
stars, sometimes also called Vergiliae. They were thus
called, as being supposed by the poets to have been the

daughters either of Atlas or his brother Hesperus, who
were translated to heaven.

ATMOSPHERE, is that invisible elastic fluid which
surrounds the earth to an unknown height, and partakes
of all its motions. — It received its name from the Greeks,
in consequence of the vapours which are continually mix-
ing with it.

Neither the properties nor the composition of the at-
mosphere seem to have occupied much of the attention of
the ancients. Aristotle considered it as one of the four
elements, situated between the regions of water and fire,
and mingled with two exhalations, the dry and the moist ;
the former of which he considered the cause of thunder,
lightning, and wind; while the second produced rain, hail,
and snow. The ancients in general seemed to have con-
sidered the blue colour of the sky as essential to the at-
mosphere ; and several of their philosophers believed that
air was the constituent principle of other bodies, or at least
that air and other bodies are mutually convertible into
each other. But these opinions floated merely in a state
of vague conjecture, till the matter was explained by the
sagacity of Hales, and of those philosophers who followed
the career that he had opened to them.

It was not till the time of Lord Bacon, who first taught
mankind to investigate natural phenomena correctly, that
the nature of the atmosphere began to be elucidated with
precision. Galileo introduced the study, by showing that
the air had heaviness, a point which was soon after in- '
vestigated completely by his pupil Torricelli, Pascal, &c.
Many other of its mechanical properties were afterwards
discovered by Boyle, Mariotte, Hooke, Newton, Derham,
&c^ but as these belong to the subject of Pneumatics, a
more particular mention of them shall be made in the ar-
ticles which they respectively constitute.

The opinions of the ancient chemists, concerning the
component parts of the atmosphere, are too vague to merit
any particular notice. Boyle however, and his contem-
poraries, put it beyond doubt, that the atmosphere, con-
tained, two distinct substances, viz, an elastic fluid distin-
guished by the name of air ; and water in the state of va-
pour. Besides these two bodies, it was supposed also that
the atmosphere contained a great variety of other sub-
stances, which were continually mixing with it, from the
earth, and which often altered its properties, and rendered
it noxious or fatal. Since the discovery however of car-
bonic acid gas, by Dr. Black, it has been ascertained that
this elastic fluid always constitutes a part of the atmo-
sphere. The constituent parts of the atmosphere, there-
fore, are, — 1. air; 2. water; 3. carbonic acid gas; 4. un-
known bodies.

That part of the composition of the atmosphere called
air, has not till very lately been perfectly recognised : it
seems to have been used at first to denote the atmosphere
in general, and was considered for many ages merely as a
simple, homogeneous, and elementary fluid; but philoso-
phers afterwards restricted it to the elastic fluid which con-
stitutes the greatest and most important part of the atmo-
sphere, excluding water and other foreign bodies, which
are occasionally found mixed with it. For many years,
all permanently elastic fluids were considered as air, from
whatever combinations they were extricated, and supposed
to possess exactly the same properties as the air of the at-
mosphere; and it was not till the discoveries of Cavendish
and Priestley had determined the peculiar properties of a

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A T M

variety of elastic fluids, that philosophers became sensible
that there existed various species of them. In consequence
of this discovery, the word air became generic, and was
applied by Priestley, and the British and Swedish philo-
sophers in general, to all permanently elastic fluids; while
the air of the atmosphere was distinguished by the epithets
of common or atmospheric air: but Macquer thought pro-
per to apply the term gas to all permanently elastic fluids
except common or atmospheric air, and to confine the
term air to this last fluid, an acceptation of the word which
is now universally adopted by all philosophers: the word
air therefore now denotes common or atmospheric air, and
all other permanently elastic fluids are denominated gases.
For the knowledge of .the component parts of air, we are
indebted to the labours of those philosophers in whose
hands chemistry advanced with such rapidity, during the
last 40 years of the 18th century. The first step was made
by Dr. Priestley in 1774, by the discovery of oxygen gas.
This gas, according to the prevailing notions of the time,
he considered as air totally deprived of phlogiston. Hence
he considered common air as oxygen gas combined with an
indefinite portion of phlogiston, varying in purity accord-
ing to that portion ; being also the purer the smaller the
quantity of phlogiston it contained. Scheele also analysed
the air, and found, from a process distinct from Priestley,
that it was composed of two different elastic fluids, similar
to those discovered by Priestley : the oxygen gas of Priest-
ley he called empyreal air, and azotic gas he denominated
foul air. Lavoisier was also conducted to the same con-
clusion by another different process; and thus was atmo-
spheric air confirmed to be a compound of oxygen and
azotic gas. _ •

This point being established, it next became a question
of considerable consequence to determine the proportion
of these two ingredients in any given bulk of air, and to
ascertain whether that proportion is in every case the same :
accordingly the most eminent philosophers engaged them-
selves in this determination. Mr. Cavendish, by improving
the eudiometer employed by Priestley and Fontana, was
enabled to ascertain correctly the constituents of air (See
Eudiometeii): he found that, by mixing together oxygen
gas and azote in various proportions, he should at last be
able. to determine the absolute quantity of oxygen in air:
for since azotic gas, one of the component parts ot that
fluid, cannot be separated by any substance with which
chemists are acquainted; the analysis of air can only be
attempted by exposing it to the action of those bodies
which have the property of absorbing its oxygen. By these
bodies, the oxygen gas is separated, and the. azote left be-
hind, and the proportion of oxygen may be ascertained by
the diminution of bulk; which being once known, it is easy
to ascertain the proportion of azotic gas, and thus to deter-
mine the relative quantity of the component parts of air.
Therefore, to find the absolute quantity of oxygen in air,
he mixed togdthe'r oxygen gas and azote in various pro-
portions, and at last found that a mixture of 5 measures of
the purest oxygen which he could procure, with If) mea-
sures of azote, was just as much diminished by nitrons gas,
as the same bulk of atmospheric air. Hence lie concluded
that air is composed of 5 parts of oxygen and 1J) of azote,
of every 24 measures of atmospheric air; which gives for
its composition percent. 7,'),10 azote, and ?0 84 oi oxygen ;
or very nearly 21 per cent, of oxygen gas. The trials made
by others, to discover the true quantity of oxygen in air,

have all ended in the same result as that obtained by Mr.
Cavendish, whenever the experiment was made with pre-
cision. They all indicate that common air consists very
nearly of 21 parts of oxygen, and 79 of azote; and prove
that in whatever place or altitude, above the surface of the
earth, the experiment is made, the same proportion of its
constituents is always manifested, let the state of the at-
mosphere be what it may. Gay Lussac examined air
brought from an altitude of 6000 feet in a balloon, and
found its constituents precisely the same as the air at the
earth's surface.

The next ingredient in the composition of the atmo-
sphere, which claims our particular notice, is water; for
since it is now fully confirmed that the atmosphere always
contains some portion of this fluid, it deserves to be explain-
ed in what state this comparatively heavier fluid exists in
the air, and the quantity which a given bulk of the atmo-
sphere contains precisely, if possible, made known. With
respect to the state in which water exists in air, two opi-
nions have been formed, each of which has been supported
by the most eminent philosophers. 1. Water may be dis-
solved in a"ir in the same manner as a salt is held in solution
by water. 2. It may be mixed with air in the state of steam
or vapour, after having been converted into vapour. The
first of these opinions was hinted at by Dr. Hooke, in his
Miqographia, and afterwards proposed by Dr. Halley; but
it was much more developed by M. Le Roy, of Montpclier,
in 1751. Dr. Hamilton of Dublin also made known the
same discoveries about the same time. The phenomena
in general coincide remarkably well with this theory. "The
quantity of water which air is capable of holding in solu-
tion, is increased by every augmentation of temperature,
and diminished by cold, which is precisely analogous to
almost all other solvents. These analogies, and several
others which will easily suggest themselves to the reader,
have induced by far the greater number of philosophers to
adopt this opinion.

The second theory, namely that water exists in a state of
vapour, is strongly confirmed by the experiments of Saus-
sure, and the reasoning of Dr. Black : but it is to Mr. Dal-
ton that we are indebted for the most precise information
on this subject, and the perfect establishment of its rea-
lity. He demonstrated that the water existing in air pos-
sesses precisely the same degree of elasticity that it docs
when in the state of vapour, in a vaeuum, at the same tem-
perature; from which it follows irresistibly that it exists
in air not in the state of water, but of an elastic fluid or
vapour.

The first who attained any thing like precision, in at-
tempting to measure the quantity of water in the atmo-
• sphere, was Saussure. This ingenious philosopher has
shown in his Hygrometrical Essays, that an English cubic
foot of air, when -saturated with water, at the temperature
of 66°, contains only about S grains troy of that liquid, or
about -p'jth of its weight. But the experiments of Mr. Dal-
ton were susceptible of far greater precision, and reflect
much credit on his talents as a philosopher. His results ate
these : — 1. That the quantity ol vapour in the atmosphere is
variable. 2. That in the torrid /.one its force of elasticity
varies from On' to one inch of mercury, :;. That in Britain
it seldom amounts to 0(i, but is often as great as O'S du-
ring summer. 4. That in winter it is frequently as low as
O'l of an inch of mercury. From these tacts, the absolute
quantity of water contained in the atmosphere, at any given

ATM

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ATM

time, might be correctly ascertained, provided we were cer- very eminent philosophers; since it is well known that, be-

tain-that the density and elasticity of vapours follow the sides those large quantities of the electrical matter, with

same law as that of gases, as is extremely likely to be the which the clouds are generally charged in a thunder storm,

case. If so, the vapour will vary from -g^th to -j-ygth part the atmosphere is never wholly destitute of the electric

of the atmosphere. Dalton supposes that the medium fluid. A person electrified negatively may satisfy him-

quantity of vapour held in solution at once in the atmo- self of this by extending his arm in the open air, and pre-

sphere, may amount to about y^th of its bulk. See Man- senting a long sharp needle with its point upwards; for

Chester Memoirs, vol. 3. the electric matter collected from the remoter air will ap-

With regard to the next constituent part. of the atmo- pear luminous as it converges to the point of the needle,

sphere, carbonic acid gas, it will be first proper to remark, Mr. Canton's balls are likewise an excellent contrivance

that its existence was first determined by Dr. Black, im- for the same purpose, and may be made use of, not only

mediately after he had ascertained the nature of that pe- for determining the electricity of the atmosphere in gene-

culiar fluid. The exposure of a pure alkali, or alkaline ral, but the positive or negative quality of it. According

earth, to the atmosphere, having been observed to be the to this ingenious philosopher, desiccated atmospheric air,

cause of a gradual conversion of it into a carbonate, by when heated, becomes negatively electric; and when

the absorption of carbonic acid gas, rendered unavoidable cooled, the electricity is of the positive kind, even when

the inference (as soon as the difference between a pure al- the air is not permitted to expand or -contract; and the

kali and its carbonate had been ascertained to depend upon expansion or contraction of atmospheric air occasions

that acid) that carbonic acid gas existed in the atmosphere, changes in its electrical state.

This fact has not only been proved for the atmosphere near Beccaria informs us, that the electricity of the atmosphere

the surface of the earth, but established for the greatest is always positive during the day, and in dry weather; but

heights accessible to man. Saussure found it at the top of always negative when a bright or serene atmosphere suc-

Mount Blanc, the highest point of theold continent; which ceeds dark and moist weather: that the quantity of it in-

is covered with eternal snow, and not exposed to the in- creases after the rising of the sun, and during his progress ;

fluence of vegetables or animals. Humbolt has more lately and that its augmentation is the more considerable as the

ascertained the existence of this gas in air brought by M. moisture of the air diminishes; but it decreases in the

Garnerin, fr.>m an aerial excursion into the atmosphere not evening. In days equally dry, the degree of electricity at

less than 4280 feet above the surface of the earth; which noon was proportional to the degree of heat; and in a

is a fact sufficient to prove that the presence of carbonic serene atmosphere with little wind, axonsiderable quan-

acid in air does not depend on its vicinity to the earth. tity of the electrical matter commonly arose after sun-set

The difficulty of separating this gas from air, has render- during the precipitation of the dew. Thick fogs were ob-

ed it no easy matter to determine, with accuracy, the rela- served, during their ascent into dry air, to carry with them a

live quantity of it in a given bulk of air. From the experi- considerable quantity of the electric matter. Andtheelec-

ments of Humbolt it appears to vary from '005 to '01. Mr. tricity was always stronger when his rods were higher, and

Dalton's experiments give the quantity much smaller; he the strings which were extended and insulated in the open

concludesthat aircontains only ^-^.th of it bulk of carbo- air were longer. See his Essay on Atmospheric Electri-

nic acid gas. Hence, and from the results of other ex-
periments, it is inferred, that the bulk of carbonic acid in
air, does not exceed much the , -500th part of the atmo-
sphere.

city annexed to the English translation of his Artificial
Electricity.

Mr. Cavallo, from numerous experiments and observa-
tions, infers, that there is in the atmosphere at all times a

From all. which we conclude, that the average propor- quantity of electric matter: — that the electricity of the at-
tion of each of these three distinct elastic fluids, forming mosphere and of fogs is always positive : — that in general

chiefly the whole body of the atmosphere, is, —
.98-9 Air.
l'O Vapour
0-l Carbonic acid.
100 0
It is indeed true that other bodies, besides the three elas-
tic fluids just mentioned, have been suspected in the at-

the strongest electricity is observable in thick fogs, and
also in frosty weather; and the weakest when it is cloudy
and warm, and rain approaches ; — that it does not seem to
be less by night than by day; and that it is stronger in
places more elevated, than in those which are lower. See
his complete Treatise on Electricity, vol. 2. p. 42. edit. 4.
Among the celebrated men who have engaged in the

mosphere; as, for instance, hydrogen gas, and carburetted cultivation- of this science, few have enjoyed the advan

hydrogen; but as no satisfactory accounts concerning tages of observing the phenomena attending atmospherical

their quantities have yet been ascertained, or that would electricity of M. Saussure, or indeed who have possessed a

justify any particular notice of them in this place, we will more extensive acquaintance with meteorology in genera),

therefore content ourselves by barely mentioning them, f°r enabling him to illustrate these phenomena by appo-

and refer our readers for the little information we have site observations. He confirms the fact, noticed by others,

been able to collect respecting them, to their proper ar- and previously known, that aerial electricity varies accord-

ticles. The electric matter, and other substances raised ing to the situation, being generally strongest in elevated

from bodies on the earth's surface, by effluvia, exhala- and insulated places, and not observable under trees, in

tions, &c, are always distinguishable; and, being ex
posed to the continual action of the sun, give rise to in-
numerable operations, sublimations, separations, composi-
tions, digestions,' fermentations, putrefactions, &c. Thom-
son's Chem. vol. 4.

Electricity of the Atmosphere. The'subject of at-

streets, houses, &c. But it is not so much the height as
the situation of the places which determines the degree of
electricity: for the projecting angle of a high hill, he says,
will often exhibit a stronger electricity than the plain at
the top of the hill, as there are fewer points of the former
to deprive the air of its electricity. The intensity of the

mospherical electricity has engaged the attention of many atmospherical electricity is also subject to a great variety
Vox.. I. ^ 2 A

ATM

[ '78 ]

ATM

of changes, of which some depend on obvious circum-
stances, while others are altogether inexplicable ; they are
sometimes so rapid, says Saussure, as to preclude all possi-
bility of noting them down. When rain falls, without a
storm, these changes are not so sudden; but with respect
to the intensity of the electric force, they are excessively
irregular; while the quality of it is more constant. Rain
or snow almost always gives positive electricity. In cloudy
weather, without rains or storms, the electricity generally
follows the same laws as in serene weather. Its intensity
is most commonly diminished by strong winds, which blend
the different strata of the atmosphere, cause them to sub-
side towards the ground, and thus distribute the electricity
uniformly between the earth and the air. The various
modifications of electricity in the atmosphere, are observed
with the greatest advantage in serene weather. M. Saus-
sure found in winter, and in such weather, that the elec-
tricity was generally weakest in the evening, when the
dew had fallen, and so continued till sun-rise; afterwards
its intensity augmented by degrees, sometimes sooner and
sometimes later ; but usually before noon it attained a
certain maximum, from which it again declined till the
fall" of the dew, when it would be sometimes stronger than
it had been during the whole day ; after which, it would
again gradually decrease during the whole night; but it
was never quite destroyed in weather perfectly serene.
Hence it may be inferred, that atmospherical electricity,
like the waters of the ocean, is subject to a flux and re-
■ flux; which produce an increase and diminution twice in
24 hours. The same author also remarks, after numerous
experiments made on evaporation, that this which ap-
peared to be the vehicle that conveys electric matter into
the atmosphere, from silver and china always produced
negative electricity'; and from iron and copper generally
positive electricity : and hence it may be inferred, that
electricity is positive with those bodies that are capable
of decomposing water, or of being decomposed themselves
by their contact with the water; and negative with all those
that are not at all decomposed or altered.

As to the producing causes or sources of atmospherical
electricity, we may observe in general, that they may be
reduced to four, viz, friction, evaporation, heat and cold,
and condensation and expansion : and with respect to the
changes and modifications to which the atmospherical elec-
tricity is continually subject, they may be attributed to the
operation of the various causes that produce them, and to
the chemical processes that are constantly carried on by
means of the various ingredients that compose the atmo-
sphere. For further observations on this subject, consult
the authors from whence these particulars were chiefly
taken; also Mr. Read's very ingenious work, entitled a Sum-
mary View of the Spontaneous Electricity of the Earth
and Atmosphere. Sec also the articles Hail, Rain, Snow,
Lightning, and those of other meteors.

Figure of the Atmosphf.iie. — As the atmosphere en-
velops all parts of the surface of our globe, it is evident,
if they both continued at rest, and were not endowed with
a diurnal motion about their common axis, that the atmo-
sphere would be exactly globular, according to the laws
of gravity; for all the parts of the surface, of a fluid in a
state of rest, must be equally removed from its centre. But
as the earth and the ambient parts of the atmosphere re-
volve uniformly together about theiraxis, thedifferentparts
of both have a centrifugal force, the tendency of which is
more considerable, and that of the centripetal less', as the

parts are more remote from the axis; and hence the figur*
of the atmosphere must become an oblate spheroid ; since
the parts that correspond to the equator are farther re-
moved from the axis, than the parts which correspond to
the poles. Besides, the figure of the atmosphere must, on
another account, represent a flattened spheroid, namely
because the sun strikes more directly the air which en-
compasses the equator, and is comprehended between the
two tropics, than that which pertains to the polar regions;
whence it follows, that the mass of air, or part of the at-
mosphere, adjoining to the poles, being less heated, can-
not expand so much, nor reach so high. Yet, notwith-
standing, as the same force which contributes to elevate
the air, diminishes its pressure on the surface of the earth,
higher columns of it about the equatorial parts, all other
circumstances being the same, may not be heavier than
those about the poles.

In the Transactions of the Royal Irish Academy for
1788, Mr. Kirwan has an ingenious dissertation on the fi-
gure, height, weight, &c, of the atmosphere. He observes
that, in the natural state of the atmosphere, that is, when
the barometer would every where, at the level of the sea-,
stand at 30 inches, the weight of the atmosphere, at the
surface of the sea, must be equal all over the globe; and
in order to produce this equality, as the weight proceeds
from its density and height, it must be lowest where the
density is greatest, and highest where the density is least;
that is, highest at the equator and lowest at the poles,
with several intermediate gradations.

Though the equatorial air however be less dense to a
certain height than the polar, yet at some greater heights
it must evidently be more dense: for since an equatorial
and polar column are equal in total weight or mass, the
lower part of the equatorial column, being more expanded
by heat &c than that of the polar, must have less mass,
and therefore a proportionably greater 'part of its mass
must be found in its superior section; so that the lower
extremity of the superior section of the equatorial column
is more compressed, and consequently denser, than the
corresponding part of the polar column. The same thing
is to be understood also of the extra-tropical columns with
respect to each other, where differences of heat prevail.

Hence, in the highest regions of the atmosphere, the
denser equatorial air, not being supported by the collate-
ral extra-tropical columns, gradually flows over, and ex-
pands itself towards the north and south.

These superior tides consist chiefly of inflammable air,
as it is much lighter than any other, and is generated in
great plenty between the tropics ; it furnishes the matter
of the aurora borealis and australis, by whose combustion
it is destroyed, otherwise its quantity would in time be-
come too great, and the weight of the atmosphere annually
increased ; but its combustion is the primary source of
the greatest perturbations of the atmosphere. See an in-
genious work on the Atmosphere by II. Robertson, M. D.
in 2 vols. 8vo, 1808.

Weight or Pressure of the Atmosphf.iie. — That the
mass of the atmosphere has weight, and consequently
presses upon bodies exposed to its action, is a fact which
has been asserted by almost all philosophers, both ancient
and modern. But it was only by means of the experi-
ments made with pumps and the barometrical tube, by
Galileo and Torricelli, that we have been convinced of its
reality, and enabled to ascertain the precise measure and
quantity of that pressure. Thus, it is found that the pres-

ATM

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ATM

sure of the atmosphere sustains a column of quicksilver,
in the tube of the barometer, of about 30 inches in height;
it therefore follows, that the whole pressure of the atmo-
sphere is equal to the weight of a column of quicksilver,
of an equal base, and 30 inches height: and because a
cubical inch of quicksilver is found to weigh nearly half a
pound avoirdupois, therefore the whole 30 inches, or the
weight of the atmosphere on every square inch of surface,
is almost equal to 15 pounds. Again, it has been found"
that the pressure of the atmosphere balances', in the case
of pumps &c, a column of water of about 34-J feet high ;
and, the cubical foot of water weighing just 1000 ounces,
or 6"2£ pounds, 34|- times 62-£, or 21581b, will be the
weight of the column of water, or of the atmosphere on a
base of one square foot ; and consequently the 144th part
of this, or 15lb nearly, is the weight of the atmosphere on
a square inch ; the same as before. Hence Mr. Cotes
computed that the pressure of this ambient fluid on the
whole surface of the earth, is equivalent to that of a globe
of lead of 60 miles in diameter. And hence also it ap-
pears, that the pressure upon the human body must be
very considerable ; for as every square inch of surface
sustains a pressure of 15 pounds, every square foot will
sustain 144 times as much, or 2l6o pounds; then, if the
whole surface of a man's body be supposed to contain 15
square feet, which is pretty near the truth, he must sus-
tain 15 times 2l60, or 32400 pounds, that is nearly 14-|-
tons weight, for his ordinary load. By this enormous pres-
sure we should undoubtedly be crushed in a moment, if
all parts of our bodies were not filled either with air or
some other elastic fluid, the spring of which is just suffi-
xient to counterbalance the weight of the atmosphere.
But whatever this fluid may be, it is certain that it is just
able to counteract the weight of the atmosphere, and no
more: for, if any considerable pressure be superadded to
that of the air, as by going into deep water, or the like, it
is always severely felt, at least when the change is made
suddenly; and if, on the other hand, the pressure of the
atmosphere be taken oft" from any part of the human body,
as the hand for instance, when put over an open receiver,
from which the air is afterwards extracted, the weight of
the external atmosphere then prevails, and we imagine the
hand strongly sucked down into the glass.

The difference in the weight of the air which our bo-
dies sustain at one time more than another, is also very
considerable; this difference in the same situation arises
from changes in the state of the atmosphere, and it most
frequently takes place in countries at some distance from
the equator; and as the barometer varies at times from
28 to 31 inches, or about one tenth of the whole quantity,
it follows that this difference amounts to about a ton and a
half on the whole body of a man, which he therefore may
sustain at one time more than at another. On the increase
of this natural weight, the weather is commonly fine, and
we feel ourselves in better spirits and more alert and ac-
tive ; but, on the contrary, when the weight of the air di-
minishes, the weather is generally moist and foggy, and
people feel a listlessness and inactivity about them. And
hence it is no wonder that persons suffer very much in
their health from such changes in the atmosphere, espe-
cially wIk'M they take place very suddenly, for it is to this
circumstance chiefly that a sensation of uneasiness and in-
disposition is to be attributed; thus, when the variations
of the barometer and atmosphere are sudden and great,
we feel the alteration and its effect on our bodies and spi-

rits very much ; but when a change of this kind is gradual,
we are scarcely sensible of it, owing, undoubtedly, to the
power with which the body is naturally endowed, of ac-
commodating itself to this alteration in the state of the
air, as well as to the change of many other circumstances
of life, the body requiring a certain interval of time to ef-
fect the alteration in its state, proper to that of the air &c.
Thus, in going up to the tops of mountains, where the
pressure of the atmosphere is diminished two or three
times more than on the plain below, little or no inconve-
nience is felt from the rarity of the air, if it be not mixed
with other noxious vapours &c ; because that, in the as-
cent the body has had sufficient time to accommodate it-
self gradually to the slow variation in the state of the at-
mosphere : but, when a person ascends with a balloon
very rapidly to a great height in the atmosphere, he feels
a difficulty in breathing and an uneasiness of body; and
the same is also true of an animal when inclosed in a re-
ceiver, and the air suddenly drawn or pumped out of it.
So likewise, on the condensation of the air, we feel little
or no change in ourselves, except when the alteration oc-
curs suddenly, as in very rapid changes in the weather,
and in descending to great depths in a diving-bell, &c. I
have often heard the late unfortunate Mr. Spalding speak
of his experience on this point : he always found it abso-
lutely necessary to descend with the bell very slowly, and
that only from one depth to another, resting a while at
each depth before he began to descend farther : he first de-
scended slowly for about 5 or 6 fathom, and then stopped
a while; he felt an uneasiness in his head and ears, which
increased more and more as he descended, till he was ob-
liged to stop at the depth above mentioned, where the
density of the air was nearly doubled ; having remained
there a while, he felt his ears give a sudden crack, and
after that he was soon relieved from any uneasiness in that
part, and it seemed as if the density of the air was not al-
tered. He then descended 30 feet more, with the same
precaution and similar sensations, being again relieved in
the same manner, after remaining awhile stationary at the
next stage of his descent, where the density of the air was
tripled. And thus he continued proceeding to a great
deptlu always with the same circumstances, repeated at
every 5 or 6 fathoms, and adding the pressure of one more
atmosphere at every period of the progress.

It is not easy to assign the true reason for the variations
that happen in the gravity of the atmosphere in the same
place. In places within the tropics, where these varia-
tions are not very considerable, the chief cause seems to
be the heat of the sun, as the barometer constantly sinks
about half an inch every day, and rises to its former sta-
tion again in the night. But in the temperate zones the-
barometer ranges from 28 to near 31 inches, showing, by
its various altitudes, the changes that are about to take
place in the weather. If we could know, therefore, the
causes by which the weather is influenced ; we should also
know those by which the gravity of the atmosphere is af-
fected. These may perhaps be reduced to immediate ones,
viz, au emission of latent heat from the vapour contained
in the atmosphere, or of electric fluid from the same, or
from the earth ; as it is observed that they both produce
the same effect with the solar heat in the tropical climates,
viz, to rarefy the air, by mixing with it, or setting loose a
lighter fluid, which did not before act in such large pro-
portion in any particular place.

Temperature of the Atmosphere. — To no part of natural
2 A 2

AT M

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A T M

and experimental philosophy, has the attention of learned
men been directed with greater assiduity, than that which
respects the various states of the atmosphere in regard to
temperature : many reasons have been adduced, as well as
theories proposed, to account fur these phenomena, and
numerous experiments made to ascertain the precise de-
grees of variation, at different seasons, and in different si-
tuations, both at sea and land. Among the writers who
have rendered themselves conspicuous in these investiga-
tions, are Dr. Hales, M. Bouguer, Marietta, Halley, De
Luc, Saussure, Mairan, Raymond, Playfair, Kirwau, and
Darwin. Theyallagree, that the principal alterationsin the
state of the atmosphere are effected by means of the sun's
heat, and that arising from the condensation of vapour : it
was from the former fact that the ancient philosophers of
Greece and Rome concluded (though too abruptly) that
the torrid zone under a vertical sun, and the frigid zone
where its rays fall very obliquely, were .uninhabitable:
time corrected this mistake, and presented new phenome-
na, which it has been found difficult to explain. The
hottest days are frequently felt in the coldest climates; and
the greatest cold, as well as perpetual snow, are found in
countries bordering on, or even immediately under, the
equator. Dr. Halley has indeed proved, that abstracting
from the intervention of fogs, mists, and mountains of ice,
the hottest weather might in summer take place even un-
der the poles; the duration of the sun's light more than
compensating for the obliquity of its direction, (see Heat) ;
but as many physical causes obstruct the activity of the
solar rays in these and other regions, it was still necessary
i to recur to some other cause. At length M. de Mairan
(Mem. Acad. Paris, 1719 and 1767) discovered, that the
rigour or intensity of the cold in winter, is tempered by
the heat imparted to the atmosphere by the earth itself;
which heat, probably possessed from its origin, is preserved
and renewed by the incessant influences of the sun, to
which one half of its surface is constantly exposed : hence
it must follow, that the/ temperature of the atmosphere de-
pends on the capacity of the earth to receive and retain
heat, and for communicating it to the surrounding me-
dium. But as the earth is composed of land and water, it
should be considered that the capacities of these consti-
tuent parts, for receiving both heat and cold, are very "dif-
ferent. Land, particularly when dry, receives heat from
the sun's rays very readily, but transmits it through its
own substance to great depths very slowly; and on the
other hand water, by reason of its transparency, receives
heat very slowly, but diffuses what it receives more rea-
dily. The experiments of Dr. Hales, and those of other
philosophers, to ascertain the degrees of heat of the earth
at different depths, and at various times of the year, ap-
pear to confirm the idea that the surface of the earth is
much heated during the summer, but that the heat de-
scends very slowly, a great part of it being communicated
to the air; and that during the winter, the earth gives out
to the air the heat which it had received during the sum-
mer; also that wet summers must be succeeded by cold
winters. Mariotte, also obtained nearly the same results:
lie found that the earth is gradually healed during the
summer, and as gradually cooled during the winter
months; and that at the distance of a few feet under the
surface, it is constantly warmer than the external air; and
the excess was found to remain till April, when the sur-
face, is again heated by the sun's rays, and slowly trans-
mits its heat downwards. Hence it appears, that at the

distance of about 80 or 90 feet below the surface, provided
there be a communication with the external air, or at a,
less depth if there be no such communication, the temper-
ature of the earth admits of very slight variation, and ge-
nerally approaches to the mean annual heat. From which
it has been inferred, that the temperature of spring is
nearly the same as the annual temperature, and varies
very little. These facts tend to prove very satisfactorily,
that the heat of the earth does not increase as we descend
below its surface, but at the greatest depths it is nearly
the same as the mean annual temperature of the latitude.
We have observed, that land is capable of receiving heat
and cold much more readily than water. To which pur-
pose Dr. Raymond found, in the neighbourhood of Mar-
seilles, land frequently heated to l60°; but he never ob-
served that the sea was hotter than 77°; and in winter he
frequently observed the earth cooled down to 14° or 15%
but the sea never lower than 44° or 45°: Mem. de la So-
ciet. de Med. de Paris, an. 1778, p. 70. From these facts
therefore, it must follow, that the atmosphere which lies
over the sea should maintain a more uniform temperature,
than that over the land ; and this is found to be the fact ;
nor is it difficult ,to give a satisfactory explication of it.
During summer, the temperature of the sea on its surface
is constantly diminished by the process of evaporation ;
and in the winter, when the superficial water is cooled, it
descends by its augmented gravity to the bottom, and its
place is occ.upied by water of a higher temperature. Thus
alternate changes of heavier and lighter air proceed, and
the winter elapses before the atmosphere has diminished the
temperature of the water below a certain degree. Between
the mean annual temperature of the atmosphere over the
ocean, and that of countries situated at a considerable di-
stance from it, there is a very perceptible difference. As-
the sea is never heated to the same degree as the land, the
mean temperature of summer over the sea may -be consi-
dered as lower than that over the land. In winter, when
the force of the sun's rays is weakened, the sea imparts its
heat to the atmosphere much more readily than the earth -y
therefore the mean temperature on sea, at this reason, is
higher than on land ; and in cold countries, this differ-
ence in the evolution of heat is so very considerable, that
it more than counterbalances the difference which takes
place in summer ; insomuch that, in high latitudes, the
mean annual temperature at sea ought to exceed that on
land.

In latitudes distant from the equator, islands are warmer
than continents, because they participate more of the tem-
perature of the sea. Countries that lie southward of any
sea, are warmer than those that have the sen to the south
of them, at least in our hemisphere, because the winds
that should cool them in winter are tempered by passing
to them from that sea; and those that are northward of the
sea, are cooled in summer by the breezes that issue from
it; but a northern or southern bearing of the sea renders
a country warmer than if it lay cither to the east or west.
Tracts of land which are covered with tries and Luxuriant
vegetables, arc much colder than (hose which have less
surface of vegetable matter; for though living vegetables
alter their temperature slowly and with difficulty, yet the
evaporation from their numerous surfaces is much greater
than from the same space of laud uncovered with vegela-
bles; and besides, when they are tall and close, as forests,
they exclude the sun's rays, and shelter the winter snows
from the wind and sun. From some experiments of Mr.

ATM [ 181 ] ATM

Williams (Philad. Trans, vol. 2, p. 150), it appe ars that fo- As to the effect of elevation on the temperature of the
rests discharge one third more vapour into the atmosphere, atmosphere, we may observe, that as heat is propaoated
than the same space of ground would do if actually covered through the atmosphere chiefly by communication and
with water. From this reasoning, it seems that woody contact with the earth, lofty mountains of limited sur-
countries are much colder than those that arc open and face cannot warm it to any considerable decree as they
cultivated. With regard to the heat afforded from the receive the sun's rays more obliquely, and communicating
condensation of vapour, we need not expatiate very con- less with the common mass of the earth, are less heated
siderably, since it is well known that vapour contains a than plains. Hence it happens that the steepest moun-
great quantity of such matter. This heat produces no tains are always the coldest. Before M. Bououer demon-
other effect but that of making it assume an aerial ex- strated and, laid open the true causes of the diminution of
panded form, until the vapour is condensed into a liquid ; heat on the tops of mountains, many hypotheses were as-
and during this condensation, a quantity of sensible heat sumed to account for this phenomenon; but this cele-
is let loose, which warms the surrounding atmosphere, brated philosopher has shown, that there is no necessity for

This condensation is frequently occasioned by the attrac
tion of an electrical cloud; and hence proceeds the sultri-
ness which we often experience before rain.

The following is a view of the annual temperature of
different places according to the order of their latitudes.

North
Lat.

Longitude.

Mean
Annual
Heat.

t> '

o '

0

Wadso in Lapland

70 i

36

Abo - - - „ -

6'0 27

22 18 e

40-00

Petersburg -

59 56

30 24 e

38-S

Upsal -

59 51

17 47 e

41-88

Stockholm -

59 2C

IS 00 e

42-39

Solyskamski -

59 oc

54 00 e

36-2

Edinburgh

55 57

3 00 w

47-7

Francker - - " .-

53 OC

. 5 42 E

52-6

Berlin -

52 32

13 31 E

49-0

Lyndon in Rutland

52 30

0 03 w

4S-03

Leyden -

52 10

4 32 e

52-25

London " - -

51 31

51-9

Dunkirk -

51 02

2 O'e

54\9

Manheim -

49 27

9 02 e

51-5

Rouen -»

49 26

1 00 w

51

Ratisbon -

48 56

12 05 e

49-35

Paris ....

48 50

2 25 e

52

Troyes in Champaigne -

48 IS

4 10 e

5317

Vienna -

48 '12

16 22 e

51-53

Dijon -

47 19

4 57 e

52 8

Nantes ....

47 13

1 28 E.

5553

Poitiers -

40" 39

0 30 e

53-8

Lausanne - - - -

46 31

6 50 e

4S-S7

Padua ....

45 23

12 OOe

5 2-2

Rhodez in Guienne

45 21

2 -39 E

52-9

Bourdeaux -

44 50

0 36 w

576

Montpelier -

43 36

3 53-e

6'0-S7

Marseilles -

43 19

5 27 e

6'l-8

Mount Louis, in Rousillon

42 00

2 40 e

44-5

Cambridge, in New England

42 25

71 00 w

50-3

Philadelphia - -

39 56

75 09 w

52.5

Pekin -"'-•--

39 54

116 29 u

55-5

Algiers ....

36 49

2 J7e

72-0

Grand Cairo - - - -

30 00

31 23 e

73

Canton -

23 00

113 OOe

75-14

Tivoli, in St. Domingo -

19 00

74

Spanish Town, in Jamaica

IS 15

76 38 w

SI

Manilla » - -

14 36

120 5Se

"S-4

Fort St. George ...

13 00

87 OOe

31-3

Pondicherry - - -

12 00
South
Lat.

67 00 e

S8-0

Falkland Islands - . -

51 00

66 00 w

17'4

Quito - - - . .

0 13

77 50 w

j'2

recurring to any dubious suppositions for an explication
of the fact, as will evidently appear from his account of
what was experienced on the mountains of Peru. He says,
" It was proper, in order to explain this subject, to consi-
der the short duration of the sun's rays, which cannot
strike the different sides of mountains but for a1 few hours,
and even this not always. A horizontal plain, when the~
sun is clear, is exposed, at mid-day to the perpendicular
and undiminished action of these rays, while they fall but
obliquely on a plain not much inclined, or on the sides of
a high pile of steep rocks. But let us conceive for a mo-
ment an insulated point, half the height of the atmosphere,
at a distance from all mountains, as well as from the clouds
which float in the air. The more a medium is transparent,
the less heat it ought to receive by the immediate action
of the sun. The free passage which a very transparent
body allows to the rays of light, shows that its small par-
ticles are hardly touched by them. Indeed, what impres-
sion could they make on it when they pass through almost
.without obstruction? Light, when it consists of parallel
rays, does not, by passing through a foot of free atmospheric
air near the earth, lose an hundred-thousandth part of its
force. From this we may judge how few rays are weak-
ened, or can act upon this fluid, in their passage through a
stratum of the diameter, not of an inch or line, but of a
particle ; yet the subtilty or transparency are still greater
at greatet heights, as was obvious in the Cordilleras when
we looked at distant objects. Lastly, the grosser air is
heated below by the contact or neighbourhood of bodies of
greater density than itself, which it surrounds and on which
it rests; and the heat may be communicated by small de-
grees to a certain distance. The inferior part of the at-
mosphere by this means contracts daily a very consider-
able degree of heat, and may receive it in proportion to its
density or bulk; but it is evident that the same thing can-
not happen at the distance of a league and a half or two
leagues above the surface of the earth, though the light
may be there something more active. The air and wind,
therefore, at this height, must be extremely cold, and colder
in proportion to its elevation."

Saussure adopts this theory, and adds the following fact,
to prove that the force of the sun's rays, considered abs-
tractedly and independently of any extrinsic source of
cold, is no less powerful on mountains than on plains, vizr
that the power of burning lenses and mirrors is the same
at all heights. For ascertaining this fact, he procured
a burning glass so weak in its effect, that at Geneva it
would just set fire to tinder. This glass -was carried to
the summit of Mount Saleve, 3000 feet high, and it there
produced the same effect, and with greater ease. Hence
he concluded that the principal source of cold on the tops
of mountains is their being perpetually surrounded by an*

ATM

[" 182. •]

ATM

atmosphere which cannot be much heated by the rays of to the works containing them ; viz, the 76th vol. of the
the sun, on account of its transparency, or by their reflec- Phil. Trans, for Mr. James Six's, and the same Trans, for
lion from the earth, by reason of its distance; but he 17S0 and 1781, for Dr. Wilson's: also see Rees's new edit;

wished also to know whether the direct solar rays had the
same power on the top of a high mountain as on the plain
below, while the body on which they acted was placed
in such a manner as to be unaffected by the surrounding
air. With this view he instituted a set of experiments,
from which he deduced the following conclusions, viz,
that a difference of 777 toises in height, diminishes the
heat which the. rays of the sun are able to communicate
to a body exposed to the external air, 14° of the thermo-
meter; that it diminishes the heat of a body partially ex-
posed only 6°; and that it augments by 1° the heat of a
third body completely defended from the air. Hence it
appears, that the atmosphere counteracts the operations
of the solar rays in producing heat, by 2. power which is
exerted at all distances, from the surface to the higher
regions.

Dr. Darwin, from his experiments and observations, in-
fers, that there is good reason for concluding that in all
circumstances where air is mechanically expanded, it be-
comes capable of attracting the fluid matter of heat from
other bodies in contact with it. Now, as the vast region
of air which surrounds our globe is perpetually moving
along its surface, rising up the sides of mountains, and
descending into the valleys ; as it passes along, it must be
perpetually varying the degree of heat, according to the
elevation of the country it traverses: for, in rising to the
summits of mountains, it becomes expanded, having so
much of the pressure of the superincumbent atmosphere
taken away; and when thus expanded, it attracts or ab-
sorbs heat from the mountains in contiguity with it; and,
when it descends into the valleys, and is compressed into
less compass, it again gives out the heat it has acquired to
the bodies it comes ill contact with. The same thing must
happen to the superior regions of the atmosphere, which
are regions of perpetual frost, as has lately been evinced by
the aerial navigators. When large districts of a^r, from
the lower parts of the atmosphere, arc raised two or three
miles high, they become so much expanded by the great
diminution of the pressure over them, and thence become
so cold, that hail or snow is produced by the precipitation
of the vapour: and as there is, in these high regions of the
atmosphere, nothing else for the expanded air to acquire
heat from after it has parted with its vapour, the same de-
gree of cold continues, till the air, on descending to the
earth, acquires its former, state of condensation and of
warmth. The Andes, almost under the line, rests its base
on burning sands: about its middle height is a most plea-
sant and temperate climate covering an extensive plain, on
which is built the city of Quito; while its forehead is en-
circled with eternal snow, perhaps coeval with the moun-
tain., Yet, according to the accounts of Don Ulloa, these
three discordant climates seldom encroach much on each
other's territories. The hot winds below, if they ascend,
become cooled by their expansion; and hence they cannot
affect the snow upon the summit; and the cold winds that
sweep the summit become condensed as they descend, and
of a moderate temperature before they reach the fertile
plains of Quito.

Many oilier experiments to this purpose were also made
by M. Pictet of Geneva, Dr. Wilson of Glasgow, and Mr.
James Six; but as our limits will not permit us to give ob-
servations of this kind in detail, we beg to refer our readers

of his Cyclopedia, art. Atmosphere.

Height and Density of the Atmosphere. Various at-
tempts have been made to ascertain the height to which
the atmosphere extends above the earth. These com-
menced soon after it was discovered, by means of the Tor-
ricellian tube, that air is endued with weight and pressure.
And had not the air an elastic power, but were every where
of the same density, from the surface of the earth to the
extreme limit of the atmosphere, like water, which is
equally dense at all depths, it would be a very easy mat-
ter to determine its height from its density and the column
of mercury which it would counterbalance in the barome-
trical tube : for, it having been observed that the weight
of the atmosphere is equivalent to a column of about 30
inches or 2i feet of quicksilver, and the density of the for-
mer to that of the latter, nearly as 1 to 11040; therefore
the height of the uniform atmosphere would be 11040 times
2{ feet, that is 27600 feet, or little more than 5 miles and
a quarter. But the air, by its elastic quality, expands and
contracts; and it being found by repeated experiments in
most nations of Europe, that the spaces it occupies, when
compressed by different weights, are reciprocally propor-
tional to those weights; or, that the more air is pressed,
so much the less space it takes up; it follows that the air
in the upper regions of the atmosphere must become con-
tinually more and more rare, as the height of it increases,
and thus, according to that law, must necessarily be ex-
tended to an indefinite height. Now, if we suppose the
height of the whole to be divided into innumerable equal
parts., the quantity of each part will be as its density; and
the weight of the whole incumbent atmosphere being alsb
as its density; it follows, that the weight of the incum-
bent air is every where as the quantity contained in the
subjacent part; which causes a difference between the
weights of each two contiguous strata of air. But, by a
theorem in the doctrine of proportion, when a magnitude is
continually diminished by the like part of itself, and the
remainders the same, they will express a series of continued
quantities decreasing in geometrical progression : therefore
if, according to the supposition, the altitude of the air, by
the addition of new parts into which it is divided, do con-
tinually increase in arithmetical progression, its density will
be diminished, or, which is the same thing, its gravity will
be decreased, in continued geometrical progression. And
hence, again, it appears that, according to the hypothesis
of the density being always proportional to the compressing
force, the height of the atmosphere must necessarily be ex-
tended indefinitely. And, farther, as an arithmetical scries
adapted to a geometrical one, is analogous to the logarithms
of the said geometrical one; it follows therefore. that the
altitudes are proportional to the logarithms of the densities
or weights of air; and that any height taken from the earth's
surface, which is the difference of two altitudes to the top
of the atmosphere, is proportional to the difference of the
logarithms of the two densities there, or to the logarithm
of the ratio of those densities, or their corresponding com-
pressing forces, as measured by the two heights of the ba-
rometer there. This law was first observed and demon-
strated by Dr. Ilalley, from the nature of the hyperbola;
and afterwards by Dr. .lames Gregory, by means of the lo-
garithmic curve. See. Philos. Trans. N" 1 81, or my Abridg.
vol. 3, p. 300, and Greg. Astron. lib. v, prop. 3.

A T M

[ 183 ]

A T M

It is now easy, from the foregoing property, and two or
three experiments, or barometrical observations, made at
known altitudes, to deduce a general rule to determine the
absolute height answering to any density, or the density an-
swering to any given altitude above the earth. And ac-
cordingly, calculations have been made upon this plan by
many philosophers, particularly by the French ; but it ha-
ving been found that the barometrical observations did not
correspond with the altitudes a9 measured geometrically,
it was suspected that the upper parts of the atmospherical
regions were not subject to the same laws with the lower
ones, in regard to the density and elasticity. And indeed,
when it is considered that the atmosphere is a heteroge-
neous mass of particles of all sorts of matter, some elastic,
and others not, it is not improbable but this may be the
case, at least in the regions very high in the atmosphere,
which it is likely may more copiously abound with the elec-
trical fluid. Be this however as it may, it has lately been
discovered that the law above given, holds very well tor all
such altitudes as are within our reach, or as far as to the
tops of the highest mountains on the earth, when a correc-
tion is made for the difference of the heat or temperature
of the air only; as was fully evinced by M. de Luc, in a
long series of observations, in which he determined the al-
titudes of hills, both by the barometer and by geometrical
measurement, from which he deduced a practical rule for
determining the allowance to be made for the difference of
temperature. See his Treatise on the Modifications of the
Atmosphere. Similar rules have also been deduced from
accurate experiments, by Sir George Shuckburgh and Ge-
neral Roy, both concurring to show, that such a rule for
the altitudes and densities holds true for all heights that
are accessible to us, when the elasticity of the air is cor-
rected on account of its density : and the result of their ex-
periments showed that the difference of the logarithms of
the heights of the mercury in the barometer, at two sta-
tions, when multiplied by 10000, is equal to the altitude
in English fathoms, of the one place above the other; that
is, when the temperature of the air is about 31 or 32 de-
grees of Fahrenheit's thermometer; and a certain quantity
more or less, according as the actual temperature is diffe-
rent from that degree.

But it may here be shown that the same rule can be de-
duced independent of such a train of experiments as those
above, and that from a single observation of the density of
the air at the surface of the earth alone. Thus, let d de-
note the density of the air at one place, and d the density
at the other; both measured by the column of mercury
in the barometrical tube: then the difference of altitude
between the two places, will be proportional to the log. of

D — the log. of d, or to the log. of -. But as this formula
° ° a

expresses only the relation between different altitudes, and
not the absolute quantity of them, assume some indetermi-
nate but constant quantity h, which multiplied into the

expression log. -, may be equal to the real difference of
altitude a, that is a = h x log. of -. Then, to determine

° d

the value of the general quantity h, let us take a case in
which we know the altitude a which corresponds to a known
density d; as for instance, a = 1 foot, or 1 inch, or some
such small altitude: then because the density D may be
measured by the pressure of the whole atmosphere, or the
uniform column of 27600 feet, when the temperature is

55°; therefore 27600 feet will denote the density d at the

lower place, and 27599 the less density d at 1 foot above

, , , , r 27600 , . , . ,

it; consequently 1 = fi x log. ot — - — , which is nearly

, M3429448 1 ,- , . , . ,

= h x or by the nature ot logarithms; and

27600 63551 J °

hence we find h = 63551 feet; which gives us this formula
for any altitude a in general, viz, a = 63551 x lo^. of —

o o d J

or« = 63551 x log. of - feet, or 10592 x log. of - fa-
thoms; where m denotes the column of mercury in the
tube at the lower place, and m that at the upper. This
formula is adapted to the mean temperature of the air 55°:
but it has been found, by the experiments of Sir Geo.
Shuckburgh and General Roy, that for every degree of the
thermometer, different from 55°, the altitude a will vary
by its 435th part; hence, if we would change the factor A
from 10592 to 10000, because the difference 592 is nearly
the 18th part of the whole factor 10592, or sufficiently so
for all practical purposes, and 18 the 24th part of 435 ;
therefore the change of temperature, answering to the
change of the factor />, is 24°, which -reduces the 55° to

31°. So that, a = 10000 x log. of - fathoms, is the ea-
0 m

siest expression for the altitude, and answers to the tempe-
rature of 31°, or very nearly the freezing point: and for
every degree above that, the result must be increased by so
many times its 435th part, and diminished when below it.
From this theorem it follows, that, at the height of 3f
miles, the density of the atmosphere is nearly 2 times rarer
than it is at the surface of the earth ; at the height of 7
miles, 4 times rarer; and so on, according to the following
table :

Height in miles. Number of times rarer.

U - - - - 2

I
14
21
28
35
•42
49
56
63
70

4

16

6*

256

- 1024
4096

- 16384

- 65536
262144

1048576

And, by pursuing the calculations in this table, it might
be easily shown, that a cubic inch of the air we breathe
would be so much rarefied at the height of 500 miles, that
it would fill a sphere equal in diameter to the orbit of Sa-
turn.

.Hence we may perceive how very soon the air becomes
so extremely rare and light, as to be utterly imperceptible
to all experience ; and that hence, if all the planets have
such atmospheres as our earth, they will, at the distances
of the planets from one another, be so extremely attenuated,
as to give no sensible resistance to the planets in their mo-
tion round the sun, for many, perhaps hundreds or thou-
sands of ages to-come. Even at the height of 50 miles, it
is so rare as to have no sensible effect on the rays of light:
for it was found by Kepler, and Lahire after him, who
computed the height of the sensible atmosphere from the
duration of twilight, and from the magnitude of the ter-
restrial shadow in lunar eclipses, that the effect of the at-
mosphere to reflect and intercept the light of the sun, is-

A T M

[ 184 ]

A T M

only sensible to the altitude of between 40 and 50 miles:
and at that altitude we may collect, from what has been
already said, that the air is above 10000 times rarer than
at the surface of the earth. It is well known that the twi-
light begins and ends when the centre of the sun is about
18 degrees below the horizon, or only 17° 27', by subtract-
ing 33' for refraction, which raises the sun so much higher
than he would be. And a ray coming from the sun in that
position, and entering the earth's atmosphere, is refracted
and bent into a curve line in passing through it to the eye.
M.Lahire took great pains to demonstrate, that, supposing
the density of the atmosphere proportional to its weight,
this curve is a cycloid ; he also says, that if the ray be a
tanoent to the atmosphere, at the point of impact, the di-
ameter of its generating circle will be the height of the at-
mosphere; and that this diameter increases, till at Mast,
when the rays are perpendicular, it becomes infinite, or the
circle degenerates into a right line, This reasoning sup-
poses that the refracting surface of the atmosphere is a
plane; but since it is in reality a curve, he observes that
these cycloids become in fact epicycloids. Herman how-
ever detected the error of M. Lahire, and showed that this
curve is infinitely extended, and has an asymptote. And
it is observed by Dr. Brook Taylor, in his Method. Increm.
pa. l68,'&c, that this curve is one of the most intricate
and perplexed that can well be proposed. The same in-
genious author also computes, that the refractive power of
the air is to the force of gravity at the surface of the. earth,
as 320 millions to 1.

The extreme rarity of the atmosphere at considerable
distances from the earth's surface, such as those of 40 or
50 miles, bounding the production of twilight, has much
perplexed philosophers in accounting for meteors which,
whatever be their origin, whether electrical or otherwise,
are observed at a much greater elevation than that to
which the refractive power of the atmosphere extends. A
very remarkable one of this kind was observed by Dr.
Halley in the month of March 1719, the altitude of which
he computed at between 69 and 73J English miles ; its di-
ameter 2800 yards, or more than a mile and a half; and
its velocity about 350 miles in a minute.- Others, ap-
parently of the same kind, but whose altitude and velocity
were still greater, have been observed ; particularly that
very remarkable one, of August 18th 17S3, whose distance
from the earth could not be less than QO miles, and its di-
ameter at least, as large as the former, while its velocity
was certainly not less than 1000 miles in a minute. Now,
from analogy of reasoning, it seems very probable, that
the meteors which appear at such great heights in the air,
are not essentially different from those which are seen on
or near the surface of the earth. The difficulty with re-
"ard to the former is, that at the great heights above-men-
tioned, the atmosphere ought not, according to computa-
tion, to have any density sufficient to support flame, or to
propagate sound ; and yet such meteors are commonly
succeeded by one explosion or more, and it is said are
even sometimes accompanied with a hissing noise as they
pass over our heads. The meteor of 1710 was not only
very bright, seeming for a short time to turn night into
day, but was attended with an explosion that was heard
over the whole island of Britain, causing a violent con-
cussion in the atmosphere, and seeming to shake the earth
itself: and yet, in the regions in which this meteor moved,
the air ought to have been 300 thousand times rarer than
the air wc breathe, or 1000 times rarer than the vacuum

commonly made by a good air-pump. Dr. Halley offers
a conjecture, indeed, that the vast magnitude of such bo-
dies might compensate for the thinness of the medium in
which they moved. But appearances of this kind are, by
others, attributed to electricity; though the circumstances
of them cannot well be reconciled to that cause ; for the
meteors move with various degrees of velocity; and though
the electrical fire easily pervades the vacuum of an air-
pump, yet it does not in that case appear in bright well-
defined sparks, as in the open air, but rather in lon<^
streams resembling the aurora borealis; and from some
late experiments it has been concluded that the electric
fluid cannot even penetrate, or be apparent in a perfect
vacuum.

Of the Refractive and Reflective Power of the Atmo-
sphere. It has been observed above, that the atmosphere
has a refractive power, by which the raj-s of light are bent
from the right-lined direction, as in the case of the twi-
light; and many other experiments manifest the same vir-
tue, which is the cause of various phenomena. Alhazen,
the Arabian, who lived about the year I 100, seems to have
been more inquisitive into the nature of refraction than for-
mer writers. But neither Alhazen, nor his follower Vitellio
were acquainted with its just quantity, nor was it known
to any tolerable degree of exactness, before Tycho Brahe,
with his accustomed diligence, settled this point. This
celebrated philosopher however did not discover, nor in-
deed did Kepler, in what manner the rays of light were
refracted by the atmosphere. Tycho thought the refrac-
tion was chiefly caused by dense vapours, very near the
earth's surface: while Kepler placed the cause wholly at
the. top of the atmosphere, which he thought was uniformly
dense ; and thence he determined its altitude to he little
more than that of the highest mountains. But the true-
constitution of the density of the atmosphere, deduced af-
terwards from theTorricellian experiment, afforded ajuster
idea of these refractions, especially after it appeared, by
a repetition of Mr. Lowthorp's experiment, that the re-
fractive power of the air is proportional to its density. By
this variation in the density and refractive power of the air,
a ray of light, in passing through the atmosphere, is con-
tinually refracted at every point, and thereby made to die-
scribe a curve, and not a straight line, as it would have done
were there no atmosphere, or were its density uniform.

The atmosphere has also a reflective power; and this
power is the means by which objects are enlightened so
uniformly on all sides. The want of this power would
occasion a strange alteration in the appearance of things ;
the shadows of which would be so very dark, and their
sides enlightened by the sun so very bright, that probably
we could see no more of them than their bright halves ;
so that for a view of the other halves, wc must turn them
half round; or if immoveable, must wait till the sun could
come round upon them. Such a pellucid unretlective at-
mosphere would indeed have been very commodious for
astronomical observations on the course of the. sun and
planets among the fixed stars, visible by day as well as by
night; but then such a sudden transition from darkness
to light, and from light to darkness, immediately on the
rising and setting of the sun, without any twilight, and
even on turning to or from the sun at noon day, would
have been very inconvenient and offensive to our eyes.
However, though the atmosphere be greatly assistant in
the illumination of objects, yet it must also be observed
that it stops a great deal of light. By M. Bouguer's ex-

ATM

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ATM

periments, it seems that the light of the moon is often 2000
times weaker in the horizon, than at the altitude of 66 de-
grees ; and that the proportion of her light at the altitudes
of 66 and 19 degrees, is about 3 to 2 ; and the lights of
the sun must bear the same proportion to each other at
those heights; which Bouguer made choice of, as being
the meridian heights of the sun, at the summer and winter
solstices, in the latitude of Croisic in France. Smith's
Optics, Rem. 95. — For the atmosphere of the sun, moon,
and planets, see the respective articles.

Atmosphere of Solid or Consistent Bodies, is a kind of
sphere formed by the effluvia, or minute corpuscles, emit-
ted from them. Mr. Boyle endeavours to show, that all
bodies, even the hardest and most coherent, as gems, &c,
have their atmospheres.

Atmosphere, in Electricity, denotes that medium
which is conceived to be diffused over the surface, as is
generally supposed, of electrified bodies, and to some di-
stance around them, and consisting of effluvia issuing from
them ; by which, other bodies immerged in it become en-
dued with an electricity contrary to that of the body to
which the atmosphere belongs. This was first noticed at
a very early period in the history of this science by Otto
Guericke, and afterwards by the academicians del Cimento,
■who contrived to render the electric atmosphere visible, by
means of smoke attracted by and uniting itself with a piece
of amber, and gently rising from it,, but vanishing as the
amber cooled. Dr. Franklin exhibited this electric atmo-
sphere with greater advantage, by dropping rosin on hot
iron plates held under electrified bodies, from which the
smoke arose and encompassed the bodies, giving them a
very beautiful appearance. But the theory of electric at-
mospheres was not well explained and understood for a
considerable time ; and the investigation led to many cu-
rious experiments and observations. The experiments of
Mr. Canton and Dr. Franklin prepared the way for the
conclusion that was afterwards drawn from them by Mess.
Wilcke and Epinus, though they retained the common opi-
nion of electric atmospheres, and endeavoured to explain
the phenomena by that means. The conclusion was, that
the electric fluid, when there is a redundancy of it in any
body, repels the electric fluid in any other body, when they
are brought within the sphere of each other's influence,
and drives it into the remote parts of the body, or quite out
of it, if there be any outlet for that purpose.

By atmosphere, Epinus says, no more is to be under-
stood than the sphere of action belonging to any body, or
the neighbouring air electrified by it. Sig. Beccaria unites
in the same opinion, that electrified bodies have no other
atmosphere than the electricity communicated to the neigh-
bouring air, and which goes with the air, and not with the
electrified bodies. Mr. Canton also, having relinquished
the opinion that electrical atmospheres were composed of
elfluvia from excited or electrified bodies, maintained that
they only result from an alteration in the state of the elec-
tric fluid contained in it, or belonging to the air surround-
ing these bodies to a certain distance ; for instance, that
excited glass repels the electric fluid from it, and conse-
quently beyond that distance makes it more dense ; where-
as excited wax attracts the electric fluid existing in the air
nearer to it, making it rarer than it was before. In the
course of experiments that were performed on this occa-
sion, Mess. Wilcke and Epinus succeeded in charging a
plate of air, by suspending large boards of wood covered
with tin, with the flat sides parallel to one another, and at

Vol. I.

some inches asunder: for they found, on electrifying
one of the boards positively, that the other was always ne-
gative ; and a shock was produced by forming a communi-
cation between the upper and lower plates. Beccaria has
largely considered the subject of electric atmospheres, in
his Artificial Electricity, p. 179 &e, Eng. edit. See also
Dr. Priestley's Hist, of Electricity, vol.ii. sect. 5; and Ca-
vallo's Electricity, p. 241.

Atmosphere, Magnetic, Sfc, is understood of thesphere
within which the virtue of the magnet, &c, acts.

Atmospheric Stones. See Aeroliths.

Atmospheric Tides, certain periodical changes of
the atmosphere, similar, in some respects, to those of the,
oCean, and produced, in a great measure, by the same
causes.

There are two kinds of motion in the atmosphere which
come under this denomination ; the first is occasioned by
the joint influence of the sun and moon on the, body of air
with which we are surrounded, in the same manner as they
act upon the mass of water, and cause the flux and reflux
of the ocean ; the second is produced by the heat of the
sun alone, which exerting upon the air its well known
power of rarefaction and expansion, gives rise to those
perpetual changes in the atmosphere, which follow him in
his course, though such changes are not altogether ob-
servable or appreciable by us. The former of these are
termed by the Abbe Mann attraction tides, and the latter
heat tides. The tides of attraction, like those of the ocean,
and from the like grounds, have at the.same time, at two
opposite ends of the globe, projecting parts, and these lie
almost in that line which might be drawn from the centre
of the earth to that of the moon. The heat tides, on the
other hand, can take place only on one point of the globe,
that is, in the point to which the sun is vertical. Their
projecting parts will be directed towards that luminary,
and nearly follow its movements.

Bacon, and the most eminent philosophers who have
since written on the wind, unanimously observe, that the
periods of the year most exposed to it are the two equi-
noxes ; that storms are most frequent at (fie times of new
and full moon, anil particularly those which happen near
the equinoxes; that at periods otherwise calm, a small
breeze always takes place atliigh water; and that a small
movement in the atmosphere is each time perceived a little
after noon and midnight.

For the investigation of this subject, the abbe lays down
the following principles. 1. The elasticity of fluids is in
the inverse ratio of their density. 2. The force or elasti-
city of the air expands and contracts itself in the direct ratio
of the weight with which it is loaded, and diffuses itself in
the inverse ratio of the force with which it is compressed.
3. The air is rarefied, or diffuses itself, in the direct ratio of
the quantity of heat which acts upon it. 4. The air, as
well as all fluids in general, has a tendency to put itself in
equilibrium, and does not rest till it has obtained it.

Our limits preclude the possibility of following the
abbe in the application of these principles to the various
phenomena of the subject in question; we therefore ie-
commend to the attention of our readers the whole of his
observations on the flux and reflux of the atmosphere, in-
serted from the Brussels Trans, in Tilloch's Phil. Mag.
vol. 5. In the 7th vol. of that work is a paper On a pe-
riodical Variation of the Barometer, &c, by LukeJIoward,
esq. to which we also refer; and conclude by noticing the
observations made by Uumbolt near the equatov : " I have

2B

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[. 186 ]

ATT

lead," says he, "in theTransactions of the Bengal Society,
that the barometer rises and falls there regularly every 24
hours. Here, in South America, its motion is more asto-
nishing: there are four atmospherical tides every 24 hours,
which depend only on the attraction of the sun ; the mer-
cury falls from 9 o'clock in the morning till 4 in the after-
noon ; it rises from 4 till 11 o'clock ; it falls from 11 till
half past 4 in the morning; and reascends from that time
till 9 o'clock; neither winds, storms, nor earthquakes have
any influence on this motion." See farther Laplace, Ex-
position du Systeme du Monde, liv. 4, ch.12.

ATOM, a particle of matter indivisible on account of
its solidity, hardness, and impenetrability; which preclude
all division, and leave no vacancy for the admission of any
foreign force to separate it into parts. As atoms are the
first matter, it is necessary they should be indissolvable,
that they may be incorruptible. Newton adds, It is also
required that they be immutable, that the world may con-
tinue in the same state, and bodies be of the same nature
now as formerly.

ATOMICAL Philosophy, or the doctrine of atoms,
a system which accounted for the origin and formation of
things, from the hypothesis that atoms are endued with
weight and motion. This philosophy was first taught by
Moschus, some time before the Trojan war ; but it was
most cultivated by Epicurus; whence it is called the Epi-
curean philosophy.

ATTRACTION, or Attractive Power, a general
term used 'to denote the cause, power, or principle, by
which all bodies mutually tend towards each other, and
cohere, till separated by some other power. The laws,
phenomena, &c, of attraction, form the chief subject of
Newton's philosophy, being the principal agent of nature,
in almost all her wonderful operations.

The principle of attraction, in the Newtonian sense of
it, appears to have been first surmised by Copernicus.
" As for gravity," says he, " I consider it as nothing more
than a certain natural appetence (appetcnlia) which the
Creator has impressed upon all the parts of matter, in or-
der to their uniting or coalescing into a globular form, for
their better preservation ; and it is probable that the same
power is also inherent in the sun and moon, and planets,
that those bodies may constantly retain that round form
in which we see them." De Revol. Oib. Ccelest. lib. i, cap. 9-
Kepler calls gravity a corporeal and mutual affection be-
tween similar bodies, in order to their union. Ast. Nov. in
Introd. And he pronounced more positively, that no bo-
dies whatever were absolutely light, but only relatively so;
and consequently that all matter was subjected to the
power and law of gravitation. Ibid,

The first in this country who adopted the notion of at-
traction was Dr. Gilbert, in his book De Magnete ; and
the next was the celebrated Lord Bacon, in his Nov. Or-
gan, lib. ii, aphor. 30, 45, 48. fylv. cent, i, e.xp. 33; also
in his treatise De Miotic, particularly under the articles of
the 9th and the 13th kinds of Motion. In France it was
received by Fermat and Roberval; and in Italy by Galileo
and Borelli.

But before Newton none appear to have had clearer
ideas of the doctrine of universal gravitation, or to have
approached so near the application of it to the laws of na-
ture, as Dr. Ilooke. This celebrated man, in his work
entitled " An Attempt to prove the Motion of the Earth,"
seems to have embraced it in nearly the whole of its ge-
nerality. The hypothesis on which the system of the world

is there explained, is grounded on the three following piin-
ciples : 1. That all the celestial bodies have not only an
attraction or gravitation towards their propel centres, but
that they mutually attract each other within their sphere
of activity. 2. That all bodies which have a simple and
direct motion, continue to move in a direct line, if some
force which operates without ceasing does not constrain
them to describe a circle, an ellipse, or some other more
complicated curve. 3. That attraction is so much the
more powerful, as the attracting bodies are nearer to each
other. Newton makes use of the word attraction, in com-
mon with the school philosophers, but at the same time
very studiously distinguishes between the ideas. The an-
cient attraction was conceived to be a kind of quality in-
herent in certain bodies themselves, and arising from their
particular or specific forms. But the Newtonian attrac-
tion is a more indefinite principle ; denoting not any par-
ticular kind or mode of action ; nor the physical cause of
such action; but only a general tendency, a conatus ac-
cedendi, to whatever cause, physical or metaphysical, such
effect be owing; whether to a power inherent in the bo-
dies themselves, or to the impulse of an external agent.
Accordingly, that author remarks, in his Philos. Nat. Prin.
Math. " that he uses the words attraction, impulse, and
propension to the centre, indifferently; and cautions the
reader not to imagine that by attraction he expresses the
modus of the action, or its efficient cause, as if there were
any proper powers in the centres, which in reality are only
mathematical points ; or as if centres could attract." Lib.
1, p. 5. So, he " considers centripetal powers as attrac-
tions; though, physically speaking, it were perhaps more
just to call them impulses, lb. p. 14". He adds, "that
what he calls attraction may possibly be effected by
impulse, though not a common or corporeal impulse,
or after some other manner unknown to us." Optics,
p. 322.

Attraction, if considered as a quality arising from the
specific forms of bodies, ought, together with sympathy,
antipathy, and the whole tribe of occult qualities, to be
exploded. But when these are set aside, there will remain
innumerable phenomena of nature, and particularly the
gravity or weight of bodies, or their tendency to a centre,
that argue a principle of action seemingly distinct from
impulse; where, at least, there is no sensible impulsion
concerned. Nay, what is more, this action, in some re-
spects, differs from all impulsion we know of; impulse
being always found to act in proportion to the surfaces of
bodies ; whereas gravity acts according to their solid con-
tent, and consequently it must arise from some cause that
penetrates or pervades the whole substance of them. This
unknown principle, (we mean in respect of its cause, for
its phenomena and effects are most obvious, with all its
species and modifications,) is called attraction ; being a
general name, under which may be ranged all mutual ten-'
dencies, where no physical impulse appears, and which
consequently cannot be accounted for on any known
laws of nature.

And hence arise divers particular kinds of attraction ;
as Gravity, Magnetism, Electricity, &c, which are so many
different principles, acting by different laws; and only
agreeing in this, that we do not perceive any physical
causes of them : but that, as to our senses, they may really
arise from some power or efficacy m such bodies, by which
they are enabled to act even upon distant bodies; though
our reason absolutely disallows of any such action.

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Attraction may be divided, with respect to the law it
observes, into two kinds.

1. That which extends to a sensible distance. As the
attraction of gravity, which is found in all bodies; and the
attractions of magnetism and electricity, found only in
particular bodies. The several laws and phenomena of
each, see under their respective articles. '

The attraction of gravity, called also among mathema-
ticians the centripetal force, is one of the greatest and most
universal principles in nature. We see and feel it operate
on bodies near the earth, and find by observation that the
same power (i. e. a power which acts in the same manner,
and by the same rules, viz, always proportionally to the
quantities of matter, and inversely as the squares of the di-
stances) does also obtain in the moon, and the other planets,
both primary and secondary, as well as in the comets; and
■even that this is the very power by which they are all re-
tained in their orbits, &c. And hence, as gravity is found
in all the bodies which come under our observation, it is
easily inferred, by one of the established rules of philoso-
phising, that it obtains in all others. And since it is found
to be proportional to the quantity of matter in any body, it
must exist in every particle of it: and hence it is proved
that every particle in nature attracts every other particle.

From this attraction arises all the motion, and conse-
quently all the mutation, in the universe. By this heavy
bodies descend, and light ones are made to ascend : by this
also projectiles are directed, vapours and exhalations rise,
and rain, &c, fall : also from the same cause rivers glide,
the ocean swells, the air presses, &c. In short, the motions
and forces arisingfrom this principle, constitute the subject of
that extensive branch of mathematics, called mechanics or
statics, with the parts or appendages of it, as hydrostatics,
pneumatics, hydraulics, &c.

2. That which does not extend to sensible distances.
Such is found to obtain in the minute particles of which
bodies are composed, attracting each other at or extremely
near the point of contact, with forces often much supe-
rior to that of gravity, but which at any distance decrease
much faster than the power of gravity. This power is ge-
nerally called the attraction of cohesion, as being that by
which the atoms or insensible particles of bodies are united
into sensible masses.

This latter kind of attraction claims Sir Isaac Newton
for its discoverer ; as the former does for its improver. The
laws of motion, percussion, &c, in sensible bodies, under
various circumstances, as falling, projected, &c, ascertain-
ed by the. later philosophers, do not reach those more re-
cluse, intestine motions in the component particles of the
same bodies, on which the changes of the texture, colour,
properties, &c, depend. So that our philosophy, if it were
only founded on the principle of gravitation, and even car-
ried as far as this would lead us, would still be deficient.

But besides the common laws of sensible masses, the mi-
nute parts they are composed of are found subject to some
others, which have but lately been noticed, and are even
yet imperfectly known. Newton himself; to whose happy
penetration we owe the hint, is satisfied with establishing
that there are such motions in the minima naturae, and that
they flow from certain powers or forces, not reducible to
any of those in the great world. He shows that, by virtue
of these powers, " the small particles act on one another
even at a distance ; and that many of the phenomena of
nature result from this action. Sensible bodies, we have
already observed, act on one another several ways; and as

we thus perceive the tenor and course of nature, it appears
highly probable that there may be other powers of the like
kind ; nature being always uniform and consistent with
herself. Those just mentioned, reach to sensible distances,
and b3r that means have been observed by vulgar eyes; but
there may be others which reach to such small distances
as have hitherto escaped observation; and it is probable
electricity may reach to such distances, even without be-
ing excited by friction."

The great author just mentioned proceeds to confirm
the reality of these suspicions by a great number of phe-
nomena and experiments, which plainly argue such powers
and actions between the particles, for example of salts and
water, oil of vitriol and water, aquafortis and iron, spirit
of vitriol and saltpetre, and most other chemical opera-
tions. He also shows, that these powers, &c, are unequally
strong between different bodies; stronger, for instance, be-
tween the particles of salt of tartar and those of aqua-
fortis than those of silver, between aquafortis and lapis
calaminaris than iron, between iron and copper
than silver or mercury. Thus spirit of vitriol acts on
water, but more on iron or copper, &c. And the other
experiments are innumerable which countenance the ex-
istence of such principle of attraction in the particles of
matter.

These actions, by virtue of which the particles of the
bodies above-mentioned tend towards each other, the au-
thor calls by a general indefinite name, attraction ; a name
equally applicable to all actions by which distant bodies
tend towards each other, whether by impulse, or by any
other more latent power : and hence he accounts for an
infinity of phenomena which would be otherwise inexpli-
cable from the principle of gravity.

"Thus," adds Newton, " will nature be found very con-
formable to herself, and very simple; performing all the
great motions of the heavenly bodies by the attraction of
gravity, which intercedes those bodies, and almost all the
small ones and their constituent parts, by some other at-
tractive power diffused through their particles. Without
such principles, there could have been no such thing as
motion ; and without the continuanceof it, motion would
soon perish, there being otherwise a great decrease or di-
minution of it, which is only supplied by these active prin-
ciples."

It need not be said how unjust it was, in the generality
of foreign philosophers, to declare against a principle which
furnishes so beautiful a view, for no other reason but bo
cause they could not conceive how one body should act
on another at a distance. It is indeed true, that philoso-
phy allows of no action but what is by immediate contact
and impulsion ; for how can a body exert any active power
where it does not exist ? Yet we see effects, without per-
ceiving any such impulse ; and where effects are observed,
there must exist causes, whether we perceive them or not.
But we may contemplate such effects, without entering into
the consideration of the causes, as indeed it seems the bu-
siness of a philosopher to do : for to exclude a number of
phenomena which we see, would be to leave a great chasm
in the history of nature; and to argue about actions which
we do not see, would be to raise castles in the air. It fol-
lows therefore, that the phenomena of attraction are mat-
ter of physical consideration, and as such entitled to a,
share in the system of physics; but that their causes will
only become so when they become sensible, that is when
they appear to be the effect of some other higher causes ;
2 B 2

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for a cause is no otherwise seen than as itself is an effect,
so that the first cause must needs be always invisible ; we
are therefore at liberty to suppose the causes of attractions
what we please, without any injury to the effects. The il-
lustrious author himself seems to be a little indetermincd
as to the causes; inclining sometimes to attribute gravity
to the action of an immaterial cause (Optics, p. 34-3, &c.)
and sometimes to that of a material one, lb. p. 3'25.

In his philosophy, the research into causes is the last
thing, and never comes under consideration till the laws
and phenomena of the effect be settled; it being to these
phenomena that the cause is to be accommodated. The
cause even of any, the grossest and most sensible action, is
not adequately known. How impulse or percussion itself
produces its effects, that is, how motion is communicated
from one body to another, or by what agency it is conti-
nued after it is communicated, confounds the deepest phi-
losophers. Yet impulse is received not only into philoso-
phy, but into mathematics ; and accordingly the laws and
phenomena of its effects make the chief part of common
mechanics.

The other species of attraction, therefore, in which no
impulse is observable, when their phenomena are suffi-
ciently ascertained, have the same title to be promoted
from physical to mathematical consideration ; and this
without any previous inquiry into their causes, to which
our conceptions may not be proportionate.

Our great philosopher, then, far from adulterating sci-
ence with any thing foreign or metaphysical, as many have
reproached him with doing, has the glory of having thrown
every thing of this kind out of his system, and of having
opened a new source of more sublime mechanics, which,
duly cultivated, might be of far greater extent than all the
mechanics yet known : it is hence therefore that we must
expect to learn the manner of the changes, productions,
generations, corruptions, &c, of natural things; with all
that scene of wonders opened to us by the operations of
chemistry.

Some of our own countrymen have prosecuted the dis-
covery with laudable zeal. Dr. Keil particularly has en-
deavoured to deduce some of the laws of this new action,
and applied them in resolving several of the more general
phenomena of matter, as cohesion, fluidity, elasticity, soft-
ness, fermentation, coagulation, &c : and Dr. Freind, se-
conding his endeavours, has made a farther application of
the same principles, to account for almost all the pheno-
mena that chemistry presents. So that some philosophers
are inclined to think that the new mechanics should seem
already raised to a complete science, and that nothing now
etui occur but what we have an immediate solution of,
from the principles of attractive forces.

Hut this seems a little too precipitate : a principle so
fertile should have been further explained ; its particular
laws, limits, &c, more industriously detected and laid
down, before we had proceeded to the application. At-
traction in the gross is so complex a thing, that it may
solve a thousand different phenomena alike. The notion
is t j vi t one degree more simple and precise than action it-
s' ll; and, till its properties be more fully ascertained, it
Wi re better to apply it less, and study it more. It may
be added', that some of Newton's followers have been
Charged with falling into that error which he industrious-
K avoided, viz, of considering attraction as a cause or
active property in bodies, not merely as a phenomenon or
, Bei r

Certain attempts have lately been made to assign the-
probable or possible cause of gravitation, but too absurd
to merit more particular notice.

For the laws, properties, &c, of the different kinds of
attraction, see their particular articles, Cohesion, Gra-
vity, Magnetism, &c.

Attraction, Centre of. See Centre of Attraction.
'Attraction of Mountains. This is but a late disco-
very, and must be esteemed a complete confirmation of the
Newtonian theory of universal gravitation. According to
that system, an attractive power is exerted, not only be-
tween those large masses of matter the sun and planets,
but likewise between all comparatively smaller bodies,
and even between the smallest particles that compose them.
Agreeably to this hypothesis, a heavy body, which gravi-
tates or tends toward the centre of the earth, ought like-
wise, though in a less degree, to be attracted and to tend
laterally towards a neighbouring mountain on the earth's
surface. So that, a plumb-line, for instance, of a quad-
rant, hanging near such a mountain, ought to be drawn a
little aside, from the perpendicular direction, in conse-
quence of the attractive power of the quantity of matter,
of which it is composed, acting in a direction different from
that exerted by the whole mass of matter in the earth, and
with a proportionally inferior degree of force.

Though Sir Isaac Newton had long ago hinted at an ex-
periment of this kind, and had remarked, that "a moun-
tain of an hemispherical figure, 3 miles high, and 6 broad,
would not by its attraction draw the plumb-line 2 mi-
nutes out of the perpendicular;" yet no attempt to ascer-
tain this matter by actual experiment was made, till about
the year 173S ; when the French academicians, particu-
larly Messrs. Bouguer and Condamine, who were sent to
Peru to measure a degree of the meridian at the equator,
attempted to discover the attractive power of Cbimboraco,
a mountain in the province of Quito. According to their
observations, made under unfavourable circumstances,
that mountain exerted an attraction equal to S". Though
this experiment was not perhaps sufficient to prove satis-
factorily even the reality of an attraction, much less the
precise quantity of it; yet it does not appear that any
steps had been since taken to repeat it.

In the year 1772, the astronomer royal, Dr. Maskelyne,
made a proposal for this purpose to the Royal Society :
and in 1 774-, he was deputed to make the trial, accom-
panied with proper assistants, and provided with accurate
instruments. The mountain Schehallien, situated nearly
in the centre of Scotland, was chosen as the most proper
then found in this island. The observations were made by
observing the meridian zenith distances of some fixed stars
near the zenith, by means of a zenith sector of 10 feet ra-
dius; first on the south, and then on the north side of the
hill, the greatest length of which was considerably extend-
ed in the east and west direction.

It is plain, that if the mass of matter in the hill should
exert any sensible attraction, it would cause the plumb-
line of the sector, through which the observer viewed a
star in the meridian, to deviate from its perpendicular si-
tuation, and would attract it contrarywise at the two op-
posite stations, thus doubling the effect. On the south side
the plummet would be drawn to the northward, by the
attractive power of the hill placed on the north side of it ;
and on the north side, a contrary and equal deflection of the
plummet would take place, by the like attraction of the
hill, now to the southward of it. The apparent zenith di-

ATT

stances of the stars must be affected contrarywise ; those
being increased at the one station, which were diminished
at the other ; and the correspondent quantities of the de-
flection of the. plumb-line must give the observer the sum
of the contrary attractions of the hill, acting on the plum-
met at the two stations ; and the half of which will of
course indicate the attractive power of the hill.

The various operations requisite for this experiment
lasted' about 4 months ; and from them it appears, that
the sum of the two contrary attractions of the hill, in the
two temporary observatories, which were successively fixed
half way up the hill (where however the effect of the at-
traction would not be the greatest), was found to be 1 1"'6;
the half of which therefore, or 5"'S, is the mean attraction
of the mountain, at that height.

The inferences drawn from these experiments may be
reduced to the following:

1. It appears that the' mountain Schehallien exerts a
sensible attraction ; therefore, by the rules of philosophi-
sing, we are to conclude, that every mountain, and indeed
every particle of the earth, is endued with the same pro-
perty, in proportion to its quantity of matter.

2. The law of the variation of this force, in the in-
verse ratio of the squares of the distances, as laid down
by Sir Isaac Newton, is also confirmed by this experiment.
For, if the force of attraction of the hill, had been only
to that of the earth, as the matter in the hill is to that in
the earth, and had not been greatly increased by the near
approach to its centre, the attraction of it must have been
wholly insensible. But now, by only supposing the mean
density of the earth to be double of that in the hill, which
seems very probable from other considerations, the attrac-
tion of the hill will be reconciled to the general law of the
variation of attraction, in the inverse duplicate ratio of the
distances, as deduced by Sir Isaac Newton from the com-
parison of the motion of the heavenly bodies, with the force
of gravity at the surface of the earth ; and the analogy of
nature will be preserved.

3. We may now therefore be allowed to admit this
law, and to acknowledge, that the mean density of the
earth is about double of that at the surface; and conse-
quently that the density of the internal parts of the earth,
is much greater than near the surface. Hence also, the
whole quantity of matter in the earth, will be at least as
great again, as if it had been all composed of matter of the
same density with that at the surface ; or will be about 4
or 5 times as great as if it were al.l composed of water. This
conclusion is totally contrary to the hypothesis of some
naturalists, who suppose the earth to be only a large hol-
low shell of matter; supporting itself from the property
of an arch, with an immense vacuity in the midst of if.
Rut, were that the case, the attraction of mountains, and
even smaller inequalities in the earth's surface, would be
very great, contrary to experiment, and would affect the
measures of the degrees of the meridian much more than
we find they do; and the variation of gravity, in different
latitudes, in going from the equator to the poles, as found
by pendulum*, would not be near so regular as experiments
have shown it to be.

4. As mountains are, by these experiments, found capa-
ble of producing sensible deflections in the plumb-lines of
astronomical instruments ; it becomes a matter of great
importance, in the mensuration of meridional degrees, ci-
ther to choose places where the irregular attractions of the
elevated parts may be small ; or where, by their situation,

[ 180 ] A T W

they may compensate or counteract each other's effects.
Sec Philos. Trans, vol. 65; or my Abridg. vol. 13, p. 702.
Besides what is above related of this grand experiment,
and its results, as it was necessary, so a great deal more
was done, than what yet appears by that account. For,
though the experiment showed the quantity of deviation
in the plumb-line, from its due position, to be 5"'8, and
thereby proved the fact of the universal attraction or gra-
vitation of matter, yet this was all that it proved ; for as
to the mean density of the whole matter in the earth, as
above noticed, much more remained to be done, what is
there said having been partly from anticipation by a near
guess, or rather from the results of accurate calculations
which I was at that very time employed in xnaking, and
which results I was in the habit of communicating to Dr.
Maskelyne, while he was composing his account of the ex-
periment. In fact, to deduce from the experiment the
real mean density of the earth, was, from the first project-
ing of the experiment, considered as a very important part
of the business ; and in order to this, it was necessary to
procure a very minute and correct survey and measure-
ment of the mountain, from which to calculate theoreti-
cally the effect of the attraction of every particle of matter
contained in its composition. Accordingly, Dr. Maske-
lyne, with his assistants, made accurate measurements of
the hill, with numerous sections, both horizontal and ver-
tical ; by means of which, an exact model of it could be
made, and the distance and position of every point in it
could be ascertained. This survey Dr. M. brought to the
committee of the R. Soc. along with the observations of
the plumb-line before mentioned; but he declined under-
taking the laborious calculations of the attractions of the
matter in the hill, which were necessary to show the real
mean density of the earth ; calculations which would pro-
bably occupy four times as much labour and time as had
been employed in making the experiment and survey.

Accordingly, at the request of the Royal Society, I un-
dertook this arduous task, and, after aboutayear's labour,
delivered to that body an account of my computations and
results, which gave universal satisfaction ; which account was
printed in the 68th volume of the Philos. Trans. It has
also been printed at large in the 2d volume of my Tracts,
where it is proved that the mean density of the whole earth
is near 5 times the density of water, or almost double the
density of the rocks near the earth's surface.

ATTRITION, the striking or rubbing of bodies against
one another, so as to throw off some of their superficial
particles : such as the grinding and polishing of bodies.
Or simply the act of rubbing: as when amber and other
electric bodies are rubbed, to make them attract, or emit
their electric force.

ATWOOD (George) F. R. S. a celebrated mathemati-
cian and philosopher, was educated at Westminster school;
after which ho studied at Trinity college, Cambridge,
where he was afterwards for some time a tutor, and many
years fellow of that college. He was very celebrated also
for his courses of lectures, on several branches of experi-
mental philosophy, which he read in the university, with
neat illustrations by means of a large and excellent col-
lection of instruments; several of which were of his own
contrivance; particularly a very ingenious one, exhibiting
the theories respecting the rectilinear motion of bodies
which are acted on by constant forces. Of these lectures,
he published a vory neat Syllabus; also, in 1784, he pub-
lished, in one large volume in Svo, A Treatise on the Reo

AVE

C 190 ]

AVI

tilinear Motion and Rotation of Bodies ; a learned and in-
genious work, but not treated in the most simple and per-
spicuous manner. Mr. A. communicated some learned
and ingenious papers to the Royal Society, which were
printed in their Transactions : viz, 1. On the Mensuration
of a certain Angle. 2.0n the Times of Vibration of Watch
Balances. 3. The Theory of Floating Bodies. 4. The
Stability of Ships : all of them in some degree liable to
the like imputation with the foregoing; as well as the two
Tracts on the subject of Arches and Bridges, in 1801 and
1804; to him apparently a new study; as most of the pro-
perties which occurred to him, though common and well
known to other writers, he unfortunately fancied to be
new discoveries. The whole composition being indeed
strongly indicative of the feeble state of body and mind, in
which Mr. A. had lingered during some years before his

death; which happened in 1807, at 62 years of age.

The late minister, Mr. Pitt, having been one of Mr. A's
auditors at Cambridge, gave him, in 1784, it is said, a si-
necure place of 500/. a-year, which he enjoyed during the
remainder of his life. It is also said, that the same mini-
ster employed him in financial calculations for the public
service, in the latter years of his life, till such time as the
declining state of his health rendered him incapable of se-
vere application.

Avant-Foss, or Ditch of the Counterscarp, in Fortifi-
cation, is a wet ditch surrounding the counterscarp, on
the outer side, next to the country, at the foot of the gla-
cis. It would not be proper to have such a ditch if it
could be laid dry, as it would then serve as a lodgment
for the enemy.

AVENPACE, a philosopher among the Saracens that
invaded Spain, who flourished in the 12th century, and
was a follower of Aristotle. He wrote a commentary on
Euclid, as well as philosophical and theological epistles.
He was also intimately conversant with the Peripatetic
philosophy, and applied it to the illustration of the Isla-
mic system of theology, and to the explanation of the Ko-
ran ; and on this account he was suspected of heresy, and
thrown into prison at Corduba. It is said that he was
poisoned at Fez, in the year of the Hegira 533, A. D.
1 138 ; or according to others, 525, A. D. 1.130.

Averuoes, or Aben-Roes, a very subtile Arabian
philosopher, who flourished about the end of the 11th
century, when the Moors had possession of part of Spain.
He was the son of the high priest and chief judge of Cor-
duba or Cordova in Spain ; but educated in the university
of Morocco, where he was professor, and where he died in
1206, having there studied natural philosophy, medicine,
mathematics, law, and divinity. After the death of his fa-
ther, he enjoyed his posts in Spain, to which was after-
ward added that of judge of Morocco and Mauritania,
where having settled deputies, he returned to his duty in
Spain. Notwithstanding he was very rich, and had a
large income, his liberality to men of letters in necessity,
whether they were his friends or his enemies, kept him al-
ways in debt. He was afterwards deprived of these means
of emolument, and thrown into prison for heresy, by the
vile instigations of his enemies ; but the oppressions of the
judge who succeeded him, caused him to be restored to
his former employments.

This philosopher has been highly celebrated for his per-
sonal virtues. Such was kis temperance, that he partook
only once a day of the plainest food. In his application,
he was unremitting and indefatigable; allowing himself

no other recreation in the course of the day than tke
change of several literary occupations, for those of poetry
or history, and spending whole nights in study. In the
exercise of forbearance, meekness, and self-command, he
was signally exemplary. When a servant employed by an
enemy intruded upon him in one of his public lectures,
and whispered some abusive language, Averroes, with per-
fect self-possession, turned round to him, and said, "Well,
well;" and proceeded with his business. This servant
waited on him next day to implore his pardon for the in-
sult he had offered him. " God forgive thee," said Aver-
roes, " thou hast publicly shown me to be a patient man ;
and as for thine injury, it is not worthy of notice." He
then gave him money, and dismissed him with this admoni-
tion ; " What thou hast done to me, do not to another."
He was extremely fond of the works of Aristotle., and
wrote commentaries upon them ; whence he was styled
the Commentator, by way of eminence. He also wrote
many other pieces ; among which is a work on the Whole
Art of Physic ; an Epitome of Ptolemy's Almagest, which
Vossius dates about the year 1 149 ; also a Treatise of
Astrology, which was translated into Hebrew by R. Jacob
Ben Samson, and said to be extant in the French regal li-
brary. He likewise wrote several poems, and many ama-
tory verses, but these last he destroyed when he grew old.
His other poems are lost, except a small piece, in which
he says, " that when he was young, he acted against his
reason ; but that when he was in years, he followed its
dictates ;" upon which he utters this wish, " Would to
God I had been born old, and that in my youth I had
been in a state of perfection!" As to religion, his opi-
nions were, that Christianity is absurd ; Judaism, the re-
ligion of children ; Mahometanism, the religion of swine.

AVICENA, Avicenne, or Avicenes, an eminent
writer, who has been accounted the prince of Arabian phi-
losophers and physicians. He was born at Assena, near
Bokhara, in 978; and died at Hamadan in 1036, being 58
years of age.

The first years of Avicena were employed on the study
of the Belles Lettres and the Koran, and at ten years of
age he was perfect master of the hidden senses of that
book. Then applying to those of logic, philosophy, and
mathematics, he made a rapid progress. After studying
under a master the first principles of logic, and the first 5
or 6 propositions of Euclid's elements, he became disgust-
ed with the slow manner of the schools, applied himself
alone, and soon accomplished all the rest by the help of
the commentators only.

Possessed with an extreme avidity to be acquainted
with all the sciences, he studied medicine also. Persuaded
that this art consists as much in practice as in theory, he
sought all opportunities of seeing the sick; and afterwards
confessed that he had learnt more from such experience
than from all the books he had read. Being now in his
l6th year, and already celebrated for being the light of
his age, he determined to resume his studies of philoso-
phy, which medicine, &c, had made him for some time
neglect : he spent a year and a half in t|iis painful indus-
try, without ever sleeping a whole night together. At the
age of 21, he conceived the bold design of incorporating,
in one work, all the objects of human knowledge; which
he carried into execution in an Encyclopedia of 20 vo-
lumes, to which he gave the title of the Utility of Utilities.

Many wonderful stories are related of his skill in medi-
cine, and the cures which he performed. Several princes

A U R

t 19' ]

AUE

had been taken dangerously ill, and Avicenes was the only
one that could discover the cause of their disorders and
the mode of curing them. His reputation increased daily,
and all the princes of the East desired to retain him in
their families, and in fact he passed through several of
them. But the irregularities of his conduct sometimes
lost him their favour, and threw him into great distresses.
His excesses in pleasures, and his infirmities, made a poet
say, who wrote his epitaph, that the profound study of
philosophy had not taught him good morals, nor that of
medicine the art of preserving his own health.

After his death however, he enjoyed so great a reputa-
tion, that till the 12th century his works of philosophy
and medicine were preferred to those of all his predeces-
sors. Even in Europe his works were the only writings in
vogue in the schools. They were very numerous, and va-
rious, the titles of which are as follow : 1. Of the Utility
and Advantage of the Sciences, in 20 books. — 2. Of In-
nocence and Criminality, 2 books. — 3. Of Health and Re-
medies, 18 books. — 4. On the Means of preserving Health;
3 books. — 5. Canons of Physic, 14 books. — 6. On Astro-
nomical Observations, 1 book. — 7. On Mathematical Sci-
ences.— 8. Of Theorems, or Mathematical and Theologi-
cal Demonstrations, 1 book. — 9- On the Arabic Language,
and its Properties, 10 books. — 10. On the Last Judgment.
— 11. On the Origin of the Soul, and the Resurrection of
Bodies. — 12. On the end we should propose to ourselves
in Harangues and Philosophical Argumentations. — 13.
Demonstration of the Collateral Lines in the Sphere. — 14.
Abridgment of Euclid. — 15. On l'inity and Infinity. — 16.
On Physics and Metaphysics. — 17 '• On Animals and Ve-
getables, &c. — 18. Encyclopedia, 20 volumes.

AUGUST, the Sth month of the year, containing 31
days. In the ancient Roman calendar it was called sexti-
lis, as being the 6th month from March, with which their
year began ; but changed to its present name by the em-
peror Augustus, who gave it his own name on account of
his having obtained many victories and honours in that
month.

AVOIRDUPOIS Weight, a weight used in England for
weighing all the larger and coarser kinds of goods ; as
groceries, cheese, butter, flesh, wool, salt, hops, &c, and
all metals except gold and silver. Avoirdupois weight is
tiius divided, viz.

16 dr. or drams make

16' oz. - - - - -

112 lb.

20 cwt. - - - -

1 ounce, marked oz.
1 pound, - - lb.
1 hundred weight, cwt.
1 ton - ton.

The Avoirdupois ounce is less than the Troy ounce, in the
proportion of 700 to 76'8, but the Avoirdupois pound is
greater than the Troy pound in the proportion of 700 to
576;

for 1 lb Avoird. is = 7000 grains Troy,
but lib Troy is = 5760 grains Troy,
also 1 oz Avoird. is = 4374 grains Troy,
and 1 oz Troy is = 480 grains Troy.
AURIGA, the Waggoner, a constellation in the nor-
thern hemisphere, consisting of 14 stars in Ptolemy's ca-
talogue; but in Tycho's, 27; in Hevclius's, 40; and in the
Britannic catalogue, 66.

This is one of the 48 old asterisms, mentioned by all the

most ancient astronomers. It is represented by the ligure

of an old man, in posture somewhat like sitting, with a

goat and her kids in his left hand, and a bridle in his right.

AURORA, the morning twilight ; or that faint light

which appears in the morning when the sun is within 18
degrees of the horizon.

AURORA BOREALIS, Northern Light, or Stream-
ers; a kind of meteor appearing in the northern part of
the heavens, mostly in the winter season, and in frosty
weather. It is usually of a reddish colour, inclining to
yellow, and sends out frequent coruscations of pale light,
which seem to rise from the horizon in a pyramidical un-
dulating form, and shooting with great velocity up to the
zenith. It appears often in form of an arch, which is
partly bright, and partly dark, but generally transparent.
And the matter of it is not found to have any effect on the
rays of light, which pass freely through it. Dr. Hamil-
ton observes, that he could plainly discern the smallest
speck in the Pleiades through the density of those clouds
which formed part of the Aurora borealis in 1763, with-
out the least diminution of its splendour, or increase of
twinkling. Philos. Essays, p. 106.

Sometimes it produces an Iris. Hence M. Godin judges,
that most of the extraordinary meteors and phenomena in
the skies, related as prodigies by historians, as battles and
the like, may probably enough be reduced to the class of
Aurora; Boreales. Hist. Acad. R. Scienc. for 1762, p. 405.

This kind of meteor never appears near the equator; but
it seems is frequent enough towards the south pole, like as
towards the north, having been observed there by voyagers.
See Philos. Trans. No. 46l, and vol. 54; also Forster"s ac-
count of his voyage round the world with Captain Cook,
where he describes their appearance as observed for several
nights together, in sharp frosty weather, which was much
the same as those observed in the north, excepting that they
were of a lighter colour.

In the Shetland Isles these phenomena are the constant
attendants of clear evenings, and afford great relief to the
inhabitants among the gloom of the long winter nights:
they are called there, the merry dancers. The same kind of ap-
pearances are also seen in the northern latitudes of Sweden
and Lapland, where they are particularly beautiful, and
afford light to travellers during the whole night. In Hud-
son's Bay the aurora borealis diffuses a variegated splen-
dour, which is said to equal that of the full moon. In the
north-eastern parts of Siberia, according to the description
of Gmelin (lieise durch Siberien, vol. 3, p. 135) cited and
translated by Dr. Blagden (Phil. Trans, vol. 74, pa. 228)
those northern lights arc observed to begin with single
bright pillars, rising in the north, and almost at the same
time in the north-east, which, gradually increasing, com-
prehend a large' space of the heavens, rush, about from place
to place with incredible velocity, and finally almost cover
the whole sky up to the zenith, producing an appearance
as if a vast tent were expanded in the heavens, glittering
with gold, rubies, and sapphire. A more beautiful spec-
tacle cannot be painted ; but whoever should see such a
northern light for the first time, could not behold it with-
out terror. For, however grand the illumination may ap*
pear, it is attended, as I have learnt from the relation of
many persons, with as much hissing, cracking, and tumult,
as if the largest fire-works were playing off. To describe
what they then hear, they make use. of the expression
" spolochi, chodjat," that is, the raging heat is passing.
The hunters who pursue the white and blue foxes in the
confines of the icy sea, are often overtaken in their courses
by these northern lights ; their dogs are then so much
frightened, that they will not move, but lie obstinately on
the ground till the noise has passed. Commonly clear and

ADR [192

calm weather follows this kind of meteor. I have heard
this account, not from one person only, but confirmed by
the uniform testimony of many, who have spent part ot
several years in these very northern regions, and inhabited
different countries, from the Yenisei to the Lena; so that no
doubt of its truth can remain. This seems, indeed, to be
the real birth-place of the aurora borealis.

It seems that meteors of this kind have appeared some-
times more frequently than others. They were so rare in
England, or else so little regarded, that none are recorded
in our annals since that remarkable one of Nov. 14, 1574,
till the surprising aurora borealis of March 6, 17l6, which
appeared for three nights successively, but by far more
strongly on the first ; except that five small ones were
observed in the year 1707 and 1708. Hence it would
seem, that the air, or earth, or both, are not at all times dis-
posed to produce this phenomenon.

1 The extent of these appearances is also very great : that
in March 17 16 was visible from the west of Ireland to the
confines of Russia and the east of Poland, extending at
least near 30 degrees of longitude, and from about the 50th
degree in latitude, over almost all the north of Europe ;
and in all places, at the same time, it exhibited the like
wondrous appearances. Father Boscovich has determined
the height of an aurora borealis, which was observed by
the Marquis of Polini the 1 6th of December 1737, to have
been S25 miles high ; and Bergman, from a mean of 30
computations, makes the average height of the aurora bo-
realis amount to 70 Swedish, or 469 English miles. But
Euler supposes the height to be several thousands of miles ;
and Mairan also assigns to them a very elevated region.

Many attempts have been made to determine the cause
of this phenomenon. Dr. Halley imagines that the watery
vapours, or effluvia, exceedingly rarefied by subterraneous
fire, and tinged with sulphureous streams, which many na-
turalists have supposed to be the cause of earthquakes, may
also be the cause of this appearance; or that it is produced
by a kind of subtile matter, freely pervading the pores of
the earth, and which, entering into it nearer the southern
pole, passes out again with a similar force into the sether,
at the same distance from the northern. This subtile mat-
ter, by becoming more dense, or having its velocity in-
creased, may perhaps be capable of producing a small de-
gree of light, after the manner of effluvia from electric
bodies, which, by a strong and quick friction, emit light in
the dark, to which sort of light this seems to have a great
affinity. Philos. Trans. No. 347. See also Mr. Cotes's de-
scription of this phenomenon, and his method of explain-
ing it, by streams emitted from the heterogeneous and fer-
menting vapours of the atmosphere, in Smith's Optics,
p. 6'9; or my Philos. Trans. Abr. vol. 5, p. 477.

The celebrated M. de Mairan, in an express treatise on
the Aurora Borealis, published in 1731, supposes its cause
to be the zodiacal light, which, according to him, is no
other than the sun's atmosphere: this light happening, on
some occasions, to meet the upper parts of our atmo-
sphere about the limits where universal gravity begins to
act more forcibly towards the earth than towards the sun,
falls into our air to a greater or less depth, as its specific
gravity is greater or less, compared with the air through
which it passes. See Tract. Phys. et Hist, de 1'Aurore Bo-
reale. Suite des Mcmoircs de l'Acad. R. des Scien. 1731.
Also Philos. Trims. No. 433.

However, Euler is of a different opinion: he does not
think that the cause of the aurora borealis is owing to

]

A U R

the zodiacal light, as M. de Mairan supposes, but to parti-
cles of our atmosphere, driven beyond its limits by the
impulse of the solar light. And on this supposition he en-
deavours to account for the phenomena observed concern-
ing this light. He supposes the zodiacal light and the tails
of comets to be owing to a similar cause.

But from the period that the identity of lightning and
the electric matter was determined, philosophers have been
naturally led to seek for the explication of aerial meteors
in the principles of electricity; and there is now no doubt
but most of them, and especially the aurora borealis, are.
electrical phenomena. Besides the more obvious and
known appearances which constitute a resemblance be-
tween this meteor and the electric matter by which light-
ning is produced, it has been observed that the aurora oc-
casions a very sensible fluctuation in the magnetic needle;
and that when it has extended lower than usual in the at-
mosphere, the flashes have been attended with various
sounds of rushing and hissing, especially in Russia and the
other more northern parts of Europe, as noticed by Sig.
Beccaria and M. Messier. Mr. Canton, soon after he had
obtained electricity from the clouds, offered a conjecture,
that the aurora is occasioned by the flashing of electric
fire from positive towards negative clouds at a great dis-
tance, through the upper part of the atmosphere, where
the resistance is least: and he supposes that the aurora
which happens at the time when the magnetic needle is
disturbed by the heat of the earth, is the electricity of the
heated air above it; and this appears chiefly in the north-
ern regions, as the alteration in the heat of the air in those
parts is the greatest. Nor is this hypothesis improbable,
when it is considered that electricity is the cause of thun-
der and lightning; that it has been extracted from the air
at the time of the aurora borealis; that the inhabitants of
the northern countries observe it remarkably strong when
a sudden thaw succeeds very cold severe weather; and
that the tourmalin is known to emit and absorb the elec-
tric fluid only by the increase or diminution of its heat.
Positive and negative electricity in the air, with a proper
quantity of moisture to serve as a conductor, will account
for this and other meteors, sometimes seen in a serene sky.
Mr. Canton has since contrived to exhibit this meteor by
means of the Torricellian vacuum, in a glass tube about 3
feet long, and scaled hermetically. When one end of the
tube is held in the hand, and the other applied to the con-
ductor, the whole tube is illuminated from end to end, and
continues luminous without interruption for a consider-
able time after it has been removed from the conductor.
If, after this, it be drawn through the hand either way,
the light will be remarkably intense through the whole
length of the tube; and though a great part of the electri-
city be discharged by this operation, it still flashes at in-
tervals, when held only at one extremity, and kept quite
still; but if at the same time it be grasped by the other
hand in a different place, strong flashes of light dart from
one end to the other; and these continue 24 hours or more,
without a fresh excitation. Sig. Beccaria conjectures that
there is a constant and regular circulation of the electric
fluid from north to south; and he thinks that the aurora,
borealis may be this electric matter performing its circu-
lation in such a state, of the atmosphere as renders it. vi-
sible, or approaching nearer than usual to the earth :
though probably this is not the mode of its operation, as
the meteor is observed in the southern hemisphere with
the same appearances as in the northern. Dr. Franklin

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supposes, that the electric fire discharged into the polar
regions, from many leagues of vaporised air raised from the
ocean between the tropics, accounts for the aurora bo-
realis; and that it appears first, where it is first in motion,
namely, in the most northern part; and the appearance
proceeds southward, though the fire really moves in the
opposite direction. Franklin's Exper. and Obs. 1769, pa.
49. Philos. Trans, vol. 48, pa. 358, 7S4; lb. vol. 51, pa.
403. LetteredeH'Ellettricismo, pa. 269; or Priestley's Hist-
of Electricity. See also an ingenious solution of this phe-
nomenon, on the same principles, by Dr. Hamilton, in his
Philos. Essays. Mr. Kirwan (in the Transactions of the
Royal Irish Academy, ann. 17SS), lias some ingenious re-
marks on the aurora borealis et australis. He gives his
reasons for supposing the rarefaction of the atmosphere in
the polar regions to proceed from them, and these from a
combustion of inflammable air caused by electricity. He
observes, that after an aurora borealis the barometer
commonly falls, and high winds from the south generally
follow.

To the same purpose it is observed by Mr. Wynn (Phil.
Trans, vol. 73), that the appearance of an aurora borealis
is a sure indication of a hard gale of wind from the south
or south-west. This occurred without fail in 23 instances;
and he thinks that the splendor of these phenomena will
enable the. observer to form some judgement concerning
the ensuing tempest. If the aurora is bright, the gale will
come on within 24 hours, but will be of no long duration :
if the light be faint and dull, the gale will be less violent,
and longer in gathering, but will continue longer. His
observations were made in the English Channel, where
such winds are very dangerous; and, by attending to the
aurora, he says, that he often escaped, when others were
nearly shipwrecked. Observations of this kind would serve
to lessen the dangers of navigation.

M. Libes has suggested a new theory, in his Nouv. Diet,
de Physique, which, it seems, is adopted by most of the
northern philosophers. In his opinion, electric light is
not the cause of the aurora borealis ; nor has electricity
itself any further influence on their existence, than as it
fixes the aeriform substances, whose combination occasions
the meteor. This philosopher's theory is founded on the
following principles: —

1. If we excite the electric spark in a mixture of azotic
and oxygen gas, there will result nitric acid, or nitrous
acid, or nitrous gas, according to the relation that subsists
between the gases which compose the mixtur*.

2. Nitric acid, when exposed to the sun, assumes more
colour and volatility. M. Libes placed a receiver over a
salver containing nitric acid, and exposed to the action of
the solar rays. Some minutes after, the acid appeared
coloured, and the receiver filled with red and volatile va-
pours, which were sustained in it a long while, and diffused
a light similar to that of the aurora borealis.

3. In flasks which contain nitrous acid, a ruddy and
volatile vapour is always perceived above the vapour.

4. Nitrous gas, in contact with atmospheric air, exhales
ruddy vapours, which fly off into the atmosphere.

5. The hydrogen, which is disengaged from the earth's
surface, rises, till it occupies, in the higher regions of the
atmosphere, a place determined by its specific gravity.

6. The solar heat has very little activity in the polar
regions.

These several principles rest on observations and expe-
riments made with great exactness, and most of them too
Vol. I.

well known to need being here described. Now it is plain,
from a simple combination of these tacts, 1st, that the pro-
duction of hydrogen must be next to nothing in the polar
regions; 2dly, that the higher regions of the polar atmo-
sphere contain very little if any hydrogen ; 3dly, that when-
ever there is a re-establishment of equilibrium of the elec-
tric fluid in the polar atmosphere, this fluid can only find
in its passage a mixture of azot and oxygen; 4 thly, that
the electric spark ought to fix and combine these gaseous
substances; 5thly, that from this combination must result
a production of nitrous acid, or of nitric acid, or of ni-
trous gas, according to the relation subsisting between the
oxygen and azot, that constitute the mixture; 6'thly, that
the production of either of those acids, or of the gas, will
give birth to red and volatile vapours, whose elevation in
the atmosphere will form the meteor known by the name
of the aurora borealis. After removing some general ob-
jections to these preliminary notions, M. Libes then ap-
plies them to the phenomena as generally observed of that
meteor.

AURUM Fulminans, a preparation from gold, which
being thrown into the fire, it explodes with a violent noise
like thunder. The matter is produced by dissolving gold
in aqua regia, and precipitating the solution by oil of tar-
tar per deliquium, or volatile spirit of sal ammoniac. The
powder being washed in warm water, and dried to the con-
sistence of a paste, is afterwards formed into small grains
of the size of hempseed.

It is inflammable, not only by fire, but also by a gentle
warmth ; and gives a report much louder than that of gun-
powder. A single grain laid on the point of a knife, and
lighted at a candle, explodes with a greater report than a
musquet: and a scruple of this powder, it is said, acts
more loudly than half a pound of gunpowder ; and yet it is
said that, by mixture, it does not increase the elastic force
of fired gunpowder. Dr. Black attributes the increase of
weight, and also the explosive property of this powder, to
adhering fixable air. — This is a very dangerous prepa-
ration, and should be used with great caution.

AUSTRAL, the same with Southern. Thus, Austral
signs, are the last 6 signs of the zodiac; and are so called
because they are on the south side of the equinoctial.
AUSTRALIS Corona. See Corona Australis.
AusxaALis Piscis, the Southern Fish, is a constella-
tion of the southern hemisphere. See Piscis Australis.

AUSTRIACASydeua, in Astronomy, a name ascribed
by Maupertuis to the spots in the sun, as supposing them
to be small stars between the sun and us.

AUTOLYCUS, a Greek mathematician and astrono-
mer, of Pitane, in /Eolia, who flourished a*bout 320 years
before Christ. He was preceptor in mathematics to Ar-
cesilaus, who was also a disciple of Theophrastus, the suc-
cessor of Aristotle. That he was eminent in mathematical
learning, appears from two of his works that are extant;
viz, a treatise On the Moveable Sphere, published by
Dasypodius, in Greek and Latin, Svo, at Strasburg, in
1572; and in a Latin translation in the Synopsis Ma-
thematica of Merscnnus, published in 4to, at Paris, in
l6'42 ; and also a treatise On the Rising and Setting of
the Stars, edited with the former work by DasypouUis.
Diog. Laert. Vit. Arcesil. Fabr. Bib. Grasc. toni. 2, p. 89.
Montucla Hist. Mathem. 1 torn. p. 192.

AUTOMATON, a seemingly self-moving machine ; or
one so constructed, by means of weights, levers, pulleys,
springs, &c, as to move for a considerable time as it it

2C

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A U T

were endued with animal life. And according to this de-
scription, clocks, watches, and all machines of that kind
are automatons.

It is said, that Archytas of Tarentum, 400 years before
Christ, made a wooden pigeon that could fly ; that Archi-
medes also made such-like automatons; that Regiomonta-
nus made a wooden eagle that flew forth from the city, met
the emperor, saluted him, and returned; also, that he
made an iron fly, which flew out of his hand at a feast,
and returned again after flying about the room; that Dr.
Hooke made the model of a flying chariot, capable of sup-
porting itself in the air. Many other surprising automa-
tons we have been eye-witnesses of, in the present age :
thus, we have seen figures that could write, and perform
many other actions in imitation of animals: M. Vaucan-
son made a figure that played on the flute; the same gen-
tleman also made a duck, which was capable of eating,
drinking, and imitating exactly the voice of a natural one;
and, what is still more surprising, the food it swallowed
was evacuated in a digested state, or considerably altered
on the principles of solution ; also the wings, viscera, and
bones, were formed so as strongly to resemble those of a
living duck; and the actions of eating and drinking showed
the strongest resemblance, even to the muddling the water
with its bill. M. Le Droz of la Chaux de Fonds, in the
province of Neufcfratel, has also executed some very cu-
rious pieces of mechanism ; one was a clock, presented to
the king of Spain, which had, among other curiosities, a
sheep that imitated the bleating of a natural one; and a
dog watching a basket of fruit, that barked and snarled
when any one offered to take it away; besides a variety of
human figures, exhibiting motions truly surprising. But
all these seem to be inferior to M. Kempell's chess-player,
which may truly be considered as the greatest master-
piece in mechanics that ever appeared in the world. " So
it certainly would have been," said my esteemed friend,
Thos. Collinson, esq. (nephew of the late ingenious Peter
Collinson, esq, F. R. S.) had its scientific movements de-
pended merely on mechanism. " Being slightly acquaint-
ed with M. Kempell when he exhibited his chess-playing
figure in London, I called on him about five years since
at his house at Vienna (another gentleman and myself be-
ing then on a tour on the continent). The baron (for I
think he is such) showed me some working models which
he had lately made — among them an improvement on
Arkwright's cotton-mill, and also one which he thought an
improvement on Boulton and Watt's last steam-engine.
I asked him after a piece of speaking mechanism which
he had shown me when in London. It spoke as before,
and I gave the same word as I gave when I first saw
it, Exploitation, which it distinctly pronounced with the
French accent. But 1 particularly noticed, that not
a word passed about the chess-player; and of course I
did not ask to sec it. In the progress of the tour I
came to Dresden, where, becoming acquainted with Mr.
Eden, our envoy there, by means of a letter given me
by his brother, lord Auckland, who was ambassador
when I was at Madrid, he obligingly accompanied me in
seeing several things worthy of attention. And he intro-
duced iny companion and myself to a gentleman of rank
and talents, named Joseph Freidrick Freyherc, who seems
completely to have discovered the vitality and soul of the
chess-playing figure. This gentleman courteously pre-
sented me with a treatise lie had published, dated at Dres-
den, Sept. 30, 1789, explaining its principles, accompanied

with curious plates neatly coloured. This treatise is in the
German language ; and 1 hope soon to get a translation of
it. A well-taught boy, very thin and small of his age (suffi-
ciently so that he could be concealed in a drawer almost
immediately under the chess-board,) agitated the whole.
Even after this abatement of its being strictly an automa-
ton, much ingenuity remains to the contriver. — This dis-
covery at Dresden accounts for the silence about it at
Vienna; for I understand, by Mr. Eden, that Mr. Freyhere
had sent a copy to Baron Kempell : though he seems un-
willing to acknowledge that Mr. F. has completely ana-
lysed the whole.

" I know that long and uninteresting letters are formi-
dable things to men who know the value of time and sci-
ence ; but as this happens to be upon the subject, forgive me
for adding one very admirable piece of mechanism to those
you have touched upon. When at Geueva, I called upon
Droz, son of the original Droz of la Chaux de Fonds,
where I also was. He showed me an oval gold snuff- box,
about (if I recollect right) 4 inches and a half long, by 3
inches broad, and about an inch and a half thick. It was
double, having an horizontal partition ; so that it may be
considered as one box placed on another, with a lid, of
course, to each box : one contained snuff — in the other, as
soon as the lid was opened, there rose up a very small
bird, of green enamelled gold, sitting on a gold stand. Im-
mediately this minute curiosity wagged its tail, shook its-
wings, opened its bill of white enamelled gold, and poured
forth, minute as it was (being only three quarters of an
inch from the beak to the extremity of the tail) such a.
clear melodious song, as would have filled a room of 20
or 30 feet square with its harmony. Droz agreed to meet
me at Florence ; and we visited the Abb6 Fontana toge-
ther. He afterwards joined me at Rome, and exhibited
his bird to the pope and the cardinals in the Vatican pa-
lace, to the admiration, I may say, to the astonishment, of
all who saw and heard it."

Another extract from a second letter on the same sub-
ject, by Mr. Collinson, is as follows: " Permit me to speak
of another automaton of Droz's, which several years since
he exhibited in England ; and which, from my personal
acquaintance, 1 had a commodious opportunity of parti-
cularly examining. It was a figure of a man, I think, the
size of life. It held in its hand a metal style ; a card of
Dutch vellum being laid under it. A spring was touched,
which released the internal clock-work from its stop,
when the figure immediately began to draw. Mr. Droz,
happening once to be sent for in a great hurry to wait
upon some considerable personage at the west end of the
town, left me in possession of the keys, which opened the
recesses of all his machinery. He opened the drawing-
master himself, wound it up, explained its leading parts,
and taught me how to make it obey my requirings, as it
had obeyed his own: Droz then went away. After the
first card was finished, the figure rested. I put a second;
and so on, to five separate cards, all different subjects; but
five or six was the ex lent of its delineating powers. The fust
card contained, I may truly say, elegant portraits and like-
nesses of the king and queen, facing each other; and it was
curious to observe with what precision the figure lifted up
his pencil, in the transition of it from one point of the
draft to another, without making the least slur whatever :
for instance, in passing from the forehead to the eye, nose,
and chin; or from the waving curls of the hair to the ear,
&C. 1 have the cards now by me, &c, &C,'"

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To the preceding account of such surprising accounts
of automata, may be added, that we have lately seen ex-
hibited at Springs-Gardens, in London, by a Mr. Maliar-
det, similar and other such mechanical contrivances; viz,
the singing-bird, and the figure of a young man that wrote
many words and lines, and drew very neatly several figures
as above described. There were also several others, equally
curious; as a tumbling figure, which performed surprising
evolutions about a horizontal, wire; also a spider resem-
bling a living one, and which crept all over the table; and
the musical lady, being one that much resembled life,
by its motions and respirations, &c, and naturally per-
forming some fine pieces on the piano-forte, &c.

AUTUMN, the third season of the year, when the har-
vest and fruits are gathered in. This begins at the de-
scending equinox, which, in the northern hemisphere, is
when the sun enters the sign Libra, or about the 22d day
of September; and it ends, when winter commences, about
the same day in December.
Autumnal, something belonging to autumn. Thus,
Autumnal Equinox, the time when the sun enters the
descending point of the ecliptic, where it crosses the equi-
noctial ; and is so called, because the nights and days are
then equal.

Autumnal Point, the point of the ecliptic answering
to the autumnal equinox.

Autumnal Signs, are the signs Libra, Scorpio, Sagit-
tary, through which the sun passes during the autumn.

AUZOUT, or Azout, (Adrian), a French mathemati-
cian of the 17th century, and one of the earliest members
of the Academy of Sciences at Paris, was born at Rouen,
and died in l6§3. Some have ascribed to him the honour
of having invented the micrometer; but he is more justly
entitled to the praise of having contributed to the improve-
ment of it, in pursuance of the ideas suggested by M.
Huygens, and the Marquis of Malvasia. He published a
treatise on this subject in 1667, which was afterwards in-
serted in the Memoirs of the Academy for 1693, torn. 6,
Auzout was also concerned with M. Picard in the impor-
tant discovery of the method of applying the telescope to
the quadrant, which has been highly useful to astrono-
mers. He published An Ephemeris of the Comet of 1665 ;
also, A Letter to the Abbe Charles on the Observations of
Campani, in 1665; and some Remarks on the Machine
of Hooke, mentioned in the article Automaton. These
two last pieces are also contained in the 6th volume of
the Memoirs of the Academy. Montucla, Hist. Mathem.
torn. 2, p. 569 — 572.

AXIOM, a self-evident truth, or a proposition immedi-
ately assented to, when the terms of it are properly un-
derstood. Such as, that the whole is greater than its part;
that a thing cannot be and not beat the same time; and
that any whole is equal to all its parts taken together.

Some axioms are in effect, strictly speaking, no other
than identical propositions. Thus, to say that all right
angles are equal to each other, is as much as to say, all
right angles are right angles; such equality being implied
in the very definition, or the very name or term itself.

Axiom is also an established principle in some art or
science. Thus, it is an axiom in physics, that nature does
nothing in vain ; that effects are proportional to their
causes ; &c. It is an axiom in geometry, that two things
equal to the same thing, are also equal to each other; that
if to two equal things equals be added, the sums will be
equal. And it is an axiom in optics, that the angle of in-

cidence is equal to the angle of reflection. In this sense
also the general laws of motion are called axioms; as, that
all motion is rectilinear, that action and reaction are
equal, &c.

These particular axioms, it may be observed, do not
however immediately arise from any first notions or ideas,
but are deduced from certain hypotheses ; this is particu-
larly observed in physical matters, wherein, as several ex-
periments contribute to make one hypothesis, so several
hypotheses contribute to one axiom.

The axioms of Euclid are general propositions ; and so
are the axioms of the Newtonian philosophy; but these
two kinds of axioms have very different origins. The
former appear true upon a bare contemplation of our
ideas; whereas the latter are the result of the most labo-
rious induction ; and therefore " ought not," as Lord Ba-
con says, " to be admitted upon conjecture or even upon
the authority of the learned ; but as they are the general
principles and grounds of all learning, they should be can-
vassed and examined with the most scrupulous attention."

AXE or AXIS, in Geometry, the straight line in a
plane figure, about which it revolves, to produce or gene-
rate a solid. Thus, if a semicircle be moved round its dia-
meter at rest, it will generate a sphere, whose axis is that
diameter. And if a right-angled triangle be turned about
its perpendicular at rest, it will describe a cone, whose
axis is that perpendicular.

Axis is yet more generally used for a right line con-
ceived to be drawn from the vertex of a figure'to the mid-
dle of the base. So the

Axis of a circle or sphere, is any line drawn through
the centre, and terminated at the circumference, on both
sides.

Axis of a cone, is the line from the vertex to the centre
of the base.

Axis of a cylinder, is the line from the centre of the
one end to that of the other.

Axis of a conic section, is the line from the principal
vertex, or vertices, perpendicular to the tangent at that
point. The ellipsis and hyperbola have each two axes,
which are finite and perpendicular to each other; but the
parabola has only one, and that infinite in length.

Transverse Axis, in the Ellipse and Hyperbola, is the
diameter passing through the two foci, and the two princi-
pal vertices of the figure. In the hyperbola it is the
shortest diameter, but in the ellipse it is the longest.

Conjugate Axis, or Second Axis, in the Ellipse and Hy-
perbola, is the diameter passing through the centre, and
perpendicular to the transverse axis ; and is the shortest
of all the diameters, as the transverse axis is the longest.

Axis, of a curve line, is still more generally used for
that diameter which has its ordinates at right angles to it,
when that is possible. For, like as in the conic sections,
any diameter bisects all its parallel ordinates, making the
two parts of them on both sides of it equal; and that dia-
meter which has such ordinates perpendicular to it, is an
Axis : So, in curves of the second order, if there be any
two parallel lines each meeting the curve in three points ;
the right line which cuts these two parallels so, that the
sum of the two parts on one side of the cutting line, be-
tween it and the curve, is equal to the third part termi-
nated by the curve on the other side, then the said line
will in like manner cut all other parallels to the former
two lines, viz, so that, of every one of them, the sum of
the two parts, or ordinates, on one side, will be equal to the
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third part or ordinate on the other side. Such cutting
line then is a diameter; and that diameter whose parallel
ordinates are at right angles to it, when possible, is an
Axis. And the same for ether curves of still higher or-
der's, Newton, Enumeratio Linearum Tertii Ordinis, sect.
2, art. I.

Axis, in Astronomy. As the Axis of the world, is an
imaginary right line conceived to pass through the centre
of the earth, and terminating at each end in the surface of
the mundane sphere. About this line, as an axis, the
sphere, in the Ptolemaic system, is supposed daily to re-
volve.

Axis of the Earth, is the line connecting its two poles,
and about which the earth performs its diurnal rotation,
from west to east. This is a part of the axis of the world,
and always remains parallel to itself during the motion of
the earth in its orbit about the sun, and perpendicular to
the plane of the equator.

Axis of a Planet, is the line passing through its centre,
and about which the planet revolves. — The Sun, Earth,
Moon, Jupiter, Mars, and Venus, it is known from obser-
vation, move about their several axes ; and the like mo-
tion is easily inferred of the other three, Mercury, Sa-
turn, and the Georgian planet.

Axis of the Horizon, Equator, Ecliptic, Zodiac, &c, are
right lines passing through the centres of those circles,
perpendicular to their planes.

Axis of a Magnet, or Magnetical Axis, is a line pass-
ing through the middle of a magnet, lengthwise; in such
manner, that however the magnet be divided, provided
the division be made according to a plane passing through
that line, the magnet will then be cut into two loadstones.
And the extremities of such lines are called the poles of
the stone.

Ax*is, in Mechanics. — The axis of a balance, is the line
upon which it moves or turns.

Axis of Rotation, of any solid, is'the line about which
the body really revolves, when it is put in motion. The
impulse given to a homogeneous sphere, in a direction
which does not pass through its centre, will cause it to re-
volve constantly round the diameter, which is perpendicu-
lar to a plane passing through its centre, and the line of
direction of the impressed force. New forces acting on
all its parts, and of which the result passes through its
centre, will not change the parallelism of its axis of rota-
tion. Thus it is that the axis of the earth remains always
nearly parallel to itself, in its revolution round the sun,
without its being necessary to suppose, with Copernicus,
an annual motion of the poles of the earth round those of
the ecliptic. If the body possess a certain figure, its axis
of rotation may change every instant. The determination
of these changes, whatever may be the forces acting on the
bodies, is one of the most interesting problems of mecha-
nics, respecting hard bodies, on account of its connexion
with the precession of the equinoxes, and the libration of
the moon. The solution of this problem has led to a cu-
. rious and very useful result, namely, that in all bodies
there exist three axes perpendicular to each other, round
which the body may turn uniformly when not solicited by
external forces. On this account these axes are properly
called the three principal axes of rotation.

Axis of Oscillation, is a line parallel to the horizon,
passing through the centre, about which a pendulum vi-
brates, and perpendicular to the plane in which it oscil-
lates.

Axis in Peritrochio, or Wheel and axle, is one of ther
five mechanical powers, or simple machines; contrived
chiefly for the raising of weights to a considerable height,
as water from a well,
&c. This machine con-
sists of a circle ab,
concentric with the
base of a cylinder, and
moveable together with
it about its axis cd.
This cylinder is called
the axis; and the cir-
cle, the peritrochium;
and the radii, or spokes,
which are sometimes
fitted immediately into
the cylinder, without any circle, the scytalas. About the
axis winds a rope, or chain, by means of which great
weights are raised by turning the wheel. — The axis in pe-
ritrochio takes place in the motion of every machine, in
which a circle may be conceived as described about a
fixed axis, concentric with the plane of a cylinder about
which it is placed; as in Crane-wheels, Mill-wheels, Cap-
stans, &c. — The chief properties of the Axis-in-peritrochio,
are as follow :

1. If the power f applied in the direction af a tangent
to the circumference, or perpendicular to the spoke, be to
a weight E, as the radius of the axis c e is to the radius of
the wheel ad, or the length of the spoke; the power will
just sustain the weight; that is, the power and the weight
will be in equilibrio, when they are' in the reciprocal pro-
portion of their distance from the centre.

2. When the wheel moves, with the power and weight ;
the velocities, of their motion, and the spaces passed over
by them, will be both in the same proportion as above,
namely, directly proportional to their distances from the
centre, and reciprocally proportional to their own weights
when they are in equilibrio.

3. A power and a weight being given, to construct an
axis-in-peritrochio, by which it shall be sustained and
raised. Let the axis be taken large enough to support the
weight and power without breaking: then, as the weight
is to the power, so make the radius of the wheel to the ra-
dius of the axis. Hence, if the power be very small in
respect of the weight, the radius of the wheel will be vastly
great. For example, suppose the weight 4050, and the
power 50; then the radius of the wheel will be SI times
that of the axis ; which would be a very inconvenient
size. But this inconvenience is obviated by increasing
the number of the wheels and axes; making One to turn
another, by means of teeth and pinions. And to find the
effect of a number of wheels and axes, thus turning one
another, multiply together, all the radii of the axes, and
all the radii of the wheels, and then it will be, as the pro-
duct of the former is to the product of the latter, so is the
power to the weight. So, if there be 4 wheels and axes,
the radius of each axis being 1 foot, and the radius of each
wheel 3 feet; then the continual product of all the wheels
is 3 x 3 x 3 x 3 or 81 feet, and that of the axis only 1 ;
therefore the effect is as 81 to 1, or the weight is 81 times
the power. And, on the contrary, if it be required to find
the diameter of each of four equal wheels, by which a
weight of 40501b shall be balanced by a power of 50lb, the
diameter of each axis being 1 foot: dividing 4050 by 50,
the quotient is 81, extract the 4th root of 81, or twice

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the square root, and it will give 3, for the diameter of the
lotir wheels sought.

Axis of a vessel, is that quiescent right line passing
through the middle of it, perpendicular to its base, and
equally distant from its sides.

Axis, in Optics. — Optic axis, or zimeil axis, is a ray
passing through the centre of the eye, or falling perpendi-
cularly on the eye.

Axis of a lens, or glass, is the axis of the solid of which
the lens is a segment. Or the axis of a glass, is- the line
joining the two vertices or middle points of the two oppo-
site surfaces of the glass.

Axis of Incidence, in Dioptrics, is the line passing
through the point of incidence, perpendicularly to the re-
fracting surface.

Axis of Refraction, is the line continued from the point
of incidence or refraction, perpendicularly to the refract-
ing surface, along the farther medium.

AZIMUTH, of the sun, or star, &c, is an arch of the
horizon, intercepted between the meridian of the place,
and the azimuth or vertical circle passing through the sun
or star; and is equal to the angle at the zenith formed by
the said meridian and vertical circle. Or it is the com-
plement to the eastern or western amplitude. — The azi-
muth is thus found by trigonometry;

As radius is to the tangent of the latitude,
So is the tangent of the altitude of the sun or star,
To the cosine of the azimuth from the south, at
the time of the equinox.

Azimuth, magnetical, an arch of the horizon contained

between the magnetical meridian, and the azimuth or ver-
tical circle of the object; or its apparent distance trom
the north or south point of the compass. This is found
by observing the sun, or star, &c, with an azimuth com-
pass, when it is 10 or 15 degrees high, either before or
after noon.

Azimuth Compass, an instrument for finding either
the magnetical azimuth or amplitude of a celestial object.
The description and use of this instrument, see under the
article Compass.

Azimuth Dial, a dial whose stile or gnomon is per-
pendicular to the plane of the horizon.

Azimuth, or Vertical Circles, are great circles of the
sphere-intersecting each other in the zenith and nadir, and
cutting the horizon at right angles. — These azimuths are
represented by the rhumbs on common sea charts; and on
the globfe by the quadrant of altitude, when screwed in the
zenith. On these azimuths is counted the height of the
sun, or stars, &c, when out of the meridian.

AZOTIC GAS, the same as the phlogisticated air of
Scheele and Priestley, or the mephitic air of Lavoisier, or '
the nitrogen gas of Chaptal, and some other French che-
mists. This gas was discovered in 1772, by Dr. Ruther-
ford of Edinburgh ; and many of its properties were ascer-
tained by Cavendish, Kirwan, and Priestley. It is nearly of
the same specific gravity as common air, or perhaps a very
little heavier, being indeed only atmospheric air deprived
of its oxygen ; and hence its noxious quality of presently
extinguishing flame and animal life ; as in unused wells
or cellars, and close places of stagnant air.

B.

B A C

TJ ACK-Staff, an instrument formerly used for taking
-*J the sun's altitude at sea; being so called because the
back of the observer was turned towards the sun when he
made the observation, ft was sometimes called Davis's
quadrant, from its inventor Captain John Davis, a Welch-
man, and a celebrated navigator, who produced it about
the year 1590.

This instrument consists of two concentric arches of
box-wood, and three vanes : the arch of the longer radius

B AC

is of 30 degrees, and the other of 60 degrees, making be-
tween them 90 degrees, 6r a quadrant: also the vane a
at the centre is called the horizon-vane, that on the arch
of 60" at B the shade- vane, and that on the other arch at
c the sight-vane.

To use the Back-Staff. — The shade-vane is to be set'
upon the 60 arch, at an even degree with some latitude,
less by 10 or 15 degrees than the complement of the
sun's altitude is judged to be; also the horizon-vane be-
ing put on at a, and the sight-vane on the 30 arch fg,
the observer then turns his back to the sun, lifts up the
instrument^ and looks through the sight-vane, raising or
falling the quadrant, till the shadow of the upper edge of
the shade-vane fall on the upper edge of the slit in the
horizon-vane ; and then if he can see the horizon through
the said slit, the observation is exact, and the vanes are
rightly adjusted : But if the sea appear instead of the ho-
rizon, the sight-vane must be moved downward towards
f; or if the sky appear, it must be moved upward to-
wards g; thus trying till it comes right : the observer next
examines how many degrees and minutes are cut by that
edge of the sight-vane that answers to the sight-hole, and
to them he adds the degrees cut by the upper edge of the
shade vane ; the sum of which is the sun's distance from
the zenith, or the complement of his altitude; that is, of
his upper limb when the superior end of the shade-vane is
used in the observation, or of his lower limb when the
lower part of that vane is used ; therefore in the former
case add l6 minutes, the sun's semidiameter, and subtract

B A C

C 198 ]

B A C

16 minutes in the latter case, to give the zenith distance
or co-altitude of the sun's centre.

Mr. f'lamsteed contrived a glass lens, or double convex,
to be placed in the middle of the shade-vane, which
throws a small bright spot on the slit of the horizon-yane,
instead of the shade ; which is a great improvement, if the
glass be truly made ; for by this means, the instrument
may be used in hazy weather, and a much more accurate
observation made at all times.

BACON (Roger), a celebrated English monk of the
Franciscan order, was born near Ilchester in Somerset-
shire, in the year 1214. He commenced his studies at
Oxford ; whence he removed to the university of Paris,
which at that time was esteemed the centre of learning;
and where it appears he made such progress in the sci-
ences, that he was esteemed the glory of that university,
and was there greatly caressed by several of his country-
men, particularly by Robert Groothead or Grouthead, af-
terwards bishop of Lincoln, his great friend and patron.
Having taken the degree of doctor of divinity, he returned
to England in 1 240, and took the habit of the Franciscan
order, being then about 2f) years of age; but according to
some he became a monk before he left France. He now
pursued his favourite study of experimental philosophy
with unremitting ardour and assiduity. In this pursuit,
in experiments, he informs us he spent, in the course of
20 years, no less than 2000/. in the purchase of instru-
ments and scarce bsoks, an amazing sum in those days,
but which was generously furnished him by some of the
heads of the university, to enable him the better to pur-
sue his noble researches. By such extraordinary talents
and rapid progress in the sciences, which in that ignorant
age were so little known to the rest of mankind, while they
raised the admiration of the more intelligent few, could
not fail to excite the envy of his illiterate fraternity, whose
malice against him was the more inflamed by the freedom
with which he treated the clergy in his writings, in which
he spared neither their ignorance nor their immorality :
these therefore found no difficulty in possessing the vulgar
with the notion of Bacon's communication with evil spi-
rits. Under this pretence he was restrained from reading
lectures to the young students in the university, and at
length so closely confined as to be debarred from all in-
tercourse with his friends, and from receiving a necessary
supply of food. These efforts of malevolence and cruelty,
whatever might have been the real or pretended causes
from which they originated, could not however deprive
this great man of the esteem and respect to which his di-
stinguished talents and character entitled him. Such was
the high opinion entertained by the cardinal bishop of Sa-
bina, who was the pope's legate in England, of his genius
and merit, that he requested from him a complete copy of
all his works. As he. was restrained, by the prohibition of
his own fraternity, from communicating any of his works
to any person whatever, he at first declined complying
with the cardinal's request ; but as soon as he heard that
the cardinal-legate was raised to the pontifical dignity,
under the name of Clement iv, he signified to him by let-
ter his readiness to perform what his holiness had desired ;
and the pope assured him of protection against any inter-
ference of his own order. Bacon immediately began to
collect, arrange, and improve the pieces he had already
written : and having digested them into one volume, under
the title of" Opus Majlis" (the greater work,) lie sent it to
the pope in the year 1267, by a special messenger, whose

name was John of Paris, and who was his own favourite
disciple. This John of Paris was a poor boy, of promi-
sing talents, taken by Bacon under his tuition, in order to
try by experience the efficacy of his -peculiar mode of in-
struction; and as the result of it he observes, " that there
was no room to conceive any high notions of the Perfection
of human wisdom, when it was possible in a year'* time to
teach a young man all that, with the utmost industry and
application, a zealous inquirer after knowledge was able
either to attain or discover in the space of 20 or even 40
years. The pope was so gratified with the present of this
learned work, that it procured for Bacon extraordinary-
favour and encouragement in his studies.

With the life of the enlightened and liberal Clement iv,
terminated the tranquillity of this philosopher ; for. under
the pontificate of Nicholas in, and with the sanction of
his authority, Jerom de Esculo, or de Ascoli, general of
the Franciscan order, prohibited the reading of his works,
and sentenced him to imprisonment. The pretended cause
of this severity has been sought by some writers in certain
tracts of Bacon on necromancy, astrology, and alchemy;
but the true reason was most probably that dread of inno-
vation, which Bacon's improvements in science caused in
the minds of bigoted or interested persons. Bacon con-
tinued in prison 10 years; but on the accession of Jerom
de Esculo to the papal see, under the name of Nicholas
iv, he attempted to conciliate the favour of the pope,
by presenting to him a treatise " On the means of avoiding
the infirmities of old age." What effect this had on the
pope does not appear; it did not at least produce an im-
mediate discharge : however, towards the latter end of his
reign, by the interposition of some noblemen, Bacon ob-
tained his liberty ; and afterwards spent the remainder of
his life in the college of his order, where he died in the
year 1294, at 80 years of age, and was buried in the
Franciscan church. Such are the few particulars which
the most diligent researches have been able to discover
concerning the life of this very extraordinary man.

Bacon's printed works are, 1. Epistola Fratris Rogeri
Baconis de Secrctis Operibus Artis et Natura;, et de Nul-
litate Magia: Paris, 1542, in 4to. Basil, 1593, in 8vo.
2. Opus Majus; London, 1733, in fol. -published by Dr.
Jebb. 3. Thesaurus Chemicus ; Francf. l603 and l620.
These printed works of Bacon contain a number of essays,
which have been considered as distinct books in the cata-
logue of his writings by Bale, Pitts, &c ; but there remain
also in different libraries several manuscripts not yet pub-
lished.

By an attentive perusal of his works, the reader is asto-
nished to find that this great luminary of the 13th century
was deeply skilled in all the arts and sciences, and in many
of them had made the most important inventions and dis-
coveries. He was, says Dr. Peter Shaw, beyond all com-
parison the greatest man of his time, and he might per-
haps stand in competition with the greatest that have ap-
peared since. It is wonderful, considering the ignorant
age in which he lived, how he came by such a depth of
knowledge on all subjects. His writings are composed
with that elegance, conciseness, and strength, and adorned
with such just and exquisite observations on nature, that,
among all the chemists, we do not know his equal. In
his chemical writings, he attempts to show how imperfect
metals may be ripened into perfect ones ; making, with
Geber, mercury the common basis of all metals, and sul-
phur the cement.

BAG

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B A C

His other physical writings show no less genius and force
of mind. In his. treatise Of the Secret Works of Art and
Nature, he shows that a person perfectly acquainted with
the manner observed by nature in her operations, would
be able to rival, and even to surpass her. In another piece,
Of the Nullity of Magic, he shows with great sagacity and
penetration, whence the notion of it sprang, and how weak
all pretences to it are. From a perusal of his works, adds
the same author, we find Bacon was no stranger to many
of the principal discoveries of the present and past ages.
The ingredients and effects of gunpowder he was doubtless
acquainted with ; and thunder and lightniflg, he tells us,
may be produced by art ; for though sulphur, nitre, and
charcoal, when separate have no sensible effect, yet when
mixed together in due proportion, and closely confined,
and fired, they yield a loud report. A more precise de-
scription of gunpowder cannot be given in words. He also
mentions a sort of unextinguishable fire prepared by art;
which shows he was not unacquainted with phosphorus:
and that he had some 'notion of the rarefaction of the air,
and the structure of an air-pump, is past contradiction.
He was the miracle, says Dr. Friend, of the age he lived
in, and the greatest genius, perhaps, for mechanical know-
ledge, that ever appeared in the world since Archimedes.
He appears also to have been master of the whole science
of optics : he has accurately described the uses of reading-
glasses, and shown the way of making them. Dr. Friend
adds, that he also describes the camera obscura, and all
sorts of glasses, which, magnify or diminish any object, or
bring it nearer to the eye, or remove it farther off. Bacon
says himself, that he had great numbers of burning-glasses :
and that there were none ever in use among the Latins, till
his friend Peter de Mahara Curia applied himself to the
making of them. That the telescope was not unknown to
him, appears from a passage where he says, that he was
able to form glasses in such a manner, with respect to our
sight and the objects, that the rays shall be refracted and
reflected wherever we please, so that we may see a thing
under what angle we think proper, either near or at a di-
stance, and be able to read the smallest letters at an incre-
dible distance, and to count the dust and sand from the
greatness of the angle under which we see the objects; and
also that we shall scarce see the greatest bodies near us,
on account of the smallness of the angle under which we
view them. His skill in astronomy was also very great:
he discovered that error which occasioned the reformation
of the calendar; one of the greatest efforts, according to
Dr. Jebb, of human industry : and his plan for correcting
it was followed by pope Gregory xm, with this varia-
tion, that Bacon would have had the correction to begin
from the birth of our Saviour, whereas Gregory's amend-
ment reaches no higher than the Nicene council.

BACON (Francis), baron of Verulam, viscount of St.
Albans, and lord high chancellor of England under king
James I. He was born in 1560, being son of Sir Nicholas
Bacon, lord keeper of the great seal in the reign of Queen
Elizabeth, by Anne daughter of Sir Anthony Cook, emi-
nent for her skill in the Latin and Greek languages. He
gave even in his infancy tokens of what he would one day
become; and Queen Elizabeth had many times occasion
to admire his wit and talents, and used to call him her
young lord keeper. He studied the philosophy of Aris-
totle at Cambridge; where he made such progress in ge-
neral literature, that at 16 years of age he had run through
the whole circle ol the liberal arts as they were then taught,

and even began to perceive those imperfections in the reign-
ing philosophy which he afterwards so effectually exposed,
and thence not only overcame that tyranny which checked
the progress of true knowledge, but laid the foundation of '
that free and useful philosophy which has since opened a
way to so many glorious discoveries. On his leaving the
university, his father sent him to France; where, before
he was 19 years of age, he wrote a general view of the state
of Europe : but his father dying, he was obliged suddenly
to return to England; where he applied himself to the
study of the common law, at Gray's-inn. His merit at
length raised him to the highest dignities in his profession,
namely, those of attorney-general, and lonl-high-chancel-
lor. But being of an easy and liberal disposition, his ser-
vants took advantage of that temper, and of their situation
under him, by accepting presents in the line of-his profes-
sion. Being abandoned by the king, he was tried by the
house of lords, for bribery and corruption, and by them
sentenced to pa}' a fine of 40,000/, and to remain prisoner
in the Tower during the king's pleasure. The king how-
ever soon after remitted the fine and punishment: but his
misfortunes had given him a distaste for public affairs, and
he afterwards mostly lived a retired life, closely pursuing
his philosophical studies and amusements, during which
time he composed the greatest part of his English and La-
tin works. Though even in the midst of his honours and
employments he forgot not his philosophy, but in l6"20
published his great work Novum Organum. After some
years spent in his philosophical retirement, he died in 1626,
being 66 years of age.

The chancellor Bacon is one of those extraordinary
geniuses who have contributed most to the advancement
of the sciences. He clearly perceived the imperfection of
the school philosophy, and pointed out the only means of
reforming it, by proceeding in the opposite way, from ex-
periments to the discovery of the laws of nature. Addison
has said of him, That he had the sound, distinct, compre-
hensive knowledge of Aristotle, with all the beautiful light
graces and embellishments of Cicero. Mr. Walpole calls
him the Prophet of Arts, which Newton was afterwards
to develope ; and adds, that his genius and his works will
be universally admired as long as science exists. He did
not yet, said another great man, understand nature, but
he knew and pointed out all the ways that lead to her.
He very early despised all that the universities called phi-
losophy; and used every exertion in his power that they
should not disgrace her by their quiddities, their horrors
of a vacuum, their substantial forms, and such-like im-
pertinencies.

He composed two works for perfecting the sciences.
The former On the Dignity and Augmentation of the Sci-"
ences. In this he shows the state in which they then were,
and points out what remains to be discovered for perfect-
ing them ; condemning the method employed by Aristotle,
of reversing the natural order of things. He here also
proposes his celebrated division of the sciences.

To remedy the faults of the common logic, Bacon com-
posed his second work, the New Organ of Sciences, above
mentioned. He. here teaches a new logic, the chief end ,
of which is to show how to make a good inference, as that
of Aristotle's is to make a syllogism. Bacon was 18 years
in composing this work, and he always esteemed it as the
chief of his compositions.

The pains which Bacon bestowed on the sciences in ge-
neral, prevented him from making any considerable appli-

B A I

[ 800 ]

B A I

cations to any one in particular: and as ho knew that na-
tural philosophy is the foundation of all the other sciences,
he chiefly endeavoured to give perfection to it. He there-
fore proposed to establish a new system of physics, reject-
ing the doubtful principles of the ancients. For this pur-
pose he took the resolution of composing every month a
treatise on some branch of physics ; and accordingly be-
gan with that of the winds ; after which he gave that of
heat ; next that of motion ; and lastly that of life and
death. But as it was impossible that one man alone could
so compose the whole circle of sciences with the same pre-
cision-after having given these patterns, to serve as a mo-
del for those who might choose to labour on his principles,
he contented himself with tracing in a few words the de-
sign of four other tracts, and with furnishing the materials,
in" his Silva Silvarum, where he has amassed a vast num-
ber of experiments, to serve as a foundation for his new
physics. In fact, no one before Bacon understood any thing
of the experimental philosophy; and of all the physical
experiments which have been made since his time, there
is scarcely one that is not pointed out in his works.

This great precursor of philosophy was also an elegant
writer, an historian, and a wit. His moral essays are va-
luable, but are formed more to instruct than to please.
There are also many excellent things in his work On the
Wisdom of the Ancients, in which he has moralized the
' fables which formed the theology of the Greeks and Ro-
mans. He wrote also The History of Henry the 7th
king of England, by which it appears that he was no Jess
a great politician than a great philosopher.

Bacon had also some other writings, published at diffe-
rent times; the whole of which were collected together,
and published at Frankfort, in the year l665, in a large
folio volume, with an introduction concerning his life and
writings. Another edition of his works was also publish-
ed at London in 1740; the enumeration of which is as
below :

1. De Uignitate et Augmentis Scientiarum.

2. Novum Organum Scientiarum, -sive Judicia vera de
Interpretatione Naturae; cum Parasceve ad Historiam
Naturalem et Experimentalem.

3. Phenomena Universi, sive Historia Naturalis et Ex-
perimentalis de Ventis ; Historia Densi et Rari ; Historia
Gravis et Levis ; Historia Sympathies et Antipathice Re-
rum; Historia Sulphuris, Mercurii, et Salis; Historia
Vita et Mortis; Historia Naturalis et Experimentalis de
Forma Calidi ; De Motus, sive Virtutis activse variis spe-
ciebus ; Ratio inveniendi causas Fluxus et Refluxus Ma-
ris ; &c, &c.

4. Silva Silvarum, sive Historia Naturalis.

5. Novus Atlas.

6. Historia Regni Henrici vn Angliae Regis.

7. Sermones Fulo.les, Ethici, Politici, Oeconomici.

8. De Sapientia Veterum.

BACULF, in Fortification, a kind of portcullis, or gate,
made like a pit-fall, with a counterpoise, and supported
by two great stakes. It is usually made before the corps
de garde, near the gate of a place.

BACULOMEI'RY, the art of measuring either acces-
sible or inaccessible linos, by the help of baculi, staves, or
rods. Schweuter has explained this art in his Goometria
Practica; and the rules of it are delivered by Wolfius, in
his Elements: Ozanam also gives an illustration of the
principles of haculometry.

BAILLY (Jean Sylvain), a celebrated French astro-

nomer, historiographer, and politician, was born at Paris
the 15th of September 1736, and has figured as one of the
greatest men of the age, being a member of several acade-
mies, and an excellent scholar and writer. He enjoyed
for several years the office of keeper of the king's pictures
at Paris. About the year 1762, the theory of Jupiter's
satellites became a particular object of his inquiries ; and,
in the competition for this prize question, he had a for-
midable rival in La Grange, generally considered as one
of the greatest mathematicians in Europe. The results
of his investigations were afterwards collected into a trea-
tise, and published in 1 vol. 4to; also in 1766, when it was
preceded by a History of the Astronomy of these Satellites.
In the Journal Encyclopedique for May and July 1773,
he addressed a letter to M. Bernoulli, astronomer royal
at Berlin, on some discoveries relative to these satellites,
which he had disputed. In 170S he published the eulogy
of Leibnitz, which obtained the prize at the Academy of
Berlin, where it was printed. In 1770 he printed at Pa-
ris, in 8vo, the eulogies of Charles the 5th, also of Lecaille,
of Leibnitz, and of Corneille. This last had the second
prize at the Academy of Rouen, while that of Moliere had
the same honour at the French Academy.

M. Bailly was admitted into the Academy as Adjunct
the 29th of January 1763, and as Associate the 14th of
July 1770. In 1771 he was a. candidate, under the pa-
tronage of Buffon, for the office of Secretary; but the in-
terest of Condorcet, and the influence of Dalembert, pre-
vailed in favour of Condorcet. Of the Academic Fran-
poise, he was chosen Secretary in 1754; and he was ad-
mitted in the following year into the Academy of Inscrip-
tions and Belles Lettres, the only instance since Fontenelle
of the same person being at once a member of all the three
Academies. Jn 1775 came out at Paris, in 4to, his His-
tory of the Ancient Astronomy, in 1 vol: in 1779 the
History of Modern Astronomy, in 2 vols: and in 1787 the
History of the Indian and Oriental Astronomy, being the
2d vol. of the Ancient Astronomy.

M. Bailly 's memoirs, published in the volumes of the
Academy, are as follow :

Memoir upon the Theory of the Comet of 1759.

Memoir upon the Epoques of the Moon's motions at
the end of the 17th century.

First, second, and third Memoirs on the Theory of Ju-
piter's Satellites; 1763.

Memoir on the Comet of 1762 ; vol. for }?G3.

Astronomical Observations, made at Noslon ; 17oV.

On the Sun's Eclipse of the 1st of April, 1/64.

On the Longitude of Polling; 1/64.

Observations made at the Louvre from 1760 to 1764 ;
1765.

On the cause of the Variation of the Inclination of the
Orbit of Jupiter's second Satellite ; l?6o.

On the Motion of the Nodes, and on the Variation of
the Inclination of Jupiter's Satellites; 1766.

On the Theory of Jupiter's Satellites, published by M.
Bailly, also according to the Tables of their Motions and
of those of Jupiter, published by M. Jeaurat ; 17 66.

Obscrvatioi.son the Opposition of the Sun ami Jupiter ;
1768.

On the Equation of Jupiter's Centre, and on some other
Elements of the Theory of that Planet ; 1768.

On the Transit, of Venus over the Sun, the 3d of J U Be
1769; and on the Solar Eclipse the 4th of June the same
year: 1709.

B A I

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B A K

i. In the beginning of the revolution in France, in 1789,
Mi Bailly took an active part in that business, and was so
popular and generally esteemed, that he was chosen the
first president of the States General, and of the National
Assembly, and was afterwards for two years together the
Mayor of Paris ; in both which offices he conducted him-
self with great spirit, and gave general satisfaction.

He soon afterward however experienced a sad reverse
of fortune; being accused by the ruling party of favour-
ing the king, he was condemned for incivism and for want-
ing to overturn the Republic ; in consequence he died by
the guillotine at Paris, November 12, 1793, at 57 years
of age.

BAINBRIDGE (John), an eminent physician, as-
tronomer, and mathematician, was born in 1582, at
Ashby-de-la-Zouch, Leicestershire. He studied at Cam-
bridge, and having taken his degrees of Bachelor and
Master of Arts, he returned to Leicestershire, where for
some years he kept a grammar-school, and at the same
time practised physic ; employing his leisure hours in stu-
dying mathematics, especially astronomy, which had been
his favourite science from his earliest years. By the ad-
vice of his friends, he removed to London, where he was
admitted a fellow of the College of Physicians. His de-
scription of the comet which appeared in 1618, greatly
raised his character, and procured him the acquaintance
of Sir Henry Savile, who, in l6l9, appointed him his first
professor of astronomy at Oxford. On his removal to this
university, he entered a master-commoner of Merton col-
lege; the master and fellows of which appointed him ju-
nior reader of Linacer's lecture in l6ol, and superior
reader in 1635. At the age of 40 he began the study of
Arabic, with an intention of publishing correct editions of
the ancient astronomers. Before completing that work
however he died, in the year 1643, at 6,1 years of age.

Dr. Bainbridge wrote many works, but most of them
have never been published ; those that were published,
were the three following, viz:

1. An Astronomical Description of the late Comet, from
the 18th of November l6l8, to the l6th of December fol-
lowing; 4to, London, l6l9- — This piece was only a spe-
cimen of a larger work, which the author intended to pub-
lish in Latin, under the title of Cometographia.

2. Procli Sphaera, Ptolemaei de Hypothesibus Planeta-
rum liber singularis. To which he added Ptolemy's Ca-
non Regnorum. He collated these pieces with ancient
manuscripts, and gave a Latin version of them, illustrated
with figures: printed in 4to, 1620.

3. Canicularia. A treatise concerning the Dog-star,
and the Canicular Days: published at Oxford in l648,
by Mr. Greaves, together with a demonstration of the he-
liacal rising of Sirius, the dog-star, for the parallel of Lower
Egypt. Dr. Bainbridge also undertook this work at the
request of Archbishop Usher, but he left it imperfect;
being prevented by the breaking out of the civil war, or
by death.

There were also several dissertations of his prepared
and committed to the press the year after his death, but
the edition of them was never completed. The titles of
them are as follow:

1st, Antiprognosticon, in quo MxvriKy; Astrological,
Coelrstium Domorum, et Triplicitatum Commentis, mag-
nisque Saturni et Jovis (cujusmodi anno l623,et 1643,
contigerunt, et vicesimo fere quoque dcinceps anno, ratis

Vol. I.

naturse legibus recurrent) Conjunctionibus innixs, Vanitas
breviter detegitur.

2nd, De Meridionorum sive Longitudinum Differentiis
inveniendis Dissertatio.

3d, De Stella Veneris Diatriba.

Besides the foregoing, there were several other tracts,
never printed, but left by his will to Archbishop Usher;
among whose manuscripts they are preserved in the library
of the college of Dublin. Among which are the following :
1. A Theory of the Sun. 2. A Theory of the Moon. 3. A
Discourse concerning the Quantity of the Year. 4. Two
volumes of Astronomical Observations. 5. Nine or ten
volumes of Miscellaneous Papers relating to Mathematical
Subjects.

BAKER (Thomas), a mathematician of some emi-
nence, was born at Ilton in Somersetshire, in 1625. He
entered on his studies at Oxford in 1640, where he remain-
ed seven years. He was afterwards appointed vicar of
Bishop's-Nymmet in Devonshire, where he lived astudious
and retired life for many years, chiefly pursuing the ma-
thematical sciences, of which he gave a proof of his criti-
cal knowledge, in the book he published, concerning the
general construction of biquadratic equations, by a para-
bola and a circle ; the title of which book at full length
is, " The Geometrical Key; or the Gate of Equations un-
locked : or a new Discovery of the Construction of all
Equations, howsoever affected, not exceeding the 4th de-
gree, viz, of Linears, Quadratics, Cubics, Biquadratics,
and the finding of all their Roots, as well False as True,
without the use of Mesolabe, Trisection of Angles, with-
out Reduction, Depression, or any other previous prepa-
rations, of equations by a circle, and any (and that one
only) Parabole, &c:" 1684, 4to, in English and Latin.

There is some account of this work in the Philos. Trans,
an. 16S4. And a little before his death, the Royal Society
sent him some mathematical queries ; to which he re-
turned such satisfactory answers, as procured the present
of a medal, with an inscription honourable and respectful.
Mr. Baker died at Bishop's-Nymmet, 1690, in the 65th
year of his age.

BAKER (Henry), an ingenious and diligent natura-
list, was born in London about the beginning of the 18th
century. He was brought up under an eminent bookseller,
but being of a philosophical turn of mind, he quitted that
line of business soon after the expiration of his apprentice-
ship, and engaged himself in teaching deaf and dumb
persons to speak and write &c, in which occupation, in
the course of his life, he acquired a handsome fortune.
For his amusement he cultivated various natural and phi-
losophical sciences, particularly botany, natural history,
and microscopical subjects, in which he especially excelled,
having, in the year 1744, obtained the Royal Society's
gold medal, for his miscroscopical experiments on the
crystallizations and configurations of saline particles. He
had also various papers published in the Philos. Trans, of
the Royal Society, of which he was a worthy member, as
well as of the Societies of Antiquaries, and of Arts. He
was author of many pieces, on various subjects, the princi-
pal of which were, his Treatise on the Water Polype, and
two Treatises on the Microscope, viz, T/ic Microscope made
easy, and Employment for the Microscope, which have gone
through several editions.

Mr. Baker married Sophia, youngest daughter of the
celebrated Daniel Defoe, by whom he had two sons, one
2D

B A L

[ 202 ]

B A L

of whom, Henry Erskine Baker, was the author of the
first edit, of the little work called, The Companion to the
Play-house. He terminated an honourable and useful life,
at his apartments in the Strand, on the 25th of November
1774, being then upwards of 70 years of age.

BAKER's Central Rule, for the Construction of
Equations; is a method of constructing all equations, not
exceeding the 4-th degree, by means of a given parabola
and a circle, without any previous reduction of them, or
first taking away their second term. See Central Rule.
BALANCE, one of the six simple powers in mechanics,
chiefly used in determining the equality or difference of
weight in heavy bodies, and consequently their masses or
quantities of matter.

The balance is of two kinds, the ancient and modern.
The ancient or Roman balance, called also Statera Ro-
mana, or Steelyard, consists of a lever or beam, moveable
on a centre, and suspended near one of its extremities.
The bodies to be weighed are suspended from the shorter
end, and their weight is shown by the division marked on
the beam, where the power or constant weight, which is
moveable along the lever, keeps the steelyard in equili-
bria This balance is still in common use for weighing
heavy bodies.

The modern balance, now commonly used, consists of a
lever or beam suspended exactly at the middle, and having
scales hung from the two extremities, to receive the
weights to be weighed.

In either case the lever is called the jugum or the beam,
and its two halves on each side the axis, the brachia or
arms; also the line on which the beam turns, or which
divides it in two, is called the axis; which when consi-
dered with regard to the length of the brachia, is esteemed
only a point, and called the centre of the balance, or
centre of motion : the extremities where the weights are
applied, are the points of application or suspension; the
handle by which the balance is held, or by which the
whole apparatus is suspended, is called trutina; and the
slender part perpendicular to the beam, by which is deter-
mined cither the equilibrium or preponderancy of bodies,
is called the tongue of the balance.

From these descriptions we easily gather the charac-
teristic distinction between the Roman balance and the
common one, viz, that in the Roman balance, there is one
constant weight used as a counterpoise, the point where
it is suspended being varied ; but, on the contrary, in the
common balance or scales, the point of suspension remains
the same, and the counterpoise is varied. The principle
of both of them may be easily understood from the gene-
ral properties of the lever, and the following observations.
See Lever.

The beam abc, the principal part of the balance, is a
lever of the first kind ; but instead of resting on a fulcrum,
it is suspended by a handle, &c, fastened to its centre of
motion b : and hence the mechanism of the balance de-
pends on the same theorems as that of the lever. Conse-
quently as the distance between the centre of motion and
the place of the unknown weight, is to the distance be-
tween the same centre and the place of the known weight,
so is the latter weight to the former. So that the un-
known weight is discovered by means of the known one,
and their distances from the common centre of motion ;
viz, if the distances from the centre be equal, then the two
weights will be equal also, as in the common balance ; but

if the distances be unequal, then the weights will also be
unequal, and in the very same proportion, alternately, the
less weight having so much the greater distance, as in the
steelyard.

The Common "Balance or Scales.

The two brachia ab, bc,
should be exactly equal in
length, and in weight also
when their scales d and e
are fixed on their ends; the
beam should hang exactly
level or horizontal in the
case of an equipoise; for
which purpose the centre of
gravity of the whole should
tail a little below the centre

of motion, and but a little, that the balance be sufficiently
sensible to the least variation of weight : the friction on the
centre should also be as small as possible.

The Steelyard.

Having taken a proper bar of steel ab, tapering at the
longer end, and very strong- at the other, suspend it by a
centre c near the shorter or thicker end, so that it may
exactly balance itself in equilibrio, and prepare a constant
weight 1 to weigh with: then hang on any weight, as one
pound for instance, at the shorter arm, and slide the con-
stant weight backwards and forwards upon the longer arm,
till it be just in equilibrio with the former; and there make
a notch and number 1, for the place of 1 pound: take off
the lib, and hang a two pound weight in its stead at the
shorter arm ; then slide the constant weight back on tfce
longer arm, till the whole come again into a state of equi-
librium, making a notch at the place of the constant
weight and the number 2, for the place of 21 b. Pro-
ceed in the same manner for all other weights, o, 4, 5, &c;
as also for the intermediate halves and quarters, &c, if it
be necessary; always suspending the variable weights at
the end of the shorter arm, shifting the constant weight so
as to balance them, and marking and numbering thcplaces
on the longer arm where the constant weight always makes
a counterpoise. The use of the steelyard is hence very
evident: the thing whose weight is required being suspend-
ed by a hook at the short end, move the constant weight
backwards and forwards on the longer arm, till the beam
is balanced horizontally: then look what notch the con-
stant weight is placed at, and its number will show the
weight of the body that was required, nc is the handle
and tongue; F the centre of motion ; eg a scale sometimes
hung on at the end by the hook u.

B A L

The Bent-Lever Balance.

f 203 ]

B A L

This instrument operates by a fixed weight, c, increas-
ing in power as it ascends along the arc fg of a circle, and
pointing by an index to the number or division of the arc
which denotes the weight of any body put into the scale at
E. And thus one constant weight serves to weigh all
others, by only varying the position of the arms of the
balance, instead of varying the places or points of suspen-
sion in the arms themselves.

The Deceitful, or False Balance. This operates in the
same manner as the steelyard, and cheats or deceives by
having one arm a little longer than the other; though the
deception is not perceived, because the shorter arm is made
somewhat heavier, so as to compensate for its shortness, by
which means the beam of the balance, when no weights
are in the scales, hangs horizontal in equilibrio. The con-
sequence of this construction is, that any commodity put
in the scale of the longer arm, requires a greater weight in
the other scale to balance it ; and so the body is fallacious-
ly accounted heavier than it really is. But the trick will
easily be detected by making the body and the weight
change places, removing them to the opposite scales, when
the weight will immediately be seen to preponderate.

The true weight of any thing may however be found
out by means of the false balance, in this manner: viz,
Weigh the body first in the one scale, next in the other;
then take a mean proportional between the two different
weights thus indicated, and that will be the. true weight re-
quired. Thus, suppose the two false weights be 7 and
&f; the product of these two is 36, the square root of
which, 6, is the true weight. Or, half the sum of the two
false weights, 7 and 5-f, which in this case is 6-^, will
commonly be very near the true weight, though always a
very little too much.

Assay-Balance. This is a very nice balance, used in de-
termining the exact weights of very small bodies. Its
structure is but little different from the common sort ; ex-
cept that it is made of the best and hardest steel, and made
to turn with the smallest weight. For an account of se-
veral balances, which have been constructed by different
persons for delicate experiments, consult the 06th vol. of
the Phil. Trans., also the same work for 17.09, part 2. See
also Nicholson's Chemistry, chap. 6, and Parkinson's
System of Mechanics.

Hydrostatical Balance. This is an instrument for de-
termining the specific gravity of bodies. See Hydrosta-
tical," and Specific Gravity.

Balance, in Astronomy, the same as Libra.

Balance of a Clock or Watch, is that part which, by its
motion, regulates and determines the beats. The circular
part of it is called the Rim, and its spindle the Verge ;
there belong to it also two. pallets or nuts, that play in the
fangs of the crown-wheel : in pocket-watches, that strong
stud in which the lower pivot of the verge plays, and in the
middle of which one pivot of the crown-wheel runs, is called
the Potence; the wrought piece which covers the balance,
and in which the upper pivot of the balance plays, is the
cock; and the small spring in the new pocket-watches is
called the Regulator.

Balance of Forces, in Mechanics. See Compound
Motion.

BALCONY, a prefecture in the front of a house, or
other building, commonly supported by pillars or consoles,
and encompassed by a ballustrade.

BALL, any spherical, globular, or round body.

Ball, in the military art, signifies all sorts of bullets
for fire-arms, from the pistol up to the largest cannon.
Cannon balls are made of cast-iron ; but the musket and
pistol-balls of lead, as these are both heavier under the same
bulk, and do not furrow the barrels of the pieces.

Ball of a Pendulum, is the weight at the bottom of it;
and is sometimes, especially in shorter pendulums, called
the Bob.

Balls of Fire, in Meteorology. See Fire Balls.

Balls, in Electricity, invented by Mr. Canton, are two
pieces of cork, or pith of elder-tree, nicely turned in a
lathe to the size of a small pea, and suspended by very deli-
cate threads. They are used as electrometers, and are of
excellent use to discover small degrees of electricity ; and
to observe its changes from positive to negative, or the re-
verse; as also to estimate the force of a shock before the
discharge, so that the operator shall always be able to tell
very nearly beforehand what the explosion will be, by
knowing how high he has charged his jars.

BALLISTA, a military engine much used by the an-
cients for throwing stones, darts, and javelins; it somewhat
resembled our cross-bows, but m'uch larger and stronger*
The word is Latin, signifying a cross-bow; but is derived
from the Greek fia,\\u>, to shoot, or throw.

Vcgetius informs us, that the ballista discharged darts
with such violence and rapidity, that nothing could resist
their force: and Alhenaeus adds, that Agistratus made one
of little more than 2 feet in length, that shot darts 500
paces. Authors have often confounded the ballista with
the catapulta, attributing to the one what belongs to the
other. According to Vitruvius, the ballista was made after
divers manners, though all were used for the same purpose :
one sort was framed with levers and bars; another with
pulleys; some with a crane; and others again with a
toothed wheel. Marccllinus describes the. ballista thus :
A round iron cylinder is fastened between two planks, from
which reaches a hollow square beam placed cross-wise, fas-
tened with cords, to which are added screws. At one end
of this stands the engineer, who puts. a wooden shaft, or
arrow, with a large head, into the cavity of the beam; this
done, two men bend the engine, by drawing some wheels;
when the top of the head is drawn to the utmost end of
the cords, the shaft is driven out of the ballista, tkc.

The ballista is ranked by the ancients among the sling
2 D 2

B A L

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BAR

kind, and its structure and effect reduced to the principles
of that instrument ; whence it is called by Hero and others,
Funda, and Fundibulus. Gunther calls it Balearica ma-
china, as a sling peculiar to the Balearic islands. — Perrault,
in his notes on Vitruvius, gives a new contrivance of a like
engine for throwing bombs without powder.

Fig. 1, Plate v, represents the ballista used in sieges, ac-
cording to Folard : where 2, 2, denote the base of the bal-
lista; 3, 4, upright beams; 5, 6, transverse beams; 7, 7,
the two capitals in the upper transverse beam, (the lower
transverse beam has also two similar capitals, which can-
not be seen in this transverse figure) ; 9, 9, two posts or
supports for strengthening the transverse beams; 10, 10,
two skains of cords fastened to the capitals; 11, 11, two
arms inserted between the two strands, or parts of the
skains; 12, a cord fastened to the two arms; 13, darts
which are shot by the ballista; 14, 14, curves in the up-
right beams, and in the concavity of which cushions are
fastened, in order to break the force of the arms, which
strike against them with great force when the dart is dis-
charged : l6, the arbor of the machine, in which a straight
groove or canal is formed to receive the darts, in order to
their being shot by the ballista; 17, the nut of the trigger,

18, the roll or windlass, about which the cord is wound;

19, a hook, by which the cord is drawn towards the cen-
tre, and the ballista cocked ; 20, a stage or table on which
the arbor is in part sustained.

Ballista, in Practical Geometry, the same, as the geo-
metrical cross, called also Jacob's Staff. See CB.oss-Staff.
BALLISTIC Pendulum, an ingenious machine invent-
ed by Benjamin Robins, for ascertaining the velocity of
military projectiles, and consequently the force of fired gun-
powder. It consists of a large block of wood, annexed to
the end of a strong iron stem, having a cross steel axis at
the other end, placed horizontally, about which the whole
vibrates together like the pendulum of a clock. This ma-
chine being at rest, a piece of ordnance is pointed straight
towards the wooden block, or ball of this pendulum, and
then discharged : the consequence is this ; the ball dis-
charged from the gun strikes and enters the block, and
causes the pendulum to vibrate more or less according to
the velocity of the projectile, or the force of the blow ; and
by observing the extent of the vibration, the force of that
blow becomes known, or the greatest velocity with which
the block is moved out of its place, and consequently the
velocity of the projectile itself which struck the blow and
urged the pendulum.

A more minute and particular description may be seen
in my Tracts, vols. 2 & 3, where are given the various rules
for using it, and for computing the velocities, with a mul-
titude of accurate experiments performed with cannon
balls, by means of which the most useful and important
conclusions have been deduced in military projectiles and
the nature of physics. In those Tracts are detailed at
great length many experiments of the same kind, by dis-
charging cannon balls at various distances from the block ;
from which have resulted the discovery of a complete se-
ries of the resistances of the air to balls passing through it
with all degrees of velocity, from 0 up to 2000 feet in a
second of time.

Other writers on this subject are Euler, Antoni, Le Roy,
Darcy, &c. Sec also Robins's Mathematical Tracts.

BALLISTICS, is used by some for projectiles, or the art
of throwing heavy bodies. Mcrsenne has published a trea-
tise on the projection of bodies, under this title.

BALLOON, or Ballon, in a general sense signifies any
spherical hollow body. Thus, with chemists, it denotes
a round short-necked vessel, used to receive what is dis-
tilled by means of fire: in architecture, a ball or globe on
the top of a pillar, &c : and among engineers, a kind of
bomb made of pasteboard, and played off in fireworks, in
imitation of a real iron bomb-shell.

^jV-Balloon. See Aerostation.

BALLUSTER, a small kind of column or pillar, used
for ballustrades.

BALLUSTRADE, a series or row of ballusters, joined
by a rail; serving for a rest to the arms, or as a fence or
inclosure to balconies, altars, staircases, &c.

BAND, in Architecture, denotes any flat low member,
or moulding, that is broad, but not very deep. The word
face sometimes means the same thing.

BANQUET, or Banquette, in Fortification, a little
foot-bank, or elevation of earth, forming a path along the
inside of a parapet, for the soldiers to stand upon to dis-
cover the counterscarp, or to fire on the enemy in the
moat, or in the covert way. It is commonly about 3 feet
wide, and a foot and a half high.

BARITONO, in Music, a voice of low pitch between a
tenor and bass.

BARLOWE (William), an eminent mathematician
and divine, of the l6"th century. He was born in Pem-
brokeshire, his father (William Barlowe) being then bishop
of St. David's. In 1500 he was entered commoner of Ba-
liol college, Oxford; and in 1564, having taken a degree
in arts, he left the university, and went to sea; but in what
capacity is uncertain : however he thence acquired consi-
derable knowledge in the art of navigation, as his writings
afterwards showed. About the year 1 573, he entered into
orders, and became prebendary of Winchester,_and rector
of Easton, near that city. In J 588 he was made preben-
dary of Litchfield, which he exchanged for the office of
treasurer of that church. He afterwards was appointed
chaplain to prince Henry, eldest son of king James the
First; and, in 1614, archdeacon of Salisbury. Barlowe
was especially remarkable for having been the first writer
on the nature and properties of the loadstone, 20 years be-
fore Gilbert published his book on that subject. He was
the first who made the inclinatory instrument transparent,
and to be used with a glass on both sides. It was Barlowe
also who suspended it in a compass-box, where, with 2
ounces weight, it was made fit for use at sea. He also
found out the difference between iron and steel, and their
tempers for magnctical uses. He likewise discovered the
proper way of touching magnetical needles; and of piecing
and cementing of loadstones; and also why a loadstone,
being double-capped, must take up so great a weight.

Barlowe died in the year 1625.- — His works are as
follow :

1. The Navigator's Supply, containing many things of
principal importance belonging to ilavigation, and use of
diverse Instruments framed chiefly for that purpose. Lond.
1597, 4to; dedicated to Robert earl of Kssex.

2. Magnetical Advertisement, or Diverse pertinent Ob-
servations and improved Experiments concerning the na-
ture and properties of the Loadstone. Lond. 10' 16", 4to.

3. A Brief Discovery of the idle Animadversions of
Mark Ridley, M. D. upon a treatise entitled Magnetical
Advertisements. Lond. I6'l8, 4to.

In the first of these pieces, Barlowe gave a demonstra-
tion of Wright's or Mercator's division of the meridian

BAR [

line, as communicated by a friend ; observing that " This
manner of carde has been publiquely extant in print these
thirtie yeares at least [he should have said 2S only], but
a cloude (as it were) and thicke miste of ignorance doth
keepe it hitherto concealed : And so much the more, be-
cause some who were reckoned for men of good knowledge,
have by glauncing speeches (but never by any one reason
of moment) gone about what they could to disgrace it."

. This work of Barlowe's contains descriptions of several
instruments for the use of Navigation, the principal of
which is an Azimuth Compass, with two upright sights;
and as the author was very curious in making experiments
on the loadstone, he treats well and fully upon the Sea-
Compass. He also treated still farther on the same instru-
ment in his second work, the Magnetical Advertisement.

BAROMETER, an instrument for measuring the weight
or pressure of the atmosphere ; and by that means the va-,
riations in the state of the air, foretelling the changes in
the weather, and measuring heights or depths, &c.

This instrument is founded on what is called the Torri-
cellian experiment, related below, and commonly consists
of a cylindrical glass tube, open at one end; which being
first filled with quicksilver, and then inverted with the open
end downwards into a bason of the same, the mercury de-
scends in the tube till it remains at about the height of 29
or 30 inches, according to the weight or pressure of the
atmosphere at the time, which is jflst equal to the weight
of that column of the quicksilver. Hence it follows that,
if by any means the pressure of the air be altered, it will
be indicated by the rising or falling of the mercury in the
tube; or if the barometer be carried to a higher station,
the quicksilver will descend in the cylinder, but when car-
ried to a lower place, it will rise higher in the tube, ac-
cording to the difference of elevation between the two places.

History of the Barometer. — About the beginning of the
17th century, when the doctrine of a plenum was in vogue,
it was a common opinion among philosophers, that the
ascent of water in pumps was owing to what they called
nature's abhorrence of a vacuum; and that in consequence
fluids might be raised by suction to any height whatever.
But an accident having discovered, that water could not
be raised in a pump unless the sucker reached to within
33 feet of the water in the well, it was conjectured by Ga-
lileo, who flourished about that time, that there might be
some other cause of the ascent of water in pumps, or at
least that the effect of this abhorrence was limited to the
finite height of 33 feet. Being unable to satisfy himself
on this head, he recommended the consideration of the dif-
ficulty to Torricelli, who had been his disciple. After some
time Torricelli fell upon the suspicion that the pressure of
the atmosphere was the cause of the ascent of water in
pumps; that a column of water 33 feet high was a just
counterpoise to a column of air, of the same base, and
which extended up to the top of the atmosphere; and
that this was the true reason why the water did not follow
the sucker any farther. And this conjecture was soon
alter confirmed by various experiments. Torricelli con-
sidered, that if a column of water 33 feet high were a coun-
terpoise to a whole column of the atmosphere, then a co-
lumn of mercury of about 2 feet and a half high would
also be a counterpoise to it, since quicksilver is nearly 14
times heavier than water, and so the 14th part of the height,
or 2 feet and a half, very nearly, would be as heavy as the
column of water. This reasoning was soon verified ; for
having tilled a glass tube with quicksilver, and inverted it

205 ] BAR

into a Tjason of the same, the mercury presently descended
till its height, above that in the bason, was about two feet
and a half, just as he had estimated. And this is what has,
from him, been called the Torricellian experiment.

The new opinion, with this confirmation of it, was
readily acquiesced in by most of the philosophers, who
repeated the experiment in various ways. Others however
still adhered to the old doctrine, and raised several pre-
tended objections against the new one ; such as that there
was a film' or imperceptible rope of mercury, extended
through the upper part of the tube, which retained the co-
lumn of mercury, and kept it from falling into that in the
bason. This and other objections were however soon over-
come by additional confirmations of the true doctrine, par-
ticularly by varying the elevation of the place. It was
hinted by Descartes and Pascal, that if the mercury be
sustained in the tube by the pressure of the atmosphere, by
carrying it to a higher situation, it would descend lower
in the tube, having a shorter column of the atmosphere
to sustain it, and vice versa. In consequence of which,
Pascal engaged his brother-in-law, M. Perier, to try that
experiment for him, being more conveniently situated for
that purpose than he was at Paris. This he accordingly
executed, by observing the height of the quicksilver in
the tube, first at the bottom of the Puy de Dome, a
mountain in Auvergne, and then at several stations, or dif-
ferent altitudes, in ascending, by which it was found that
the mercury fell lower and lower all the way to the top of
the mountain ; by this means confirming the truth of the
doctrine relating to the universal pressure of the atmo-
sphere, and the consequent suspension of the mercury in the
tube of the barometer. Thus, by the united endeavours of
Torricelli, Descartes, Pascal, Mersenne, Huygens, and
others, the cause of the suspension of the quicksilver in the
tube of the barometer became pretty generally established.

It was some time however after this general consent, be-
fore it was known that the pressure of the air was various
at different times, in the same place. This could not how-
ever remain long unknown, as the frequent measuring of
the column of mercury, must soon show its variations in
altitude; and experience and observation would presently
show that those variations in the mercurial column,
were always succeeded by certain changes in the weather,
as rain, wind, frosts, &c. Hence this instrument soon
came into use as the means of foretelling the changes of
the weather; and on this account it obtained the name of
the weather-glass, as it did that of barometer from its be-
ing the measure of the weight or pressure of the air. We
may now proceed to take a view of its various forms and
uses.

The Common Barometer. This is represented at fig. 1,
plate iv, in the same form as it was invented by Torricelli.
ab is a glass tube, of |, or |, or i inch wide, the more the
better, and about 34 inches long, being hermetically seal-
ed at the top a, and the open end b immersed in a bason
of quicksilver cd, which is also the better the wider it is.
To fill this, or any other barometer ; take a clean new
glass tube, of the dimensions above named, and pour into
it well purified quicksilver, with a small funnel, either of
glass or paper, in a fine continued stream, till it wants
about half an inch or an inch of being full ; then stop-
ping it close with the finger, invert it slowly, and the air
in the empty part will ascend gradually to the other end,
collecting in its ascent such other small air-bubbles as
unavoidably get mto the tube among the mercury, wheu

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filling it with the funnel : and thus continue to invert it the tube may Lie afterwards emptied, and filled even with
several times, turning the two ends alternately upwards, cold mercury, and will be found nearly as free from the
till all the air-bubbles are collected, and brought up to air as before : the mercury in the tube thus prepared, will
the open end of the tube, or till the part filled shall ap- rise higher than in those of the common sort, and the ba-
pear, without speck, like a fine polished steel rod. This rometers will more nearly correspond with each other,
done, pour in more quicksilver, to fill the empty part whereas there will be a difference of 6 or 8 lines in the
entirely, and so exclude all air from the tube: then, stop- ascent of the mercury in the common barometers. When
ping the orifice again with the finger, invert the tube, and this operation is completed, the mercury generally re-
immerse the finger and end, thus stopped, into. a bason of mains suspended at the top, and will not descend to its
like purified quicksilver; in this position withdraw the proper level without shaking the tube to bring it down,
finder, so shall the mercury descend in the tube to some The tubes, which should not be chosen less than 3 feet long,
place, as e, between 28 and 31 inches above that in the may now be filled to their proper length. This barometer
bason at f, these being the limits between which it always is commonly fitted up in a neat mahogany case, together
stands in this country on the common surface of the earth, with a thermometer and hygrometer, as represented in
Then measure, from the surface of the quicksilver in the plate 4, fig. 13.

bason at F, 28 inches to g, and 31 inches to h, dividing As the scale of variation is but small, being included
the space between them into inches and tenths, which are within 2 inches in the common barometer, several contri-
marked on a scale placed against the side of the tube; vances have been devised to enlarge the scale, or to render
and the tenths are subdivided into hundredth parts of an the motion of the quicksilver more sensible.
inch by a sliding index carrying a vernier or nonius. (See Descartes first suggested a method of increasing the sen-
art. Vernier.) These 3 inches, between 28 and 31, so sibility, which" was executed by Huygens. This was ef-
divided, will answer for all the ordinary purposes of a sta- fected by making the barometrical tube end in a pretty
tionary or chamber barometer ; but for experiments on large cylindrical vessel at top, into which was inserted al-
altitudes and depths, it is proper to have the divisions car- so the lower or open end of a much finer tube than the
ried on somewhat higher, and a great deal lower down, former, which was partly filled with water, to give little
In the proper filling and otherwise fitting up of the baro- obstruction by its weight to the motion of the "mercury,
meter, several circumstances are to be carefully noted ; while it moved through a considerable space of the very
as, that the bore of the tube be pretty wide, to allow the fine tube by a small variation of the mercury below it, and
freer motion of the quicksilver, without being impeded so rendered the small changes in the state of the air very
by an adhesion to the sides ; that the bason below it be sensible. But the inconvenience was this, that the air con-
also tolerably large, at least 10 times larger in diameter tained in the water gradually disengaged itself, and escaped
than that of the tube, in order that the surface of the mer- through into the vacuum in the top of the small tube, till
cury at f may not sensibly rise or fall with that in the tube; it was collected in a body there, and by its elasticity pre-
fer the numbers marked in the scale annexed to the tube, venting the free rise of the fluids in the tubes, spoiled the
denote their distance from a certain fixed point, and there- instrument as a true barometer. And this, it may be ob-
fore cannot truly indicate the height of the column above served by-the-bye, is the reason why a water barometer
the mercury in the vessel, unless its surface coincides with cannotsucceed. This barometer ishere represented in fig. 2,
this point, and be immoveable; also, that the bottom of where CD is the vessel, in which are united the upper or
the tube be cut off rather obliquely, that when it rests on small water tube AC, with the lower or mercurial one cb.
the bottom of the bason there may be a free passage for To remedy this inconvenience, Huygens thought of pla-
the quicksilver; and that, to have the. quicksilver very cing the mercury at top, and the water at bottom, which
pure, it is best to boil it in the tube, in order to expel the he thus contrived, adg (fig. 3) is a bent tube hermeti-
air from it, which may be done by the following simple cally sealed at a, but open at G, of about one line in di-
process. Having procured a tube of 21 lines or 3 lines in ameter, and passing through the two equal cylindrical ves-
bore, and about half a line in thickness, fill it with mer- sels bc, ef, which are about 20 inches apart, and of 15
cury within' two inches of the. top, and hold it with the lines diameter, their length being 10. The mercury being
sealed end lowest, in an inclined position, over a chafing- put into the tube, will stand between the middle of the
dish of burning charcoal, presenting first the sealed end vessels ef and bc, the remaining space to a being void
to the fire, and moving it obliquely over the chafing-dish, both of air and mercury. Lastly, common water, tinged
As the mercury is heated, the air-bubbles will assume an with a 6th part of aqua regia, to prevent its freezing, is
appearance like so many studs on the inner surface of the poured into the tube fg, till it rises a foot above the mer-
tube, and gradually rushing into one another, ascend to- cury in bf. To prevent the water from evaporating, a
wards the higher parts of the tube, which are not heated; drop of oil of sweet almonds floats on the top of it. But
here they are condensed and almost disappear, and after the column of water will be sensibly affected by heat and
repeated emigrations they acquire a bulk by their union, cold, and consequently the accuracy of the instrument will
which enables then) at length to escape. When the mer- be spoiled, For which reason other contrivances have
cury boils, its parts Strike against each other, and been made, as below.

against the sides of the tube, with such violence, that a The Horizontal or Rectangular Barometer, fig. 4, was in-

person unaccustomed to this operation is ready to appre- vented by .). Bernoulli and Cassini ; where All is a pretty

hend their force to bo sufficient to break the tube. The wide cylindrical part at the top of the tube, which tube

mercury is thus freed from all heterogeneous particles is bent at right angles at C, the lower part of it en being

contained in it, together with their surrounding atmo- turned into the horizontal direction, and closed above at

spheres, and the air which lines the inside of the tube, a, but open at the lower end D, where the mercury can-

which cannot be expelled in any other way, is discharged ; not run out, being opposed by the pressure of the atmo-'

when this last-mentioned stratum of air is thus expelled, sphere. This and the foregoing contrivance of Huygens

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are founded on a theorem in hydrostatics, viz, that fluids
of the same base, press according to their perpendicular
altitude, not according to the quantity of their matter; so
that the same pressure of the atmosphere sustains the
quicksilver that fills the tube acd, and the cistern b, as
would support the mercury in the tube alone. Hence,
having fixed upon the size of the scale, as suppose the ex-
tent of 12 inches, instead of the 3, in the common baro-
meter from 28 to 31, that is 4 times as long; then the
area of a section of the cylinder .ab must be 4 times that
of the tube, and consequently its diameter double, since
the areas of circles are as the squares of their diameters :
then for every natural variation of an inch in the cylinder
ab, there will be a variation of 4 inches in the tube CD. —
But on account of the attrition of the mercury against the
sides of the glass, and the great momentum from the quick
motion in CD, the quicksilver is apt to break, and the rise
and fall is no longer equable ; and besides, the mercury is
apt to be thrown out of the orifice at d by sudden mo-
tions of the machine.

The Diagonal Barometer of Sir Samuel Moreland, fig. 5,
is another method of enlarging the natural scale of 3 inches
perpendicular, or cd, by extending it to any length bc in
an oblique direction. This is liable in some degree to the
same inconvenience, from friction and breaking, as the
horizontal one: and hence it is found that the diagonal
part bc cannot properly be bent from the perpendicular
more than in an angle of 45°, which only increases the
scale nearl3' in the proportion of 7 to 5.

Dr.Hooke's Wheel Barometer, fig. 6. This was invented
about l6oS, and is meant to render the alterations in the
air more sensible. Here the barometer-tube has a large
ball ab at top, and is bent up at the lower or open end,
where an iron ball G floats on the top of the mercury in
the tube, to which is connected another ball h by a cord,
hanging freely over a pulley, turning an index kl about
its centre. When the mercury rises in the part l'G, it raises
the ball, and the other ball descends and turns the pulley
with the index round a graduated circle from N towards
m and p; and the contrary way when the quicksilver and
the ball sink in the bent part of the tube. Hence the scale
is easily enlarged 10 or 12 fold, being increased in pro-
portion of the axis of the pulley to the length of the index
kl. But then the friction of the pulley and axis is some
obstruction to the free motion of the quicksilver. Con-
trivances to lessen the friction &c, may also be seen in the
Philos. Trans, vol. .52, art. 29, and vol. ()0, art. 10.

The Steelyard Barometer, for so that may be called
which is represented by fig. 7, which enlarges the scale in
the proportion of the shorter to the longer arm of the steel-
yard, ab is the barometer tube, closed at a and open at
b, immersed in a cylindrical glass cistern cd, which is but
very little wider than the tube ab is. The barometer
tube is suspended to the shorter arm of an index like a

tons, in l6$5. It consists of a single conical tube ab,
hung up by a thread, the larger or open end downwards,
and having no vessel or cistern, because the conical figure
supplies that, and the column of mercury sustained is al-
ways equal to that in the common barometer tube; which
is effected thus : when the pressure of the air is less, the
mercury sinks-down to a lower and wider part of the tube,
and consequently the altitude of its column will be less ;
and on the contrary, by a greater pressure of the atmo-
sphere, the mercury is forced up to a higher and narrower
part, till the length of the column cd be equal to that in
the tube of the common barometer. — The inconvenience
of this barometer is, that as the bore must be made very
small, to prevent the mercury from falling out by an ac-
cidental shake, the friction and adhesion to the sides of the
tube prevent the free motion of the mercury.

Mr. Roivning's Compound Barometers. This gentleman
has several contrivances for enlarging the scale, and that
in any proportion whatever. One of these is described
in the Philos. Trans. No. 427, and also in his Nat. Philos.
part 2; and another in the same part,, which is here re-
presented at fig. 0. abc is a compound tube, hermeti-
cally sealed at a, and open at c; empty from a to d, fill-
ed with mercury from thence to b, and from hence to e
with water. Hence by varying the proportions of the two
tubes af and fc, the scale of variation may be changed in
any degree.

The Marine Barometer. This was first invented by Dr.
Hooke, to be used at sea, being contrived so as not to be
affected or injured by the motion of the ship. His con-
trivance consisted of a double thermometer, or a couple
of tubes half filled with spirit of wine ; the .one sealed at
both ends, with a quantity of air included ; the other seal-
ed at one end only. The former of .these is affected only
by the warmth of the air; but the other is affected both
by the external warmth and by the variable pressure of
the atmosphere. Hence, considering the spirit thermo-
meter as a standard, the excess of the rise or fall of the
other above it will show the increase or decrease of the
pressure of the atmosphere. This instrument is described
by Dr. Halley, in the Philos. Trans. No. 269, where he
says of it, " I had one of these barometers with me in my
late southern voyage, and it never failed to prognosticate
and give early notice of all the bad weather we had, so
that I depended thereon, and made provision accordingly;
and from my own experience I conclude, that a more use-
ful contrivance hath not for this long time been offered for
the benefit of navigation." — Dr. Hooke also contrived an-
other barometer, similar to that of Mr. Rowning above
described. The Rev. J. Wilson has also described a me-
thod of insuring the sensibility of the barometer ad libi-
tum. See Nich. Journal, No. 9, new series.

Mr. Nairne, an ingenious artist in London, has lately
invented a new kind of Marine Barometer; which differs

steelyard, moving on the fulcrum e, and the extremity of from the common barometer by having the lower part of

its longest arm pointing to the divisions of a graduated
arch, with which index the tube is nearly in cquilibrio.
When 'the pressure of the atmosphere is lessened, the mer-
cury descends out of the tube into the cistern, which raises
the tube and the shorter arm of the index, and conse-
quently the extremity of the longer moves downwards,
and passes over a part of the graduated arch. And on the
contrary this moves upwards when the pressure of the
atmosphere increases.

TIlc Pendant Barometer, fig.8, was invented by M.Amon-

the tube, for about 2 feet in length, made very small, to
check the vibrations of the mercury, which would other-
wise arise from the motions of the ship. This is also assist-
ed by being fixed in gimbals, by a part which subjects it
to be the least affected by such motions.

Another sort of Marine Barometer has also been in-
vented by M. Passementc, an ingenious artist at Paris.
This contrivance consists only in twisting the middle of
the tube into a spiral of two revolutions; by which contri-
vance the impulses which the mercury receives from the

BAR [ 208 ]

motions of the ship are destroyed, by being transmitted
in contrary directions.

Messrs. W. and S. Jones of London, have also con-
structed two excellent Marine Barometers, which seem to
be well adapted to the purposes intended.

The Statical Baroscope, or Barometer, of Mr. Boyle, &c.
This consists of a large glass bubble, blown very thin, and
then balanced by a small brass weight. Hence these two
bodies being of unequal bulk, the larger will be. very much
affected by a change of the density of the medium, but the
less not at all that is perceptible : so that, when the atmo-
sphere becomes denser, the ball loses more of its weight,
and the brass weight preponderates ; and contrariwise
when the air grows lighter.

Mr. Cabell's Baroscope, or Barometer. This is described
in the Philos. Trans, vol. 24, and seems to be the most
sensible and exact of any. It is thus described : Suppose
Aucn (Fi<*. 10) a bucket of water, wherein is placed the
baroscope xrezyosm, which consists of a body xrsm, and a
tube ezyo, which are both concave cylinders, made of tin,
or rather glass, and communicating with each other. The
bottom of the tube zy has a leaden weight to sink it, so
that the top of the body may just swim even with the sur-
face of the water by the addition of some grain weights on
the top. When the instrument is forced with its mouth
downwards, the water ascends into the tube to the height
yu. To the top is added a smalt concave cylinder, or pipe,
to keep the instrument from sinking down to the bottom :
md is a wire: and mS, de are two threads oblique to the
surface of the water, which perform the office of diago-
nals : for while the instrument sinks more or less by an
alteration in the gravity of the air, there where the surface
of the water cuts the thread is formed a small bubble,
which ascends up the thread while the mercury of the
common baroscope ascends, and vice versa.

It appears from a calculation which the author makes,
that this instrument shows the alterations in the air 1200
times more accurately than the common barometer. He
observes, that the bubble is seldom known to stand still
even for a minute ; that a small blast of wind, which can-
not be heard in a chamber, will sensibly make it sink; and
that a cloud passing over it always makes it descend, &c.

While some have been increasing the sensibility of the
barometer by enlarging the variations, others have endea-
voured to make it more convenient by reducing the length
of the tube. M. Amontons, in l688, first proposed this
alteration in the structure of barometers, by joining seve-
ral tubes to one another, alternately filled with mercury
and with air, or some other fluid ; and the number of
these tubes may be increased at pleasure : but the con-
trivance is perhaps more ingenious than useful.

M. Mairan's reduced Barometer, which is only 3 inches
long, serves the purpose of a manometer, in showing the
dilatations of the air in the receiver of an air-pump; and
instruments of this kind are now commonly applied to this
use.

The Portable Barometer, is so contrived that it may be
carried from one place to another without being disor-
dered. The end of the tube is tied up in a leathern bag
not quite full of mercury ; which being pressed by the air,
forces the mercury into the tube, and keeps it suspended
at its proper height. This bag is usually inclosed in a
box, through the bottom of which passes a screw, by
means of which the mercury may be forced up to the top
of the tube, and prevented from breaking it by dashing

BAR

against the top when the instrument is removed from one
station to another. It seems Mr. Patrick first made a
contrivance of this kind : but the portable barometer has
received various improvements since; and the most com-
plete of this kind has been described by M. De Luc? in
his Recherches, vol. 2, pa. 5 &c, together with the appa-
ratus belonging to it, the method of construction and use,
and the advantages attending it. Improvements have also
been suggested by Sir George Shuckburgh, and Col. Roy,
which have been carried into execution, with farther im-
provements also, by Mr. Ramsden, and other ingenious
artists in London.

Sir Henry Englefield has likewise much simplified the
portable barometer, reducing its weight to less than a
pound and a half. See Nicholson's Journal, No. 55; Phi-
los. Mag. No. 117; or Retrospect of Discoveries, No. 13.
Fig. 11 represents this instrument, as inclosed in its
mahogany case by means of three metallic rings aaa. This
case is a hollow cone, so shaped within as to contain
steadily the. body of the barometer, and is divided into
three branches from b to c, forming three legs or supports
for the. instrument when observations are making, and
sustaining it at the part d of the case, as it appears in Fig.
12, by an improved kind of gimbals, in which its own
weight renders it sufficiently steady at any time.

A thermometer is always attached to the instrument, as
a necessary appendage to it, being fastened to the body at
h, and sunk into the surface of the frame, to preserve it
from injury; the degrees of this thermometer are marked
on two scales, one on each side of it, viz, the scale of
Fahrenheit, and that of Reaumur; the freezing point of
the former being at 32, and of the latter at 0. Also on
the right-hand side of these two scales there is a third,
called a scale of correction, placed oppositely to that of
Fahrenheit, with the word add and subtract marked, which
shows the necessary correction of the observed altitude of
the mercury at any given temperature of the air, indicated
by the thermometer.

The Common Chamber Weatherglass, is also usually fitted
up in a neat mahogany frame, and other embellishments,
to make it an ornamental piece of furniture. It consists
of the common tube barometer, with a thermometer by
the side of it, and an hygrometer at the top, as exhibited
in fig. 13.

To the foregoing may be added a new sort of Barome-
ter, or Weather Instrument by the Sound of a Wire. This is
mentioned by M. Lazowski in his tour through Switzer-
land : it is as yet but in an imperfect state, and was lately
discovered thereby accident. It seems that a clergyman,
though near-sighted, often amused himself with firing at
a mark, and contrived to stretch a wire so as to draw the
mark to him to see how he had aimed. He observed that
the wire sometimes sounded as if it vibrated like a musical
chord ; and that after such soundings, a change always en-
sued in the state of the atmosphere ; from whence he came
to predict rain or fine weather. On making farther ex-
periments, it was found that the sounds were most distinct
when extended in the plane of the meridian. And ac-
cording to the weather which was to follow, it was found
that the sounds were more or less soft, or more or less con-
tinued ; also fine weather, it is said, was announced by the
tones of counter-tenor, and rain by those of bass. It has
been said that M. Volta mounted 15 chords in this way at
Pavia, to bring this method to some precision, but no ac-
counts have yet appeared of the success of his observations.

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[

]

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Humbolt, the celebrated traveller, contrived a portable
barometer, composed of several parts, which could be ea-
sily put together, and which should correct itself at each
observation, like an astronomical instrument. In this ba-
rometer the tube is distinct from the scale; and when this
breaks, its place can be supplied by another, even on the
top of a mountain.

The Phenomena and Observations of the Barometer. — The
phenomena of the barometer are various; but authors are
not yet agreed upon the causes of them ; nor is the use of
it, ,as a weather-glass, yet perfectly ascertained, though
daily observations and experience lead us still nearer to
precision. Mr. Boyle observes, that the phenomena of
the barometer are so precarious, that it is exceedingly
difficult to form any certain general rules concerning the
rise and fall of the mercury. Even that rule fails which
seems to hold the most generally, viz, that the mercury is
low in high winds. The best rules however that have been
tleduced by several authors are as follow :

Dr. Halley's Rules for judging of the Weather.

1. In cairn weather, when the air is inclined to rain,
the mercury is commonly low.

2. In serene, good, and settled weather, the mercury is
generally high.

3. Upon very great winds, though they be not accom-
panied with rain, the mercury sinks lowest of all, accord-
ing to the point of the compass the wind blows from.

4. The greatest heights of the mercury are found upon
easterly or north-easterly winds, other circumstances alike.
— The above four observations made by Dr. Halley in
England appear to be most universal, as they were found
by Mr. Melander to lat. 39°, and by M. deLuc to lat. 46°.

5. In calm frosty weather, the mercury commonly
stands high.

6. After very great storms of wind, when the mercury
has been very low, it generally rises again very fast.

7. The more northerly places have greater alterations
of the barometer than the more southerly, near the
equator.

8. Within the tropics, and near them, there is little or
no variation of the barometer, in all weathers. For in-
stance, at St. Helena, it is little or nothing, at Jamaica
but 3-10ths of an inch, and at Naples the variation sel-
dom exceeds an inch ; whereas in England it amounts to
2 inches and a half, and at Petersburgh to 3-j nearly.

Dr. Beal, who followed the opinion of M. Pascal, ob-
serves that, casteris paribus, the mercury is higher in cold
weather than in warm : and in the morning and evening
usually higher than at mid-day. — That in settled and fair
weather, the mercury is higher than either a little before
or after, or in the rain; and that it commonly descends
lower after rain than it was before it. And he ascribes
these effects to the vapours with which the air is charged
in the former case, and which are dispersed by the falling
rain in the latter. If it chance to rise higher after rain,
it is usually followed by a settled serenity. And that there
are often great changes in the air, without any perceptible
alteration in the barometer.

Mr. Patrick's Rules for judging of the Weather. — These
are esteemed the best of any general rules hitherto made.

1. The rising of the mercury presages, in general, fair
weather ; and its falling, foul weather, as rain, snow, high
winds, and storms.

2. In very hot weather, the falling of the mercury in-
dicates thunder.

Vol. I..

3. In winter, the rising forebodes frost: and in frosty
weather, if the mercury fall 3 or 4 divisions, there will
certainly follow a thaw. But in a continued frost, if the
mercury rises, it will certainly snow.

4. When foul weather happens soon after the falling of
the mercury, expect but little of it; and on the contrary,
expect but little fair weather when it proves fair shortly
after the mercury has risen.

5. In foul weather, when the mercury rises much and
high, and so continues for 2 or 3 days before the foul
weather is quite over, then expect a continuance of fair
weather to follow.

6. In fair weather, when the mercury falls much and
low, and thus continues for 2 or 3 days befors the rain
comes ; then expect a great deal of wet, and probably
high winds.

7. The unsettled motion of the mercury, denotes un-
certain and changeable weather.

8. You are not so strictly to observe the words engraved
on the plates, as the mercury's rising and falling ; though,
in general it will agree with them. For if it stands at
much rain, and then rises up to changeable, it presages fair
weather ; though not to continue so long as if the mercury
had risen higher. And so, on the contrary, if the mer-
cury stood at fair, and falls to changeable, it presages foul
weather; though not so much of it as if it had sunk lower.

'Upon these rules of Mr. Patrick, the following remarks
are made by Mr. Rowning; who also observes, that it is
not so much the absolute height of the mercury in the
tube that indicates the weather, as its motion up and
down : wherefore, to pass a right judgment of what
weather is to be expected, we ought to know whether the
mercury is actually rising or falling ; to which end the
following rules are of use.

1. If the surface of the mercury be convex, standing
higher in the middle of the tube than at the sides, it is a
sign that the mercury is then rising.

2. But if the surface be concave, or hollow in the mid-
dle, it is then sinking. And,

3. If it be plain, or rather a little convex, the mercury
is stationary : for mercury being put into a glass tube, es-
pecially a small one, naturally has its surface a little con-
vex, because the particles of incrcury attract one another
more forcibly than they are attracted by glass. Farther,

4. If the glass be small, shake the tube; then if the air
be grown heavier, the mercury will rise about half a 10th
of an inch higher than it stood before ; but if it be grown
lighter, it will sink as much. And, it may be added, in
the wheel or circular barometer, tap the instrument gently
with the finger, and the index will visibly start forwards
or backwards according to the tendency of the fluid to
rise or fall at that time. This proceeds from the mer-
cury's adhering to the sides of the tube, which prevents
the free motion of it till it be disengaged by the shock :
and therefore when an observation is to be made with such
a tube, it ought to be first shaken ; for sometimes the mer-
cury will not vary of its own accord, till the weather is
present which it ought to have indicated.

To the foregoing rules may be added the following ad-
ditional ones, deduced from later and more close observa-
tion of the motions of the barometer, and the consequent
changes in the air in this country.

1. In winter, spring, and autumn, the sudden falling of
the mercury, and that for a large space, denotes high
winds and storms ; but in summer it presages heavy
2E

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c

3

BAR

showers, and thunder: and it always sinks lowest of all
for great winds, though not accompanied with rain ; though
it falls more for wind and rain together than for either of
them alone. Also, if, after rain, the wind change into any
part of the north, with a clear and dry sky, and the mer-
cury rise, it is a certain indication of fair weather.

2. After very great storms of wind, when the mercury
has been low, it commonly rises again very fast. In settled
fair and dry weather, except the barometer sink much,
expect but little rain; for its small sinking then, is only
for a little wind, or a few drops of rain; and the mercury
soon rises again to its former station. In a wet season,
suppose in hay-time and harvest, the smallest sinking of
the mercury must be regarded; for when the constitution
of the air is much inclined to showers, a little sinking in
the barometer then denotes more rain, as it never then
stands very high. And, if in such a season it rises sud-
denly, very fast, and high, expect not fair weather more
than a day or two, but rather that the mercury will fall
again very soon, and an immediate rain follow : the slow
gradual rising, and keeping on to do so for 2 or 3 days, are
most to be depended on for a week's fair weather. And
the unsettled state of the quicksilver always denotes un-
certain and changeable weather, especially when the mer-
cury stands any where about the word changeable on the
scale.

3. The greatest heights of the mercury, in this country,
are found upon easterly and north-easterly winds ; and it
may often rain or snow, the wind being in these points,
and the barometer sinks little or none, or it may even be
in a rising state, the effect of those winds counteracting.
But the mercury sinks for wind, as well as rain, in all the
other points of the compass; but rises as the wind shifts
about to the north or the east, or between those two
points : but if the barometer should sink with the wind in
that quarter, expect it soon to change from thence; or
else, should the fall of the mercury be much, a heavy rain
is then likely to ensue, as it sometimes happens.

Cause of the Phenomena of the Barometer.

To account for the foregoing phenomena of the baro-
meter, many hypotheses have been framed, which may be
reduced to two general heads, viz, mechanical and chemi-
cal. The chief writers on these causes, are Pascal, Beal,
Wallis, Garcin, Garden, Lister, Halley, Garsten, Labile,
Mariotte, Le Cat, Woodward, Leibnitz, De Mairan, Bam-
berger, 1). Bernoulli, Muschenbroek, Chambers, Ue Luc,
Black, &c : and an account of most of their hypotheses
may be seen at large in M. De Luc's Recherches sur les
Modifications de l'Atmosphere, vol. 1, chap. 3 ; see also
the Philos. Trans, and various other works on this subject.
It may suffice to notice here slightly a few of the principal
of them.

Dr. Lister accounts for the changes of the barometer
from the alterations by heat and cold in the mercury it-
selfi; contracting by cold, and expanding by heat. But
this, it is now well known, is quite insufficient to account
for the whole of the effect.

The changes in the weight or pressure of the atmosphere
must therefore be regarded as the principal cause of those
in the barometer. But then, the difficulty will be to as-
sign the cause of, that cause, or whence arise those alter-
ations that take place in the atmosphere, which are some-
times so great as to alter its pressure by the 10th part of
the whole quantity. It is probable that the winds, as
driven about in different directions, have a great share in

them; vapours and exhalations, rising from the earth, may
also have some share; and some perhaps the flux and re-
flux occasioned in the air by the moon ; as well as some
chemical causes operating between the different particles
of matter.

Dr. Halley thinks the winds and exhalations sufficient;
and on this principle gives a theory, the substance of which
may be comprised in what follows:

1st, That the winds must alter the weight of the air in
any particular country; and this, either by bringing toge-
ther a greater quantity of air, and so loading the atmo-
sphere of any place ; which will be the case as often as
two winds blow from opposite parts, at the same time, to-
wards the same point : or by sweeping away some part of
the air, and giving room for the atmosphere to expand it-
self; which will happen when two winds blow opposite
ways from the same point at the same time : or lastly, by
cutting off the perpendicular pressure of the air ; which
is the case when a single wind blows briskly any way ; it
being found by experience, that a strong blast of wind,
even made by art, will render the atmosphere lighter; and
hence the mercury in a tube below it, as well as in others
more distant, will considerably subside. See Philos. Trans.
No. 292.

2dly, That the cold nitrous particles, and even the air
itself condensed in the northern regions, and driven else-
where, must load the atmosphere, and increase its pressure.

3dly, That heavy dry exhalations from the earth must
increase the weight of the atmosphere, as well as its elastic
force; as we find the specific gravity of menstruums in-
creased by dissolved salts and metals.

4thly, That the air being rendered heavier by these and
the like causes, is thence better able to support the va-
pours; which being also intimately mixed with it, make
the weather serene and fain Again, the air being made
lighter from the contrary causes, it becomes unable to sup-
port the vapours with which it is replete ; these therefore
precipitating, are collected into clouds, the particles of
which in their progress unite into drops of rain.

Hence, he infers, it is evident that the same causes which
increase the weight of the air, and render it more able to
support the mercury in the barometer, do likewise pro-
duce a serene sky, and a dry season; and that the same
causes which render the air lighter, and less able to sup-
port the mercury, do also generate clouds and rain.

But these principles, though well adapted to many of
the particular cases of the barometer, seem however to
fall short of some of the principal and most obvious ones,
besides being liable to several objections.

For first, If the wind were the sole agent in effecting
these alterations, we should have no sensible alterations
without a sensible wind, nor any wind without some alter-
ation of the barometer; both which are contrary to ex-
perience.

2dly, If two winds be supposed blowing from the same
place, viz, London, opposite ways, viz, N. l". and s. w.
there will be two others blowing from opposite points, viz,
N. w. and s. E. to the same place; which two last will
balance tho first, and bring as much air towards the point
as the others swept from it. This follows from the laws
of fluidity.

3dly, If the wind were the sole agent, the alterations
in the height of the mercury would only be relative, or
topical ; there would be still the same quantity supported
at several places taken collectively; thus, what a tube at

BAR

[ 211 ]

BAR

London lost, another at Paris, or at Pisa, or at Zurich,
would gain. But we find the very contrary true in fact ;
for, from all the observations hitherto made, the barome-
ters in several parts of the globe rise and fall together; so
that it must be some alteration in the absolute weight of
the atmosphere that accounts for the rise and fall of the
mercury.

Lastly, setting aside these popular phenomena, the mer-
cury's fall before, and rise after rain, seem to be inexpli-
cable on the ground of this hypothesis : for suppose two
contrary winds sweeping the air from over London ; we
know that few if any of the winds reach above a mile
high ; all therefore that they can do, will be to cut off a
certain part of the column of air over London : if the
consequence of this be the fall of the mercury, yet there
is no apparent reason for the rain's following it. The va-
pours indeed may be let lower, but it will only be till they
come into an air of the same specific gravity with them-
selves ; and there they will be suspended as before.

Leibnitz accounted for the fall of the mercury before
rain on another principle, viz, That as a body specifically
lighter than a fluid, while it is sustained by it, adds more
weight to that fluid than when, by being reduced in bulk,
it becomes specifically heavier, and descends; so the va-
pour, after it is reduced into the form of clouds, and de-
scends, adds less weight to the air than it did before ; and
hence the mercury sinks in the tube. — But here, granting
that the drops of rain formed from the vapours always in-
creasing in size as they fall lower, were continually acce-
lerated also in their motion, and<so the air suffer a conti-
nued loss of their weight as they descend ; it may however
be objected, that by the descent of the mercury the rain
is foretold a much longer time before it comes, than the
vapour can be supposed to take up in falling: trjat many
times, and in different places, there falls a great deal of
rain, without any sinking of the mercury at all ; as also
that there often happens a fall of the mercury, without any
rain ensuing: and that sometimes the mercury will sud-
denly sink, in a short space of time, half an inch or more,
which answers to 7 inches of rain, or about one third of
the whole quantity falling in the whole year.

M. De Luc supposes that the changes observed in the
pressure of the atmosphere, are chiefly produced by the
greater or less quantity of vapours floating in it: others
have attributed them to the same cause, but have given a
different explanation of it. His opinion is, that vapours
diminish the specific gravity, and consequently the abso-
lute weight of those columns of the atmosphere into which
they are received, and which, notwithstanding this admix-
ture, still remain of the same height with adjoining co-
lumns that consist! of pure or dry air. He afterwards vin-
dicates and more fully explains this theory, and applies it
to the solution of the principal phenomena of the barome-
ter, as depending on the varying density and weight of the
atmosphere.

Dr. James Hutton, in his Theory of Rain, printed in
the Transactions of the Royal Society of Edinburgh, vol. 1,
gives ingenious and plausible reasons for thinking that the
diminution of the weight of the atmosphere by the fall of
rain, is not the cause of the descent of the mercury in the
barometer tube; but that the principal, if not the only
cause, arises from the commotions in the atmosphere, which
are chiefly produced by sudden changes of heat and
cold in the air. " The barometer," he observes, " is an
instrument necessarily connected with motions in the at-

mosphere; but it is not equally affected with every motion
in that fluid body. The barometer is chiefly affected by
those motions by which there are produced accumulations
and abstractions of this fluid, in places or regions of suffi-
cient extent to affect the pressure of the atmosphere on the
surface of the earth. But, as every commotion in the at-
mosphere may, under proper conditions, be a cause of rain,
and as the want of commotion in the atmosphere is natu-
rally a cause of fair weather, this instrument may be made
of great importance for the purpose of meteorological ob-
servations, though not in the certain and more simple man-
ner in which it has been, with the increase of science, so
successfully applied to the measuring of heights." See
Rain.

In the Encyclopedia Britannica there is another theory
of the changes in the barometer, as depending on the heat
in the atmosphere, not as producing commotions there, but
as altering the specific gravity of the air by the changes of
heat and cold. The principles of this theory are, — 1st,
That vapour is formed by an intimate union between the
elements of fire and water, in consequence of which the
fire or heat is so totally enveloped, and its action so per-
fectly suspended by the aqueous particles, that it not only
loses its properties of burning and of giving light, but be-
comes incapable of affecting the most sensible thermome-
ter; in which case it is said by Dr. Black to be in a latent
state: and 2d, That if the atmosphere be affected by any
unusual degree of heat, it thence becomes incapable of sup-
porting so long a column of mercury as before; for which
reason the barometer sinks.

From these axioms it would follow, that as vapour is
formed by the union of fire with water, whether by attrac-
tion or a solution of the water in the fire, the vapour can-
not be condensed till this union, attraction, or solution, be
at an end. Hence the beginning of the condensation of
the vapour, or the first signs of approaching rain, must be
the separation of the fire which is latent in the vapour.
In the beginning, this may be either slow and partial, or
it may be sudden and violent : in the first case, the rain
will come on slowly, and fall after a considerable time ;
but in the other, it will accumulate very quickly, and in
a great quantity. But Dr. Black has proved, that when
fire quits its latent state, however long it may have lain
dormant and insensible, it always reassumes its proper
qualities, and affects the thermometer just the same as if
it had never been absorbed. The consequence of this must
be, that in proportion as the latent heat is discharged from
the vapour, those parts of the atmosphere into which it is
discharged must be sensibly affected by it; and in propor-
tion to the heat communicated to those parts, they will
become specifically lighter, and the mercury will sink of
course.

In the Memoirs of the Literary Society of Manchester,
vol. 4, is also a curious paper on this subject, viz. Me-
teorological Observations made on different Parts of the
Western Coast of Great Britain : arranged by T. Garnett,
M. D. This paper is composed of materials furnished by
several observers ; those of Mr. Copland, surgeon at Dum-
fries, are of special importance. This gentleman is of opi-
nion that the changes of the barometer indicate approach-
ing hot and cold weather, with much more certainty than
dry and wet. " Every remarkable elevation of the baro-
meter," he says, " where it is of any duration, is followed
by very warm or dry weather, and moderate as to wind,
or by all of them : but heat seems to have most influence
*E2

BAR

[ -a 12 ]

BAR

and connexion; and when it is deficient, the continuance has got to the west, or rather a little to the northward of

of the other two will be longer and more remarkable ; there- it, when, it may be added, it commonly blows with some

fore the calculation must be in a compound ratio of the violence.'

excess and deficiency of the heat, and of the dryness of Many other observations on the barometer, the weather,

the weather in comparison of the medium of the season ; &c, may be seen in various parts of the Philos. Trans.

and with regard to the want of strong wind, it appears to And for some curious papers on the same, and other sub-

be intimately connected with the last, as they show that
no precipitation is going on in any of the neighbourin
regions."

In his 14th and 15th remarks, he had said,

jects connected with the barometer, see the Gentleman's
Magazine for 1789, p- 317; also Greu's Journal of Nat.
Philos. printed at Leipzig 1792, for the influence of the
sun and moon on the barometer. See also a very ela-
14th, That the barometer being lower, and continuing borate paper on this subject in vol. 2, p. 49, &c, of the
so longer than what can be accounted for by immediate Royal Irish Trans, by Mr. Kirwan.

falls, or stormy weather, indicates the approach of very
cold weather for the season ; and also, cold weather, though
dry, is always accompanied by a low barometer, till near
its termination.'

' 15th, That warm weather is always preceded and
mostly accompanied by a high barometer; and the rising
of the barometer in the time of broken or cold weather, is
a sign of the approach of warmer weather ; and also if the
wind is in any of the cold points, a sudden rise of the ba-
rometer indicates the approach of a southerly wind, which
in winter generally brings rain with it.'

In the two following remarks, Mr. Copland had ex>

The Barometer applied to the Measuring of Altitudes.
The secondary character of the barometer, namely, as
an instrument for measuring accessible heights or depths,
was first proposed by Pascal and Descartes, as has been
before observed ; and succeeding philosophers have been
at great pains to ascertain the proportion between the fall
of the barometer and the height to which it is carried; as
Halley, Mariotte, Maraldi, Scheuchzer, J. Cassini, D.Ber-
noulli, Horrebow, Bouguer, Shuckburgh, Roy, and more
especially by De Luc, who has given a critical and histo-
rical detail of most of the attempts that have at different
times been made for applying the motion of the mercury

plained certain phenomena from a principle similar to that in the barometer to the measurement of accessible heights.

on which Dr. Darwin has so much insisted: Botanic And for this purpose serves the portable barometer, before

Garden, I. notes p. 79, &c. described, (fig. 11 and 12, plate 4,) which should be made

' That the falling of the barometer may proceed from with all possible accuracy. Various rules have been given

a decomposition of the atmosphere occurring around or by the writers on this^subject, for computing the height

near that part of the globe where we are placed, which ascended from the given fall of the mercury in the tube of

will occasion the electricity of the atmosphere to be re- the barometer, the most accurate of which was that of Dr.

pelled upwards in fine lambent portions; or driven down- Halley, till it was rendered much more'so by the indefati-

wards or upwards in more compacted balls of fire; or gable researches of De Luc, by introducing into it the cor-

lastly, to be carried along with the rain, &c, in an imper- rections of the columns of mercury and air, on account of

ceptible manner to the surface of the earth: the precipi- heat. Other corrections and modifications of the same

tation of the watery parts generally very soon takes place, may be seen inserted under the article Atmosphere,

which diminishes the real gravity of the atmosphere, and
also by the decomposition of some of the more active parts,
the air loses part of that elastic and repulsive power which
it scr eminently possessed, and will therefore press with less
force on the mercury of the barometer than before, by
which means a fall ensues.

' That the cause of the currents of air, or winds, may
also be this way accounted for: and in very severe storms,
where great decompositions of the atmosphere take place,
this is particularly evident, such as generally occur in one cury in the barometer tube at the bottom, and m that at
or more of the West-India islands at one time, a great loss the top of the hill, or other eminence; which lengths may

where the most correct rule, is deduced from one single ex-
periment only.

The Rule for computing Altitudes, is this,

Viz, 10000 x log. of - is the altitude in fathoms, in the

' ° m

temperature of 31°: and for every degree of the thermo-
meter above that, the result must be increased by so many
times its 435th part, and diminished when below it: in
which theorem m denotes the length of the column of mer-

of real gravity, together with a considerable diminution of

the spring of the air immediately ensues ; hence a current

commences, first in that direction whence the air has most

gravity, or is most disposed to undergo such a change;

but it being soon relieved of its superior weight or spring

on that side, by the decomposition going on as fast as the

wind arrives on the island, it immediately veers to another

point, which then rushes in mostly with an increase of of any height or depth, proposed to be measured; together

force; thus it goes on till it has blown more than half way with the temperature of the mercury by means of the ther-

round the points of the compass during the continuation momcler attached to the barometer, and also the temper*

of the hurricane. For in this manner the West-India phe- ture of the air in the shade by another thermometer which

be expressed in any one and the same sort of measures,
whether feet, or inches, or tenths, &c, and either English,
or French, or of any other nation; but the. result in this
case is always in fathoms, of 0 English feet each.

And the Precepts, in words, for the practice of measuring
altitudes by the barometer, are these following:

1st, Observe the height of the barometer at the bottom

nomena, as well as the alteration of the wind during heavy
rains in this country, can only be properly accounted for.'
See remark No. 4.

Mr.C.'s 4th aphorism is, ' That the heaviest rains, when
of long continuance, generally begin with the wind blow

is detached from the barometer.

2dly, Let the same thing be done also at the top of the
said height or depth, and as near the time of the former as
possible. And let those altitudes of mercury be reduced
to the same temperature, if it be thought necessary, by cor-

ing easterly, when it gradually veers round to the south ; retting either the one or the other, viz, augmenting the
and that the rain does not then begin to cease til) the wind height of the mercury in the colder temperature, or dimi-

BAR

[ 213 ■]

BAR

nishing that in the warmer, by its 9600th part for every
decree of difference between the two ; and the altitudes of
mercury so corrected, are those which are denoted by m
and m in the algebraic formula above.

3dly, Take out the common logarithms of the two heights
of mercury, so corrected, and subtract the less from the
greater, cutting oft" from the right hand side of the remain-
der three places for decimals; so shall those on the left be
fathoms in whole numbers, the tables of logarithms being
understood to be such as have 7 places of decimals.

4thly, Correct the number last found, for the difference
of the temperature of the air, as follows : viz, Take half
the sum of the two temperatures of the air, shown by the
detached thermometers, for the mean one ; and for every
degree which this differs from the standard temperature of
31°, take so many times the 435th part of the fathoms above
found, and add them if, the mean temperature be more
than 31°, but subtract them if it be below 31°; so shall
the sum or difference be the true altitude in fathoms, or
being multiplied by 6, it will give the true altitude in
English feet.

Example 1. Let the state of the barometers and thermo-
meters be as follows, to find the altitude: viz.

Thermometers.

Baromet

detached.

attached.

57

57

29'68 lower

42

43

25'28 upper

ean 49£ dif. 14

As 96OO

mean 49-|
stand. 31
dif. Til

14

29-68
cor. "04

•U4-

m = 29-64 - 471S7S2
m = 25-28 - 4027 771
As 435 : 18i : : 6'91'01 1 : 29-38S
cor. add 29'3S8
the altitude^
sous

altitude")
ight is j

720-399 fathoms,
or 4322'394 feet.

Example 2. To find the altitude of a hill, when the state
of the barometer and thermometer, as observed at the bot-
tom and summit of the same are as follows: viz,
Thermometers. Barometers.

detached.
35
31

mean 33

attached.
41
38
dif.ir

2945
26-82

As 96OO

mean 33

stand. 31

dif. 2

29-45
•01

M = 29-44

m = 26-82
As 435 : 2 : :
cor. add
the altitude
sought is

•01
logs.

4689378

4284588
404-790

rs6

1-86

") 406-65 fathoms.

j or 2439-90 feet.
See this rule investigated under the article Pneuma-
tics, at the end.

N. B. The mean height of the barometer in London,
on an average of two observations in every day of the
year, kept at the house of the Royal Society, for many years
past, is 29-88; the medium temperature, or height of the
thermometer, according to the same, being 58°. But the
medium height at the surface of the sea, according to Sir
Geo. Shuckburgh (Philos. Trans. 1777, p. 586) is 30-04
inches, the heat of the barometer being 55°, and of the air

62°. See a learned paper in vol. 1 of the Trans, of the
R. Soc. of Edinburgh, " on the Causes which affect the
Accuracy of Barometrical Measurements; by John Play-
fair, A. M. F. R. S. Edin. and Professor of Mathematics in
the University of Edinburgh ;" also, another by Dr. Da-
men, late Professor of Mathematics and Philosophy in the
University of Leyden, entitled " Dissertatio Physica et
Mathematica de Montium Altitudine Barometro Metienda.
Accedit Refractionis AstronomicasTheofia," in Svo, at the
Hague, 1783. Also other corrections and modifications
that have been given by several philosophers, as Robison,
Trembley, Englefield, Prony, Laplace, &c.

We shall now conclude this subject by an account of
the height of some of the most remarkable mountains, &c,
on the earth, above the surface of the ocean, in feet.
Mount Puy de Dome, in Auvergne, the first? ,„„„

> Alps )

5ic de los Reyes } C

^ic du Medi - f „ V -

>icd'Ossano -f Pyre™ees -j _
^anegou -J {_

mountain measured by the barometer,
Mount Blanc
Monte Rosa

Aiguille d'Argentine - »
Monastery of St. Bernard
Mount Cenis -
Pic de los Reyes
Pic du Medi - .
Picd'Ossann -Y ty^ees

Cane^

Lake of Geneva

Mount ./Etna -

Mount Vesuvius

Mount Hecla, in Iceland

Snowden -

Ben Moir

Ben Laurs -

Ben Gloe - - ■

Shihallien

Ben Lomond - - •

Tinto - - - -

Table-Hill, Cape of Good Hope,

Gondar city, in Abyssinia,

Source of the Nile

Pic of Teneriffe

Chimboracou - - -

Cayambourou

Antisana -

Pichinha

City of Quito -

Caspian Sea below the Ocean

15662

15084

13402

794:*

9212

7620

9300

11700

8544

1232

10954

3938

4S87

3555

3723

385S

3472

3461

3180

2342

3454

8440

8082

14026'

19595

19391

19290

15070

9977

306

BAROSCOPE, a machine for showing the alterations in
the weight or pressure of the atmosphere. See Barome-
ter.

BARREL, an English vessel or cask," containing 36
gallons of beer-measure, or 32 gallons of ale-measure. The
barrel of beer, vinegar, or of liquor preparing for vinegar,
ought to contain 34 gallons, according to the standard of
the ale quart.

Barrel, in clock-work, is the cylinder about which
the spring is wrapped.

BARRICADE, or Barricado, a military term for a
fence, or retrenchment, hastily made with vessels, or bas-
kets of earth, carts, trees, stakes, or the like, to preserve
an army from the shot or assault of an enemy.

BARRIER, a kind of fence made at a passage, retrench-
ment, gate, or such like, to stop it up against an enemy.

BARROW (Isaac), avery eminent mathematician and
divine of the 17th century, was born at London in Octo-

BAR

[ 214 ]

BAR

ber, 1630, being the son of Thomas Barrow, then a linen-
draper of that city, but descended from an ancient family
in Suffolk. He was first placed at the Charter-house school
for two or three years; where his behaviour afforded but
little hopes of success in the profession of a scholar, being
much addicted to fighting, and promoting it among his
school-fellows ; but on removal to Felsted in Essex, his
disposition took a different turn, and he applied himself
with great assiduity to his studies, in which he made very
rapid advancement. He was first admitted a pensioner of
Peter-house in Cambridge; but when he came to join the
university in Feb. 1645, he was entered at Trinity-college.
In 1647 he was chosen a scholar of the house., and sub-
scribed the engagement; but afterwards repenting what
he had done, he went back and struck out his name from
the list. At this period he began to apply himself with
great diligence to the study of all parts of literature, espe-
cially natural philosophy. He afterwards turned his at-
tention to the profession of physic, and made considerable
progress in anatomy, botany, and chemistry : he next stu-
died divinity; then chronology, astronomy, geometry, and
the other branches of the mathematics ; with what success,
his writings afterwards most eminently showed. In l6'4-9
he was chosen fellow of his college.

When Dr. Duport resigned the chair of Greek professor,
he recommended his pupil Mr. Barrow for his successor,
who, in his probationary exercise, showed himself equal
to the character that had been given of him by this gen-
tleman; but being suspected of favouring Arminianism, he'
was not preferred. This disappointment, it appears, de-
termined him to quit the college, and visit foreign coun-
tries ; but such were his. finances, that he was obliged to
dispose of his books, to enable him to execute that design.

He left England in June 1655, and visited France, Italy,
Turkey, &c. At several places, in the course of this tour,
he met with much kindness and liberal assistance from the
English ambassadors, &c, which enabled him to benefit
the more from it, by protracting his stay, and prolonging
his journey. He spent more than a year in Turkey, and
returned to England by way of Venice, Germany, and Hol-
land, in 1659. At Constantinople he read over the works
of St. Chrysostom, once bishop of that see, whom he pre-
ferred to all the other fathers.

On his return home Barrow was episcopally ordained
by bishop Brownrig ; and in 1660, he was chosen to the
Greek professorship at Cambridge. In 1 662 he was elected
professor of geometry in Gresham-college ; in which sta-
tion he not only discharged his own duty, but supplied
also the absence of Dr. Pope the astronomical professor.
Among his lectures, some were on the projection of the
sphere and perspective, which are lost; but his Latin ora-
tion, previous to his lectures, is still extant. About this
time Mr. Barrow was offered a good living; but the con-
dition annexed, of teaching the patron's son, made him
refuse it, as thinking it too likeasimoniacal contract. On
the 20th of May 1663 he was elected a fellow of the Royal
Society, in the first choice made by the council after their
charter. The same year the executors of Mr. Lucas hav-
ing, according to his appointment, founded a mathemati-
cal lecture at Cambridge, they selected Mr. Barrow for
the first professor; and though his two professorships were
not incompatible with each other, he chose to resign that
of Gresham-college, which he did May the 20th, 1664 ;
and at the same time his Greek professorship also. In
1669 he resigned the mathematical chair to his learned

friend Mr. Isaac Newton, having determined to quit the
study of mathematics altogether for that of divinity. On
leaving his professorship, he had only his fellowship of
Trinity-college, till his uncle gave him a small sinecure in
Wales; and Dr. Seth Ward, bishop of Salisbury, conferred
on him a prebendary in his church. In the year 1670 he
was created doctor in divinity by mandate; and, on the
promotion of Dr. Pearson, master of Trinity-college, to
the see of Chester, he was appointed to succeed him by
the king's patent, bearing date the 13th of Feb. 1672; on
which occasion the king was pleased to say, " he had given
it to the best scholar in England." In this his majesty did
not speak from report, but from his own knowledge; the
doctor being then his chaplain, he used often to converse
with him, and, in his humorous way, would call him an
"unfair preacher," because he exhausted every subject,
and left no room for others to come after him. In l6'75
he was chosen vice-chancellor of the university; and he
omitted no endeavours for the good of that society, nor in
the line of his profession as a divine, for the promotion of
piety and virtue : but his useful labours were abruptly
terminated by a fever on the 4th of May, l677, in the
47th year of his age. He was interred in Westminster
Abbey, where a monument, adorned with his bust, was
soon after erected by the contribution oi his friends.

Dr. Barrow's works are very numerous, and indeed va-
rious— mathematical, theological, poetical, &c, and such
as do honour to the English nation. They are principally
as follow:

1. Euclidis Elementa. Cantab. 1655, in 8vo.

2. Euclidis Data. Cantab. 1657, in 8vo.

3. Lectiones Optica? xviii. Lond. 1669, 4to.

4. Lectiones Geometrica xiii. Lond. 1670, 4to.

5. Archimedis Opera, Apollonii-Conicorum libri iv,The-
odosii Sphericorum lib.iii;nova methodo illustrata,et suc-
cincte demonstrata. Lond. 1675, in 4to.

The following were published after his decease, viz:

6. Lectio, in qua theoremata Archimedis de sphaira et
cylindro per methodum indivisibilium investigata, ac bre-
viter investigata, exhibentur. Lond. l67S, 12mo.

7. Mathematica3 Lectiones habitae in scholis publicis
academisCantabrigiensis, an. 1664, 5, 6, &c. Lond.l683.

8. All his English works, in 3 volumes. Lond. l683,
folio. — These are all theological, and were published by
Dr. John Tillotson.

9. lsaaci Barrow Opuscula; viz, Determinationes, Con-
dones ad Clerum, Orationes, Poemata, &c, volumen quar-
tum» Lond. l6'87, folio.

Dr. Barrow left also several curious papers on mathe-
matical subjects, written in his own hand, which were
communicated by Mr. Jones to the author of" The Lives
of the Gresham Professors," a particular account of which
may be seen in that book, in the Life of Barrow.

Several of his works have been translated into English,
and published : as the Elements and Data of Euclid; the
Geometrical Lectures; the Mathematical Lectures; and
accounts of some of them were also given in several vo-
lumes of the Phil OS. Trans.

It is remarkable, that Dr. Barrow's method of drawing
tangents to curves is exactly similar to that by the method
of fluxions, the difference being only in the notation: via,
he considered the elemental triangle, formed by the dis-
tance and the difference of two indefinitely near ordinatcs,
with the element of the curve, as similar to the triangle
formed by the ordinate, the tangent, and the subtangent;

BAR

[ 2'5 ]

B A S

which thence he found by this proportion : As the difference
of the ordinates is to their distance, so is the ordinate to
the subtangent.

Dr. Barrow must ever be esteemed, in all the subjects
which exercised his pen, a person of the clearest percep-
tion, the finest fancy, the soundest judgment, the pro-
foundest thought, and the closest and most nervous reason-
ing. " The name of Dr. Barrow (says the learned Mr.
Granger) will ever be illustrious for a strength of mind and
a compass of knowledge that did honour to his country.
He was unrivaled in mathematical learning, and especially
in the sublime geometry ; in which he has been excelled
only by his successor Newton. The same genius that
seemed to be born only to bring hidden truths to light, and
to rise to the heights or descend to the depths of science,
would sometimes amuse itself in the flowery paths of po-
etry, and compose verses both in Greek and Latin. He
at length gave himself up entirely to divinity; and parti-
cularly to the most useful part of it, that which has a ten-
dency to make men wiser and better."

Several good anecdotes are told of Barrow, as well of
his great integrity, as of his wit, and bold intrepid spirit
and strength of body. His early attachment to fighting
when a boy is some indication of the latter; to which may
be added the two following anecdotes: In his voyage be-
tween Leghorn and Smyrna, the ship was attacked by an
Algerine pirate, which, after a stout resistance, they com-
pelled to sheer off, Barrow keeping his post at the gun to
the last. And when Dr. Pope in their conversation asked
him, " Why he did not go down into the hold, and leave
the defence of the ship to those to whom it did belong?"
He replied, " It concerned no man more than myself: I
would rather have lost my life, than to have fallen into the
hands of those merciiess infidels."

There is another anecdote told of him, which showed
not only his intrepiditv, but an uncommon goodness of dis-
position, in circumstances where an ordinary share of it
would have been probably extinguished. Being once on a
visit at a gentleman's house in the country, where a large
mastiff was kept, as he was going out one morning before
day (for he w;is a very early riser) the fierce animal that
used to be chained up during the day, and let loose at
night for the security of the house, perceiving a strange
person in the garden at that unusual time, set upon him
with great fury. The doctor caught him by the throat,
grappled with him, and throwing him down, lay upon him :
once he thought of killing him ; but he altered his reso-
lution, on recollecting that this would be unjust, since the
dog did only his duty, and he himself was in fault for
rambling out of his room before it was light. At length
he called out so loud, that he was heard by some of the
family, who came presently out, and released the doctor
and the dog from their disagreeable situation.

BARS, in Music, are the spaces quite through any com-
position, separated by upright lines drawn across the five
horizontal lines, each of which either contains the same
number of notes of the same kind, or so many other notes
as will make up a like interval of time; for all the bars,
in any piece, must be of the same length, and played in
the same time.

BARS, in Architecture, are long slender piecesof wood,
or iron, used to keep things secure, and at their proper
assigned distances.

BARTER, or Truck, is the exchanging of one com-
modity for another; and forms a rule in the commercial

part of arithmetic, by which the commodities are properly
calculated and equalled, by computing first the value of
the commodity which is given, and then the quantity of
the other which will be adequate to the same sum.

Example. How much wine at 70/ per pipe must be given
in barter for 500 loads of timber at 2/ 5s per load ?

First find the value of the commodity whose quantity
is given; thus,

500 loads at 2/ 5s per load.
2

1125 the value of the timber: hence,
to find what quantity of wine at 70/ per pipe will amount
to this sum, say, as 701 : 1 pipe : : 1 125/ : l6 pipes 9 gal-
lons the quantity sought.

BARTHOLINE (Erasmus), a mathematician and phi-
losopher, in the latter part of the 17th century, who died
in 1698, at 73 years of age. He was successively pro-
fessor of mathematics and philosophy at Copenhagen ; and
attained the rank of counsellor of state. He published
De Cometis, an. 1664 et 1665, Opusculum ex observa-
tion! bus Hafniae habitis adornatum. Hafniffi, 4to. Also,
Experimenta Crystalli Islandici, 1670, in 4to; and De
Aere Hafniensi, l679> >n 8vo.

BASE, or Basis, in Chemistry, is a term applied bythe
old chemists to designate those substances of a fixed, inert,
passive nature, which combined with, and were acted on,
by more volatile or active menstrua. Thus, the alkalies,
earths, and metallic oxyds, which form compound salts, by
uniting with acids, were called the bases of these salts. Mo-
dern chemists, though they maintain, that in every combi-
nation the nisus or force of affinity between two ingredients
is mutual and equal, have yet retained the term for the sake
of precision, to express either species or families of salts
which differ with regard to the acid, but agree as to the'
alkali, earth, or metallic oxyd which they contain. Thus,
salts with a base of potash, include all those species which
are formed by the combination of the various acids with
the particular alkali, potash. Again, salts with an alka-
line base comprehend the three families of salts with bases
of potash, soda, or ammonia, as distinguished from the other
salts with earthy or metallic bases.

The utility, therefore, of this mode of expression is evi-
dent; for though the compound salts are usually divided
into genera according to their acids, as sulphats, nitrats,
muriats, &c, yet it is often desirable to arrange them ac-
cording to their oth?r element or base, for which the La-
voisierian nomenclature has not particularly provided.

The term base is also used on other occasions as a me-
thod of denoting species; as when we say sulphuric acid
is composed of oxygen united with the base of sulphur;
the vegetable acids, of oxygen and a compound base of hy-
drogen and carbon. Sometimes also the word base is ap-
plied in a more indefinite manner, as in the expressions, phos-
phat of lime is the base of animal bone, azot is the base of
muscular fibre, where it means merely the characteristic
or principal part.

BASE, Basis, in Architecture, denotes the lower part
of a column or pedestal. The base of columns is diffe-
rently formed in different orders : Thus, the Tuscan base
consists only of a single tore, besides the plinth : the
Doric has an astragal more than the Tuscan : the Ionic
has a large tore over two slender scotias, separated by two

B A S

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B A S

astragals : the Corinthian has two tores, two scotias, and
two astragals : the Composite has an astragal less than the
Corinthian : the Attic base has two tores and a scotia, and
is proper for either the Ionic or Composite columns.

Base, in Geometry, the lowest side of any figure. Any
side of a figure may be considered as its base, according to
the position in which it may be conceived as standing; but
commonly it is understood of the lowest side : as the base
of a triangle, of a cone, cylinder, &c.

Base Line, in Perspective, denotes the common section
of the picture and the geometrical plane.

Base Ring, of a cannon, is the great ring next behind
the vent or touch-hole.

Base, Alternate. See Alternate.

BASEMENT, in Architecture, a continued base, ex-
tended a considerable length, as about a house, a room,
or other piece of building.

BASILIC, in the ancient Architecture, was a large hall,
or "court of judicature, where the magistrates sat to ad-
minister justice.

BASILICA, or Basilicus, the same asRegulns, or Cor
Leonis, being a fixed star of the first magnitude in the con-
stellation of Leo.

BASILISK, in the older artillery, was a large piece of
ordnance so called from its resemblance to the supposed
serpent of that name. It threw an iron ball of 200 pounds
weight; and was in great repute in the time of Solyman
emperor of the Turks, in the wars of Hungary; but it is
now grown .out of use in most parts of Europe. Paulus
Jovius relates the terrible slaughter made in a Spanish
ship by a single ball from one of these basilisks; after pass-
ing through the beams and planks in the ship's head, it
killed upwards of 30 men. And Maffeus speaks of basi-
lisks made of brass, each of which required 100 yoke of
oxen to draw them. — More modern writers also give the
name basilisk to a much smaller and sizeable piece of ord-
nance, made of 15 feet long by the Dutch, but of only 10
by the French, and carrying a ball of 48 pounds. The
largest size of cannon now used by the English, are the 32-
pounders.

BASIS, in Geometry, the same as Base.

BASS, in Music, that part of a concert which is the
most heard, which consists of the gravest, deepest, and
longest sounds : it is played on the largest pipes or strings
of common instruments ; or on instruments larger than
ordinary for the purpose ; by some it is esteemed the'basis
and principal part in music, and by others as scarcely ne-
cessary in some tunes.

BASSANTIN (James), a Scotch astronomer of the
l6th century, born in the reign of Janles the 4th of Scot-
land. He was a son of the Laird of Bassantin in the
Merse. After finishing his education at the university of
Glasgow, he travelled through Germany and Italy, and
then settled in the university of Paris, where he taught
mathematics with great applause. Having acquired some
property in this employment, he returned to Scotland in
1562, where he died 6 years after.

From his writings it appears he was no inconsiderable
astronomer, for the age in which he lived ; but, according
to the. fashion of the times, he was greatly addicted to ju-
dicial astrology. It was doubtless to our author that Sir
.Tames Mclvil alludes in his Memoirs, when he says that
his brother Sir Robert, when he was using his endeavours
to reconcile the two queens Elizabeth and Mary, met with
one Bassantin, a man learned in the high sciences, who told

him, " that all his travels would be in vain ; for," said he,
" they will never meet together; and next, there will never
be any thing but dissembling and secret hatred for a while,
and at length captivity and utter wreck to our queen from
England." He added, " that the kingdom of England at
length shall fall, of right, to the crown of Scotland : but
it shall cost many bloody battles ; and the Spaniards shall
be helpers, and take a part to themselves for their labour."
Aprediction inwhich Bassantin partly guessed right, which
it is likely he was enabled to do from a judicious conside-
ration of probable circumstances and appearances.
Bassantin's works are,

1. Astronomia Jacobi Bassantini Scoti, opus absolu-
tissimum, &c; ter. edit. Latine et Gallice. Genev. 1599-
fol. This is the title given it byTornoesius, who translated
it into Latin from the French, in which language it was
first published.

2. Paraphrase de 1' Astrolabe, avec une amplification de
l'usage d'Astrolabe. Lyons 1555. Paris l6l?, Svo.

3. Mathematica Genethliaca.

4. Arithmetica.

5. Musica secundum Platonem.

6. De Mathesi in Genere.

BASTION, in the modern fortification, a large mass of
earth at the angles of a work, connecting the curtains to
each other; and answers to the bulwark of the ancients.
It is formed by two faces, two flanks, and two demigorges.
The two faces form the saliant angle, or angle of the bas-
tion ; the two flanks form with the faces, the epaules or
shoulders; and the union of the other two ends of the flanks
with the curtains forms the two angles of the flanks.

Curtain-

The first introduction of bastions into fortification, is
not precisely defined. However, they were well known
soon after the year 1500; for Tartalea gives a plan of
Turin, which had been completely fortified for some time
with four bastions, in his Quesiti et Inventioni diverse,
published in 1546.

If the angle of the bastion be less than 60°, it will be
too small to give room for guns; and besides so acute as
to be easily beaten down by the enemy's guns : to which
may be added, that it will either render the line of de-
fence too long, or the flanks too short; it must therefore
be more than 6o° ; but whether or no it should be a right
angle, some intermediate angle between 60 and 90°, or
even whether or no it should exceed 90°, is still disputed,
though those are generally preferred which are not much
less than 90°, and not exceeding 120 or 130°. Hence it
follows that a triangle can never be fortified ; because
either some or all of the angles will be cither 60°, or less
than 60.

Bastions are of various kindjj solid, void, regular, &c.

Solid Bastions, are those that are entirely filled up
with earth to the. height of the rampart, without any void
space towards the centre.

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Void or Hollow Bastion, has the rampart and parapet
ranging only round the flanks and spaces, so that a void
space is left within towards the centre, where the ground
is so low that if the rampart he taken, no retrenchment
can be made in the centre, but what will lie under the fire
of the besieged.

Regular Bastion, is that which has its due proportion
of faces, flanks, and gorges.

Deformed ov Irregular Bastion, iswhen the irregularity
of the lines and angles throws the bastion out of shape; as
when it wants one of the demigorges, one side of the in-
terior polygon being too short, &c.

Demi Bastion, or Half Bastion, also otherwise called
an Epaulment, has but one face and flank.

Double Bastion, is when one bastion is raised within,
and upon the plane of another bastion.

Flat Bastion, is one built in the middle of the curtain,
whenit is too long to be- defended by the usual bastions at
the extremities.

Composed Bastion, is when the twosides of the interior
polygon are very unequal, which makes the gorges also
unequal.

Out Bastion, is that which has a re-entering angle at the
point, and is sometimes called a Bastion with a Tenail/e,
whose point is cut off, making an angle inwards, and two
points outwards. This is used when the saliant angle would
be too sharp, or when water or some other impediment
prevents it from being carried out to its full extent.

BASTON, or Batoon, in Architecture, a moulding in
the base of a column, called also a Tore, or Torus.

BATTARDEAU, or Cofferdam, in bridge-building,
is a case of piling, &c, without a bottom, fixed in the bed
of the river, by which to lay the bottom dry for a space
large enough to build the pier on. When it is fixed, its
sides reaching above the surface of the river, the water is
drawn out by pumps, &c, and it is in like manner kept
dry till the pier is built up ; and then the materials of the
cofferdam are drawn up. See my Tracts, vol. i, p. 100.

BATTEN, a name given by workmen to a scantling or

piece of wooden stuff, about an inch thick, and from 2 to

4 inches broad; of a considerable butindeterminatelength.

BATTERING, the attacking a place, work, or the like,

with heavy artillery.

Battering-Raw, a military engine used for beating
down walls before the invention of gunpowder and the mo-
dern artillery. It was no other than a long heavy beam of
timber, armed with an iron head, something like the head
of a ram. This being pushed violently with constant suc-
cessive blows against a wall, gradually shakes it with a
vibratory motion, till the stones are disjointed and the wall
falls down. There were several kinds of battering-rams :
the first was rude and plain, which the soldiers carried in
their arms by main force, and so struck the head of it
against the wall. The second was slung by a rope about
the middle to another beam lying across upon a couple of
posts; which was the kind described by Joscphus as used
at the siege of Jerusalem. A third sort was covered with
a shell or screen of boards, to defend the men from the
stones and darts of the besieged upon the walls, and thence
called Testudo Arictaria. And Felibien describes a fourth
sort of battering-ram, which ran upon wheels; and was
the most perfect and effectual of them all.

Vitruvius affirms, that the battering-ram was first in-
vented by the Carthaginians, while they laid siege to Cadiz ;
yet Pliny assures us, that the ram was invented or used at
Vol. I.

the siege of Troy; and that it was this that gave occasion
to the fable of the wooden-horse. The ram soused by the
Carthaginians was of the most simple construction, being
the one first mentioned. Pephasmenos, a Tyrian, after-
wards contrived to suspend it with ropes; and lastly, Po-
lydus, the Thessalian, mounted it on wheels, at the siege of
Byzantium, under Philip of Macedon. It has been sus-
pected that the walls of Jericho, mentioned in the Book of
Joshua, were beaten down with this instrument ; the ram's
horns there mentioned, and by means of which they were
overthrown, being no other than the horns of the battering-
ram.

Plutarch relates, that Marc Anthony, ,m the Parthian
war, made use of a ram SO feet long: and Vitruvius af-
firms that they were sometimes 106, and even 120 feet
long ; which must have given an immense force to this en-
gine^ The ram required 100 soldiers to work* and ma-
nage it atone time; who being exhausted, another century
relieved them ; by which means it was kept playing con-
tinually. See fig. 2, plate v, which represents the batter-
ing-ram suspended in its open frame; in which 3 denotes
the form of the head, fastened to the enormous beam 2,
by three or four bands (4) of iron, of about four feet in
breadth. At the extremity of each of these bands was an
iron chain (5), one end of which was fastened to a hook
(6), and to the last link at the other extremity was firmly
bound a cable, which ran along the whole length of the
beam to the end of the ram 7, where these cables were
bound together as fast as possible with small ropes. To
the end of these cables was likewise fastened another, that
consisted of several strong cords platted together to a cer-
tain length, and then running single (8), at each of which
were placed several men, to balance and work the machine.
10 is the chain or cable by which the ram was hung to the
cross beam ( 1 1), fixed on the top of the frame ; and 12 is
the base of the machine.

The unsuspended ram differed from this only in the
manner of working it; as it moved on small wheels upon
another large beam, instead of being slung by a chain or
cable.

Mr. Altwood, in his Mechanics, p. 26*5, has compared
the effects of a battering-ram to that of a 24-pounder ;
' the metal extremity of the ram, or the head of it, he sup-
poses to be of the same magnitude as the shot: "In or-
der," says he, " that the ball may have, the same effect in
making a breach in a wall, the weight of the ram must
exceed that of the ball in the proportion of about 1700',
to the square of the velocity with which this engine could
be made to impinge against a wall, expressed in feet; if
this may be estimated at about 10 feet in a second, the pro-
portion of the weights will be that of about 28^0000 to
100, or 2S.O00 to 1 : the- weight of the battering-ram must
therefore be equal to 34-6 ton. In this case the battering-
ram and the cannon-ball, moving with the velocities of
10 feet and 1700 feet respectively in a second, would have
the same effect in penetrating the substance of an opposed
obstacle; but it is probable that the weight of the ram
never amounted to so much as is above described, and con-
sequently the effects of the cannon-ball to cut down* walls
by making a breach in them, must exceed those of the
ancient battering-rams : but the momentum of these, or
the impetus whereby they communicated a shock to the
whole building, was far greater than the utmost force of
cannon-balls; for if the weight of the battering-ram were
no more than 170 times greater than that of a cannon-

2F

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BAT

ball, cacti moving' with iis respective velocity; the mo-
menta of both would be equal : but as it is certain, that
the weight of these ancient machines was far more than
170 times our heaviest cannon-balls," it follows that their
momentum or impetus, to shake or overturn walls, &c,
was far superior to that which is exerted by modern ar-
tillery. And since the strength of fortifications will in ge-
neral be proportioned to the means which can be used for
their demolition, the military walls of the moderns have
been constructed with less attention with regard to their
solidity and massy weight, than the ancients thought a
necessary defence against the power of the ram ; that sort
of cohesive firmness of texture which resists the penetration
of bodies being now more necessary than in ancient times:
but it is manifest that even now, solidity or weight in for-
tifications also is of material consequence, to the effectual
construction of a wall or battery."

BATTERY, in the Military Art, a place raised to plant
cannon upon, to play with more advantage on the ene-
my. It consists of an epaulment or a breast-work, of about
8 feet high, and 18 or 20 feet thick.

In all batteries, the open spaces through which the muz-
zles of the cannon are pointed, arc called Embrazures, and
the distances between the embrasures, Merlons. The guns
are placed on a platform composed of planks &c, ascend-
ing a little from the parapet, to check the recoil, and that
the gun may be the easier brought back again to the pa-
rapet : they are placed from 12 to 16 feet distant from one
another, that the parapet may be strong, and the gunners
have room to work.

Mortar Batteries differ from the others, in that the
slope of the parapet is inwards, and is without embra-
sures, the shells being fired quite over the parapet, com-
monly at an angle of 45 degrees elevation.

Open Battery, is nothing more than a number of can-
non, commonly field-pieces, ranged in a row abreast of
one another, perhaps on some small natural elevation of
the ground, or an artificial bank a little raised for the
purpose.

Covered or Masked Battery, is when the carinon and
gunners are covered by a bank or breast-work, commonly
made of brush-wood, faggots, and earth, called a fascine
battery.

Sunk or Buried Battery, is when its platform is sunk,
or let down into the ground, so that trenches must be cut
into the earth opposite the muzzles of the guns, to serve
as embrasures to fire through. This is mostly used on
the first making of approaches in besieging and battering
a place.

Ricochet Battery, so called by its inventor M. Vau-
ban, and first used at the siege of Aeth in 1697. It is a
method of firing with a very small quantity of powder,
and a little elevation, so as just to fire over the parapet;
and then the shot will roll along the opposite rampart,
dismounting the cannon, and driving or destroying the
troops. In a siege, they are generally placed at about 300
feet before the first parallel, perpendicular to the faces
produced which they are to enfilade. Ricochet practice
is not confined to cannon alone; small mortars and how-
itzers may effectually be used for the same purpose. They
are of singular use in action to enfilade the enemy's ranks;
for when the men perceive the shells rolling about with
their fuses burning, expecting them to burst every mo-
ment, the bravestamong them will not havecourage to wait
their approach, and face the havoc of their explosion.

Cross Batteries, are two batteries playing athwart
each other upon the same object, funning an angle there,
and battering to more effect, because what one battery
shakes, the other beats down.

Battkry d'Enfi'adc, is one that scours or sweeps the
whole length of a straight line.

Battery en Ec/iarpe, is one that plajs obliquely.
Battery de Meverse, or Murdering Buttery, is one that
plays upon the enemy's back.

Camerade or Joint Battery, is when several guns plav
upon one place at the same time.

Battery, in Electricity, is a combination of coated
surfaces of glass, commonly jars, so connected together
that they may be charged at once, and discharged by a
common conductor. Mr. Gralath, a German electrician,
first contrived to increase the shock by charging several
phials at the same time. — Dr. Franklin, having analyzed
the Leyden phial, and found that it lost at one surface the
electric fire received at the other, constructed a battery
of eleven large panes of sash window glass, coated on both
sides, and so connected that the whole might be charged
together, and with the same labour as one single pane;
then by bringing all the giving sides into contact with one
wire, and all the receiving sides with another, he contrived
to unite the force of all the plates, and by that means to
discharge them at once. — Dr. Priestley describes a still
more complete battery. This consists of (14 jars, each 10
inches long, and 2J inches in diameter, all coated within
an inch and a half of the top, forming in the whole about
31 square feet of coated surface. A piece of very fine
wire is also twisted about the lower end of the wire of
each jar, to touch the inside coating in several places ; and
it is put through a pretty large piece of cork, within the
jar, to prevent any part of it from touching the side, by
which a spontaneous discharge might be made. Each wire
is turned round so as to make a loop at the upper end ;
and through these loops passes a considerably large brass
rod with knobs, each rod serving for one row of the jars ;
and these rods are made to communicate together by a
thick chain laid over them, or as many of them as may be
wanted. The jars stand in a box, the bottom of which is
covered with a tin plate; and a bent wire touching the
plate passes through the box, and appears on the outside.
To this wire is fastened any conductor designed to com-
municate with the outside of the battery ; and the dis-
charge is made by bringing the brass knob into contact
with any of the knobs of the battery. When a very great
force is required, the size or number of the jars must be
increased, or two or more batteries may be used. — But the
largest and most powerful battery of all, is that employed
by Dr. Van Marum, to the great electrical machine, con-
structed for Teyler's museum at Haarlem. This grand
battery consists of a great number of jars coated as above,
to the amount of about 130 square feet; and the effects
of it, which are truly astonishing, are related by Dr. Van
Marum in his description of this machine, and of the ex-
periments made with it, at Haarlem 1785, &c. See also
Franklin's Expcr. and Observ. and Priestley's History of
Electricity.

Battery or Pile, in Galvanism, is an apparatus em-
ployed for accumulating the electricity of galvanism, which
is produced by the mutual agencies of certain metallic and
carbonaceous substances, and peculiar fluids. It was in-
vented by the celebrated Volta, from whose labours the.
new science of galvanism has derived many advantages,

BAY

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B E A

and much improvement. Mr. Henry, in his valuable
Epitome of Chemistry, gives the following easy directions
for the construction of this pile : " Procure at a brazier's
or coppersmith's,. 30, 40, or 50 pieces of zinc, or speltre,
cast in sand, of the size of half-crowns or shillings,
but rather thicker. A corresponding number of half-
crowns or shillings will also be required, according to the
sizes of the pieces of zinc that may be employed. Let an
equal number of pieces of woollen cloth be cut, of a cir-
cular shape to correspond with the pieces of zinc, and
steep these in a strong solution of common salt in water.
Then dispose the three substances alternately, in the fol-
lowing order ; silver, zinc, moistened cloth ; silver, zinc,
&c ; till a sufficient number of these triplicates, not less
than 20 or 30, have been thus arranged, the silver termi-
nating the pile at top. In order to facilitate the touching
of the bottom piece of silver, it may be well to put under
it a slip of tinfoil, or Dutch leaf, which may project a few
inches. Next let the hands be moistened with salt and
water; and on touching the piece of tinfoil with one hand,
the uppermost piece of silver with the other, a shock will-
pass through the arms, which will be strong in proportion
to the number of pieces of zinc &c employed." Of late,
copper has been used instead of silver, on account of its
being cheaper ; and solutions of muriate of ammonia (sal-
ammoniac), of nitrous acid and of muriatic acid, have
been substituted for the solution of common salt with in-
creased effect. Any two metallic substances which are
perfect conductors of electricity will answer the purpose,
on condition that the interposed fluid is capable of oxy-
dating at least one of them.

Various forms of this battery have been adopted by
different philosophers ; but none have been brought into
such general use as the galvanic trough invented by Mr.
Cruicshank, of Woolwich. " It consists of a box of baked
wood, in which plates of copper and zinc, or of silver and
zinc, soldered together at their edges, are cemented in
such a manner as to leave a number of water-tight cells
corresponding to the number of the series." The common
voltaic pile, on account of the loss of moisture, generally
loses its electrical action in a few days, and this cannot
be renewed without the trouble of reconstruction ; but by
Mr. Cruieshank's contrivance, which becomes active on
merely filling the cells with the proper saline fluids, and
freeing it when necessary from oxyd by muriatic acid,
greater permanency is secured, much trouble is prevented,
and much time saved.

A very powerful galvanic pile is that of M. Pepys, ju-
nior; for a description of which the reader may consult
the Phil. Mag. No. 57. See also Galvanism.

BATTLEMENTS, in Architecture, are notches or in-
dentures in the top of a wall or other building, like em-
brasures, to look through.

BAY, in Geography, denotes a small gulf, or an arm
of the sea stretching up into the land ; being larger in the
middle within, than at its entrance, called the mouth of
the bay.

BAYER (John), a German lawyer and astronomer,
flourished the latter part of the l6th and beginning of the
17th century, but in what particular year or place he was
born, is not known; his name however will be ever me-
morable in the annals of astronomy, on account of that
great and excellent work which he first published in the
year 1603, under the title of Uranometria, being a com-
plete celestial atlas, or large folio charts of all the con-

stellations, with a nomenclature collected from the different
tables of astronomy, ancient and modern, together with,
the useful invention of denoting the stars in every constel-
lation by the letters of the Greek alphabet, in their order,
and according to the order of their magnitude in each con-
stellation. By means of these marks, the stars of the hea-
vens may, with as great facility, be distinguished and re-
ferred to, as the several places of the earth are by means
of geographical tables ; and as a proof of the usefulness
of this method, our celestial globes and atlasses have ever
since retained it ; and hence its general use through all
the literary world. Astronomers, in speaking of any star
in the constellation, denote it by saying it is marked by
Bayer, a, or j3, or y, &c.

Bayer lived many years after the first publication of this
work, which he greatly improved and augmented by his
constant attention to the study of the stars. At length, in
the year 1(>27, it was republished under a new title, viz,
Coelum Stellatum Christianum, or the Christian Stellated
Heaven, or the Starry Heavens Christianized : far in this
work, the heathen names and characters, or figures of the
constellations are rejected, and others, taken from the
Scriptures, inserted in their stead, to circumscribe the re-
spective constellations. This was the project of one Ju-
lius Schiller, a civilian of the same place. But this at-
tempt was too great an innovation, to find success, or a
general reception, which might occasion great confusion.
And, we even find in the latter editions of this work, that
the ancient figures and names were restored again; at
least so I find them in two editions, of the years 1654, and
l66T, whi<?h are now before me.

BAYLY (Wm.), a respectable astronomer and mathe-
matician, was born at Bishop-Carions in Wiltshire, 1737,
where the early part of his youth was employed in the line
of his father, on a small farm in that parish ; till accident
threw in his way books and assistance, by which means
he became qualified for an usher in a school. He after-
wards passed a few years as assistant with Dr. Maskelyne,
the astronomer royal at Greenwich ; by whose recom-
mendation Mr. B. in 1769, was sent out by the Royal So-
ciety to the North Cape, to observe the transit of Venus
of that year; of which his account was published in the
Philos. Trans. In. 1772 Mr. B. as one of the astronomers,
saih d with Capt. Cook, on his second voyage of discovery
in the southern hemisphere; and again, in 1776, in the
third and last voyage, which unfortunately terminated the
useful life of that brave and excellent commander. On
the decease of Mr. Witchell, the first master in the Royal
Naval Academy at Portsmouth, in 17S5, Mr. B. was ap-
pointed his successor, which situation he held, with much
credit to himself, and benefit to the institution, until the
new establishment of the Royal Naval College, in 1807;
when Mr. B. was allowed to retire on a handsome pen-
sion, which he enjoyed till the time of his death, which
happened at Portsea the latter part of 1S10, at 73 years
of age.

BEAD, in Architecture, is a round'moulding, carved
in short embossments, like beads in necklaces; and some-
times an astragal is thus carved. There is also a sort of
plain bead often set on the edge of each fascia of an archi-
trave ; as also sometimes on the lining board of a door-
case, the upper edge of skirting boards, &c.

Bead, in Assaying, the small ball or mass of pure me-
tal separated from the scoria, and seen distinct and pure
in the middle of the coppel while in the fire.
2F2

B E

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B E A

BEAM, in Architecture, the largest piece of wood in a
building; being laid across the walls, and serving to sup-
port the principal rafters of the roof. Several ingenious
authors have considered the force or strength of beams,
as supporting their own weight and any other additional
weight; particularly Varignon, and Parent in the Me-
moir. Acad. R. Scien. an. 1708, and Mr. Emerson, on the
Strength and Stress of Timber,in his Mechanics. Mr. Pa-
rent makes the proportion of the depth to the breadth ot
a rectangular beam to be as 7 to 5 when it is strongest.

The proportions of beams near London, are fixed by
statute, as follows : a beam 15 feet long must be, at least,
7 inches on one side its square or end, and 5 on the other;
if l6feet long, the ends must be 8 inches by 6; if 17 feet
long, 10 inches by 6 : in country places they are usually
made stronger. Sir H. Wotton advises these to be of the
strongest and most durable timber. Some of the best au-
thors have examined the force or strength of beams, and
brought their resistance to a precise calculation ; and the
chief results of their investigations are the following :

1. The strengths of two beams, of the same wood, and
of different dimensions, are to each other, as the products
of their sections multiplied by the distances or heights of
their centres of gravity from the base, or underside, and
divided by the lengths.

2. When the lengths are equal, the strengths are as the
sections multiplied by the distances of the centres of gra-
vity above the base.

3. When the sections are equal, the strengths are as the
heights of their centres of gravity divided by their lengths.

4. When the heights of the centres of gravity are equal,
the strengths are as their sections divided by their lengths.

5. A rectangular beam with its narrower face upwards,
is stronger than with its broader face, in proportion as the
broader face is to the narrower.

6. A rectangular beam will bear a greater weight when
■it is fixed so, that its diagonal is in a vertical position, than
when it is placed flat on one of its sides.

7. The strength of a cylindric beam, is to that of the
circumscribed square one, as their sections, or as the area
of a circle is to its circumscribed square, that is, as '7854
to 1, or as 11 to 14.

8. The strongest rectangular beam which can be cut
out of a cylindrical tree, is that whose breadth is to its
depth, as 1 to ^/2, or very nearly as 5 to 7.

9. The strength of a triangular beam when laid flat on
its base, is double the strength of it when laid the reverse
way, or with the base uppermost.

10. The stress upon a beam, to break it, by a weight
laid upon the middle, is to the stress when laid nearer one
end than the other, as the square of half the length, is to
the product of the distances of the weight from each end.
Therefore a beam will sustain the least weight in the mid-
dle, but gradually more and more towards one end.

11. When abeam is laid in an oblique position, its
strength is to the strength in a horizontal position, as ra-
dius is to the cosine of the inclination.

12. Abeam supported near both ends on two props,
will sustain twice as much when the ends beyond the props
are kept from rising, as when it rests loosely on the props.
See Carpentrv,

13. When a bar or beam is sustaining any very great
compression endways, it is much easier broken by any
transverse strain. Several experiments have been made
on this kind of strain : a piece of white marble, for in-

stance, of i an inch square, and 3 inches between the
props, bore 38lbs; but when compressed endways with
300lbs, it broke with 14£. The difference is much more
remarkable in timber and soft bodies, but it is consider-
able in all.

14. As the strength of the same kind of wood varies very
much, it is impossible ever to come to an exact knowled°e
of the just proportion between the strengths of different
kinds. Thus much however may be said, with regard to
the two kinds commonly used for beams, &c, namely oak
and fir, that the former is stronger than the latter, nearly
in the proportion of 9 to 8.

Beams of a ship, are the large, main, cross timbers,
stretched from side to side, to support the decks, and keep
the sides of the ship from falling together.

Beam of a balance, is the horizontal piece of wood or
iron supported on a pivot in the middle, and at the extre-
mities of which the two scales are suspended, for weiohing
any thing.

BriAM-Compass, an instrument consisting of a wooden
or brass square beam, having sliding sockets carrying steel
or pencil points; and are used for describing large circles,
the radii of which are beyond the extent of the common
compasses.

BEAR, in Astronomy, a name given to two constella-
tions, called the greater and the lesser bear, or Ursa nia-
jor and minor. The pole star is in the tail of the little
bear, and is within less than 2 degrees of the north pole.
See Ursa, major and minor.

BEARD, of a Comet, the rays which it emits in the di-
rection in which it moves; as distinguished from the tail,
or the rays emitted or left behind it as it moves along, be-
ing always in a direction from the sun.

BEARER, in Architecture, any subsidiary or interme-
diate support in aid of the principal supports, as the small
joists or brackets which bear a gutter or the covering
board of a cornice ; the piers and blockings under the
joints of a ground-floor; or the joists, &c, which bear any
thing independent of or unconnected with the building;
as the bearers of a cistern, of a vat, of a platform.

BEARING, in Geography and Navigation, the situation
of one place from another, with regard to the points of
the compass; or an arch of the horizon between the me-
ridian of a place and a line drawn through this and ano-
ther place, or the angle formed by a line drawn through
the two placesand their meridians. — The bearings of places
on the ground are usually determined by the magnetic
needle.

BEATS, in a Clock or Watch, arc the strokes made by
the fangs or pallets of the spindle of the balance; or of
the pads in a royal pendulum. For the number and use
of the beats, see Derham's Artificial Clock Maker, pa. 14
and scq.

BEAT, in Music, is a grace marked thus ", or thus;/.
Its effect is just the contrary of a transient shake in rapid
movements, where it can neither be prepared nor turned.

BEAUNE (Florimone), counsellor or presidiaj of
Blois, was born in 1601, and died in 1051. He illustrated
with notes the most difficult passages in the geometry of
Descartes, all the mysteries of which he understood. In-
deed that great man had such a respect for Beaune, that
in several of his letters he declares, that he relied more' on
his learning and approbation, than on those of all the
other mathematicians then in France. — Beaune proposed a
problem which gave rise to the inverse method of tangents.

B E I

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B E L

BECCARIA (James Bartholomew), a celebrated
mathematician and philosopher of the 17th century, was
born at Bononia, in 16*82, and received the early part of
his education among the Jesuits. Turning his attention
to th&study of natural philosophy, he soon became distin-
guished for the variety and depth of his knowledge in phy-
sics and in mathematics, of which he was made public
professor, and in conjunction with Morgagni, and other
celebrated characters at Bononia, assisted in forming an
academy there for teaching mathematics, natural history,
chemistry, anatomy, and medicine. He was a frequent
correspondent with the Royal Society of London, of which
he was made an honorary member. Among other com-
munications from Beccaria, which appear in the Phil.
Trans, are his " Observations on the Weather," " On the
Ignis Fatuus," and " On the power some persons have
enjoyed of supporting life for a great length of time with-
out food." This was afterwards published at Padua, un-
der the title of" Dejejuniis longis Dissertatio," folio 1748.
He died Jan. 1766", being S4 years of age. Among his
publications are the following:

1. Dissertatio Meteorologica Medica, in quaaeris tem-
peries et morbi Bononia? grassantes annis 1729 etsequenti
describuntur.

2. De quamplurimis phosphoris nunc primum detec-
tis Commentarius, in 4to, 1744 at Bonon.

3.*Scriptura Medico-legalis, 1749.
Besides numerous other compositions, .the titles of which
may be seen in Gen. Biog. and Hal. Bib. Amat.

BECCARIA (Giambattista), an eminent philoso-
pher, and a monk of Ecoles-Pies, was a native of Men-
dovi, in Piedmont, and became a professor of mathema-
tics and philosophy, first at Palermo, and next at Rome.
He afterwards occupied the chair of experimental philo-
sophy at Turin, and was appointed preceptor to the
princes of Sardinia. His chief works are :

1. Dell' Elettricismo Artificiale e Naturale, in 4to,
1753, and of which an English translation was published
also in 4to, 1776".

2. Lettere dell' Elettricismo, folio 175S.

3. Gradus Taurinensis, 1774, in 4to, being an ac-
count of his measurement r>f a degree of the meridian
near Turin, which he carefully executed between the years
176'0 and 1764.

He also published Essays on the Cause of Storms and
Tempests on the meridian of Turin, &c. This philosopher,
no less respected for his virtues than his knowledge, died
at Turin in an advanced age, May 22, 1781.

BED, of a Gredt Gun, a plank laid between the cheeks
of the carriage, on the middle transum, for the breech of
the gun to rest upon.

Bed, or Stool, of a mortar, a thick and strong planking
on which a mortar is placed, hollowed a little to receive
the breech and trunions,

BED-MOULDING, in Architecture, a term used by
workmen for those members in a cornice, which are placed
below the coronet, or crown. It usually consists of these
four members,. an ogee, a list, a large boultinc, and an-
other list under the coronet.

BEES, in Naval Architecture, denote pieces of elm
plank bolted to the outward ends of bowsprits.

BRIGHTON (Henry), f. h.s., was a respectable ma-
thematician and mechanist, being well skilled in machi-
nery, and followed the profession of a civil-engineer, be-
ing indeed the most eminent one of his time. He was the

intimate friend of Dr. Desaguliers, who received from him
many valuable communications in mechanics, engines, &c,
some of which may be seen in several parts of the 2d vol.
of the Doctor's Course of Experimental Philosophy, parti-
cularly at pp. 43 1 , 437, 449, 46 1 , 472, 497, 534, 539, &c.
He communicated to the Royal Society also a curious let-
ter, containing a Description of the Water- Works at Lon-
don Bridge; printed in the Philos. Trans, vol. 37, p. 5;
or my Abridg. vol. 7, p. 442.

It is probable that Mr. B. died in 1743 or 1744, as it
appears he conducted the Ladies' Diary for the Stationers'
Company, from 1714 to 1744 inclusively; discharging
that trust with such satisfaction to the company, that they
permitted his widow to enjoy it for many years afterwards,
by employing a deputy to compile that useful annual lit-
tle book. In this almanac, for the year 1721, Mr. B. in-
serted a curious table of calculations on the steam-engine.

BEL1DOR (Bernard Forest de), an engineer in
the service of France, but born in Catalonia in lo'9S. He
was professor in the new school of artillery at la Fere,
where he published his course of mathematics for the use
of the artillery and engineers. He was the first who seri-
ously considered the quantity of gunpowder proper for
charges, and reduced it to 2-3ds the quantity. He was
named Associate in the Academy of Sciences in 1751; and
died Sept. 8, 176l, at 63 years of age.

His works that have been published, are :

1. Sommaire d'un Cours d'Architecture militaire, civile,
et hydraulique, in 12mo, 1720.

2. Nouveau Cours de Mathematiques, &c. in 4to, 1725.
3\ La Science des Inu6nieurs, in 4to, 1729.

4. Le Bombardier Francois, in 4to, 1734.

5. Architecture Hydraulique, 4 vols, in 4to, 1737-

6. Dictionnaire pbrtatif de l'lngenieur, in 8vo.
7- Traite des Fortifications, 4 vols, in 4to.

Besides several pieces inserted in the volumes of the
Memoirs of the Academy of Sciences, for the years 1737,
1750, 1753, and 1756.

BELLATRIX, in Astronomy, a ruddy, glittering star
of the 2d magnitude, in the left shoulder of Orion. lis
name is from the Latin helium, as being anciently sup-
posed to have great influence in kindling wars, and form-
ing warriors.

BELLOWS, a machine so contrived as to exspire and
inspire the air by turns, by enlarging and contracting its
capacity. This machine is used in chambers and kitchens,
in forges, furnaces,, and founderies, to blow up the fire:
it serves also for organs and other pneumatic instruments,
to give them a proper supply of air. All these are of va-
rious constructions according to" their different purposes;
but in general they are composed of two flat boards, some-
times of an oval, sometimes of a triangular figure: two
or more hoops, bent according to the figure of the boards,
are placed between them ; a- piece of leather, broad in
the middle, and narrow at both ends, is nailed on the
edges of the boards, which it thus unites together; as also
in the hoops which separate the. boards, that the leather
may the easier open and fold again : a tube of iron, brass,
or copper, is fastened to the undermost board, and there
is a-valve within that covers the hole in the underboard,
to keep the air from escaping!

BELTS, Fascite, in Astronomy, two zones or girdles
surrounding the planet Jupiter, brighter than any of the
rest of his body, and terminated by parallel lines. They
arc observed however to be sometimes broader and some-

B E R

[

]

B E R

times narrower, and not always occupying exactly the
same part of the disc. Jupiter's belts were first observed
and described by Huygens, in his Syst. Saturn. Dark
spots have often been observed on the belts of Jupiter ;
and M. Cassini observed a permanent one on the northern
side of the most southern belt, by which he determined
the length of Jupiter's days, or the time in which the pla-
net revolves upon its axis, which is 9h- 56m. Some astro-
nomers suppose that these belts are seas, which alternately
cover and leave bare large tracts of the planet's surface :
and that the spots are gulfs in those seas, which are
sometimes dry and sometimes full. But Azout conceived
that the spots are protuberances of the belts; and others
again are of opinion that the transparent and moveable
spots are the shadows of Jupiter's satellites.

Cassini also speaks of the belts of Saturn ; being three
dark, straight, parallel bands, or fascia, on the disc of that
planet. But it does not appear that Saturn's belts adhere
to his body, as those of Jupiter do ; but rather that they
are large dark rings surrounding the planet at a distance.
Some imagine that they are clouds in the atmosphere of
Saturn, though it would seem that the middlemost is the
shadow of his ring.

BENDING, the reducing a body to a curved or crook-
ed form. The bending of boards, planks, &c, is effected
by means of heat, whether by boiling or otherwise, by which
their fibres are so relaxed that they may be bent into any
figure. Bernoulli has a discourse on the bending of springs,
or elastic bodies. And Amoutons gives several experiments
concerning the bending of ropes. The friction of a rope
bent or wound about an immoveable cylinder, is sufficient,
with a very small power, to sustain very great weights.
BERENICE'S Hair; see Coma Berenices.
BERKELEY (George), the virtuous and learned bi-
shop of Cloyne in Ireland, was born in that kingdom, at
Kilcrin, the 12th of March l6S4. After receiving the first
part of his education at Kilkenny school, he was admitted
a pensioner of Trinity College, Dublin, at 15 years old ;
and chosen fellow of that college in 1707.

The first public proof he gave of his literary abilities
was in his Arithmetica absque Algebra aut Euclide demon-
strata: which from the preface it appears he wrote before
he was 20 years old, though he did not publish it till 1707.
It is followed by a Mathematical Miscellany,' containing
observations and theorems inscribed to his pupil Samuel
Molineux.

In 1709 came out the Theory of Vision ; which of all his
works appears to do the greatest honour to his sagacity ;
being the first attempt that ever was made to distinguish
the immediate and natural objects of sight, from the con-
clusions we have been accustomed from infancy to draw
from them. The boundary is here traced out between the
ideas of sight and touch ; and it is shown, that though ha-
bit has so connected these two classes of ideas in the mind,
that they are not without a strong effort to be separated
from each other, yet originally they have no such con-
nexion: insomuch, thata person born blind, and suddenly
made to see, would at first be utterly unable to tell how
any object that affected his sight would affect his touch ;
and particularly would not from sight receive any idea of
distance, or external space, but would imagine all objects
to be in his eye, or rather in his mind.

Next to his Theory of Vision, in the year 1710 appeared
The Principles of Human Knowledge; and in 1713 Dia-
logues between Ilylas and Philonous: the object of both

which pieces is, to prove that the commonly received no-
tion of the existence of matter, is false ; that sensible ma-
terial objects, as they are called, are not external to the
mind, but exist in it, and are nothing more than impres-
sions made upon it by the. immediate act of God, accord-
ing to certain rules called laws of nature.

For acuteness of parts and beauty of imagination, few
authors can vie with Berkeley: his writings excited uni-
versal admiration, and his company was courted even by
men of opposite parties, who also concurred in recommend-
ing him. For Steele he wrote several papers in the Guar-
dian, and at his house became acquainted with Pope, with
whom he always lived in friendship. Swift recommended
him to the celebrated earl of Peterborough, who being ap-
pointed ambassador to the king of Sicily and the Italian
States, took Berkeley with him as chaplain and secretary
in 1713, with whom he returned to England the year fol-
lowing.

His hopes of preferment expiring with the fall of queen
Anne's ministry, he some time after embraced an offer
made him by Ashe, bishop of Clogher, of accompanying
his son in a tour through Europe. At Paris he visited the
illustrious Malebranche, whom he found in his cell cook-
ing in a small pipkin a medicine for an inflammation of
the lungs, with which he was afflicted ; and as they en-
gaged in a conversation onBerkeley's system, Malebranche
in the heat of disputation, raised his voice so high, and in-
dulged the natural impetuosity of his temper to such a de-
gree, that he brought on an increase of his disorder, which
carfied him off a few days after. During four years' ab-
sence from England, he not only visited those places which
' fell within the grand tour, but some that are less frequent-
ed, and with great industry collected materials for a natu-
ral history of those parts, but which were unfortunately
lost in his passage to Naples. He arrived at London in
1721 ; and being much concerned for the miseries of the
nation, brought on by the disastrous South-sea scheme of
1720, he published the same year An Essay towards pre-
venting the ruin of Great Britain : reprinted in his Miscel-
laneous Tracts.

The path leading to the highest company being now
open, Pope introduced him to Lord Burlington, by whom
he was recommended to the Duke of Grafton, then appoint-
ed lord-lieutenant of Ireland, who took Berkeley over as
one of his chaplains in 1721. The latter part of this year
he obtained the degrees of bachelor and doctor in divinity :
and the year following he had a very unexpected increase
of fortune from the death of Mrs. Vanhomligh, the cele-
brated Vanessa, to whom he had been introduced by Swift :
this lady had intended Swift for her heir; but perceiving
herself to be slighted by him, she left her fortune, of 8000/,
between her two executors, of whom Berkeley was one. In
172-1 he was promoted to the deanery of Deny, worth
1100/ a year.

In 1725 he published A Proposal for converting the sa-
vage Americans to Christianity, by a college to be erected
in the Summer Isle«, otherwise called the Isles of Bermuda,
which lias also been reprinted in his Miscellaneous Tracts.
The proposal was well received, at least by the king; and
he obtained a charter lor founding the college, with a par-
liamentary grant of 20,000/ towards carrying it into exe-
cution: but he could never get the money, it being other-
wise employed by the ministry: so that after two years'
stay in America on this business, he was obliged to ro-
turn, and the scheme consequently expired.

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In 1732 he published The Minute Philosopher, in 2
volumes Svo, against Freethinkers. In 1733 he was made
bishop of Cloyne ; and might have been removed in 1745,
by Lord Chesterfield, to Clogher; but declined it. Here-
sided constantly at Cloyne, where he faithfully discharged
all the offices of a good bishop, yet continued his studies
with unabated attention.

About this time he engaged in a controversy with the
mathematicians, which excited the highest interest in the
literary world ; the occasion of it was this: Mr. Addison
had given the bishop an account of the behaviour of their
common friend, Dr. Garth, in his last illness, which was
equally unpleasing to both these advocates.of revealed re-
ligion. For when Addison went to see the doctor, and
began to discourse, with him seriously about another
world, "Surely, Addison," replied he, "I have good rea-
son not to believe those trifles, since my friend Dr. Halley,
who has dealt so much in demonstration, has assured me,
that the doctrines of Christianity are incomprehensible,
and the religion itself an imposture." The bishop there-
fore took up arms against Halley, and addressed to him,
as to an Infidel Mathematician, a discourse called The
Analyst; with a view of showing that mysteries in faith
were unjustly objected to by mathematicians, who he
thought admitted others much greater, and even falsehoods
in science, of which he endeavoured to prove that the doc-
trine of Fluxions furnished a clear example. This attack
gave occasion to Robins's Discourse concerning the Me-
thod of Fluxions, to Maclaurin's Fluxions, and to other
smaller works on the same subject ;■ but the direct answers
to the Analyst were made by a person under the name of
Philalethes Cantabrigiensis, but commonly supposed to be
Dr. Jurin, whose first piece was, Geometry no Friend to
Infidelity, 1734. To this the bishop replied in A Defence
of Freethinking in Mathematics ; with an Appendix con-
cerning Mr. Walton's Vindication, 1735 ; which drew a se-
cond answer the same year from Philalethes, under the
title of The Minute Mathematician, or The Freethinker no
just Thinker, 1735. Other writings in this controversy,
besides those before mentioned, were,

1. A Vindication of Newton's Principles of Fluxions
against the objections contained in the Analyst, by J. Wal-
ton, Dublin, 1735.

2. The Catechism of the Author of the Minute Philoso-
pher fully answered, by J.Walton, Dublin, 1735.

3. Reasons for not replying to Mr. Walton's Full Answer,
in a letter to P. T. P. by the author of the Min.ute Philo-
sopher, Dublin, 1735.

4. An Introduction to the Doctrine of Fluxions, and
Defence of the Mathematicians against the objections of
the author of the Analyst, &c. Lond. 1736.

5. A newTreatise of Fluxions; with answers to the prin-
cipal objections in the Analyst, by James Smith, A. M.
Lond. 1737-

6. Mi-. Robins's Discourse of Newton's Methods of Flux-
ions, and of Prime and Ultimate Ratios, 1735.

7- Mr. Robins's Account of the preceding Discourse, in
the Repub. of Letters, for October 1735.

8. Philalethes's Considerations upon some passages con-
tained in two letters to the author of the Analyst &c, in
Repub. of Letters, Novemb. 1735.

9. Mr. Robins's Review of some of the principal objec-
tions that have been made to the doctrine of Fluxions &c,
Repub. of Letters for Decern. 1735.

10. Philalethes's Reply to ditto, in the Repub. of Letters,
Jan. 1736.

1 1. Mr. Robins's Dissertation, showing that the account
of the doctrines of Fluxions &c, is agreeable to the real
sense and meaning of their great Inventor, &c, Repub. of
Letters, April 1736.

12. Philalethes's Considerations upon ditto, in Repub.
of Letters, July 1736.

13. Mr. Robins's Remarks on ditto, in Repub. of Let-
ters, Aug. 1736.

14. Mr. Robins's Remaihderof ditto, in an Appendix to
the Repub. of Letters, Sept. 1736.

15. Philalethes's Observations upon ditto, in an Appendix
to the Repub. of Letters, Nov. 1736.

16. Mr. Robins's Advertisement in Repub. of Letters,
Decemb. 1736.

17. Philalethes's Reply to ditto, in an Appendix to the
Repub. of Letters for Decern. 1736.

IS. Some Observations on the Appendix to the Repub.
of Letters for Decern. 1736, by Dr. Pemberton, in the
Works of the Learned for Feb. 1737. With some smaller
pieces in the same.

In 1736 Bishop Berkeley published "The Querist, a dis-
course addressed to magistrates, occasioned by the enor-
mous licence and irreligion of the times;" and manypther
things afterward of a smaller kind. In 1744 appeared his
celebrated and curious book, "Siris; a Chain of Philoso-
phical Reflections and Inquiries concerning the virtues of
Tar-water." It received a second impression, with addi-
tions and emendations, in 1747; and was followed by Far-
ther Thoughts on Tar-water, in 1752. In July the same
year he removed, with his lady and family, to Oxford,
partly to superintend the education of a son, but chiefly
to indulge the passion for learned retirement, which had
always strongly possessed him. He would have resigned
his bishopric for a canonry or headship at Oxford; but it
was not permitted him. Here he lived highly respected,
and collected and printed the same year all his smaller
pieces in Svo. But this happiness did not long continue,
being suddenly cut off by a palsy of the heart, Jan. 14,
1753, in the 69th year of his age, while listening to a ser-
mon that his lady was reading to him.

The excellence of Berkeley's moral character is fully
attested by his writings : he was certainly a very amiable
as well as a very learned man ; and it is thought that Pope
scarcely said too much, when he ascribed

" To Berkeley every virtue under heaven."

BERME, in Fortification, a small space of ground, 4 or
5 feet wide, left without the rampart, between it and the
side of the moat, to receive the earth that rolls down from
the rampart, and prevent its 'falling into and filling up the
ditch. — Sometimes, for greater security, the berme is pal-
lisadoed.

BERNARD (Dr. Edward), a learned English astro-
nomer, critic, and linguist, was born at Perry St. Paul,
near Towcester, the 2d of May 16'3S, and educated at Mer-
chant-Taylor's school, whence he was removed in l655 to
St. John's college, Oxford. Having laid in a good fund of
classical learning at school, in the Greek and Latin lan-
guages, he applied himself very diligently at the university
to the study of history, the Eastern languages, and mathe-
matics under the celebrated Dr. Wallis. In 1668 he went
to Leydcn, to consult some Oriental manuscripts left to that
university by Joseph Scaliger and Levin Warner, and espe-

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cially the 5th, 6th, and 7th books of Apollonius Pergseus's
Conies, brought from the East by the celebrated Golius.
These he transcribed and brought to Oxford, with a view
of publishing them, but was obliged to relinquish that de-
sign for want of proper encouragement. This however was
afterwards carried into effect by Dr. Halley in 1710, with
the addition of the Sth book, which he supplied by his own
ingenuity and industry.

On his return to Oxford, he examined and collated the
most valuable manuscripts in the Bodleian library. In
1669, the celebrated Christopher Wren, Savilian professor
of astronomyat Oxford, having been appointed surveyor-ge-
neral of his majesty's works, and being much detained at
London by thisemployment, obtained leave to namea deputy
at Oxford. He accordingly fixed on Mr. Bernard, which
engaged the latter in a more particular application to the
study of astronomy. But in l673 he was appointed to the
professorship himself, which Wren was obliged to resign,
as, by the statutes of the founder, Sir Henry Saville, the pro-
fessors are not allowed to hold any other office either ec-
clesiastical or civil. About this time a scheme was set on
foot at Oxford for collecting and publishing the works of
the ancient mathematicians. Mr. Bernard, who had first
formed the project, collected all the old books published
on that subject since the invention of printing, together
with all the manuscripts he could get from the Bodleian
and Savilian libraries, which he arranged in order of time,
and according to the matter they contained; of these he
drew up a synopsis; and as a specimen he published a few
sheets of Euclid, containing the Greek text, and a Latin
version, with Proclus's commentary in Greek and Latin,
and learned scholia and corollaries. The synopsis itself
was published by Dr. Smith, under the title of Veterum Ma-
thematicorumGra;corum,Latinorum,et Arabum, Synopsis.

Mr. Bernard undertook also an edition of the Parva
Syntaxis Alexandrina ; in which, besides Euclid, are con-
tained the small treatises of Theodosius, Menelaus, Ari-
starchus, and Hipsicles ; but it never was published.

In l6'76 he was sent to France, as tutor to the dukes of
Grafton and Northumberland, sons to king Charles the 2d
by the duchess of Cleveland, who then lived with their
mother at Paris: but the simplicity of his manners not
suiting the gaiety of the duchess's family, he returned
about a year after to Oxford, and pursued his studies with
his usual ardour and great success. In 1691, being pre-
sented to the rectory of Brightwell in Berkshire, hequitted
his professorship at Oxford, in which he was succeeded by
David Gregory, professor of mathematics at Edinburgh.

In the year 1696, he undertook a voyage to Holland,
to attend the sale ofGolius's manuscripts, as he had once
before done at the sale of Heinsius's library ; and soon
after his return to England, he fell into a languishing con-
sumption, which put an end to his life, the 12th of January
]6'g6', in the 58th year of his age.

Besides his works already mentioned, he was author .of
many other compositions. He composed tables of the
longitudes, latitudes, right-ascensions, ecc, of the fixed
stars: and wrote Observations on the Obliquity of the
Ecliptic; and other pieces inserted in the Philosophical
Transactions. I Ie wrote also,

1. A Treatise of the Ancient Weights and Measures.

2. Chronologise Samaritanas Synopsis, in two tables.
;5. Testimonies of the Ancients concerning the Greek

Version of the old Testament by the Seventy.

And several other learned works. Besides a great num-
ber of valuable manuscripts left at his death.

BERNARD (Dr. James), professor of philosophy and
mathematics, and minister of the Walloon church at Ley-
den, was born September the 1st l65S, at Nions in Dau-
phine. Having studied at Geneva, he returned to France
in 1679) and was chosen minister of Venterol, a village in
Dauphin^; but some time after he was removed to the
church of Vinsobres in the same province. To avoid the
persecutions against the protestants in France, he went
into Holland, where he was appointed one of the pensi-
onary ministers of Ganda. He here published several po-
litical and historical works. And in l699 he began the
Nouvelles de la Republique des Lettres, which continued
till December J710. In 1705 he was chosen minister of
the Walloon church at Leyden ; and about the same time,
Mr. de Voider, professor of philosophy and mathematics
at Leyden, having resigned, Mr. Bernard was appointed
his successor; upon which occasion the university also
presented him with the degrees of doctor of philosophy
and master of arts. In 1716 he published a supplement
to Moreri's dictionary in 2 volumes folio. The same year
be resumed his Nouvelles de la Republique des Lettrfes ;
which he continued till his death, which happened the
27th of April 17 18, in the 60th year of his age.

BERNOULLI (James), a celebrated philosopher and
mathematician, born at Basil the 27th of December 1654-
Having taken his degrees in that university, he applied
himself to divinity agreeably to the entreaties of his father,
but against his own inclination, which led him to astro-
nomy and mathematics. He gave very early proofs of his
genius for these sciences, and soon became a geometrician,
without a preceptor, and almost without books; for if
one by chance fell into his hands, he was obliged to con-
ceal it, to avoid the displeasure of his father. This situa-
tion induced him to choose for his device, Phaeton dri-
ving the chariot of the sun, with these words, Invito patre
sidera verso, " I traverse the stars against my father's will :"
alluding particularly to astronomy, to which he then
chiefly applied himself.

In 1676 he began his travels. When he was at Geneva,
he fell upon a method to teach a young girl to write who
had been blind from her infancy. At Bourdeaux he com-
posed universal gnomonic tables ; but they were never pub-
lished. He returned from France to his own country in
16"S0. About this time there appeared a comet, the re-
turn of which he foretold ; he wrote also a small treatise
upon it. Soon after this he went into Holland', where, he
applied himself to the study of the new philosophy. Hav-
ing visited Flanders and Brabant, he passed over to Eng-
land ; where he formed an acquaintance with the most,
eminent men in science, and attended their philosophical
meetings. He returned to his native country again in
16*82; and exhibited at, Basil a course of experiments in
natural philosophy and mechanics, which consisted of a
variety of new discoveries. The same year he published his
Essay on a new System of Comets ; and the year follow-
ing, his Dissertation on the Weight of the Air. About this
time Leibnitz having published, in the Ada Lnulitoruni
at Leipsic, some essays on his new Calculus Differentialis,
but concealing the art and method of it, Bernoulli and his
brother John soon discovered, from what they saw, its
beauty anil extent: this induced them to endeavour to un-
ravel the secret; which they did with such success, that.

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Leibnitz declared that the invention belonged to them as
much as to himself.

In l6"87 James Bernoulli succeeded to the professorship
of mathematics at Basil; a trust which he discharged with
great applause; and his reputation drew many foreigners
from all parts to attend his lectures. In 1699 he was ad-
mitted a foreign member of the Academy of Sciences of
Paris; and in 1701 the same honour was also conferred
on him by the Academy of Berlin: in both of which he
published several ingenious compositions, about the years
1702,-3, and-4. He wrote likewise several pieces in the
Acta Eruditorum of Leipsic, and in the Journal des Spa-
vans. His intense application to study brought on him
the gout, and by degrees a slow fever, which terminated
his life the 16th of August 1705, in the 51st year of his
age — Archimedes having found out the proportion of a
sphere and its circumscribing cylinder, ordgred them to be
engraven on his monument: in imitation thereof Bernoulli
appointed that a logarithmic spiral curve should be in-
scribed on his tomb, witli these words, Eadem mutata re-
surgo; in allusion to the hopes of the resurrection, which
are in some measure represented by the properties of that
curve, which he had the honour to discover.

James Bernoulli had an excellent genius for invention
and elegant simplicity, as well as a close application. He
was eminently skilled in all the branches of the mathema-
tics, and contributed much to the promotion of the new
analysis, infinite series, &c. He carried to a great extent
the theory of the quadrature of the parabola ; the geo-
metry of curve lines, of spirals, of cycloids, and.epicycloids.

His works, that had been published, were collected, and
printed in 2 volumes 4to, at Genevain 1744. At the time
of his death he was occupied on a great work entitled
De Arte Conjectandi, which was published in 4to in 1713.
It contains an elegant and most excellent introduction to
Infinite Series, &c. This posthumous piece is omitted in
the collection of his works above mentioned.

BERNOULLI (John), the brother of James, last men-
tioned, and a celebrated mathematician, was born at Basil
the 7th of August l66'7. His father intended him for
trade; but his own inclination was at first for the belles-
lettres, which however, like his brother, he forsook for
that of mathematics. He laboured with his brother to
discover the method used by Leibnitz in his essays on the
Differential Calculus, and gave the first principles of the
Integral Calculus. Our author, with Messieurs Huygens
and Leibnitz, was the first who gave the solution of the
problem proposed by James Bernoulli, concerning the ca-
tenary, or curve formed by a chain suspended by its two
extremities.

John Bernoulli had the degree of doctor of physic at
Basil, and two years afterward was named professor of ma-
thematics in the university of Groningen. It was here that
he discovered the mercurial phosphorus or luminous baro-
meter, occasioned by friction of mercury against glass in
a vacuum ; and whore he also resolved the problem pro-
posed by his brother concerning Jsoperiinqtricals.

On the death of James, the professor at Basil, our au-
thor returned to his native country, against the pressing in-
vitations of the magistrates of Utrecht to come to that city,
and of the university of Groningen, who also wished to re-
tain him. The Academic Senate of Basil soon appointed
him to succeed his brother, without assembling competi-
tors, and contrary to the established practice: an appoint-
ment which he held during his whole life.

Vol. I.

In 1714 was published his treatise on the management
of ships; and in 1730, his memoir on the elliptical figure
of the planets which gained the prize of the Academy of
Sciences at Paris. The same academy also divided the
prize, for their question concerning the inclination of the
planetary orbits, between our author and his son Daniel.

John Bernoulli was a member of most of the academies
of Europe, and received as a foreign associate of that -of
Paris in 1699. After a long life spent in constant study
and the improvement of all the branches of the mathema-
tics, he died full of honours the 1st of January 1748, in
the 81st year of his age. Of five sons which he had, three
pursued the same study with himself. One died before
him ; the two others, Nicholas and Daniel, he lived to see
become eminent and respected in mathematical science.

The writings of this great man were dispersed through
the periodical memoirs of several academies, as well as in
many separate treatises. And the whole of them were care-
fully collected and published at Lausanne and Geneva,
1742, in 4 volumes, 4to.

BERNOULLI (John), son of the preceding, was born
at Basil, Jan. 17, 1695, and died at Petersburg!], July 26,
1 726. He was licentiate of law, professor of law at Berne ;
and afterwards professor of mathematics at Petersburg,
and member of the Institute of Bologna.

BERNOULLI (Nicholas), nephew of the two first
named, was professor of mathematics at Padua, after-
wards of logic, and then of law at Basil. He was member
of the Academy of Sciences and Belles-Letters at Berlin,
and also of the Royal Society of London, and of the Insti-
tute of Bologna ; was born at Basil, Oct. 10, l687, where
he also died the 29th of Nov. in the year 1759.

BERNOULLI (Daniel), a celebrated physician and
philosopher, son of John, and nephew of James, was
born at Groningen Feb. the 9th, 1700, where his fa-
ther was then professor of mathematics. He was first de-
signed for trade, but his genius led him to other pursuits.
He passed some time in Italy ; and at 24 years of age he
declined the honour offered him of becoming president of
an academy intended to have bet-n established at Genoa.
He spent several years with great credit at Petersburg;
and in 1733 returned to Basil, where his father was then
professor of mathematics; and here our author successively
filled the chair of physic, of natural and of speculative phi-
losophy.

In his work Exercitationes Mathematics, 1724, he took
the only title he then had, viz, " Son of John Bernoulli,"
and never would suffer any other to be added to it. This
work was published in Italy, while he was there on his
travels; and it classed him in the rank of inventors. His
work entitled Hydrodynamica, was published in 4to at
Argentoratum or Strasbourg, in 1738.

Daniel Bernoulli wrote a multitude of other pieces,
which have all been published either in the Mem. of the
Acad, of Sciences at Paris, or in those of other Academies.
He gained and divided tin prizes from the Academy of
Sciences at Paris, which were contended for by the most
illustrious mathematicians in Europe. The only person
who has had similar success is Euler, his countryman, dis-
ciple, rival, and friend. His first prize he gained at 24
years of age. In 1734 he divided that with his father re-
specting the inclination of the orbits of the planets; which
hurt the family union ; for the father considered the con-
test itself as a want of respect ; and the son did not suf-
ficiently conceal that he thought (what was really the case)
2G

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his own piece better than his father's. And besides, he de-
clared for Newton, against whom his father had contend-
ed all his life. Jn 1740 our author divided the prize, " On
the Tides of the Sea," with Euler and Maclaurin. The
Academy at the same time rewarded a fourth piece, the
chief merit of which was that of being Cartesian; but this
was the last public act of adoration paid by the Academy
to the authority of the author of the Vortices, which it
had already obeyed too long. In 1 748 Daniel Bernoulli
succeeded his father John in the Academy of Sciences, who
had succeeded his brother James ; this place, since its first
erection in 1699, having never been without a Bernoulli to
fill it.

Ou,r author was extremely respected at Basil ; and to
bow to Daniel Bernoulli, when they>met him in the streets,
was one of the first lessons which every father gave to his
child. He was a man of great simplicity and modesty of
manners. He used to tell two little adventures, which he
said had given him more pleasure than all the other ho-
nours he had received. Travelling with a learned stran-
ger, who, being pleased with his conversation, asked his
name; " [ am Daniel Bernoulli," answered he with great
modesty; "and I," said the stranger(who thought he meant
to laugh at him) " am Isaac Newton." Another time,
having to dine with him the celebrated Koenig, the mathe-
matician, who boasted, with some degree of self-compla-
cency, of a difficult problem he had resolved with much
trouble, Bernoulli went on doing the honours of his table,
and when they went to drink coffee he presented Koenig
with a solution of the problem more elegant than his own.

After a long, useful, and honourable life, Daniel Ber-
noulli died the 17th of March, 1782, in the 83d year of
his age.

BERNOULLI (James), another mathematical branch
of the foregoing celebrated family. He was born at Basil
in October 1759; being the son of John Bernoulli, and
grandson of the first John Bernoulli, before mentioned, and
the nephew of Daniel Bernoulli last noticed. Our author's
elder brother John, who still lives at Berlin, is also well
known in the republic of science, particularly for his as-
tronomical labours.

The gentleman to whom this article relates, was educated,
as most of his relations had been, for the profession of the
law : but his genius led him very early into the study of
mathematics; and at 20 years of age he read public lec-
tures on experimental philosophy in the university of Basil,
for his uncle Daniel Bernoulli, whom he hoped to have
succeeded as professor. Being disappointed in this view,
he resolved to leave his native place, and to seek his for-
tune elsewhere; hence he accepted the office of secretary
to Count Breuner, the emperor's envoy to the republic of
Venice; and in this city he remained till the year 1786,
when, on the recommendation of his countryman, M.Fuss,
he was invited to Petersburg to succeed M. Lexell in the
academy there, where he continued till his death, which
happened the 3d of July 1789, at not quite 30 years of
age, and when he had been married only two months, to
the youngest daughter of John Albert Euler, the son of the
celebrated Leonard Euler.

BETELGELSE, a fixed star of the first magnitude in
the right shoulder of Orion.

BICV1S (John, m. d.), an able astronomer, was born
October 31, 1695, near Old Sarum in Wiltshire. He was
entered at Christ-Church College, Oxford, where he ap-
plied himself diligently, not only to the study of physic,

for which he was designed, but to other sciences, particu-
larly astronomy and optics, in which he became a consi-
derable proficient. After taking the degree of m. d. he
left the university, and devoted himself almost entirely to
pursuits connected with astronomy. In 1738 he made an
excellent collection of astronomical instruments for a new
observatory, which he had erected at Newington Green, in
Middlesex. Here he became an indefatigable observer,
often observing the transits of 160 stars in one night. —
About 1745 the doctor undertook the laborious task of
arranging a work called Uranographia Britannica, or an
Exact View of the Heavens, on 52 plates, similar to Bay-
er's work representing the constellations, &c. The plates,
which would have done great honour as well to the nation
as Dr. Bevis, were engraven but never published, the doc-
tor having consigned the management of them to a person
who became a bankrupt. Dr. Bevis was the real author
of a great many works which have been well received by
the public, but which his modesty prevented him from
taking the merit of. It is to him we are indebted for the
publication of Dr. Halley's Astronomical Tables, after
they had been printed more than 20 years: having supplied
some auxiliary tables, and the precepts for using them, he
favoured the public with the whole in the year 1749. In
Mr. Simpson's Essays, p. 10, are delivered practical rules
for finding the aberration, which were drawn up and given
him by Dr. Bevis, with examples of the correction applied
to several stars, which he had carefully observed with pro-
per instruments; by which he proved, first of any one,
that the phenomena are as conformable in right ascension,
as Dr. Bradley, who made this great discovery, found they
were in declination. Several pieces by the doctor were
inserted in the few numbers that were published of a work
called the Mathematical Magazine, by Messrs. Whitchell
and Moss; particularly a curious paper on the satellites
of Venus, and several sheets of a new mathematical dic-
tionary. Dr. Bevis enriched the Philosophical Transactions
with 27 valuable papers, from vol.40 to vol. 59 inclusive:
The only pieces which appeared separately with his name,
besides these papers in the Transactions, were two pam-
phlets, the one entitled The Satellite's Sliding Rule, for de-
termining the immersions and emersions of Jupiter's four
satellites. The other was An Experimental Enquiry con-
cerning the Contents, Qualities, and Medicinal Virtues, of
the Two Mineral Waters lately discovered at Bagnigge-
Wells, near London, 1760. He was the first who appro-
priated the term achromatic to the curious telescopes in-
vented by Mr. Dollond. On the death of Mr. Bliss in
1765, Dr. Bevis was a candidate with Dr. Maskelyne for
the honourable and important situation of Astronomer
Royal. Soon after which, he removed from Newington to
the Temple, London, where he followed his occupation as
a physician, and continued to make astronomical obser-
vations. Here he died, Nov. 6, 177', at 76 years of age;
his death having been occasioned by a fall he received a
short time before, in going too hastily from his instrument
to the clock wljen observing the sun's meridian altitude.

BEZOUT (Stephen), a celebrated Trench mathema-
tician, Member of the Academies of Sciences and the Ma-
rine, and Examiner of the Guards of the Marine and of
the Elevc-s of Artillery, was born at Nemours the 31st of
March, 1730. In the course of his studies he met with
some books of geometry, which gave him a taste, for that
science; and the Elogos of Fontcnellc, which showed him
the honours that attended tuknts and the love of the sci-

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ences. His father in vain opposed the strong attachment
of young Bezout to the mathematical sciences. April 8,
1758, he was named adjoint-mechanician in the French
Academy of Sciences ; having before thut sent them two
ingenious memoirs on the integral calculus, and given other
proofs of his proficiency in the sciences. In 1763 he was
named to the new office of Examiner to the Marine, and
appointed to compose a Course of Mathematics for their
use ; and in 1768, on the death of M.Camus, he succeeded
as Examiner of the Artillery Eleves.

Bezout fixed his attention mtore particularly to the reso-
lution of algebraic equations; and he first found out the
solution of a particular class of equations of all degrees.
This method, different from all former ones, was general
for the cubic and biquadratic equations, and just became
particular at those of the fifth degree. At this work of
finding the roots of equations, our author laboured from
1762 till 1779, when he published it. He composed
two courses of mathematics; the one for the Marine, the
other for the Artillery. The foundation of these two works
was the same; the applications only being different, ac-
cording to the two different objects : these courses have
every where been held in great estimation. In his office
of Examiner, he discharged the duties with great atten-
tion, care, and tenderness. A trait of his justiee and zeal
is remarkable in the following instance: During an exami-
nation which he held at Toulon, he was told that two of
the pupils could not be present, being confined by the
small-pox : he himself had never had that disease, and he
was greatly afraid of it; but as he knew that if he did not
see these two young men, it would much impede their ad-
vancement, he ventured therefore to their bed-sides, to ex-
amine them, and was happy to find them so deserving the
hazard he had made for their benefit.

M. Bezout lived thus several years, beloved of his fa-
mily and friends, and respected by all, enjoying the fruits
and the credit of his labours. But the trouble and fatigues
of his offices, with some personal chagrins, had reduced
his strength and constitution; he was attacked by a malig-
nant fever, of which he died Sept. 27, 1783, in the 54th
year of his age, regretted by his family, his friends, the
young students, and by all his acquaintance in general.

The books published by him were :

1. Course of Mathematics for the use of the Marine,
with a Treatise on Navigation; 6 vols, in 8vo. Paris, 1764.

2. Course of Mathematics for the Corps of Artillery; 4
vols, in 8vo, 1770.

3. General Theory of Algebraic Equations, 1779.

His papers printed in the volumes of the Memoirs of the
Academy of Sciences, are:

1. On curves whose rectification depends on a given
quantity, in the vol. for 1758.

2. On several classes of equations that admit of an al-
gebraic solution, 1762.

3. First vol. of a course of mathematics, 17(54.

4. On certain equations, &c. 1764.

5. General resolution of all equations, 1765.

6. Second vol. of a course of mathematics, 1765.

7. Third vol. of the same, 1766.

8. Fourth vol. of the same, 1767.

£)■ Integration of differentials, &c. vol. 3, Sav. Etr.
10. Experiments on cold, 1777.

BIANCHINI (Francis), a very learned Italian philo-
sopher and mathematician of the J 7th century, was born

at Verona the 13th of December, 1662. He was highly
esteemed by the learned, and was a member of several
academies; and was even the founder of that at Verona,
called the Academy of Aletofili, or Lovers of Truth. He
went to Rome in l684; and was made librarian to Cardi-
nal Ottoboni, who was afterwards Pope by the name of
Alexander the Sth. He entered into the church, and he-
came canon of St. Mary de la Rotondo, and afterward
of St. Lawrence in Damaso.

Bianchini was author of several learned and ingenious
dissertations. In 1 697 he- published La Istoria universale
provata con monumenti, et figurata con simboli de gli an.-
tichi. In 1701 Pope Clement the 11th named him secre-
tary of the conferences for the reformation of the calendar ;
in which situation he published in 1703, De Calendario et
Cyclo Caesaris, ac de Canone Paschali sancti Hypoliti,
Maityris, Disscrtationes duas. Bianchini was employed
likewise to construct the large gnomon in the church of the
Chartreux at Rome, on which he published an ample
dissertation entitled, De Nummo et Gnomone Clementino.
The research concerning the parallax and the spots of Ve-
nus occupied him a long time; but his most remarkable
discovery is that of the parallelism of the axis of Venus in
her orbit. He was employed 8 years in preparing means
for tracing a meridian line through the whole extent of
Italy, but he never commenced this undertaking. He
was admitted a foreign Associate in the Paris Academy of
Sciences, in 1706; and had many astronomical disserta-
tions inserted in their Memoirs, partieularly in those of the
years 1702, 1703, 1704, 1706, 1707, 1708, 1713, and 17 18.
— Bianchini died the 2d of March, 1729, in the 67th year
of his age.

BILLION, in Numeration, the same as a million mil-
lions; as much as to say bismillion or bimillion, or by fur-
ther contraction, billion.

BILLY (James De), of Compeigne, a Jesuit, died in
1679 at Dijon, aged 77- He was a respectable mathema-
tician, and published some good works in that science; as,

1. Nova Geometrise Clavis Algebra. Paris, 1643, in 4to.

2. Tabulse Lodoiceas de Doctrina Eclipseon. Divions,
1656, in 4to. 1

3. De Proportionc Harmonica Tractatus. Paris, 1658,
in 4to.

4. Diophantis Geometria, ex Grasco versa, in qua per ra-
tioncs Euclidianas efficiuntur Problemata, qua? ille Ratio-
nibus Algebraicis enodavit. Paris, l660, in 4to.

5. Opus Astronomicum. In which he explained the hy-
potheses and motions of the stars, and every thing relating
to the theory and practice of astronomy. Dijon, l66l, in
4to.

6. Crisis Astronomica, de Motu Cometarum. In which
he refuted the rectilinear hypothesis, and discovered a new
system of comets. Dijon, 1666, in 8vo.

7. Diophantes Redivivus. Lyons, 1670, in 8vo.

8. Doctrina; Analyticae inventum novum. Contained in
the Commentaries on Diophantus, composed by M. Fer-
mat. Toulouse, 1671, in folio.

BIMEDIAL Line, is the sum of two medials. Euclid
reckons two of these bimcdials, in pr. 38 and 39, lib. x; the
first is when the rectangle is rational, which is contained by
the two medial lines whose sum makes the bimedial; and
the second when that rectangle is a medial, or contained
under two lines that are commensurable only in power.

BINARY Number, that which is composed of two units.
2 G 2

B I N

[ 228 ]

B I N

Binary Arithmetic, that in which two figures or cha-
racters, viz. 1 and 0, only are used; the cipher multiplying
every thing by two, as in the common arithmetic by 10:
thus, L is one, 10 is 2, 1 1 is 3, 100 is 4, 101 is 5, 110 is 6,
111 is 7, 1000 is 8, 1001 is 9, 1010 is ten; being founded
on the same principles as common arithmetic. This kind of
arithmetic was invented by Leibnitz, who pretended that it is
better adapted than the common arithmetic, for discovering
certain properties of numbers, and for constructing tables ;
but he does not venture to recommend it for ordinary use,
•on account of the great number of places of figures requi-
site to express all numbers, even very small ones. Jos. Pe-
lican, of Prague, has more largely explained the principles
and practice of the binary arithmetic, in a book entitled
Arithmeticus Perfcctus, qui tria numerare nescit : 1712.
And De Lagni proposed a new system of logarithms, on the
plan of the binary arithmetic; which he finds shorter, and
more easy and natural than the common ones.

BINOCLE, or Binocular Telescope, is one by
which an object is viewed with both eyes at the same time.
It consists of two tubes, each furnished with glasses of the
same power, by which means it has been said to show ob-
jects larger and more clearly than a monocular or single
telescope; though this is probably only an illusion, occa-
sioned by the stronger impression which two equal images,
alike illuminated, make upon the eyes": they arc however
more embarrassing than useful. This telescope has been
chiefly treated of by the fathers Reita and Cherubin of
Orleans. — There are also microscopes of the same kind,
though but little, used, being subject to the same inconve-
niences as the telescopes.

BINOMIAL, a quantity consisting of two terms or mem-
bers connected by the sign 4- or — , viz, plus or minus;
as a -t- b, or 3a — 2c, or az ■+■ b, or x" — 2^/c, &c ; de-
noting the sum or the difference of the two terms. When
either term is negative, the binomial is sometimes called
a residual, and by Euclid an apotome. The term bino-
mial was first introduced by Lucas de Burgo. See his Arith.
carta 120, edit. 1404.

Binomial Line, or Surd, is that in which at least one
of the parts is a surd. Euclid enumerates six kinds of bi-
nomial lines or surds, in the 10th book of his Elements,
which are exactly similar to the 6 residuals or apotomes
there treated of also, and of which an account is given un-
der the art. Apotojie, which see. Those apotomes be-
come binomials by only changing the sign of the latter,
term from minus to plus, which therefore are as below.
Euclid's 6 Binomial Lines.

1st binomial 3 -+- ^/5,

2d binomial ^/lS ■+• 4,

3d binomial v/24 ■+- ^/IS,

4th binomial 4 -+■ ^/3,

5th binomial */6 -+- 2,

6th binomial ^/6 ■+■ »/2.
To extract the Square Root of a Binomial, as of a -+- ^/b,
or </c •+• \/b. — Various rules have been given for this
purpose. The first is that of Lucas de Burgo, in his
Suinma de Arith. &C, which is this : When one part, as a,
is rational, divide, it into two parts such, that their product
may be equal to^ of the number under the radical b; then
shall the sum of the roots of those parts be the root of the
binomial sought: or their difference is the root when the
(juantity is residual ; that is, if c ■+- e = a, and c e =■ -^ b;
then is s/c + *Je = v/(n ■+• \/b) ihe root sought. For

example, let the binomial be 23 + \/44S; then the parts
of 23 are 16 and 7, and their product is 1 12, which is ith
of 448 ; therefore the sum of their roots 4 ■¥ v/7 is the
root sought of 23 -+- ^448.

Lucas de Burgo gives also another rule for the same ex-
tractions, which is this: The given binomial being, for ex-
ample, s/c ■+■ ^/b, its root will be \/[i-/c -+■ {\/(c — b)]

-*~ */ [» \/c ~~ 5 s/ (c — &) ]• — So, in the foregoing exam-
ple, 23 -+- ^448, here y'c = 23, and s/b = i/iii; hence

i*/c = 11|, and ij{c - b) = 1^/(23- - 448) =

lv/81 =41;

therefV[fv/c + W(c ~ b)l = v/(l If -+- 4|) =\6 =4,
and v/ [i^/c - |^/(c - b)J = ^/(lll-4{) =z </7;

conseq. 4 -+■ */7 is the root sought, as before.

Again, if the binomial be ^/ 18 -+■ ^/10; here c =s 18, and

b= 10; theref. 1^0 = 1^/18=1^2, and J v/(c -b) =

i\/8 = </2; hence,

ViWc+ iv/(c-6)]=A/(|v/2 + v/2) = -/zv/2 =

J/B and

consequently, y/~ -+- %/-, or ^-rp-_ — is the root of a/ IS ■+■

-v/10 sought. This latter rule has been used by all au-
thors, down to the present time.

To extract the Cubic and other higher Roots of a Binomial.
This is useful in resolving cubic and higher equations, and
was introduced with the resolution of those equations by
Tartalea and Cardan. The rules for-such extractions are
in great measure tentative; some of the principal ones of
which are the following.

Tartalea' s Rule for the Cube Root of a Binomial p -+■ q.
This rule is given in his Qth book of Miscellaneous Ques-
tions, quest. 40; being made out from either of the terms,
p or q, of the binomial, taken singly, in this manner : Se-
parate either term, asp, into two such parts that one of
them may be a cubic number, and the other part divisible
by 3 without a remainder; then the cube root of the said
cubic part will be one term of the root, and the other term
will be the square root of the quotient arising from divi-
ding the aforesaid third part by the first term just found.

Thus Up be divided into r3 -+- 3s, then the root is r -+- ^/-.

For example, to extract the cube root of v/108 ■+■ 10.
Suppose the part 10 to be taken: this separates into the
parts 1 and Q, the fortner of which is a cube, and the latter
divisible by 3; that isr3 = 1, and 3* = *); hence r= 1, and

s = 3; consequently r -+- *J- = 1 -+■ \/3 is the cube root

of v/108 -4- 10 sought. Again, to use the other term
V'lOS: this divides into s/27 -*- y/27, of which the for-
mer is a cube, and the latter divisible by 3; that is, r3 =
■v/27, and 3s = </'27; therefore r = </3, and * = \\/^7
=\/3 also: hence r -f- y/~ = >/3 + */jft — v^ + 1
the cube root, the same as before.

Bombelli's Rule for the Cube Root of the Binomial a +
t/~ b. First find l/(a'! ■+■ b); then, by trials, search out
a number c, and a square root */d, such that the sum of
their squares c' ■+■ d may be =3v/(n -h b), and also c3 —
3cd =: a; then shall c ■+- >J — d be the cube root of a ■+■
*/ — b sought. For example, to find the cube root of 2 ■+■
•v/ ' — 121 1 herey/(«i ■+■ b) =^/12.r> = 5; then taking c =
2, and d = 1, it is C* -+- d = 5 = \/ (a' -t- b) and c3 — 3cd

B I N

[ 229 ']

B I N

&c.

•'0

a/5

■v/20

— s — 6 = 2 = a, .as it ought ; therefore 2 + ^ — 1 is
the cube root of 2 -t- ^/ — 121 . — Bombelli gave also a rule
for the cube root of the binomial a -+- ^/b, but it is good
for nothing.

Albert Girard's Rule for the Cube Root of a Binomial.
This is given in his Invention Nouvelle en l'Algebre, and is
explained by him thus: Let 72 ■+■ ^/5I20 be the given
binomial whose cube root is sought.

The square of 72 the greater term is 5184
and of the less term is 5120
their difference 64
its cube root 4,
which 4 must be the difference between the squares of the
two terms of the root sought; and as the rational part 72
of the given binomial is the greater term, therefore the ra-
tional part of the required root will be the
greater part also ; consequently the root
sought must be one of the binominals here set
in the margin, where the difference of the
squares of the terms is always 4, as required ;
and to find out the true one, proceed thus:
The first, 2 -+- */0 must be rejected, because one term of it
is 0 or nothing; also because 5 exceeds the cube root of
72, or ^/'20 exceeds the cube root of v/5120, therefore 5
-t- v^20, and all after it must be rejected likewise ; so that
the root must be either 3 ■+• */5 or 4 -+- \/l'2, if the given
quantity has a binomial root : to know which of these is to
be taken, it must be considered that the rational term of
the root must measure the rational term given ; and also
the irrational term of the root must measure the irrational
term given; then, on examination it is found that both -5
and 4 measure or divide the 72 without a remainder, but
that only the </5, and not <i/l2, measures .y/5120; con-
sequently none but 3 4- \/ 5 can be the cube root of the
given quantity 72 -<- */ 5 120; which is found to answer, by
cubing the said root 3 -t- ^/5.

Dr. Wallis's Rule for the Cube Root of the Binomial a j^
m^/b ora± m*/ — b. In these forms the greatest rational
part m is extracted out of the radical part, leaving only b
the least radical part possible under the radical sign. He
first observes that if the given quantity have a binomial
root, it must be of this form c ± n^/b, with the same radi-
cal 6. Then to find the value of c and n, he raises this root
to the3d power, which givesc3 -+- 3cnb j- 3c'n -+- nJb • ^ ' b,
vhjch must be = a ±m^/b the given quantity; hence

putting the rational part of the one quantity equal to that of
the other, and also the radical part of the one equal to that
of the other, gives c3 ■+■ 3cri*b = a, and Stfn -t- ri*b = m. Then
assuming several values of n, from the last equation he finds
the value of e; hence if these values ofc and n, substituted
in the first equation, make it obtain, they are right ; but if
not, another value of n js assumed, and so on, till the first
equation hold true. And it is to be noted thatn is always
an integer, or else the half of an integer. For example, if
the cube root of 135 ± v/1825 be required, or 135 ± 78
y 3; here a = 135, m = 78, and b = 3 ; hence 3c'n -+■
n'b = m is 3c2ra ■+- 3n3 = 78, or c'n ■+■ n3 = 26; then as-
suming n = 1, this last equation becomes c2 -+- 1 = 26,
from which c is found = 5 ; which values of c and n being
substituted in the first equation c3 -1- 3cn2b^za, make 53 -+-
3 . 5.3 = 170, but ought to be 135, showing that c is too
great, and consequently n taken too little. Let n therefore
be assumed = 2, so shall 1c1 ■+■ 8 = 26, and c ■= 3 ; and
the first equation becomes 3s -+- 3 . 3 . 22. 3 = 27. 5 = 135

= a as it ought, which shows that the true value of n is 2,
and that of c ie 3 ; hence then the cube root of 135 ± 78
^/3 or c + n^/b is 3 ± 2^/3 or 3 ± y'12. And in like
manner is the process instituted when the number in the ra-
dical is negative, as the cube root of 8 1 ± 30*/ — 3, which
•sf!fcfv/-S.

Another rule for extracting the cube root of an imaginary
binomial was also given by Demoivre, at the end of Saun-
derson's Algebra, by means of the trisection of an arc or
angle.

Sir I. Newton's Rule for any Root of a Binomial a ± b.
In his Universal Arith. is given a rule for the square root
of a binomial, which is the same as the 2d by Lucas de
Burgo, before given ; and also a general rule for any root
of a binomial, which I have not met with elsewhere; it is
this : Of the given quantity a ± b, let a be the greater term,
and c the index of the root to be extracted. Seek the least
number n whose power ne can be divided by aa — bb with-
outa remainder, and let the quotient be q ; Compute £/((a
-+■ b)^/q) in the nearest integer number, which call r ; di-
vide a ^/ q by its greatest rational divisor, calling the quo-
tient s : and let the nearest integer number above — r be

t: so shall

t s ±_ t/lf s- — n)

be the root sought, if the ex-
And this rule is demonstrated

■yi ••

traction of it be possible,
by Gravesande in his commentaiy on Newton's Arithmetic.
Many numeral examples, illustrating this rule, are given
in Gravesande's Algebra, abovementioned, pa. 160, as also
in Newton's Univers. Arith. pa. 53, 2d edit, and in Mac-
laurin's Algebra, pa. 1 1 8. Other rules may be found in
Schooten's Commentary on the Geometry of Descartes, and
elsewhere.

Impossible or Imaginary Binomial, isa binomial which
has one of its terms an impossible or an imaginary quantity:
as a +'^/ — b.

In the foregoing article are given several rules for the
roots of Binomials. Dr. Maskelyne, the late Astronomer
Royal, has also given a method of finding any power of an
Impossible Binomial, by another like Binomial. This rule
is given in his Introduction prefixed to Taylor's Tables of
Logarithms, pa. 56 ; and is as follows.

The logarithms of a and b being given, it is required
to find the power of the Impossible Binomial a ± */ — b2

whose index is — , that is, to find (a i ^/— b'~)" by an-
other Impossible Binomial; and thence the value of (a -+-

v/— 62)" ■+■ (a — aJ — fc*)° , which is always possible, whe-
ther a or b be the greater of the two.

Soluti

Then

Put - — tang. z.
ia)°= (as -1- b2)ln x (cos.- 2± \/— sin. — z).
</- b2)»+ (a - v/- V-y = (a* + b2f" x

(«±v/-

Hence (a

2 cos. - 2 = (a x, sec. r)n x 2 cosin. - z = (b x cosec. z) "

x 2 cosin. -2, where the first or second of these two last

expressions is to be used, according as z is an extreme or

mean arc ; or rather, because - is not only the tangent of

2, but also of 2 -1- 360°,2 4- 720°, &c ; therefore the factor
in the answer will have several values, viz,

B I N

[ 230 J

B I N

2 cos. -z; 2 cos.- (z -t- 360°) ; 2 cos. -(s + 7200) ; &C;
n ft ' n

the number of which, if m and n be whole numbers, and
the fraction - be in its least terms, will be equal to the de-
nominator n; otherwise infinite.

By Logarithms. Put log. b + 10 — log. a = log. tan. z.

Then log. [(a + ,/ -b*)° + (ft - v/-&i)"J = ™ * (!•«

•+■ 10 - 1. cos. z) -+- 1. 2 + 1. cos. ^ s - 10 = ^ x (1. J +

10 — 1. sin. z) h-1.2+1. cos.- s — 10; where the first or

second expression is to be used, according as z is an ex-
treme or mean arc. Moreover by taking successively, 1.

cos.- z; l.cos.- (z ■+■ 360<>); 1. Cos.- (z + 720°) ; &c,
there will arise several distinct answers to the question,
agreeably to the remark made above.

BiieoMiAL Curve, is a curve whose ordinate is expressed
by a binomial quantity ; as the curve whose ordinate is
x3- x b ■+- dxfy. Stirling, Method. Diff. pa. 58.

Binomial Theorem, is Used to denote the celebrated
theorem given by Sir I. Newton for raising a binomial to
any power, or for extracting any root of it by an approxi-
mating infinite series. It was known by Stifelius, and
others, about the beginning of the l6th century, how to
raise the integral powers, not barely by a continued mul-
tiplication of the binomial given, but also by a table of
numbers formed by Stifelius^ which showed by inspection
the coefficients of the terms of any power of the binomial,
contained within the limits of the table ; but still they could
not independent of a table, and of any of the lower powers,
raise any power of a binomial at once, by determining its
te|tns one from another only, viz, the 2d term from the 1st,
the 3d from the 2d, and so on as far as we please, by a ge-
neral rule ; and much less could they extract general al-
gebraic roots in infinite series by any rule whatever.

For though the nature and construction of that table,
which is composed of figurate numbers, was so early known,
and employed in raising powers, and extracting roots ; yet
it was only by raising the numbers one from another by
continual additions, and then taking them from the table
when wanted ; till Mr. Briggs first pointed out the way of
raising any line in the table by itself, without any of the
preceding lines; and thus teaching to raise the terms of
any power of a binomial, independent of any of the other
powers; and so gave the substance of the binomial theo-
rem in words, wanting only the algebraic notation in sym-
bols ; as is shown at large at pa. 78 of the historical intro-
duction to my Mathematical Tables. Whatever was known
however of this matter, related only to pure or integral
powers, no one before Newton having thought of extract-
ing roots by infinite series. He happily discovered that,
by considering powers and roots in a continued series, roots
being as powers having fractional exponents, the same bi-
nomial series would equally serve for them all, whether
the index should be fractional or integral, or whether the
series be finite or infinite. The truth of this method how-
ever was long known only by trial in particular cases, and
by induction from analogy ; nor does it appear that even
Newton himself ever attempted any direct proof of it :
however, various demonstrations of the theorem have since
been given by the more modern mathematicians, some of
which are by means of the doctrine of fluxions, and others,

more legally, from the pure principles of algebra only :
for a full account of which, see pa. 228, &c, of my Mathe-
matical Tracts, vol. 1.

This theorem was first discovered by Sir I. Newton in
1669, and sent in aletter of June 13, 1676, to Mr. Olden-
burg, Secretary to the Royal Society, to be by him com-
municated to Mr. Leibnitz ; and it was in this form :

m m m in — n , m — Qn „ .

(p -*-pq)n =pu n — aq h — bq ■+■ — - — cq -+- &c: where

p + pq signifies the quantity whose root, or power, or root
of any power, is to be found ; p being the first term of that
quantity; q the quotient of all the rest of the terms divided

by that first term ; and - the numeral index of the power

or root of the quantity p ■+■ pq, whether it be integral or
fractional, positive or negative ; and lastly a, b, c, d, &c,
are assumed to denote the several terms in their order as

they are found, viz, a = the first term pT, b = the 2d term

-aq, c =: the third term bq, and so on. As Newton's

u ' in 3

general notation of indices was not commonly known, he
takes this occasion to explain it; and then he gives many
examples of the application of this theorem, one of which
is the following.

T

Ex. 1. To find the value of t/(cx -+■ Xs) or (c5 ■+■ a:2)5,
that is, to extract the square root of c1 •+■ x2 in an infinite

series. Here p = cx,q =^, m = 1, and n = 2; therefore

™ . -| , m x" in — n — x"

" " ^ ' ' n " 2c' 2k " 8c3 '

&c ; therefore the root sousrht isc h s + m ■-

0 2C Be3 16c5 12SC

-f- &c.

A variety of other examples are also given in the same
place, by which it is shown that the theorem is of univer-
sal application to all sorts of quantities whatever. — This
theorem is sometimes represented in other forms, as

Pnx :1h q -i • q ■+■

n " n in * n

■+■ &c ; which conies to the same thing

OP

Or thus j?u ■+■
m ~ " P TT* &c; where the binomial is

On * ' '

r)n.

In another letter to Mr. Oldenburg, of Oct. 24, 1676"*
Newton explains the train of reasoning by which he ob-
tained the said theorem, as follows : " In the beginning of
my mathematical studies, when I was perusing the works
of the celebrated Dr. Wallis ' (see his Arith. of Infinites,
prop. J 18, and 121, also his Algebra, chap. 82), and con-
siderili" the series by the interpolation of which he exhi-
bits the area of the circle and hyperbola; for instance, in
this scries of curves, whose common base or axis is r,
and the ordinates respectively (1 — ar*J* (1 — J*) ,
(1 — xx)\ (1 —-")', (I — xx)\ &c; I perceived that if
the areas of the alternate curves, which are

x - -J~r3,

x — fr3 -t- -fx5,

x — far3 ■+■ ft' - j-r7,

&c; could be interpolated, we should be able to ob-
tain the areas of the intermediate ones; the first of which,

(I — xx) , is the area of the circle: now in order to this,
it appeared that in all the series the first term was x; that
the 2d terms Ji3, }x3, |.r3, }x\ &c, were in arithmetical

B I N

[• 231 ]

B I N

progression ; and consequently that the first two terms of (1 — if)", will be produced jay the continued multiplier

all the series to be interpolated would be

X — — , X——, X — — , &C.
8 ' 3 3 '

" Now for the interpolation of the rest, I considered
that the denominators, 1, 3, 5, 7, &c, were in arithme-
tical progression ; and that therefore only the numeral
coefficients of the numerators were to be investigated.
But these in the alternate areas, which are given, were the
same with the figures of which the several powers of 11
consist; viz, of 11°, 11", ll2, ll3, &c ; that is,

the first 1 ,

the second 1 , 1

the third 1, 2, 1

the fourth 1, 3, 3, 1

the fifth 1, 4, 6, 4, 1
&c.
" I enquired therefore how, in these series, the rest of the
terms may be derived from the first two being given ; and
I found that by putting ra for the 2d figure or term, the
rest would be produced by the continued multiplication
of Che terms of this series,

m — 0 m— 1 m — 1 m — 3 m — £ „

I * ~~ 2 3~~ 4 5

2d term m = 4 ; then shall
-, or 6, be the 3d term; and 6 x — - — , or 4,

-a 3

the 4th term; and 4 x - or 1, the 5th term; and

1 x — ■ — , or 0, the 6th; which shows that in this case

the series terminates.

" This rule therefore I applied to the series to be in-
terpolated. And since, in the series for the circle, the

2d term was —- I put m = ■£, which produced the terms

i-3

" Kor instance, if the

m — 1

or ; x

or -t- — ; -+- — x

or — — ; and so on ad infinitum. And hence I found

128

that the required area of the circular segment is

_ if! _ ill _ ~hiL — t^1" _ k.c

X 3 5 7 9

" And in the same manner might be produced the in-
terpolated areas of the other curves : as also the area of
the hyperbola and the other alternates in thisseries

(i + xxy, (i ■+- xxy, (i -+- xxy, (1 + xx)T, &c.

And in the same way also may other series be interpolated,

tion of the terms of this series

m — 1 m — 2 m nr? 3

—j- x 3 x — — &c.

" Thus, for example, there would be found

(1 - xx)^ = l-Ii!-|r«- _^.T* &c.

(1 - xx)* - l - | x2 *■ | x4 .«- ,ij x6 &c.

(1 -xx)i = i-^x\-^xy-^xsScc.

" Thus then I discovered a general method of reducing
radical quantities into infinite series, by the theorem which
I sent in the beginning of the former letter, before I knew
the same by the extraction of roots.

" But having discovered that way, this other could not
long remain unknown: for to prove the truth of those
operations, I multiplied

1 — £x2 — §x4 — -j-'-jX6 &c, by itself, and the product was
1 — x2, all the rest of the terms vanishing after these, in
infinitum. In like manner,

1 t ix' — %x4 — -JT*6 &c' twice multiplied by itself, pro-
duced 1 — x2. But as this was a certain proof of those
conclusions, so I was naturally led to try conversely whe-
ther these series, which were thus known to be the roots
of the quantity 1 — x2, could not be extracted out of it
after the manner of arithmetic; and upon trial I found
it to succeed. The process for the square root is here set
down

1 - x2 (1 - 4.x2 - |x4 - -^x« &c.

1

0 - x2

— X2 -*- ix4

— i

— r r*

— -i X" —

"These methods being found, I laid aside the other
way by interpolation of series, and used these operations
only as a more genuine foundation. Neither was I igno-
rant of the reduction by division, which is so much easier."
See Collins's Commercium Epistolicum.

And this is all the account that Newton gives of the
invention of this theorem, which is engraved on his mo-
nument in Westminster Abbey, as one of his greatest dis-
coveries.

Francis Maseres, Esq. (Cursitor Baron of the Exchequer)
has communicated the following observations on the Bino-
mial theorem, and its demonstration; viz, About the year

and that too if they should be taken at the distance of iu'6'6 the celebrated Sir Isaac Newton discovered that, if

two or more terms. m were put for any whole number whatsoever, the coeffr

" This was the way then in which I first entered upon cients of the terms of the »ith power of 1 *■- x would be

these speculations; which I should not have remembered,
but that in turning over my papers a few weeks since, I
chanced to turn my eyes on those relating to this matter.
" Having proceeded so far, I considered that the terms

(1 - xx)°, (1 - xx)*, (1 - xx)% (1 - xx)T, &c,
that is, 1

1 - x1

1 — 2x2 -+■ x4 "

1 — 3x2 -+- 3x* — x", &c, might be interpolated
in the same manner as the areas generated by them : and
for this, nothing more was required but to omit the de-
nominators 1, 3, 5, 7» &c, in the terms expressing the
areas; that is, the coefficients of the terms of the quantity

to be interpolated (1 — xx)^, or (1 — xx)r, or generally strate it.

m m m — l

m — 1 m — 2

&c;

(m - 1)

, which will be the

1}>

till we come to the term

last term. But by what particular means he discoTered
this proposition, he has not told us, nor has he even at-
tempted to give a demonstration of it. Dr. John Wallis,
of Oxford, informs us (in his Algebra, chap. 85, pa. 319)
that he had endeavoured to find this manner of generating
these coefficients one from another, but without success;
and he was greatly delighted with the discovery, when he
found that Mr. Newton had made it. But he likewise has
omitted to give a demonstration of it, as well as Sir Isaac
Newton ; and probably he did not know how to demon-

b r n

[ 232 ]

B I N

Newton, after having discovered this rule, for generating
the coefficients of the powers of 1 ■+■ x when the indices of
those powers were whole numbers, conjectured that the
same might possibly obtain also when the indices were
fractions. He therefore resolved to try whether it would
or not, by applying it to such indices in a few easy in-
stances, and particularly to those of \ and \, which, if the
rule served in the case of fractional indices, would enable
him to find series equal to the values of (1 -+- i)T and
(1 ■+■ xfs, or the square-root and the cube-root of the bi-
nomial quantity 1 ■+■ x. And, when he had in this man-
ner obtained a series for (1 + x)T, which he suspected to

be equal to (1 •+■ x)T, or the square root of I + x, he mul-
tiplied the said series into itself, and found that the pro-
duct was l+i; and when he had discovered in like man-
ner a series for (1 -+- x)? he multiplied the same twice into
itself, and found that the product was 1 ■+- x; and thence
he concluded that the former series was really equal to
the square-root of 1 -t- x, and that the latter series was
really equal to its cube-root. And from these and a few
more such trials, in which he found the rule to answer, he
concluded universally that the rule was always true, whe-
ther the index m stood for a whole number or a fraction
of any kind, as %, f, f , h f, f , or, in general, t

After the discovery of this rule by Sir Isaac Newton,
and the publication of it by Dr. Wallis, in his Algebra,
chap. 85, in the year 1685, (which I believe was the first
time it was published to the world at large, though it was
inserted in Sir Isaac Newton's first letter to Mr. Olden-
burg, the secretary to the Royal Society, dated June 13,
1676, and the said letter was shown to Mr. Leibnitz, and
probably to some other of the learned mathematicians of
that time) it remained for some years without a demon-
stration, either in the case of integral powers or of roots.
At last however it was demonstrated in the case of integral
powers by means of the properties of the figurate numbers,
by that learned, sagacious, and accurate mathematician
Mr. James Bernoulli, in the 3d chapter of the 2d part of
his excellent treatise De ArteConjectandi, or, On the Art
of forming reasonable Conjectures concerning Events that
depend on Chance ; which appears to me to be by far the
best-written treatise on the doctrine of Chances that has
yet been published, though M. Demoivre's book on the
same subject may have carried the doctrine something fur-
ther. This treatise of James Bernoulli's was not published
till 8 years after his death, which happened in August
1705; but there is reason to think that it was composed
in the latter 5 or 6 years of the preceding century, and
even that some parts of it, or some of the propositions in-
serted therein, had been discovered by the author in the
years 1689, 169O, 1691, and 1692. For the first part of
his very curious tract, entitled, Positiones Arithmetica; de
Seriebus Infinitis, was published at Basil or Basle in Swit-
zerland in the year l689; and the second part of the said
Positiones (in the 19th Position of which those properties
of the figurate numbers from which the binomial theorem
may be deduced, are set down) was published at the same
place in the year 1692. But the demonstrations of those
properties of the figurate numbers, and of the binomial
theorem, which depends upon them, were never I believe
communicated to the public till the year 1713, when the
luthor's posthumous treatise De Arte Conjcctandi made

its appearance. These demonstrations are founded on
clear and simple principles, and afford as much satisfaction
as can well be expected on this subject. But the full dis-
play and explanation of these principles,and the deduction
of the said properties of the figurate numbers, and ulti-
mately of the Binomial theorem, from them, is a matter
of considerable length. It will therefore not be improper
to give a shorter proof of the truth of this important
theorem, that shall not require a previous knowledge
of the properties of the figurate numbers, but yet shall
be equally conclusive with that which is derived from
those properties. Now this may be done in the manner
following.

Let us suppose that the coefficients of the terms of the
first six powers of the Binomial quantity 1 -1- x have been
found, upon trial, to be such as would he produced by the1
general expressions

m ?n m — 1 m ra — 1 m — 2 .
' l' 1 2 ' 1 2 " 3 ' '

by substituting in them first 1, then 2, then 3, then 4,
then 5, and lastly 6 instead of m. This may easily be
tried by first raising 1 + 1 to the 6th power, by repeated
multiplications in the common way, and afterwards finding
the terms of the powers by means of the said general ex-
pressions above ; which will be found to produce the very
same terms as arose from the multiplications. After these
trials we shall be sure that those general expressions are
the true values of the coefficients of the powers of 1 •+- x
at least in the said first six powers. And it will therefore
only remain to be proved that, since the rule is true in the
said first six powers, it will also be true in the next fol-
lowing, or the 7th power, and consequently in the 8th,
9th, and 10th powers, and in all higher powers whatso-
ever.

Now, if the coefficients of the 1st, 2d, 3d, 4th, and

other following terms of 1 +r) be denoted by the letters
a, b, c, d, &c, respectively, it is evident from the nature
of multiplication, that the coefficients of the 1st, 2d, 3d,
4th, and other following terms of the next higher power

of 1 ■+■ x, to wit, 1 -+- x\ will be equal to a, a ■+■ b,

b ■+■ C, c + d, &c, respectively, or to the sums of every
two contiguous coefficients of the terms of the preceding

series which is = 1 -+- x1 . This will appear from the
operation of multiplication, which is as follows:
a ■+■ b'x,+ ex'2 -+- dx3 -+- ex* ■+- &c

1 -+- x

bx

dx*

bx*

dx'

Ike

&c.

Therefore, if 1 -+- xj

bx

be equal to the series
- ex2 ■+■ dx3 ■+• ex* -<- &c,

then 1

will be equal to the scries
a ■+■ a ■+- b . r ■+- b ■+- c . x- ■+■ c •+• d . x3 ■+• &c.
Now let n be = m •+- 1. We shall then have to prove
that, if the coefficients a, b, c, d, &c, be respectively

, m m m — 1 m 7)1 — 1 vi — 1 o

equal to ),-,-• -5-1 - ■ -3- • — -' &c-

the coefficients «, a -*- b, b ■+■ c, &c, will be respectively

n n n — 1 n n — 1 n — 2
equal to 1, -, T . — , T . — • — , cW.

And to do which, there is nothing more required than
to collect together every two terms of the former of these
two series, and then substitute into these sums, n instead

BIQ C

of m •+- 1, when there will immediately come out the terms
of the latter series, as above, viz,

— \n , n n n — 1 „ a

1-1- .r =1-1 X -\ . ' X' ■+- &C Q. E. D.

' 112

For several other demonstrations of the Binomial Theo-
rem, see my new Tracts, vol. 1, pa. 228, &c.

BIPARTIENT, is a number that divides another into
two equal parts without a remainder. So 2 is a bipartient
to 4, and 5 a bipartient to 10.

BIPART1TION, is a division into two equal parts.

BIQUADRATE, or Biquadratic Power, is the squared
square, or 4th power of any number or quantity. Thus
l6 is the biquadrate or 4th power of 2, or it is the square
of 4 which is the 2d power of 2.

Biquadratic Root, of any quantity, is the square
root of the square root, or the 4th root of that quantity.
So the biquadratic root of l6 is 2, and the biquadratic
root of 81 is 3.

Biquadratic Equation, is that which rises' to 4 di-
mensions, or in which the unknown quantity rises to the
4th power ; as x4 ■+- ax3 -+- bx~ •+■ ex -+■ d = 0.

Any biquadratic equation may be conceived to be ge-
nerated or produced from the continual multiplication of
four simple equations,

as (x — p) x (x — q) x (x — r) x (x - s) = 0 ;
or from that of two quadratic equations,
as (x2 -+- px -+- q) x (x2 ■+- rx -t- s) = 0 ;
or, lastly, from that of a cubic and a simple equation,
as (x — p) x (r3 -+- qx* ■+• rx -t- s) = 0 : which was the
invention of Harriot. And on the contrary, a biquadratic
equation may be resolved into four simple equations, or
jnto two quadratics, or into a cubic and a simple equa-
tion, having all the same roots with it.

1. Ferrari's Method for Biquadratic Equations.

The first resolution of a biquadratic equation was given
in Cardan's Algebra, chap. 39, being the invention of his
pupil and friend Lewis Ferrari, about the year 1540. This
is effected by means of a cubic equation, and is indeed a
method of depressing the biquadratic equation to a cubic,
which Cardan demonstrates, and applies in a great variety
of examples. The principle is very general, and consists
in making one side of the equation a square, by help of
some multiples or parts of its own terms and an assumed
unknown quantity ; which it is always easy to do; and
then the other side is made to be a square also, by as-
suming the product of its 1st and 3d terms equal to the
square of half the 2d term ; for it consists only of three
terms, or three different denominations of the original let-
ter; then this equality will determine the value of the as-
sumed quantity by a cubic equation: other circumstances
depend on the artist's judgment. But the method will be
farther explained by the following examples, extracted
from Cardan's book.

Ex. 1. Given x4 h- 6x2 ■+■ 36 = 60x, to be resolved.
Add 6x2 to both sides of the equation, so shall
r1 + I 2x2 + 36' or (x2 -i- 6)2 = 6V ■+- 60x.
Assume^, and add (x2 ■+■ 6) 2y ■+- y2 to both sides, then
is (x2 + 6)2 + 2j/(x2 + 6) -i- f = 6x2 + 60x + 2y{x2 + 6)
+ y\ or (x2 ■+- 6 ■+■ y)" = (6 ■+• 2y)x2 •+- 60x + 12y
■+•/.

Make now the 1st x 3d term = isq. 2d,
this gives (6 + 2y) . {\2y +/) x2 = 900x2,
or/ -i- 15/ -i- 36y = 450; and hence
y =£/(287i + ^80449^) +>/(287i - y/SQU9i) - 5.
From which x may be found by a quadratic equation.
Vol. I.

233 ]

BIQ

For, because (6 ■+■ 2y) . (I2y ■+■ y) . x2 = 900 x\
(6 ■+■ 2y) . x2 ■+■ 60x ■+■ \2y + y", becomes a square, whose
root is evidently equal to x^/(6 -+• 2y) ■+■ *J (12y -t- /) ;
and since (x'2 -+• 6 ■+■ y)'2 = (6 -i- 2y)x2 -i- 60x -+■ I2y ■+- y1,
their roots are also equal, that is x2 -t- 6 ■+■ y = x-y/(6
■+■ 2y) ■+• \/(12y -t-yz), a quadratic equation.

Ex. 2. Given x4 = x -+- 2.
Before applying Ferrari's method to this example, Cardan
resolves it by another way as follows: subtract I,' from
both sides of the equation, then is x4 — 1 = x -+- 1 ; di-
vide by x -+- 1, then is x3 — x2 -t- x ~ 1 = 1, or x3 -+- x

= x2 ■+- 2; and hence x =l/(^/

v w 2916

-i7) + I.

54y 3

But to resolve it by Ferrari's rule :
Because x4 = x ■+- 2 ; therefore

x4 -+- 2yx2 ■+- y"- or (x2 ■+- y)- = 2yx" ■+- x •+- (2 ■+- y1) ;
hence 2y3 -+- 4y, = £, or y3 -+- 2y = \ ; and the root is

, ' .2075 1 . .„, .2075 1 .
a v w 6912 W . v w 6912 16'

by means of which x is found by a quadratic equation.

Ex. 3. Given x4 -+■ 32x* + 16 = 48x — Add 240, then
x4 ■+- 32X1 + 250 or (x2 + l6)2 = 4Sx +. 240; complete
square again, then

(x2 ■+- l6)z + 2j/ . (x1 4- 16) + y* = 2yxz + 48x + (j/1
■+- 32y ■+- 240); make the last side a sq. by the rule,
which gives y3 + 32y* + 240,y = 24 x 12 = 288.
Put now z =y h- lOf , and the last transforms to
z3 = 101J-2 ■+- 420^£; then the value of z found from
this, gives the value of y, and hence the value of x as
before.

2. Descartes's Rule for Biquadratic Equations.
Another solution was given of biquadratic equations by

Descartes, in the 3d book of his Geometry. In this solu-
tion he resolved the given biquadratic equation into two
quadratics, by means of a cubic equation, in this manner:
First, let the 2d term or 3d power be taken away out of
the equation, after which it will stand thus,

x1 ■+ px~ ■+■ qx ■+■ r = 0. Find y in this

cubic equation y° + 2/y4 ".

the values off , T ,,

i \x- — yx + iy' + ip + — = 0
x in these 1 J *J r 2y

two quadra-) „ . d q

tics. (*2+^ + f/ + i/>-^ = o,

so shall these values of x be the roots of the given biqua-
dratic equation.

Ex: Let the equ. be xA — 17x2 — 20x — 6 = 0.
Here p = — 17, </ = — 20, and r= — 6;
and the cubic equ. is x" — 34y* •+■ 313/ — 400 = 0,
the root of which is j/2 = l6, or y = 4;
hence the two qua- f x2 — 4x — 3 = 0
dratics are - \ x2 + 4x + 2 = 0,
the four roots of which are 2 ± </7 and — 2 ± >J2.

3. Euler's Method for Biquadratic Equations.

The celebrated Leonard Eulcr has also given, in the 6th
volume of the Petersburg Ancient Commentaries, for the
year 1738, an ingenious and general method for resolving
equations of all degrees, by means of the equation of the
next lower degree, and among them that of the biquadratic
equation by means of the cubic; and this last was also
given more at laige in his treatise of Algebra, translated
from the German into French in 1774, in 2 volumes Svo.
The method is this: Let x' — ax2, — bx — c = 0, be the
2 H

. . . „. i ...„ j j

_> y1 — q1 = 0 ; and then

b r q

I C34 ]

BI Q

given biquadratic equation, wanting the 2d term. Take
/= \a, g=-^aa -+- \c, and k±s'*J)h ; with which values
of/, g, h, form the cubic equation e3 — fzl -+- gz — h = 0.
Then find the three roots of this cubic equation, and let
them be called p, q, r. So shall the four roots of the pro-
posed biquadratic be these following, viz,

AVhen |i is negative :
y/P + V'l ~ \/r>

• v7- - s/q -*■ \A>

— x/p ■+- ^/q ■+- x/r,

- Vp - y/q - vr-

25x2 ■+- 60x - 36 = 0.

When |6 is positive :
1st. a/p +■ x/q -+- */r
2d. y7> -+• l/q — ^/r
3d. y> — v/g -i- </r
4th. -//> - ■/? — \/r
Ex. Let the equ. be x4 ■
Here a == 25, & = — 60, and c = 36;

. , - 25 62S 769 , , 225

therefore /= — , g = — ■+- 9 = — , and ft = — .
210 10 4

Consequently the cubic equation will be
23_2i2* + 7-£?~_2if = 0

2 16 4

The three roots of which are

z = - = p, and s = 4 = o, and z=z — =.r\ and the roots
4 r 4

of these are vO5 — lb a/? = 2 or §, x/7" = t-

Hence, as the value of \b is negative, the four roots are

let r — Ji-i-4 .1 — 1

2d. x = |_4+^.= 2;
3d. !c=T| + f + (= 3,
4th. i=-|-f-|.= -6.
4. Simpson s Rule for Biquadratic Equations.
Mr. Simpson gave a general rule for the solution of bi-
quadratic equations, having all its terms, in the 2d edit, of
hisAlgebra, pa. 150, published in 1755, in which the given
equation is also resolved by means of a cubic equation, as
well as the two former ways ; and it is investigated on the
principle, that the given equation is equal to the difference
between two squares; being indeed a kind of generaliza-
tion of Ferrari's method.

Thus, he supposes the given equation, viz,
x4 -+- px3 ■+■ qx2 •+■ rx -+- s = (x2 -+- \px -+- a)2 — (bx ■+- c)2;
then from a comparison of the like terms, the values of the
assumed letters are found, and the final equation becomes
a3 — I9A2 ■+■ k\ — \l = 0,
where k = fpr — s, and I .-= fr2 ■+■ s . (£ p2 — q).
The value of a being found in this cubic equation, those
of B and c will be had from their general values, viz,

b = \/{2\ -+- %p2 — q), and c = . Hence, finally,

the root x will be obtained from the assumed equation
(x2 •+- \px -+- a)2 — (bx -+- c)2 = 0, or x2 -+- %px ■+- a s=
fuic (which always is the case when the value of x
is taken such, that the given biquadratic becomes equal to
nothing), in four several values.

Ex. Given the equ. x4 — 6x3 — 58x2 — 1 14x — 11=0.
Here p = — 6, q = — 58, r = — 114, and s = — 11,
whence

k or Ipr — $ = 182, I or }r2 ■+■ * . ($ p2 — q) = 2512;
and therefore the cubic equation becomes
a3 -1- 29a2 + 182a - 1256 = 0,
the root of which is a = 4.

Hence then b or V2a -+- §p2 — q = \/7$, and c or
v—- = v/27 :

2B v

then the quadratic equation becomes
x2 — 3x ■+■ 4 =.± *\/75 ± ^/27 ; which solved gives
x= U±2{v/3 + V(17±lli*/3),
or = U±2£V3-\/(l7 ±11*^3),

exhibiting all the 4 roots of the given equation according
to the variations of the signs.

Mr. Simpson here subjoins an observation which is to-
tally erroneous, viz, that f The value of a, in this equation,
will be commensurate and rational (and therefore the easier
to be discovered), not only when all the roots of the given
equation are commensurate, but when they are irrational
and even impossible ; as will appear from the example
subjoined." It must indeed appearsingular that he should
have adduced such a reason for the proof of a proposition
without any foundation; and it is wonderful that he fell
upon no examples that refuted it, as the instances in which
it holds true, are so few, in comparison with the number
of those in which it fails.

By methods similar to most of the foregoing ones, Dr.
Waring gave very general investigations of such rules for
biquadratics, in his Meditationes Algebraica?, published in
1770.

Note. In any biquadratic equation having all its terms,
if ■!■ of thesquare of the coefficient of the 2dterm be greater
than the product of the coefficients of the 1st and 3d terms,
or |- of the square of the coefficient of the 4th term be
greater than the product of the coefficients of the 3d and
5th terms, or ■* of the square of the coefficients of the 3d
term greater than the product of the coefficients of the
2d and 4th terms; then all the roots of that equation will
be real and unequal ; but if either of the said parts of
those squares be less than cither of those products, the
equation will have imaginary roots.

For the construction of biquadratic equations, sec Con-
struction. See also Descartes's Geometry, with theCom-
mentaries of Schooten and others; Baker's Geometrical
Key; Slusius's Mcsolabium; 1' Hospital's Conic Sections;
Wolfius's Elementa Matheseos ; &c.

Biquadratic Parabola, a curve of the 3d order, ha-
ving two infinite legs, and expressed by one of these three
equations, viz,

a3x =j/4, as in fig. 1,

asx =y — a2y2, as in fig. 2,

ra3x =y — (a + b) . y2 — aby2, as in fig. 3 ;
where x = ap the absciss, y = pq the ordinate, b = ab,
c = ac, and a = a certain given quantity.

But the most general equation of this curve is the fol-
lowing, which belongs to fig. 4, viz,

'R A

ay = x4 4- bx3 ■+■ cx2 -t- dx + c;

where x = Ap or ap the absciss, and — j/ or ■+■ y is the
ordinate put or I'M, also a, b, c, d, e, are constant quanti-
ties'; the beginning of the absciss being at any point a in
the indefinite line A p."

But if the beginning of the absciss a be at the point
where this line intersects the curve, as in fig. 5, then the

BIS

[ 335 ]

B L A

nature of the curve will be defined by this equation a x
pm =Ap x sp x c/> x Dp, wherever the point p is taken
in the infinite line rs.

When the curve has no serpentine part, as fig. 6, the
equation is more simple, being in this case barely a x pm
= Apl x ps2. See CwRVE-Lmes, and Geometrical
Lines.

BIQUINTILE, an aspect of the planets when the di-
stance between them is 144 degrees, or twice the 5th part
of 360 degrees.

BISECTION, or Bissection, the division of a quan-
tity into two equal parts, otherwise called bipartition.

BISSEXTILE, or Leap-year, a year consisting of 366
days, and happening every 4th year, in consequence of the
addition of 1 day to the month of February, when that
month consists of 29days. And this is done to compensate
for the 6 hours which the sun takes up nearly in his course,
more than the 365 days commonly allowed for it in other
years.

The day thus added was by Julius Caesar appointed to
be the day before the 24th of February, which among
the Romans was the 6th of the calends, and which on this
occasion was reckoned twice ; whence it was called the
Bissextile.

By the statute De anno bissextile, 21 Hen. 3d, to prevent
misunderstandings, the intercalary day and that next be-
fore it are to be accounted as one day.

To find what year of the period any given year is ; di-
vide the given year by 4 ; then if 0 remains, it is leap-year;
but if any thing remain, the given year is so many after
leap-year.

But the astronomers concerned in reforming the calendar
in 1582, by order of Pope Gregory the 1 3th, observing that
in 4 years the bissextile added 44 minutes more than the
sun spent in returning to the same point of the ecliptic;
and computing that in 1 33 years these supernumerary mi-
nutes would form a day; to prevent any changes being
thus insensibly introduced into the seasons, directed, that
in the course of 400 years there should be three scxtiles
retrenched ; so that every centesimal year (which is a leap-
year according to the Julian account), is a common year
in the Gregorian account, unless the number of centuries
can be divided by 4 without a remainder. So 1600 and
2000 are bissextile; but 1700, 1800, and lf)00 are com-
mon years.

The Gregorian computation has been received in most
foreign countries ever since the reformation of the calendar
in 1582; excepting some northern countries, as Russia,
&c. And by act of parliament, passed in 1751, it com-
menced in all the dominions under the crcwn of Great
Britain in the year following ; it being ordered by that act
that the natural day next following the 2d of September,
should be accounted the 14th; omitting the intermediate
] 1 days of the common calendar. The supernumerary
day, in leap-years, being added at the end of the month
of February, and called the 29th of that month.

The centesimal year 1800 having been a common year,
instead of a bissextile, from that time there are now 12
days difference between the old and new style, and so will

continue till the year 1900, when the difference between
will be one day more, and so on, at every centesimal year
which is not exactly divisible by 4.

BLACK, a colour so called, or rather a privation of all
colour. This, it appears, arises from such a peculiar tex-
ture and situation of the superficial parts of a black body,
that they absorb all or most of the rays of light, reflecting
little or none to the eye ; and hence it happens that black
bodies, thus imbibing the rays, are always found to be
hotter than those of a lighter colour. Dr. Franklin observes
that black clothes heat more, and dry sooner in the sun
than white clothes ; that therefore black is a bad colour
for clothes in hot climates ; but a fit colour for the linings
of ladies' summer hats; and that a chimney painted black,
when exposed to the sun, will draw more strongly. Frank-
lin's Experim. &c. — Dr. Watson, the present bishop of
Landaff, covered the bulb of a thermometer with a black
coating of Indian ink, and the consequence was that the
mercury rose 10 degrees higher. Philos. Trans, vol. 63,
pa. 40; and the same thing has since been evinced by the
experiments of Mr. Leslie and other philosophers. — And
a virtuoso of some credit assured Mr. Boyle, that in a hot
climate by blacking the shells of eggs, and exposing them
to the sun, he had seen them thus well roasted.

BLACKNESS, the quality of a black body, as to co-
lour; arising from its stifling or absorbing the rays of light,
and reflecting little or none. In which sense it stands di-
rectly opposed to whiteness; which consists in such a tex-'
ture of parts, as indifferently reflects all the rays thrown
upon it, of whatever colour they may be.

Descartes, it appears, first rightly distinguished these
causes of black and white, though he might be mistaken
with respect to the general nature of light and colours. —
Sir Isaac Newton shows, in his Optics, that to produce
black colours, the corpuscles must be smaller than for ex-
hibiting those other colours ; because, where the sizes of
the component particles of a body arc greater, the light
reflected is too much for constituting this colour; but
when they are somewhat smaller than is requisite to reflect
the white, and very faint blue of the first order, they will
reflect so small a portion of light, as to appear intensely
black; and yet they may perhaps reflect it variously to and
fro within them so long, till it be stifled and lost.

And hence it appears, why fire, or putrefaction, by di-
viding the particles or substances, turn them black ; why
small quantities of black substances impart their colours
very freely, and intensely, to other substances, to which
they are applied ; the minute particles of these, on account
of their very great number, easily overspreading the gross
particles of others. Hence it also appears, why glass ground
very elaborately on a copper-plate with sand, till it be well
polished, makes the sand, with what is rubbed off from the'
copper and glass, become very black ; also why blacks
commonly incline a little towards a blueish colour; as may
be seen by illuminating white paper with light reflected
from black substances, when the paper usually appears of
a blueish white; the reason of which is, that black borders
on the obscure blue of the first order of colours, and there-
fore reflects more rays of that colour than of any other;
and lastly, why black substances sooner than others be-
come hot in the sun's light, and burn; an effect which rriay
proceed partly from the multitude of refractions in a little
space, and partly from the easy commotion among such
minute particles.

BLAGRAVE (John), an eminent mathematician, who
2 II 2

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flourished in the l6th and 17th centuries. He was the
second son of John Blagrave, of Bulmarsh-court, near
Sunning in Berkshire, descended from an ancient family
in that country. From a grammar-school ut Reading he
was sent to St. John's-college in Oxford, where he applied
himself chiefly to the study of mathematics. From hence
he retired to his patrimonial scat of Southcote-lodge, near
Reading, where he spent the rest of his life, in a retired
manner, without marrying, that he might have more lei-
sure to pursue his favourite studies; which he did with
great application and success. After a life thus spent in
study, and in acts of benevolence to all around him, he
died in the year l6ll ; and was buried at Reading in the
church of St. Lawrence, where a sumptuous monument
was erected to his memory.

He left the bulk of his fortune to the posterity of his
three brothers, which were very numerous. There have
been mentioned various acts of his beneficence in private
life, for the encouragement of learning, the reward of merit,
and the relief of distress. Some of these were the result
of a quaint, humorous disposition, discovered chiefly in
his legacies. One of these was 10 pounds left to be annually
disposed of in the following manner: On Good-friday, the
churchwardens of each of ihe three parishes of Reading
send to the town-hall " one virtuous maid who has lived
five years with her master;" there, in the presence of the
magistrates, these three virtuous maids throw dice for the
10 pounds. The year following the two losers are re-
turned with a fresh one, and again the third year, till each
has had three chances. He also left an annuity to SO
poor widows, who should attend annually on Good-friday
also, and hear a sermon, for the preaching of which he left
10 shillings to the minister. He took care also for the
maintenance of his servants, rewarding their diligence and
fidelity, and providing amply for their support. Thus it
appears he was not more remarkable for his scientific know-
ledge, than for his generosity and philanthropy. His works
are,

1. A Mathematical Jewel. Loud. 15S5, folio.

2. Of the Waking and Use of the Familiar- Stall'. Lond.
1590, 4to.

3. Astrolabium Uranicum Generale. Lond. 1596, 4 to.

4. The Art of Dyalling. Lond. 1609, 4to.

BLAIR (John), an eminent chronologist, was born
and educated at Edinburgh. Afterward, coining to Lon-
don, he was for some time usher of a school in Hedge-lane.
In 1754 he first published The Chronology and History
of the World, from the Creation to the year of Christ 1753,
illustrated in 56' tables. In 1755 he was elected a fellow
of the Royal Society, and in 1761 of the Society of Anti-
quaries. In 1756' he published a 2d edition of his Chro-
nological Tables, and in 1768 an improved edition of the
same, with the addition of It Maps of Ancient and Mo-
dern Geography, for illustrating the Tables of Chronology
and History; to which is prefixed a Dissertation on the
Progress of Geography. In 1/57 he was appointed chap-
lain to the Princess Dowager of Wales, and Mathematical
Tutor to the Duke of York ; whom he attended in 170'3
111 a tour to the continent, from which they returned the
year after. Dr. Blair had successively several good church
preferments: as a prehendal stall at Westminster; tin- vi-
carage ol Hinckley, and the rectory of Burton CqggleS in
Lincolnshire, all in lj6l; the vicarage ol St. Pride's in
London, in 1771, in exchange for that of Hinckley; the
rectory of St. Jcibn the Evangelist in Westminster, in 177(>,

in exchange for the vicarage of St. Bride's; in the same
year the rectory of Horton near Colebrooke, Bucks. Dr.
Blair died the 24th of Juue, 1782.

BLIND, an epithet applied to an animal deprived of the
use of eyes; or one from whom light, colours, and all the
glorious objects of the visible creation, are intercepted by
some natural or accidental cause.

BLINDNESS; a privation of the sense of sight. The
ordinary causes of blindness are, some external violence,
vicious conformation, growth of a cataract, gutta serena,
small-pox, &c; or a decay of the optic nerve; an instance
of which we have in the Academy of Sciences, where, on
opening the eye of a person long blind, it was found that
the optic nerve was extremely shrunk and decayed, and
without any medullain it. The more extraordinary causes
of blindness are malignant stenches, poisonousjuices drop-
ped into the eye, baneful vermin, long confinement in the
dark, &c.

We find various compensations for blindness, or substi-
tutes for the use of eyes, in the wonderful sagacity of many
blind persons, related by Zaphnius in his Oculus Artifici-
alis, and others. In some the defect has been supplied by
a most excellent gift of memory; others by a delicate nose
or the sense of smelling ; others by a very nice ear ; and
others again by an exquisite touch, or sense of feeling,
which they have possessed in great perfection.

Some have been able to perform many sorts of curious
works in the nicest and most dexterous manner. Aldro-
vandus speaks of a sculptor, who had become blind at 20
years of age, and yet lOyearsafterwards he made a perfect
marble statue of Cosmo 11 de Medicis, and another of clay
like pope Urban viii. Bartholin speaks of a blind sculp-
tor in Denmark, who, by mere touch, distinguished perfectly
well every kind of wood, and even colours; and father
Grimaldi relates an instance of the same kind; besr t; the
blind organist, lately living in Paris, who it was said did
the same thing. What seems more extraordinary still, we
are told by authors of good report, of a blind guide, who
used to conduct the merchants through the sands and de-
serts of Arabia; and a not less marvellous instance is now
existing in this country, in oue John Metcalf, near Man-
chester, who became quite blind at a very early age; and
yet passed many years of his life as a waggoner, and occa-
sionally, as a guide in different roads during the night, or
when the paths were covered with snow; and, what is
stranger still, his present occupation is that of surveyor and
projector of highways in difficult and mountainous parts,
particularly about Buxton, ami the Peak in Derbyshire.

There are also many instances of blind men who have
been highly distinguished for their mental and literary ta-
lents, not to speak of the poets Homer, Milton, Ossian, &c;
of these we have a remarkable instance in ihe late Dr.
Sanderson, professor of mathematics in the uniwisiry of
Cambridge, and in Dr. Henry Mpyes, public lecturer in
philosophy, who both of them lost their sight by tlieMiiall-
pox at an age before they had any recollection ; these men
were well skilled in all branches of tin' matheniatic* phi-
losophy, optics, .Sic, which they taught will) 'hi ■■ 1 , ..test.

reputation ; In siiii s the n oiuini, 111 ol fame w hich ihe for-
mer has left behind him in his mathematical and philoso-
phical works. Another instance ol these verj extraordi-
nary characters is likewise evinced in the present day,
in Mr. John Cough of Middleshow, near Kendal, who,
though blind from his infancy, has nevertheless acquir-
ed not only a very extensive knowledge in almost all the

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branches of the mathematics, but also in botany and che-
mistry.

The effects of a sudden recovery of sight in such as have
been born blind, are also1 very remarkable. Devoid of the
experience of distance and figure arising from sight, they
are liable to the greatest mistakes in this respect; so much
that it has been said they could not distinguish by the mere
sight between a cube and a globe, without first touching
them. Mr. Boyle mentions a gentleman that was blind,
who having been restored to sight at IS years of age, was
nearly distracted with joy. See Boyle's works abridg. vol. 1,
p. 4. See also a remarkable case of this kind in the Tatler,
No. 55, vol. 1. And the gentleman couched by Mr. Che-
selden had no ideas of colour, shape, or distance: though
he knew the colours asunder in a good light during his
blind state, yet when he saw them after he had been
couched, the faint ideas he had of tliem before, were not
sufficient for discriminating them afterwards. As to dis-
tance, his ideas were so deficient, that he thought all the
objects he saw touched his eyes, as what he felt did his
skin; and it was a considerable time before he could re-
member which was the dog and which the cat, though
often informed, without feeling them.

BLINDS, or Blindes, in Fortification, a kind of de-
fence usually made of oziers or branches interwoven, and
laid across between two rows of stakes, about a man's
height, and 4 or 5 feet asunder. They are used particu-
larly at the heads of trenches, when these are extended in
front towards the glacis, serving to defend the workmen,
and prevent the enemy from overlooking them.

BLOCKADE, is the blocking up of a place by posting
troops in all the avenues and roads leading to it, to prevent
farther supplies of men and provisions to the besieged, and
thus starving them out, without forming any regular siege
or attacks.

BLONDEL (Francis), a celebrated French mathe-
matician and military engineer, was born at Ribemond
in Picardy in 1 6' X 7 - While he was yet but young, he was
chosen Regius Professor of Mathematics and Architecture
at Paris. Not long after he was appointed governor to
Lewis-Henry de Lomenix, Count de Brienne, whom he ac-
companied in his travels from 1652 to 16'55, of which he
published an account. He enjoyed many honourable em-
ployments, both in the navy and army; and was intrusted
with the management of several negociations with foreign
princes. He arrived at the dignity of marshal de camp,
and counsellor of state, and had the honour to be appoint-
ed mathematical preceptor to the dauphin. He was a
member of the Royal Academy of Sciences, director of the
Academy of Architecture, and lecturer to the Royal, Col-
lege, in all which lie supported his character with dignity
and applause. Blondcl was no less versed in the knowledge
of the belles lettres than in the mathematical sciences, as
appears by the comparison he published between Pindar
and Horace. He died at Paris the 22d of February l6"86\
in the 69th year of his age. His chief mathematical works
were,

1. Cours d' Architecture. Paris, 1675, in folio.

2. Resolution des Quarre I'rincipaux Problemes d'Ar-
chitecture. Paris, \6?6, in folio.

3. Hi3toired'u Calcndrier Elomain. Paris, IfJ82, in 4to.

4. Cours de Mathema/tiqucs. Paris, 1083, in4to.

5. L'Art ile jcttcr (lis Bombes. La llaye, 1()S5, in 4to.
Besides A New Method of fortifying Places, and other

works.

Blonde! had also many ingenious pieces inserted in the
Memoirs of the French Academy of Sciences, particularly
in the year 1666.

BLOW, in a general sense, denotes a stroke given either
with the hand, a weapon, or an instrument. The effect of
a blow is estimated like the force of percussion, and ex-
pressed also by the velocity of the body multiplied by its
weight.

BLOWING of a Fire-arm, is when the vent or touch-
hole is run or gullied, atid becomes wide, so that the powder
will flame out.

BLUE, one of the seven primitive colours of the rays of
light, into which they are divided when refracted through
a glass prism. See Newton's Optics, &c. See also Chro-
matics.

BLUENESS, that quality of a body, as to colour, from
which it is called blue; depending on such a size and tex-
ture of the parts that compose the surface of a body, as
disposes them to reflect only the blue or azure rays of light
to the eye.

The biueness of the sky is thus accounted for by De la
Hire, after Da Vinci; viz, that a black body viewed
through a thin white one, gives the sensation of blue, like
the immense-expanse viewed through the air illuminated
and whitened by the sun. For the same reason he says it
is, that soot mixed uith white, makes a blue; for that
white bodies, being always somewhat transparent, when
mixing with a black behind, give the perception of blue.
■From the same principle too he accounts for the biueness
of the veins on the surface of the skin, though the blood
they are filled with be a deep red.

In the same manner was the blue colour of the sky ac-
counted for by many other of the earlier writers, as Fro*
mondus, Funccius, Otto Guericke, and many others, with
several of the more modern writers, Wolfius, Muschen-
broek, &c. But in the explication of this phenomenon by
Newton, he observes, that all the vapours, when they begin
to condense and coalesce into natural particles, become
first of such a magnitude as to reflect the azure rays, be-
fore they can constitute clouds of any other colour. This
being therefore the first colour they begin to reflect, must
be that of the finest and most transparent skies, in which the
vapours are not yet arrived at a grossness sufficient to re-
flect other colours.

Bouguer however ascribes this biueness of the sky to the
constitution of the air itself, being of such a nature that
these fainter-coloured rays are incapable of making their"
way through any considerable tract of it. And as to the
blue shadows which were first observed by Buffon in the
year 1742, he accounts for them by the aerial colour of
the atmosphere, which enlightens these shadows, and in
which the blue rays prevail ; while the red rays are not re-
flected so soon, but pass on to the remoter regions of the
atmosphere. And the Abbe Mazeas accounts for the phe-
nomenon of blue shadows by the diminution of light; ob*
serving that, of two shadows which were cast upon a white
wall from an opaque body, illuminated by the moon and
by a candle at the same time, that from the candle was,
somewhat red, while the other from the moon was blue.
See Newton's Optics, pa. 22S ; Bouguer Traits d'Optique
pa. 308; Kdinb. F.ss. vol. % pa. 75; or Priestley's Hist, of
Vision &c, pa. 436".

BLUNDERBUSS, in the Military Art, ashortkind of
fire-arm, with a large bore, contrived to carry a number of
pistol or musket bullets at once. The blunderbuss is proper

BOD [238

to do execution in a crowd, or to make good any narrow

1

BOD

BLUNDEVILLE (Thomas), of Newton Flotman, a
writer on navigation near the end of the 16th century. In
his Briefe Description of Universal Mappes and Cardes,
first printed in 1589, he gives an account of Gerrard Mer-
cator's universal map, observing that Bernard Putean of
Bruges had published in 1579, one altogether like it. And
though Blundeville is so particular, as to set down num-
bers expressing the distances between each parallel of lati-
tude in those maps, yet beseems to slight them, by saying,
that no better rules than those given by Ptolemy can be
devised : but, in the after editions of his book, Blundeville
omitted the commendation he had given of Ptolemy's me-
thod of delineating an universal map. But Blundeville's
chief work seems to be his Exercises, written in 1593, the
4th edit, in 1613, 4to, 800 pages, containing treatises of
Arithmetic ; tables of Sines, Tangents, and Secants ; Cos-
mographie; Description and Use. of the Globes, Mappes,
Cardes, Astrolabe, Navigation, &c.

BOB of a Pendulum, the same as the ball or weight;
which see.

BODY, in Geometry, is a figure conceived to be ex-
tended in all directions, or what is usually said to consist
of length, breadth, and thickness ; being otherwise called a
Solid." A body is conceived to be formed or generated by
the motion of a surface ; in the same manner as a surface
is by the motion of a line, and a line by the motion of a
point. — Similar bodies, or solids, are in proportion to each
other, as the cubes of their like sides, or linear dimensions.

Body, in Physics, or Natural Philosophy, is a solid, ex-
tended, palpable substance; of itself merely passive, or ra-
ther being tenacious either of motion or rest, but capable
of any sort of motion or figure.

Body is composed, according to the Peripatetics, of mat-
ter, form, and privation; according to the Epicureans and
Corpuscularians, of an assemblage of hooked, heavy atoms ;
according to the Cartesians, of a certain extension ; and
according to the Newtonians, of a system or association
of solid, massy, hard, impenetrable, moveable particles,
ranged or disposed in a certain manner; from which arise
bodies of various kinds, and distinguished by particular
names: these elementary or component particles of bodies,
they assert, must be perfectly hard, so as never to wear or
break in pieces; which, Newton observes, is necessary, in
order to the world's persisting in the same state, and bodies
continuing of the same nature and texture in several ages.
Body of a piece of Ordnance, the part contained between
the centre of the trunnions and the cascabel. This should
always be more fortified or stronger than the rest, to sus-
tain ihe first or greatest effort of the powder. SeeCANNON.
Bony of the Place, in Fortification, denotes either the
buildings inclosed, or more generally the inclosure itself.
Thus to construct the body of the place, is to fortify or in-
close the place with bastions and curtains.

Body of a Pump, the thickest part of the barrel or pipe
of a pump, within which the piston moves.

Bodies, Regular or Platonic, are those which have all
their sides, angles, and planes, similar and equal. Of these
there are only 5 ; viz,

the Tetraedron, contained by 4 equilateral triangles;
the Hexaedron or cube, by 6 squares;
the Octaedron, by 8 triangles;
the Dodecaedron, by 12 pentagons; and
the Icosaedroi), by 20 triangles.

To form the five Regular Bodies.
Let the annexed figures be exactly drawn on pasteboard,
or stiff paper, and cut out by the extreme or bounding
lines: then cut the others, or internal lines, only half
through, so that the parts may be turned up by them, and
then glued or otherwise fastened together with paste, seal-
ing-wax, &c ; so shall they form the respective body
marked with the corresponding number ; viz, n° 1 the te-
traedron, k° 2 the hexaedron or cube, u°3 the octaedron,
n°4 the dodecaedron, and n°5 theicosaedron.

To find the Superficies or Solidity of the Regular Bodies.

1 . Multiply the proper tabular area (taken from the fol-
lowing table) by the square of the linear edge of the solid,
for the superficies.

2. Multiply the tabular solidity by the cube of the linear
edge, for the solid content.

Table of the Surfaces and Solidities of the five Regular
Bodies, the linear edge being 1.

No. 61 Faces

Names

Surfaces.

Solidities

4

6

8

12
| 20

Tetraedron
Hexaedron

Octafdron

Dodecaedron

Icosaedron

1 73205

6-00000

:>-46'4 1 0

20-64573

S'()o'025

0-11785

1 -ooooo

0-47140

7-66313
2-18169

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For more particular properties, see each respective word.
See also my large Mensuration, pa. 185, edit. 4.

These bodies were called Platonic, because they were
said to have been invented, or first treated of, by Plato, who
conceived certain mysteries annexed to them.

Bodies, Descent of. See Acceleration.

BOETH1US, or BOETIUS, (a.m.t.s.), a Roman
philosopher and mathematician, was descended of a patri-
cian family, and born at Padua in the 5 th century . He
studied at Athens, where he adopted the opinions of Ari-
stotle ; and, returning to Rome, was raised to the rank of
senator and patrician, and even to the consulate, in 4S7.
On the entry of Theodoric, king of the Goths into Rome,
500, Boethius pronounced his panegyric, in the name of
the senate. He was consul in 510 and 511 ; but, in 523,
having remonstrated against the violence of Theodoric, he
was accused to that prince of having conspired with the
emperor Justin against the Goths; and, being seized with
his father-in-law Symeuacbus, he was conducted to Padua,
and there beheaded by order of Theodoric, in 524, after
6' months' imprisonment. — Besides some books on mystical
divinity, Boethius wrote 5 books on the Consolation of
Philosophy, said to have been composed to soften the
rigour of his confinement ; a work which has been trans-
lated into most other languages, particularly into Anglo-
Saxon, by the illustrious Alfred. It seems that in his time
they had the use of the Arabian arithmetic, with their ten
numeral characters, which he employed, and from which
our present numerals have been derived by several succes-
sive variations.

BOFFRAND (Germain), a celebrated French archi-
tect and engineer, was born at Nantes in Bretagne, in
1667. He was brought up under Harduin Mansarad,who
trusted him with conducting his greatest works. Boffrand
was admitted into the French Academy of Architecture in
1709. Many German princes chose him for their archi-
tect, and raised considerable edifices on his plans. His
manner of building approached that of Palladio ; and
there was much of grandeur in all his designs. As engi-
neer and inspector-general of bridges and highways, he
directed and constructed a number of canals, sluices,
bridges, and other mechanical works. He published a
curious and useful book, containing the general principles
of his art; with an account of the plans, profiles, and ele-
vations of the principal works which he executed in
France and other countries. Boffrand died at Paris in
1755, dean of the Academy of Architecture, first engineer
and inspector-general of the bridges and highways, and
architect and administrator of the general hospital.

BOILING, or Ebullition, the bubbling up of any
fluid by the application of heat. This is, in general, oc-
casioned by the discharge of an elastic vapour through the
fluid, which is then said to boil; and the same appearance
is observed, whether it be that of common air, fixed air,
or steam, &c. It is proved by Dr. Hamilton of Dublin,
in his Essay on the ascent of vapour, that the boiling of
water is occasioned by the lowermost particles of it being
heated and rarefied into vapour, or steam ; in consequence
of this diminution of their specific gravity, they ascend
through the surrounding heavier fluid with great velocity,
agitating and throwing up the body of water in their ascent,
and giving it the tumultuous motion called boiling.

That this is occasioned by clastic steam, and not by par-
ticles of fire or air, as some have imagined, iseasily proved
by the following simple experiment: Take a common

drinking-glass filled with hot water, and invert it into a
vessel of the same; then, as soon as the water in the vessel
begins to boil, large bubbles will be seen to ascend in the
glass, by which the water in it will be displaced, and there
will soon be a continued bubbling from under its edge:
but if the glass be then drawn up, >.o that its mouth may
just touch the water, and a cloth wetted in cold water be
applied to the outside, the elastic steam within it will be
instantly condensed, on which the water will ascend so as
nearly to fill it again. Some small parts of air &c, that
may happen to. be lodged in the fluid, may also perhaps
be expelled, as well as the rarefied steam. And this is
particularly recommended as a method of purifying quick-
silver, for more accurately making barometers and ther-
mometers. If a vessel containing water be placed over a,
steady fire, the water will grow continually hotter, till it
reaches the limit of boiling; after which, the regular ac-
cessions of heat are wholly spent in convertingit intosteam,
or the steam carries off the heat as fast as it is generated
by the fire. The water therefore remains at the same de-
gree of temperature, however fiercely it may boil ; the
only difference is, that with a strong fire it sooner boils,
and more quickly evaporates. Hence the reason why a vessel
full of water, and plunged into the centre of a larger one,
which is likewise filled with that fluid, barely acquires the
boiling heat, but will never actually boil.

We commonly annex the idea of a certain very great
degree of heat to the boiling of liquids, though often with-
out reason; for different liquids boil with different degrees
of heat ; and any one given liquid also, under different
pressures of the atmosphere. Thus, a vessel of tar being
set over the fire till it boils, it is said a person may then
put his hand into it without injury: and by putting
water under the receiver of an air-pump, and applying the
flame of a candle or lamp under it, it will be found, by
gradually exhausting the receiver, that the water is made
to boil with less and less degrees of heat : and without ap-
plying any heat at all, the water, oreven the moistureabout
the bottom or edges of the receiver, will rise in an elastic
vapour up into it, when the exhaustion is nearly completed.

Spiritof wine boils still sooner in vacuo than water. And
Dr. Freind gives a table of the different times required to
make several fluids boil by the same heat. See also Philos.
Trans. N° 122.

Every particular liquid has a fixed point of the ther-
mometer, at which boiling takes place, and this is called
the boiling point of the liquid. Thus, water begins to boil
when heated to 212 degrees of Fahrenheit's scale. A
strong heat makes it boil more rapidly, but does not in-
crease its temperature: a fact which wasfirst observed by

Dr. Hooke. The annexed table

i_ .ii-i- L c Bodies, Boilinir point,

shows the boiling point 01 a num- „ , D1 „„

ber of liquids, under the common

Ammonia

140

pressure ot 30 inches of the baro- . , , , ,_,-

1 . e- " « . ii-i Alcohol - - - 1/0

meter; tor the heat and boiling .., „.„

point are found to depend on the ^^ ^.^ " 23Q

degree of pressure upon the 1,- Nitric aci() . . 248

quid. When the pressure is dimi- c , , • ■ , .,,_

1 , , ., ,■ • , , •, . ! Sulphuric acid - oOO

111s bed, the liquid boils at a lower mi ??,

1 , . , Phosphorus - - 554

temperature; when it is increased, ,Vi c . .- ?rn

.\ , ' . ' Oil of turpentine 560

a limber temperature is neccs- c , , ' .__

0 ' . ..... p Sulphur - - - 570

sary to produce ebullition. From jj^.^j . . g00

the experiments of Professor Ro- Mc _ . . gg

binson, it appears that, in a J

vacuum, all liquids boil at about 145° lower than in open

BOM

[ 240 J

BON

air, under a pressure of 30 inches of mercury : therefore
water would boil in vacuo at 6l° and alcohol at 34-°. In
Papin's digester, the temperature of water may be raised
to 300 or 400 degrees, without ebullition ; but the mo-
ment that the pressure is removed, the boiling commences
with the greatest violence.

The mixture of various salts with water affects its boiling
point considerably ; as appears from Mr. Achard's expe-
riments recorded in the Berlin Memoirs for 1785. He
found thata saturated solution of muriat of soda raised the
boiling point 10'35 degrees; one of muriat of ammonia
raised it 979 degrees; one of carbonat of -potass raised it
1 f -2 degrees ; while a small quantity of borax lowered it
1-35 degrees; ditto ofsulphat of magnesia lowered it2-47
degrees ; and sulphat of lime in any proportion lowered it
2-02 degrees.

BOMB, in Artillery, a shell or hollow ball of cast-iron,
having a large vent, by which it is filled with gun-powder,
and which is fitted with a fuze or hollow plug to give fire
by, when thrown out of a mortar, &c. About the time
when the shell arrives at the intended place, the composi-
tion in the pipe of the fuze sets fire to the powder in the
shell, which blows it in pieces, to the great annoyance of
the enemy, by killing the people, or firing the houses, &c.
They are now commonly called Shells simply, in the Eng-
lish artillery.

These shells, or bombs, are of various sizes, from that of
17 or 18 inches diameter downwards. But these very large
ones are not used by the English, that of 13 inches diame-
ter being the greatest size now employed by them; the
weight, dimensions, and other circumstances of the 13-inch
shells, and of those downwards, are as in the following
table.

Diameter of
the Shell.

Weight of
the Shell.

Powder to
fill them.

Powder to burst

them into most

pieces.

13 inch

10
8

5f Royal
4f Cohorn

lbs.

195
89
46
14i
A

lb. oz.
9 4§
4 14§
2 3§
1 1§
0 8

lb. 02.
7 8
3 4
2 0
0 14
0 7

Mr. Muller gives the following proportion for all shells.
Dividing the diameter of the mortar into 30 equal parts,
then the other dimensions, in SOths of that diameter, will
be thus:

Diameter of the bore, or mortar 30

Diameter of the shell .... 29*

Diameter of the hollow sphere - - 21

Thickness of metal at the fuze hole - - . - 3 J

Thickness at the opposite part - - 5

Diameter of the fuze hole - 4

Weight of shell empty .... T»T°.rf

Weight of powder to fill it - » - - xri^
where d denotes the cube of the diameter of the bore in
inches. — But shells have also lately been made of the same
thickness of metal throughout.

In general the windage, or difference between the di-
ameter of the shell and mortar, is -^ of the latter ; also
the diameter of the hollow part of the shell is T75 of the
same.

Bombs are thrown out of mortars, or howitzers ; but
they may also be thrown out of cannon; and a very small
sort are thrown by the hand, which are called granados :
and the Venetians at the siege of Candia, when the Turks
had possessed themselves of the ditch, used large bombs

without any piece of ordnance, barely rolling them down
upon the enemy along a plank set aslope, with ledges on
the sides to keep the bomb directly forward.

Mr.Blondel,in his Artdejetter des Bom bes, says the first
bombs were those thrown into the city of Watchtendonch
in Guelderland, in 1588 ; and they arc described by our
countryman Lucar, in his book on Artillery published this
same year 15SS; though it is pretended by others that
they were practised near a century before, namely at the
siege of Naples in 1495. They only came into common
use, however, in 1&34, and then only in the Dutch and
Spanish armies. It is said that one Malthus, an English
engineer, was sent forfrom Holland by Lewis the 14lh, who
used them for him with much success, particularly- at the
siege of Cohoure in l6"42.

The art of throwing bombs, or shells, constitutes one of
the chief branches of Gunnery, founded on the theory of
projectiles, and the quantities and laws of the force of gun-
powder. And the principal writers on this art are Mess.
Blondel, Guisnee, De Ressons, De La Hire, &c.

Bomb-Chest, is a kind of chest usually filled with
bombs, and sometimes only with gunpowder, placed under
ground, to be blown up into the air with those who stand
upon it; being set on fire by means of a saucissee fastened
at one end. But they are now much out of use.

BOMBARD, an ancient piece of ordnance, now out of
use. It was very short and thick, with a large mouth ;
some of which threw balls of 300 pounds weight, requi-
ring the use of cranes to load them. The Bombard is by
some called basilisk, and by the Dutch donderbus.

To bombard, is to attack by throwing of bombs, or
shells.

BOMBARDIER, a person em ployed in throwing bombs
orshells. He adjusts thefuze, and loads and tires the mortar.

BOND (Henry), was an intelligent mathematician,
and a teacher of navigation at Ratcliff, near London, in
the early part of the 17th century In the course of his
nautical concerns he interested himself much about the
magnetical needle, its variation and inclination. The varia-
tion of the magnetic needle was first observed by John
Moir, then by Ecim. Gunter, next by Hen. Gallibrand,
and then by our author Bond. In the year 1657 there
was no variation at London : before that time it had been
eastwards, as observed by several of those persons, but gra-
dually decreasing, or the needle moving westwards, till it
pointed due north in that year l6~57; ever since it has been
travelling westwards, but at unequal rates, till now, in
1S12, it has arrived at above 24° west variation, and ap-
pears to be nearly stationary, after which it will probably
return eastward again ; according to which rate of motion,
the magnetic needle, or pole, appears to perform an entire
circular revolution ill about 620 years. Besides the
changeable variation east and west, the needle was also
found to have a changeable inclination, or dip below the
horizon; and on this last change it was that Mr. Bond
founded his method of finding the longitude; win, h how-
ever has failed, as might be expected. On this method he
published a book, called The Longitude found, printed at
London 1676, in 4to. He here states, as bis prediction,
after many boasting and empirical pretensions, what would
be the latitude, and longitude, and the inclination of the
needle, in many of the most noted places on the earth, in
the year lo'7<>; also another table of the latitude of the
magnetic pole, answering to every 5 minutes in the incli-
nation of the needle. He also had inserted in the Philos.

BOO

C 241 ]

BOR

Trans. No. 40, of the year l6'68, or pa. 282, vol. 1, of my
Abridgment, a table of the needle's variation at London,
for every year from l668 to 17l6", when he states it at

9° i?i'.

Dr. Birch states that Sir Charles Cavendish, brother to
William, marquis, and afterwards duke of Newcastle, in an
original letter of his from Antwerp, Nov. 13, 1()48, to Mr.
John Pell, then professor of mathematics at Breda, men-
tions Mr. Bond as an old mathematician at London-, a
humble man, who speaks, says he, very meanly of himself,
and " yet he found an easy and short demonstration of
that proposition concerning spherical triangles, which Mr.
Dughtred demonstrated first, who told me, Mr. Bond's
demonstration was shorter." Sir Charles adds, that Mr.
Bond was in hopes of finding the longitude by the load-
stone; and his treatise on that subject, entitled "The
Longitude found," was printed at London 1676, in 4to.

Bond's scheme for the longitude was soon opposed, a
book being published at London in l6"78, called "The
Longitude not found," written by one Bcckborrow. And
indeed as Bond's hypothesis did not in anywise answer its
author's sanguine expectations, Dr. Halley also undertook
this affair ; and from a multitude of observations concluded,
that the magnetic needle was influenced by four poles. His
speculations on this subject are in the Philos. Trans. No.
148 and 195. But this wonderful phenomenon seems to
have hitherto eluded all researches. However, Dr. Halley,
in 1700, published a general map, on which were deli-
neated curve lines, showing the paths where the magnetic
needle had the same variation. The positions of these
curves will indeed continually suffer alterations, requiring

Areas; Arctophylax, Arcturus-Minor, Bubulcus, Buhu-
lus, Canis-Latrans, Clamator, Jcarus, Lycaon, Philome-
tus, Plaustri-Custos, Plorans, Thegnis, and Vociferator ;
by Hesychius it is called Orion, and by the Arabs, Ara-
mech, or Archamech. Schiller, instead of Bootes, makes
the figure of St. Sylvester; Schickhard, that of Nimrod ;
and Weigehus, the three Swedish crowns. See Wolf Lex
Math. p. 266.

BORDA (Charles), a celebrated French mathema-
tician and philosopher, died in 1799, hi the 64th year of
his age; in whom the National Institute, and the Parisian
Board of Longitude, lost one of their most learned and
active members. Borda very early gave proofs of his
eminent mathematical talents ; as appears by his many ex-
cellent tracts inserted in the Memoirs of the Royal- Aca-
demy of Sciences, on a variety of abstruse subjects; on hy-
draulics, on the resistance of fluids, on water-wheels, on
pumps, on the projection of military shells, and their re-
sistance by the air, &c.

In 1771and 1772, by order of the kins, he accompanied
La Crenne and Pingre, on a literary voyage to various
coasts of Europe, Africa, and America, for improving the
science of geography, and for trying several new nautical
instruments, time-pieces, and methods of finding the longi-
tude. The three travellers afterwards published coniointfy
in 2 vols 4to, 1778, an account of the fruits of their nu-
merous researches ; in which Borda's share was certainly
not the smallest. The results of this expedition are also
recorded in the Memoirs of the Academy for the year 1773
In the year 1787, he published his much-esteemed De-

scnption and Use of the Circle of Reflection ; in which hc
to be corrected from time to time, as they have since been revived and recommended the use of the specular circle
for the years 174* and 1756, by two ingenious persons, which had been proposed in 1756 by Tobias Mayer R /'

William Mountaine and James Dodson; the latter of
whom died not long after he had been chosen, for his me-
rit, mathematical master at Christ's Hospital, in London.

RONES, Napier's. See Napier.

BONING, iir Surveying and Levelling, &c, is the pla-
cing three or more rods or poles, all of the same length, in
or upon the ground, in such a manner, that the tops of
them be all in one continued straight line, whether it be
horizontal or inclined, so that the eye can look along the
tops of them all, from one end of the line to the other.

BONNET, in Fortification, a small work of two faces,
having only a parapet, with two rows of palisadoesat about
10 or 12 feet distance. It is commonly placed before the
salliant angle of the counterscarp, and having a communi-
cation with the covered way, by means of a trench cut
through the glacis, and palisadoes on each side.

Bonnet d PrStre, or Priest's Cap, is an outwork, hav-
ing three salliant angles at the head, besides two inwards.
It differs from the double tenaille only in this, that its
sides, instead of being parallel, become narrower, or closer,
at the gorge, and opening at the front ; from whence it is
tailed Queue d'aronde, or swallow's tail.

BOOTES, a constellation of the northern hemisphere,
and one of the 48 old ones ; having 23 stars in Ptolemy's
catalogue, 28 in Tycho's, 34 in Bayer's, 52 in Hevelius's,
and 54 in Flamstecd's ; of which one, in the skirt of the
coat, is of the first magnitude, and called Arcturus.

Bootes is represented as a man in the posture of walking;
his light hand grasping a club, and his left extended up-
wards, and holding the cord of the two dogs which seem
barking at the Great Bear.

This constellation is called by various other names ; as
Vol. I.

by Tobias Mayer. Borda
was the founder of the schools of naval architecture in
France: he first conceived the project, aud formed the
plan of instruction, with the regulations for those semi-
naries. By his exertions too, a uniformity in the building
of the ships was introduced, according to the principles of
Euler, by which improvement, an equality of sailing was
effected in all the ships of the royal navy. Hence, the form
of the French ships, being constructed on mathematical
principles, is clearly preferable to that of the ships of the
other naval powers ; being the most advantageous, and the
best adapted for fast sailing, and for manoeuvring.

Borda also brought into use Mayer's old method
of measuring terrestrial angles, applied it to astro-
nomical observations, and, for that purpose, invented a
new construction of circles, with double moveable tele-
scopes, which have been used in the new measurement of
degrees in France. He was also the inventor of the in-
genious mensuration rod, with which the new station-lines
were measured; and he had the chief share in the reform
of weights and measures; of which he was so zealous a
promoter, that he printed, at his own expense, Tables of
Sines, &c, on the decimal system. In 1792 he determined,
with an accuracy perhaps never before attained, the length
of the pendulum vibrating seconds at Paris. In 1797 %e
find his name in the list of candidates for the office of
Director of the French Republic.

BORE, of a gun, or other piece of ordnance, is the chase
cylinder, or hollow part of the piece.

BORliAL Signs, are the first six signs of the Zodiac
or those on the northern side of the equinoctial; viz the
signs <V aries, « taurus, n gemini, ss cancer, SI loo. m
virgo.

21

B O R

[ 242 ]

BOS

BOREAL1S, Aurora. See Aurora Boreulis.

BORELLI (Johi* Alphonso), a celebrated philoso-
pher and mathematician, born at Naples the 28th of Jan.
1608. He was professor of philosophy and mathematics
in some of the most celebrated universities of Italy, par-
ticularly at Florence and Pisa, where he became highly in
favour with the princes of the house of Medicis. But hav-
ing been concerned in the revolt of Messina, he was
obliged to retire to Rome, where he spent the remainder

5. Osservatione intorno alia vistu ineguali degli Orci. —
This piece was inserted in the Journal of Rome, for the year
1669.

€. De Motionibus Naturalibus de Gravitate pendentibus.
Regio Julio, 16'70, 4to.

7. Meteorologia Aetnea, &c. Regio Julio, 1 670, 4to.

8. Osservatione dell' Ecclissi Lunare, 11 Gennaro 1675.
— Inserted in the Journal of Rome 1675, p. 34.

9. Elementa Conica Apollonii Pergsi, et Archimedis

of his life under the protection of Christina queen of Opera, nova et breviori methodo demonstrata. — Printed at
Sweden, who honoured him with her friendship, and by
her liberality towards him softened the rigour of his hard
fortune. He continued two years in the convent of the
regular clergy of St. Pantaleon, called the Pious Schools,
where heinstructed the youth in the mathematical sciences.
And this study he prosecuted with great diligence for many
years afterward, as appears by his correspondence with
several ingenious mathematicians of his time, and the fre-
quent mention that has been made of him by others, who
have endeavoured to do justice to his memory; He wrote
a letter to Mr. John Collins, in which he discovers his
great desire and endeavoul-s to promote the improvement
of those sciences: he also speaks of his correspondence
■with, and great affection for, Mr. Henry Oldenburg, Se>
cretary of the Royal Society ; of Dr. Wallis; of the then
iate learned Mr. Boyle, and lamented the loss sustained
by his death to the commonwealth of learning. Mr. Bax-
ter, in his Enquiry into the Nature of the Human Soul,
makes frequent Use of our author's book De Motu Anima-
lium, and tells us, that he was the first who discovered
that the force exerted within the body prodigiously ex-
ceeds the weight to be moved without, or that nature em-
ploys an immense power to move a small weight. He ac-
knowledges, however, that Dr. James Keil had shown that
Borelli was mistaken in his calculation of the force of the
muscle of the heart ; but that he nevertheless ranks him
with the most authentic writers, and Says he is seldom mis-
taken : and, having remarked, that it is so far from being
true that great things are brought about by small powers,
that, on the contrary, a stupendous power is manifest in
the most ordinary operations of nature, he observes that
the ingenious Borelli first observed this in animal motion ;
and that Dr. Stephen Hales, by a course of experiments in
his Vegetable Statics, had shown the same in the force of
the ascending sap in vegetables. And this circumstance,
that nature employs a very great power, or force, in the
animal frame, to move a small weight, we may here ob-
serve, is owing to the levers in the animal frame being all
necessarily of the third kind, the power acting very near
the centre of motion, and the weight at a comparatively
great distance from it.

Aftera course of unceasing labours, Borelli died at Pan-
taleon of a pleurisy, the 31st of December l679> at 72
years of age.

Beside several books 'on physical subjects, Borelli pub-
lished the following mathematical ones : viz,

1. Apollonii Pergaei Conicoruin Lib. 5, C, and 7- Floren.
1661, fol.

2.TheoriseMedicorurn Planetarum ex causisphysicisde-
ductae. Fior. 1666, 4to.

3. De ViPercussionis. Bologna, l66'7, 4to. — This piece
Was reprinted, with his celebrated treatise De Motu Ani-
malium, and that other De Motionibus Naturalibus, in
1686.

4. Euclidcs Rcstitutus, &c. Pi>a, 1668, 4to.

Rome in 1679, in 12mo, at the end of the 3d edition of
his Euclides Restitutus.

10. De Motu Animalium. Pars prima in 1680, and
Pars altera in 1681, 4to. — These were reprinted at Leyden
16S5, revised and purged from many errors ; with the ad-
dition of John Bernoulli's Mathematical Meditations con-
cerning the Motion of the Muscles.

11. At Leyden, I686, in 4to, a more correct and accu-
rate edition, revised by J. Broen, M. d. of Leyden, of his
two pieces, De Vi Percussionis, et De Motionibus de Gra-
vitate pendentibus.

BORELLI (Peter), who flourished about the middle
of the l6th century, appears to have been a native of Mid-
dleburg, capital of the island of Waleheren and all Zea-
land; and was counsellor and physician in ordinary to his
most Christian majesty. He wrote two treatises on the
telescope and microscope, printed at the Hague, inl655
and 165(3. The former is a very full and particular trea-
tise on the history of the invention of those very useful in-
struments; which, by a train of strong evidence he ascribes
to Zachariah Jansen, a spectacle-maker of Middleburg, in.
1590. He gives a particular account of the pretensions-,
that have been ascribed to all the other persons that have
been named as the inventors; the whole forming a very,
interesting history of the invention. The chief body of evi-
dence it seems he received from a Wm. Borelli, of the same-
city, some time ambassador in England; but whether any.
relation of Peter's does not appear.

The first volume consists of two parts : the former treat-
ing of the invention itself; and the latter of the fabric,,
construction, and uses of the instruments. The second
volume contains the account and appearances of one hun-
dred minute objects, as viewed by the microscope. The
volumes are both neatly printed in 4to, by Adrian Vlacq,
at the Hague.

BOSCOVICH (Roger Joseph), an eminent mathe-
matician and philosopher, was born May 11th, 1711, at
Ragtisa. He studied Latin grammar in the schools which
were taught by the Jesuits in his native city, until 1725,
when, in consistence with a maxim of the Jesuits to send
their most eminent pupils to Rome for the completion of
their education, lie was removed to that city. After this
he soon acquired very great reputation for his eminent at-
tainments in divinity and science: at three successive pe-
riods he became professor of mathematics and astronomy
at Rome, at Pavia, and at Milan. When the order of Je-
suits was suppressed he was invited to Paris, and received
the place of director of the optical instruments of the ma-
rine. Previous to this however he had been employed, in
conjunction with Mayer, in measuring a degree of the meri-
dian in Italy, and correcting the maps of the Papal estate.
He published, in 1755, an interesting account of the expe-
dition in which these objects were effected, lie had also
been employed in adjusting a disagreeable! affair between
the republic of Lucca and the regency of Tuscany, and in a

B O TJ

[ 243 ]

Boa

similar business between the republic of Ragusa and the
court of Great Britain, which brought him to London,
where he soon became acquainted with the most eminent
of the English philosophers. During his stay he was
elected a fellow of the Royal Society, and he dedicated to
that learned body his Poem on Eclipses, which contains a
neat compendium of astronomy. He remained at Paris 10
years, where he was much respected ; but, being a fo-
reigner, his celebrity was envied: this; together with the
irreligion which then prevailed among many of the French
philosophers, was disagreeable to him, so that he obtained
leave of absence to revisit his friends in Italy. The first
place he resided at for any considerable length of time in
Italy, was Bassano, where he published his Opuscules, in
five volumes 4to, composed in Latin, Italian, and French;
and containing a variety of elegant and ingenious disqui-
sitions connected with astronomical and optical science.
From Bassano our author went to Rome, and thence to
Milan, where he took up his abode; being in the neigh-
bourhood of his favourite observatory at Brera. Here
he continued to enjoy the pleasures of study, and occasi-
onally the society of many respected friends. His unwil-
lingness to leave Italy, and at the same time a solicitude
to avoid the charge of ingratitude from the French
nation, occasioned him great perplexity of mind, which
was followed by deep melancholy, a disordered imagina-
tion, and at length direct madness. During the heat of
delirium, he frequently exclaimed that he would die poor
and inglorious. He had indeed some short lucid moments,
and once there were hopes of his recovery ; but he soon re-
lapsed, and an imposthume, breakirtg in his breast, put an
end to his mortal life the 13th of Feb. 1787, in his 76th
year. He was interred decently, but without pomp, in
the parochial church of S. Maria Pedrone. " Such was
the exit," says Fabroni, " of this sublime genius, whom
Rome honoured as her master; whom all Italy regarded
as her ornament; and to whom Greece would have erected
a statue, had she for want of space been obliged even to
throw down some of her heroes." The works of this me-
morable man are the following: — 1. Elements of Mathe-
matics, with a Treatise on Conic Sections. 2. His many
Dissertations published during his professorship in the Ro-
man College. 3. His Account of the. Survey of the Pope's
Estate. 4. A curious and elegant Poem on Solar and Lu-
nar Eclipses. 5. The five volumes published at Bassano.
S. His Hydronamical Pieces. 7. A Theory of Natural
Philosophy.

BOUGUER (Peter), a celebrated French mathemati-
cian, was born at Croisic, in Lower Bretagne, the 10th of
February 1698. He was the son of John Bouguer, Pro-
fessor Royal of Hydrography, a tolerably good mathema-
tician, and author of A complete Treatise on Navigation.
Young Bouguer was accustomed to learn mathematics from
his father, from the time he was able to speak, and thus
became :i proficient in those sciences while he was yet a
child. He was sent very early to the Jesuits' college at
Valines, where he had the honour to instruct his regent in
the mathematics, at eleven years of age.

Two year9 after this he had a public contest with a pro-
fessor of mathematics, on a proposition which the latter
had advanced erroneously ; and he triumphed over him ;
on which the professor, unable to bear the disgrace, left
the country.

Two years after this, when young Bouguer had not yet

finished his studies, he lost his father; whom he was ap-
pointed to succeed in his office of hydrographer, after a
public examination of his qualifications; being then only
15 years of age; an occupation which he discharged with
great respect and dignity at that early age.

At the age of 28 in 1727 he obtained the prize proposed
by the Academy of Scienpes, for the best way of mastinc
of shjps. This first success of Bouguer was soon after fol-
lowed by two others of the same kind ; he successively
gained the prizes of 1729 and 1731; the former, for the
best manner of observing at sea the height of the stars; and'
the latter, for the most advantageous way of observing the
declination of the-magnetic needle, or the variation of the
compass.

In 1729, he gave an Optical Essay upon the Gradation
of Light; a subject quite new, in wr^ch he examines the
intensity of light, and determines its degrees of diminution
in passing through different pellucid mediums, and parti-
cularly that of the sun in traversing the earth's atmosphere.
Mairan gave an extract of this first essay in the Journal
des Savans, in 1730.

During this year, 1730, he was removed from the port
of Croisic to that^of Havre, which brought him into ft,
nearer connection with the Academy of Sciences, in which
he obtained, in 1731, the place of associate geometrician,
vacant by the promotion of Maupertuis to that of pen-
sioner; and in J735 he was promoted to the office of pen-
sioner-astronomer. The same year he was likewise sent on
the commission to South America, along with Messieurs
Godin, Condamine, and Jeussieu, to determine the mea-
sure of the degrees of the meridian, and the figure of the
earth. In this painful and troublesome business, of 10
years' duration, chiefly among the lofty Cordelier moun-
tains, our author determined many other new circum-
stances, besides the main object of the voyage; such as the
expansion and contraction of metals and other substances,
by the sudden and alternate changes of heat and cold
among those mountains; observations on the refraction of
the atmosphere from the tops of the same, with the singu-
lar phenomenon of the sudden increase of the refraction,
when the star can be observed below the line of the level ;
the laws of density of the air at different heights, from ob-
servations made at different points of these enormous moun-
tains; a determination that the mountains have an effect
on a plummet, though he did not assign the exact quan-
tity of it; a method of estimating the errors committed by
navigators in determining their course; a new construction
of the log for measuring a ship's way ; with several other
useful improvements.

Other inventions of Bouguer, made upon different oc-
casions, were as follow : The heliometer, being a telescope
with two object glasses, affording a good method of mea-
suring the diameters of the larger planets with ease and
exactness: his researches on the figure in which two lines
or two long ranges of parallel trees appear : his experiments
on the famous reciprocation of the pendulum : and those;
upon the manner of measuring the force of light: &c, Sze.

The close application which Bouguer made to study,
undermined his health, and terminated his life the 15th of
August 1758, at 60 years of age. — His chief works, that
have been published, arc,

1. The Figure of the Earth, determine/:l by the observa-
tions made in South America; 1749, in 4to.

2. Treatise on Navigation and Pilotage; Paris, 1752,

212

B O U

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BOU

in 4to, This work has been abridged by M. La Caille, in
1 volume, 8vo, 1768.

3. Treatise on Ships, their Construction and Motions;
in 4to, 1756.

4. Optical Treatise on the Gradation of Light; first in
1729 ; then a new edition in 1760, in 4to.

His papers that were inserted in the Memoirs of the
Academy, are very numerous and important : as, in the
Memoirs for 172o, On the Comparison of the force of the
solar and lunar light with that of candles. — 1731, Obser-
vations on the curvilinear motion of bodies in mediums. —

1732, On the new curves called the Lines of Pursuit. —

1733, To determine the species of conoid, to be constructed
on a civen base which is exposed to the action of a fluid,
so that the impulse may be the least possible. — Determi-
nation of the orbit of comets. — 1734, Comparison of the
two laws which the earth and the other planets must ob-
serve under the figure which gravity causes them to take. —
On the curve lines proper to form the arches in domes. —
1735, Observations on the equinoxes. — On the length of
the pendulum. — 1736, On the length of the pendulum in
the torrid zone. — On the manner of determining the figure
of the earth by the measure of the degrees of latitude and
longitude. — 1739, On the astronomical refractions in the
torrid zone. — Observations on the lunar eclipse of the Sth
of September 1737, made at Quito. — 1744, Short account
of the vo3'age to Peru, by the members of the Royal Aca-
demy of Sciences, to measure the degrees of the meridian
near the equator, and from thence to determine the figure
of the earth. — 1745, Experiments made at Quito, and di-
vers other places in the torrid zone, on the expansion and
contraction of metals by heat and cold. — On the problem
of the masting of ships. — 1746", Treatise on ships, their
structure and motions. — On the impulse of fluids upon
the fore parts of pyramidoids having their base a trape-
zium.— Continuation of the short account given in 1744,
of the voyage to Peru for measuring the earth. — 1747, On
a new construction of the log, and other instruments for
measuring the run of a ship. — 1748, On the diameters of
the larger planets. The new instrument called a Helio-
meter, used for determining them ; with observations of
the sun. — Observation on the eclipsie of the moon the 8th
of August 1748. — 1749, Second memoir on astronomical
refractions, observed in the torrid zone, with remarks on
the manner of constructing the tables of them. — Figure of
the earth determined by MM. Bouguer and Condamine,
with an abridgment of the expedition to Peru. — 1750, Ob-
servation of the lunar eclipse of the 13th of December
1750. — 1751, On the form of bodies most proper to be
assumed for their revolution, when they are urged at one
of their extremities, or any other point. — On the moon's
parallax, with the estimation of the changes caused in the
parallaxes by the figure of the earth. — Observation of the
lunar eclipse, the 2d of December 1751. — 1752, On the
operations made by seamen, called Corrections. — 1753,
Observation of the passage of Mercury over the sun, the
6th of May 1753. — On the dilatations of the air in the at-
mosphere.— New treatise of navigation, containing the
theory and practice of pilotage, or working of ships. —
1754, Operations, &c, for distinguishing, among the dif-
ferent determinations of the degree of the meridian near
Paris, that which ought to be preferred. — On the direction
winch the string of a plummet takes. — Solution of the
chief problems in the working of ships. — 1755, On the ap-

parent magnitude of objects. — Second memoir on the chief
problems in the working of ships. — 1757, Account of the
treatise on the working of ships. — On the means of mea-
suring the light. — 1758, His Eulogy.

In the volumes of the prizes given by the academy, are
the following pieces by Bouguer:

In vol. 1, on the masting of ships. — Vol. 2, On the me-
thod of exactly observing at sea the height of the stars;
and the variation of the compass. Also on the cause of
the inclination of the planets' orbits.

BOULTINE, in Architecture, a convex moulding, of a
quarter of a circle, and placed next below the plinth in
the Tuscan and Doric capital.

BOURNE (Wm.), was a useful and ingenious writer on
navigation about the middle of the l6th century. In 1567
was published what he calls " Rules of Navigation," as ap-
pears from his Almanac, printed in 1571. And in 1577
was published his " Regiment for the Sea," designed as a
supplement to Cortes, whom he often quotes. Besides
many things common with others, Bourne gives a table of
the places and declinations of 32 principal stars, by which
to find the latitude and hour; as also a larger tirie-table
than that published by Leonard Digges, in 1556. He
shows, by considering the irregularities in the moon's mo-
tion, the errors of thesailorsin findingher age bytheepact;
as also in their determining the hour from observing on
what point of the compass the sun and moon appeared.
He advises, in sailing towards high latitudes, to keep the-
reckoning by the globe; as there the plane chart errs most.
He despairs of our ever being able to find the longitude by
any instrument, unless the variation of the compass should
be caused by some such attractive point as Cortes had
imagined. Though of this he doubts: and as he 'had
shown how to find the variation of the compass at all
times, he advises to keep an account of the observations,
as useful to discover the place of a ship : which advice the
famous Simon Stevin prosecuted at huge, in a treatise pub-
lished at Leyden in 1599, entitled "Portuum investigan-
dorum Ratio Metaphrasto Hugone Grotio;" the substance
of which was the same year printed at London, in English,
by Edw. Wright, entitled " The Haven-finding Art."

But the most remarkable thing in this ancient tract is,
the describing of the way by which our sailors estimated
the rate a ship made in her course, by an instrument called
the Log ; an instrument so named from the wooden log
that floats in the water, while the time is reckoned during
which the line fastened to it is veering out. The origin of
this device is not known; and it seems no further mention
was made of it till 1607, inan East-India voyage, published
by Purchas; but from that time its name occurs in other
voyages, among his collections. And henceforward it was
often noticed, by foreigners as well as our own authors;
as by Gunter in 1623, Snellius in 1 624, Metius in Hi31,
Oughtred in l6'33, Herigone in 1634, Saltonstall in 1636,
Norwood in 1637, Fournier in 1643; and indeed by most
succeeding writers on navigation of every country. And
it still continues to be in use as at first, though attempts
have often been made to improve it, and other contrivances
proposed to supply its place. Many of these have suc-
ceeded in quiet water, but proved useless in a troubled sea.
See our article Log.

A following edition of this book was revised by the au-
thor; among the additions to which, he enlarges on the
account of the log-line; and at the end subjoins a " Hy-

BOY

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BOY

drographical Discourse touching the (ive several Passages sue their studies: here our author resumed his acquaint-

into Cathay." Bourne published other tracts also; as, one ance with the elements of the mathematics, which he had

entitled '* Inventions or Devises;" where he describes a commenced at Eton when 10 years old, on account of an

method, by wheel- work, of measuring the velocity of a ship illness which prevented his other usual studies.
at sea; an artifice which he attributes to one HumfreyCole. In the autumn of 1641 he quitted Geneva, and travelled

BOW, an offensive weapon made of wood, horn, steel, through Switzerland and Italy to Venice, whence he re-

or other elastic matter, by which arrows are thrown with turned again to Florence, where he spent the winter stu-

great force. This instrument was of very general use dying the Italian language and bistorj, and also the works

among the ancients, and is still found among all savage of the celebrated astronomer Galileo, who died in a village

nations unacquainted with the use of fire-arms. There are near this city during Mr. Boyle's residence here,
two species of the Bow, the Long and the Cross Bow. About the end of March l6'42,he set out from Florence,

The Long Bow is simply a bow, or a rod, with a string visited Rome and other places in Italy, then returned 4o'

fastened to each end of it, to the middle of which the end the south of France. At Marseilles, in May 164,2, Mr.

of an arrow being applied, and then drawn by the hand, Boyle received letters from his father, which informed him

on suddenly quitting the hold, the bow returns by means that the rebellion had broken out in Ireland, and with what

of its elasticity, and impels the arrow from the string with difficulty he had procured 250/ then remitted to help him

great violence. The old English archers were famous for and his brother home. This remittance however never

the long bow, by means of which they gained many vkto- reached them, and they were obliged to return to Geneva

ries in France and other countries. with iheir governor Mr. Marcombes, who contrived on his

The Cross Bow, called also Arbalest or Arbalet, is a bow own credit, and by selling some jewels, to raise money

strung and set in a shaft of wood, and furnished with a enough to bring them to England, where they arrived in

trigger; serving to throw bullets, darts, and large arrows, 1644. On their arrival theyVound that their father was

&c.' The ancients had large machines for throwing many dead, and had left our author ihe manor of Stalbridec in

arrows at once, called Arbalets, or Balistoe.

The force of a bow may be calculated on this principle,
that its spring, i. e. the power by which it restores itself to
its natural position, is always proportional to the space or
distance it is bent or removed from it.

England, with some other considerable estates in Ireland.
From this time Mr. Boyle's chief residence, for some
years at least, was at his manor of Stalbridge, whence he
made occasional excursions to Oxford, London, &c ; ap-
plying himself with great industry to various kinds of stu-

Bow, a mathematical instrument formerly used at sea dies, but especially to philosophy and chemistry; and

for taking the sun's altitude. It consisted of a large arch seizing every opportunity of cultivating the acquaintance

divided into 90 degrees, fixed on a statf, and furnished with of the most learned men of his time. He was one of the

three vanes, viz, a side vane, a sight vane, and another members of that small but learned body of men, in l645,

called the horizon vane. who, when all academical studies were interrupted by the

Bow-Compass, an instrument for drawing arches of very civil wars, secreted themselves ; and held private meetings,

large circles, for which the common compasses are too first in London, afterwards at Oxford, to cultivate subjects

small. It consists of a beam of wood or brass, with three of natural knowledge on that plan of experiment which

long screws that govern or bend a lath of wood or steel to Lord Bacon had delineated. They styled themselves then

any arch. The Philosophic College; but after the restoration, when

BOX and Needle, the small compass of a theodolite, they were incorporated, and distinguished openly, they took

circumferentor, or plain-table. the name of the Royal Society.

BOYAU, in Fortification, a ditch covered by a para- In the summer of 1654 he retired to settle at Oxford,

pet, and serving as a communication between two trenches, the Philosophical Society being removed from London to'

It runs parallel to the works of the body of the place, and that place, that he might enjoy the conversation of the

serves as a line of contravallation, both to hinder the sal- other learned members/his friends, who had retired thither,

lies of the besieged, and to secure the miners. When it is such as Wilkins, Wallis, Ward, Willis, Wren, &c. It was

a particular cut running from the trenches, to cover some during his residence here that he improved that admirable

spot of ground, it is drawn so as not to be enfiladed or engine the air-pump; and by numerous experiments was

scoured by the enemy's shot. enabled to discover several qualities of the air, so as to lay

BOYLE (Robert), one of the greatest philosophers, as a foundation for a complete theory. He declared against

well as best men, that any country has ever produced, was the philosophy of Aristotle, as having in it more of words

the 7th son and the 14th child of Richard earl of Cork, than things; promising much, but performing little; and

and was born at Lismorc in the province of Munster in giving the inventions of men for indubitable proofs, in-

Ireland, the 25th of January, 1026-7; the very year of the stead of building on observation and experiment. He was

death of the learned Lord Bacon, whose plans of experi- also so zealous for this true method of learning by experi-

. mental philosophy our author afterwards so ably seconded, ment, and so careful about it, that though the Cartesian

While very young, he was instructed in his father's house philosophy then made a great interest in the world, yet be

to read and write, and to speak French and Latin. In could never be persuaded to read the works of Descartes,

1635, when only 8 years old, he was sent over to England for fear he should be amused and led away by plausible

to be educated at Eton school. Here he soon discovered accounts of things founded on conjecture, and merely hy-

an extraordinary force of understanding, with a disposition pothetical. But philosophy, and inquiries into nature

to cultivate and improve it to the utmost. though they engaged his attention deeply, did not occupy

After remaining at Eton between 3 and 4 years, his fa- him entirely;- as he still continued to pursue critical and

ther sent o'ur author and his brother Francis, in 1638, on theological studies. He had offers of preferment to enter

their travels upon the continent. They passed through into holy orders, by the government, after the restoration.

France to Geneva, where they settled for some time to pur- But he declined the offer, choosing rather to pursue his.

BOY

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BOY

.studies as a layman, in such a manner as might be most
effectual for the support of religion ; and began to com-
municate to the world the fruits of these studies. These
productions were very numerous and important, as well
as'various: the principal of which, as well as of some
other memorable occurrences of his life, were nearly in
the following order :

In ld60 came out, I. New experiments, physico-me-
chanical, touching the spring of the air and its effects. —
2. Seraphic love ; or some motives and incentives to the
love of God, pathetically discoursed of in a letter to a
friend. A work which it has been said was owing to his
courtship of a lady, the daughter of Cary earl of Mon-
mouth ; though our author was never married. — 3. Cer-
tain physiological essays and other tracts, in l66l. — 4.
Sceptical chemist, 1662; reprinted about the year 1 679,
with the addition of divers experiments and notes on the
producibleness of chemical principles.

In the year 1660, the Royal Society being incorporated
hy king Charles the 2d, Mr. Boyle was named one of the
council; and as he might justly be reckoned among the
founders of that learned body, so he continued one of the
most useful and industrious of its members during the
whole course of his life. His next publications were, 5.
Considerations touching the usefulness of experimental na-
tural philosophy, 1663. — 6. Experiments and considera-
tions upon colours ; to which was added a letter, contain-
ing Observations on a diamond that shines in the dark,
1663. This treatise is full of curious and useful remarks
on the hitherto unexplained doctrine of light and colours;
in which he shows great judgment, accuracy, and pene-
tration; and which may be said to have led the way for
Newton, who made such brilliant discoveries in that
branch of physics. — 7. Considerations on the style of the
holy Scriptures, 1663. This was an extract from a larger
work, entitled An essay on Scripture; which was afterwards
published by Sir Peter Pett, a friend of Mr, Boyle's.

In l664 he was elected into the company of the royal
miners -, and was all this year occupied in prosecuting va-
rious good designs, which was probably the reason that
he did not publish any works in this year. Soon after
came out, 8. Occasional reflections upon several subjects,
1665. This piece exposed our author to the censure of
the celebrated Dean Swift, who, to ridicule these dis-
courses, wrote A Pious Meditation upon a Broomstick, in
the Style of the honourable Mr. Boyle. — 9- New experi-
ments and observations upon cold, 1665. — 10. Hydrosta-
tical paradoxes made out by new experiments, for the
most part physical and easy, 1666. — 11. The origin of
forms and qualities, according to the corpuscular philoso-
phy, 1666*.— Both in this and the former year, our author
Communicated to his friend Mr. Oldenburg, then secre-
tary to the Royal Society, several curious and excellent
short pieces of his own, on a great variety of subjects, and
others transmitted to him by his learned friends, which
are printed in the Philosophical Transactions.

In the year 1668 Mr. Boyle resolved to settle in London
for life ; and for that purpose he removed to the house of
his sister, the Lady Ranelagh, in Pail-Mall. This removal
was to the great benefit of the learned in general, and par-
ticularly of the Royal Society, to whom he gave great and
continual assistance;, as abundantly appears by the several
pieces communicated to them from time to time, and
printed in their Transactions. To avoid improper waste
of time, he had set hours in the day appointed for receiv-

ing such persons as wanted to consult him, either for their
own assistance, or to communicate new discoveries to him i
and he besides kept up an extensive correspondence with
the most learned men in Europe ; so that it is wonderful
how he could bring out so many new works as he did.
His next publications were, 12. A continuation of new
experiments touching the weight and spring of the air ;
to which is added, A discourse of the atmosphere of con-
sistent bodies, l66"9- — 13. Tracts about the cosmical qua-
lities of things ; cosmical suspicions ; the temperature of
the subterraneous regions; the bottom of the sea; to
which is prefixed an introduction to the history of par-
ticular qualities, I669. — 14. Considerations on the useful-
ness of experimental and natural philosophy, the 2d part,
I67I. — 15. A collection of tracts on several useful and
important points of practical philosophy, 1671. — 16. An
essay on the origin and virtues of gems, 16'72. — 17- A cot-
lection of tracts on the relation between flame and air ;
and several other useful and curious subjects, 16"72 ;
besides furnishing, in this and the former year, a number
of short dissertations on various topics, addressed to the
Royal Society, and inserted in their Transactions. — 18.
Essays on the strange subtilty, great efficacy, and deter-
minate nature, of effluvia; with a variety of experiments
on other subjects, 1673. — 19. The excellency of theology
compared with philosophy, 1673- This discourse was
written in the year 1665, while our author, to avoid the
great plague which then raged in London, was forced to
go from place to place in the country, having little or no
opportunity of consulting his books. — 20. A collection of
tracts on the saltness of the sea, the moisture of the air,
the natural and. preternatural state of bodies; to which is
prefixed a dialogue concerning cold, 1674. — 21. A col-
lection of tracts containing suspicions about hidden qua-
lities of the air; with an appendix touching celestial mag-
nets ; animadversions on Mr. Hobbes's problem concern-
ing a vacuum ; a discourse of the cause of attraction and
suction, 1674. — 22. Some considerations about the rea-
sonableness of reason and religion ; by T. E. (the final
letters of his names). To which is annexed a discourse
about the possibility of the resurrection ; by Mr. Boyle,
1675. The same year several papers communicated to
the Royal Society, among which were two upon quick-
silver growing hot with gold. — 23. Experiments and notes
about the mechanical origin or production of particular
qualities, in several discourses on a great variety of sub-
jects, and among the rest on electricity, 1676. — He then
communicated to Mr. Hook a short memorial of some ob-
servations madeon an artificial substance that shines with-
out any preceding illustration ; published by Hook in his
Lectiones Cutlerianaj. — 24. Historical account of a de-
gradation of gold made by an anti-elixir. — 25. Aerial
noctiluca; or some new phenomena, and a process of a
factitious self-shining substance, 1680. This year the
Royal Society, as a proof of the just sense of his great
worth, and of the constant and particular services which
through a course of many years he had done them, made
choice of him for their president ; but he being extremely,
and, as he says, peculiarly tender in point of oaths, de-
clined that honour. — 26. Discourse of things above rea-
son; inquiring, whether a philosopher should admit any
such, 1681. — 27- New experiments and observations on
the icy noctiluca; to which is added_a chemical paradox,
grounded on new experiments, making it probable that
chemical principles are transmutable, so that out of 0110

BOY

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BRA

of them others may be produced, 1682. — 28. A continu-
ation of new experiments, physico-mechanical, touching
the spring and weight of the air, and their effects, 1682. —
29. A short letter to Dr. Beale, in relation to the making
of fresh water out of salt, 1683. — 30. Memoirs for thena-
tural history of human blood, especially the spirit of that
liquor, 16*84. — 31. Experiments and considerations about
the porosity of bodies, 1684. — 32. Short memoirs for the
natural experimental history of mineral waters, &c,l6'85.^ —
33. An essay on the great effects of even languid and un-
heeded motion, &c, 16"8>. — 34. Of the reconcileableness
of specific medicines to tile corpuscular philosophy, &c,
1685. — 35. Of the high veneration man's intellect owes to
God, peculiarly for his wisdom and power, 16S5. — 36'.
Free inquiry into the vulgarly received notion of nature,
l686.-^37. The martyrdom of Theodora and Didymia,
16"S7 ; a work he had drawn up in his youth. — '38. A dis-
quisition about the final causes of natural things, and
about vitiated light, lG'SS.

Mr. Boyle's health declining very fast, he abridged
greatly his time given to conversations and communica-
tions with other persons, that he might ha\e more time to
prepare for the press some others of his papers,, before his
death, which were as follow : — 39- Medicina Hydrostatica,
&c, I69O.— 40. The Christian virtuoso, &c, 16*90.— 4L
Experimentaet Observationes Phvsiea:, &c, 1691 ; which
is the last work that he published.

Mr. Boyle died on the last day of December of the
same year, 1691, in the 6*5th year of his age, and was bu-
ried in St. Martin's church in the Fields, Westminster; his
funeral sermon being preached by Dr. Gilbert Burnet, bi-
shop of Salisbury; in which he displayed the excellent
qualities of our author, with many circumstances of his
life, &c. But as the limits of this work will not allow us
to follow the bishop in the copious and eloquent account
he has given of this great man's abilities, we must content
ourselves with adding the following short eulogium by the
celebrated physician, philosopher, and chemist, Dr. Bocr-
haave; who, after having declared Lord Bacon to be the
father of experimental philosophy, asserts, that, " Mr.
Boyle, the ornament of his age and country, succeeded to
the genius and inquiries of the great chancellor Verulam.
Which," says he, " of Mr. Boyle's writings shall I recom-
mend? All of them. To him we owe the secrets of fire, air,
water, animals, vegetables, fossils : so that from his works
may be deduced the whole system of natural knowledge."
Mr. Boyle left also several papers behind him, which
have been published since his death. Beautiful editions
of all his works have been printed at London, in 5 volumes
folio, and 6 volumes 4to. Dr. Shaw also published in 3
volumes 4to, the same works " abridged, methodized, and
disposed under the general heads of Physics, Statics,
Pneumatics, Natural History, Chemistry, and Medicine;"
to which he has prefixed a short catalogue of the philo-
sophical writings, according to the order of time when
they were first published, as follows:

Physico-mechanical experiments on the spring

and weight of the air - - l66l

The Sceptical Chemist - - 16*6*1

Physiological Lssays - - 1662

History of Colours - - l66"3

Usefulness of Lxperimcntal Philosophy - 1663

History of Cold - - 16*6'5

Historical Paradoxes - - 1666

Origin of Forms and Qualities - 166*6"

Cosmical Qualities - - 1670

The admirable Rarefaction of the Air - 1670
The Origin and Virtues of Gems - 1672

The Relation betwixt Flame and Air - 1672

Effluviums - - - 1673

Saltness of the Sea - - 1674

Hidden Qualities of the Air - - 16*74

The Excellence, &c, of the Mechanical Hypo-
thesis ... 1674
Considerations on the Resurrection - 1675
Particular Qualities - - J 676
Aerial Noctiluca - - 16S0
Icy Noctiluca - - - 16'SO
Things above Reason - - 16*81
Natural History of Human Blood - 1684
Porosity of Bodies - - 1684
Natural History of Mineral Waters - 1684
Specific Medicines - - l685
The High Veneration due to God - l6"85
Languid Motion - - - l685
The Notion of Nature - - 1685
Final Causes -•-.'>- 16S8
Medicina Hydrostatica - - , 16*90
The Christian Virtuoso - - 169*0
Experimenta et Observationes Physical I69I.
Natural History of the Air - - 16*92
Medicinal Experiments - -"• 1718
BRACE, in Architecture, denotes a piece of timber
framed in with bevel-joints; serving to keep the building
from swerving either way.

BRACHMINS, or Braciimans, a branch of the an-
cient Gymnosophists, or philosophers of India, remarkable
for the severity of their lives and manners.

BRACKET, in Building, denotes a kind of wooden stay,,
in form of a knee or shoulder, serving for the support of
shelves, &c, or a cram ping-iron, which serves as a stay in
timber-work.

BRADLLY (Dr. James), a celebrated English astro-
nomer, the third son of William Bradley, was born at Sher-
borne in Gloucestershire in the year l6'92. He was fitted
for the university at Northleach, in the same county, ax
the boarding-school of Mr. lies and Mr. Bricc. From:
thence he was sent to Oxford, and admitted a commoner
of Baliol college, March 15, 1710; where he took the
degree of bachelor the 14th of October 1714, and of mas-
ter of arts the 21st of January, 17l6. His friends intend-
ing him for the church, his studies were regulated with
that view; and as soon as he was of a proper age to receive
holy orders, the bishop of Hereford,. who had entertained
great esteem for him, gave him the living of Bridstow, and:
soon after he was inducted to that of Landewy Welfrey in;
Pembrokeshire.

He was nephew to Mr. Pound, a gentleman well known
in the learned world, by many excellent astronomical and.
other observations, and who would have enriched it much,
more, if the journals of his voyages had not been burnt at
Pudor Condor, when the place was set on fire, and the
English who were settled there cruelly massacred, Mr..
Pound himself very narrowly escaping with his life. With
this gentleman, at Wanstcad, Mr. Bradley passed all the
time that he could spare from the duties of his function,,
being then sufficiently acquainted witb the mathematics to
improve by Mr. Pound's conversation. It may easily be
imagined that the example and conversation of this gentle-
man did not render Bradley more fond of his profession, to ■

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which he had before no great attachment: he continued
however as yet to fulfil the duties of it, though at this time
he bad made such observations as laid the foundation of
those discoveries which afterwards distinguished him as one
of the greatest astronomers of his age. These observations
gained him the notice and friendship of the lord-chancel-
lor Macclesfield, Mr. Newton, afterwards Sir Isaac, Mr.
Halley,and many other members of the Royal Society into
which he was soon after elected a member.

Soon after, the chair of Savilian professor of astronomy
at Oxford became vacant, by lhe death of the celebrated
Dr. John Keil; and Mr. Bradley was elected to succeed
him on the 31st of October, 1721, at 29 years of age: his
colleague being Mr. Halley, who was professor of geometry
on the same foundation. On this appointment Mr. Brad-
ley resigned his church livings, and applied himself wholly
to the study of his favourite science. In the course of his
observations, which were innumerable, he discovered and
settled the laws of the alterations of the fixed stars, from
the progressive motion of light, combined with the earth's
annual motion about the sun, and the nutation of the earth's
axis, arising from the unequal attraction of the sun and
moon on the different parts of the earth. The former of
these effects is called the Aberration of the fixed stars, the
theory of which he published in 1727; and" the latter the
Nutation of the earth's axis, the theory of which appeared
in 1747, deduced from 20 years' assiduous observations,
by which he communicated to the world two of the finest
discoveries in modern astronomy, making a memorable
epoch in the history of that science. See Aberration
and Nutation.

In 1730 Bradley succeeded Mr. Whiteside, as lecturer
in astronomy and experimental philosophy in the Museum
at Oxford; which was a. considerable emolument to him,
and which he enjoyed till within a year or two of his death,
when the ill state of his health induced him to resign it.

Our author always preserved the esteem and friendship
of Dr. Halley; who, being worn out by age and infirmities,
thought he could not do better for the service of astrono-
my, than procure for Mr. Bradley the place of regius pro-
fessor of astronomy at Greenwich, which he himself had
many years possessed with the greatest reputation. With
this view he wrote many letters, desiring Mr. Bradley's
permission to apply for a grant of the reversion of it to
him, and even offered to resign it in his favour, if it should
be thought necessary; but Dr. Halley died before he could
accomplish this kind object. Bradley however obtained
the place, by the interest of lord Macclesfield, who was
afterwards president of the Royal Society; and upon this
appointment the university of Oxford conferred on him a
diploma.

The appointment of astronomer royal at Greenwich,
which was dated the 3d of February, 1741-2, placed our
author in his proper sphere; and he pursued his observa-
tions with unwearied diligence. However numerous the
collection of astronomical instruments at that observatory,
it was impossible that such an observer as Dr. Bradley
should not desire to increase them, as well to answer those
particular views, as in general to make observations with
greater exactness. In the year 1748, therefore, he took
the opportunity of the visit of the Royal Society to the
observatory, annually made to examine the instruments and
receive the professor's observations for the year, to repre-
sent so strongly the necessity of repairing the old instru-
ments and providing new ones; that the society thought

proper to make application to the king, who was pleased
to order 1000 pounds for that purpose. This sum was
laid out under the direction of Bradley, who, with the as-
sistance of the l?ro celebrated Mr. Graham and Mr. Bird,
furnished the observatory with as complete a collection
of astronomical instruments, as the most skilful and dili-
gent observer could desire. Our author being thus fur-
nished with such assistance, pursued his observations with
great assiduity during the rest of his life; an immense num-
ber of which was found after his death, in 13 folio vo-
lumes, and were presented to tht, university of Oxford in
1776, on condition of their printing and publishing them.

During Dr. Bradley's residence at the Royal Observa-
tory, the living of the church at Greenwich became va-
cant, and was ottered to him: upon his refusing to accept
it, from a conscientious scruple, " that the duty of a pas-
tor was incompatible with his other studies and necessary
engagements," the king was pleased to grant him a pension
of 250/ over and above the astronomer's original salary
(100/) from the Board of Ordnance, " in consideration
(as the sign manual, dated the 15th Feb. 1752, expresses it)
of his great skill and knowledge in the several branches of
astronomy and other parts of the mathematics, which have
proved so useful to the trade and navigation of this king-
dom ;" a pension which has been regularly continued to
the astronomers royal ever since.

About 1748, our author became entitled to bishop Crew's
benefaction of 30/ a year, to the lecture reader in experi-
mental philosophy at Oxford. He was elected a member
of the academy of Sciences at Berlin in 1747 ; of that at
Paris, in 1748; of that at Petersburg, in 1754; and of
that at Bologna, in 1757. He was married in the year
1744, but never had more than one child, a daughter.

By his unremitting application to study and obser-
vations, he became afflicted, for near two years before his
death, with a violent oppression on his spirits, which inter-
rupted his useful labours. This distress arose chiefly from
an apprehension that he should outlive his rational facul-
ties; but this so much dreaded evil never came upon him.
In June 1762 he was seized with a suppression of urine,
occasioned by an inflammation in the reins, which termi-
nated his existence the 13th of July following. His death
happened at Chalford in Gloucestershire, in the 70th year
of his age; and he was interred at iMinchinhampton in the
same county.

As to the character of Dr. Bradley, he was remarkable
for a placid and gentle modesty, very uncommon in per-
sons of an active temper and robust constitution. Though
he was a good speaker, and possessed the rare but happy
art of expressing his ideas with the utmost precision and
clearness, yet no man was a greater lover of silence, for he
never spoke but when he thought it absolutely neces-
sary. Nor was he more inclined to write than to speak,
as he has published very little: he bad a natural diffidence,
which made him always afraid that his works might injure
his character; so that he suppressed many which might
have been worthy of publication.

His papers, which have been inserted in the Philos.
Trans, arc,

1. Observations on the Comet of 1723, vol. 33, p. 41. —
2. The Longitude of Lisbon and of the Fort of New York
from VVansted and London determined by the Eclipse of
the First Satellite of Jupiter, vol. 34, p. 85. — 3. An Ac-
count of a new discovered Motion of the Fixed Stars, vol.
35, p. 637. — 4. On the Going of Clocks with Isochronical

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Pendulums, vol. 38, p. 302. — 5. Observations on the Co-
met of 1736-7, vol. 40, p. 111. — 6. On the Apparent Mo-
tion of the Fixed Stars, vol. 45, p. 1. — 7- On the Occu.l-
tation of Venus by the Moon, the 15th of April, 1751,
vol. 46, p. 201.— 8. On the Comet of 1757, vol. 50, p.408.
— 9- Directions for Using the Common Micrometer, vol.
62, p. 46.

BRADWARDIN (Thomas), a skilful mathematician
and divine, was born at Hartfield in Sussex, about the close
of the 13th century. He was educated at Merton-college,
Oxford, where he took the degree of doctor of divinity.
It has been said he was professor of divinity at Oxford ;
that he was chancellor of the diocese of London, and con-
fessor to Edward the 3d, whom he constantly attended
during his war with France. After his return from the
field, he was made prebendary of Lincoln, and afterward
archbishop of Canterbury. He died at Lambeth in the
year 1349, forty days after his consecration. His works
a.10, 1. De Causa Dei, printed in London, l6l8, published
by J. H. Savil. — 2. De Geometria Speculativa, &c. Pa-
ris, 1495, 1512, 1530.— 3. De ArithmeticaPractica. Paris,
1502, 1512* — 4. De Proportionibus. Paris, 1495. Venice,
1505, folio. — 5. De QuadrataturaCirculi. Paris, 1495, fol.
BRAHE' (Tycho), a celebrated astronomer, descended
from a noble family, originally of Sweden, but settled
in Denmark, was born the 14th of December 1546, at
Knudstorp in the count)' of Schonon, near Helsimbourg.
He was taught Latin when 7 years old, and studied 5 years
under private tutors. His father dying while he was very
young, his uncle, George Brahe, having no children, adopted
him, and sent him, in 1559, to study philosophy and rhe-
- toric at Copenhagen. The great eclipse of the sun, on the
21st of August 1650, happening at the precise time the
astronomers had foretold, he began to consider astronomy
as something divine; and purchasing the tables of Stadius,
he gained some notion of the theory of the planets. In
1562 he was sent by his uncle to Leipsic to study the law,
where his acquirements gave manifest indications of ex-
traordinary abilities. His natural inclination however
was to the study of the heavens, to which he applied him-
self so assiduously, that, notwithstanding the care of his
tutor to keep him close to the study of the law, he made
use of every means in his power for improving his know-
ledge of astronomy; he purchased with his pocket-money
whatever books he could meet with on this subject, and
read them with great attention, procuring assistance in
difficult cases from Bartholomew Scultens, his private
tutor ; and having procured a small celestial globe, he took
opportunities, when his tutor was in bed, and when the
weather was clear, to examine the constellations in the
heavens, to learn their names from the globe, and their
motions from observation.

After a course of 3 years study at Leipsic, his uncle
dying, he returned home in 1565. In this year a differ-
ence arising between Brahe and a Danish nobleman, they
fought, and our author lost part of his nose cutoff by a
blow in the contest. This defect he so artfully supplied
with one rriade of precious metal and wax, that it was
not perceivable. About this time he began to apply him-
self to chemistry, proposing nothing less than to obtain the
philosopher's stone. But becoming greatly disgusted to
see the liberal arts despised, and finding his own relations
and friends uneasy that he applied himself to astronomy,
as thinking it a study unsuitable to a person of his quality,
Vol. I.

he went to Wirtemberg in 1566, from which the breaking
out of the plague soon occasioned his removal to Rostock"
and in 1569 to Augsburg, where he was visited by Peter
Ramus, then professor of astronomy at Paris, and who
greatly admired his uncommon skill in this science.

In 1571 he returned to Denmark; and was favoured
by his maternal uncle, Steno Billes, a lover of learning,
with a convenient place at his castle of Herritzvad near
Knudstorp, for making his observations, and buildin" a
laboratory. And here it was he discovered, in 1573, a
new star in the constellation Cassiopeia. But soon after,
his marrying a country girl, beneath his rank, occasioned'
so violent a quarrel between him and his relations, that
the king was obliged to interpose to reconcile them.

In 1574, by the king's command, he read lectures at
Copenhagen on the theory of the planets. The year fol-
lowing he began his travels through Germany, and pro-
ceeded as far as Venice. He then resolved to remove his
family, and settle at Basil ; but Frederick the 2d, king of
Denmark, being informed of his design, and unwilling to
lose a man who was capable of doing so much honour to
his country, he promised to enable him to pursue his stu-
dies, and bestowed on him for life the island of Huen in
the Sound, and engaged that an observatory and labora-
tory should be built for him, with a supply of money for
carrying on his designs: accordingly the first stone of the
observatory was laid the 8th of August 1576, under the
name of Uranibourg. The king also gave him a pension of
2000 crowns out of his treasury, a fee in Norway, and a.
canonry of Roshild, which amounted to 1000 more. This
situation he enjoyed about 20 years, pursuing his obser-
vations and studies with great industry: here he kept al-
ways in his hoiise ten or twelve young men, who assisted
him in his observations, and whom he instructed in astro-
nomy and mathematics. And it was also at this place
that he received a visit from James vi, king of Scotland,
afterwards James 1 of England, having come to Denmark
to espouse Anne, daughter of Frederick 11. This king
made our author some noble presents, and wrote a copy
of Latin verses in his praise.

Brahe's tranquillity however in this happy situation
was at length fatally interrupted. Soon after the death
of King Frederick, by the aspersions of envious and male-
volent ministers, he was deprived of his pension, fee, and
canonry, in 1596. Being thus rendered incapable of sup-
porting the expenses of his establishment, he quitted his
favourite Uranibourg, and withdrew to Copenhagen, with
some of his instruments, and continued his astronomical
observations and chemical experiments in that city, till
the same malevolence procured from the new king, Charles
iv, an order for him to discontinue them. This induced
him to fall upon means of being introduced to the em-
peror Rodolphus, who was fond of mechanism and chemi-
cal experiments: and to smooth the way to an interview,
Tycho now published his book, Astronomia instaurata
Mechanica, adorned with figures, and dedicated it to the
emperor. That prince received him at Prague with great
civility and respect; gave him a magnificent house, till
he could procure one for him more fit for astronomical
observations; he also assigned him a pension of 3000
crowns; and promised him a fee for himself and his de-
scendants. Here then he settled in the latter part of
1598, with his sons and scholars, and among them tho
celebrated Kepler, who had joined him. But he did not

2 K

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long enjoy this happy situation; for, about 3 years alter,
he died, on the 24th of October l601, of a retention of
urine, in the 55th year of his age, and was interred in a
magnificent manner in the principal church at Prague,
where a noble monument was erected to his memory;
leaving, besides his wife, two sons and four daughters. On
the approach of death, he enjoined his sons to take care
that none of his works should be lost; exhorted the students
to attend closely to their exercises ; and recommended to
Kepler the finishing of the Rodolphine tables, which he
had constructed for regulating the motion of the planets.

Brahe's skill in astronomy is universally known; and
he is famed for being the inventor of a new system of
the planets, which he endeavoured, though without suc-
cess, to establish on the ruins of that of Copernicus. He
was very credulous with regard to judicial astrology and
presages. If he met an old woman when he went out of
doors, or a hare upon the road on a journey, he would
turn back immediately, being persuaded that it was a bad
omen : also, when he lived at Uranibourg, he kept in his
house a madman, whom he generally placed at his feet at
table; for as he imagined that every thing spoken by mad
persons presaged something, he carefully observed all that
this man said ; and because it sometimes proved true, he
fancied it might always be depended on. He was of a
very irritable disposition : a mere trifle would put him in
a passion ; and against persons of the first rank, whom he
thought his enemies, lie openly discovered his resentment.
He was very apt to rally others, but highly provoked
when the same liberty was taken with himself. — The prin-
cipal part of his writings, according to Gassendus, are,

1. An account of the New Star, which appeared Nov.
11th, 1572, in Cassiopeia; Copenh. 1573, in 4to. — 2. Ah
Oration concerning the Mathematical Sciences, pro-
nounced in the university of Copenhagen, in the year
1574 : published by Conrad Aslac, of Bergen in Norway.
— 3. A treatise on the Comet of the year 1577, immedi-
ately after it disappeared. Nine years afterwards he re-
vised it, and added a 10th chapter; printed at Urani-
bourg, 15S9- — 4. Another treatise on the New Phenome-
na of the heavens. In the first part of which he treats of
the Restitution, as he calls it, of the sun, and of the fixed
stars. And in the 2d part, of a New Star, which had then
made its appearance. — 5. A collection of Astronomical
Epistles: printed in 4to, at Uranibourg in 1596; at Nu-
remberg in 1602 ; and at Frankfort in 1610. It was de-
dicated to Maurice Landgrave of Hesse ; because there is
in it a considerable number of letters of the Landgrave
William his father, and of Christopher Rothmann, the ma-
thematician of that prince, to Tycho, and of Tycho to
them. — 6. The Mechanical Principles of Astronomy re-
stored : Wandesburg, 1598, in folio. — 7. An Answer to
the Letter of a certain Scotchman, concerning the comet,
in the year 1577- — S. On the composition of an Elixir
for the plague; addressed to the emperor Rodolphus. —
9. An Elegy on his Exile: Rostock, 1614, 4to.— 10. The
Rodolphihe Tables ; which he had not finished when he
died; but were revised, and published by Kepler, as Ty-
cho had desired. — 11. An Accurate Enumeration of the
Fixed Stars: addressed to the emperor Rodolphus.— 12.
A complete Catalogue of 1000 of the fixed Stars; which
Kepler has inserted in the Rodolpihe Tables. — 13. Histo-
na Ccelestis ; or a History of the Heavens; in two parts.
The ist contains the Observations he had made at Urani-

bourg, in 16 books; the latter contains the Observations
made at Wandesburg, Wittemberg, Prague, &c. ; in 4
books. — 14. Is an Epistle to Caster Pucer; printed at Co-
penhagen l6()8.

BRANCKER, or BRANKER (Thomas), an eminent
mathematician of the 17th century, son of Thomas
Brancker some time bachelor of arts in Exeter-college,
Oxford, was born in Devonshire in 1 636, and was ad-
mitted butler of the said college, Nov. 8, 1652, in the
17th year of his age. In 1 655, June the 15th, he took
the degree of bachelor of arts, and was elected probatio-
nary-fellow the 30tli of the same month. In 1658, April
the 22d, he took the degree of master of arts, and became
a preacher ; but after the Restoration, refusing to conform
to the ceremonies of the church of England, he quitted
his fellowship in 1-662, and retired to Chester: but not
long after, he became reconciled to the service of the
church, took orders from a bishop, and was made a mini-
ster of Whitegate. He had however, for some time, en-
joyed great opportunity and leisure for pursuing the bent
of his genius in the mathematical sciences; and his skill
both in the mathematics and chemistry procured him the
favour of Lord Brereton, who gave him the rectory of Til-
Stan. He was afterward chosen master of the well-en-
dowed school at Macclesfield, in that county, where he
spent the few remaining years of his life, which was ter-
minated by a short, illness in 1676, at 40 years of age;
and was interred in the church at Macclesfield.

Brancker wrote a piece on the Doctrine of the Sphere,
in Latin, which was published at Oxford in 1662; and in
1668 he published at London, in 4to, a translation of
Rhonius's Algebra, with the title of An Introduction to
Algebra; which treatise having been communicated to Dr.
John Pell, he received from him some assistance towards
improving it; which he generously acknowledges in a let-
ter to Mr. John Collins ; with whom, and some other gen-
tlemen, proficients in this science, he continued a cone-
spondence during his life.

BREACH, in Fortification, a gap or opening made in
any part of the works of the town, by the cannon or mines
of the besiegers, with intent to storm or attack the place.

BREAKING Ground, in Military Affairs, is beginning
the works for carrying on the siege of a place ; more es-
pecially the commencement of digging trenches, or ap-
proaches.

BREAST-WORK, in the Military Art, is an elevation
of earth thrown up around a fortified place or post, to
conceal or protect the garrison, and which is at the same
time so strong that the enemy's shot cannot pierce through
it. See Pa it a pet.

BREECH of a Gun, the hinder part, from thecascabel
to the lower part of the bore.

BREREWOOD (Edward), a learned mathematician
and antiquary, was the son of Robert Brerewood, a repu-
table tradesman, who was three times mayor of Chester,
Our author was born in that city in 156*5, where he was
educated in grammar erudition at the free-school ; and
was afterward admitted, in 15SI, of Brazen-nose college,
Oxford, where he soon acquired the character of a hard
student; as he has shown in liis commentaries on Aristo-
tle's Ethics, which he produced about the age of 21.

In the year 1596 he was chosen the fust professor of
astronomy in Gn slum-college, being one of the two who,
at the desire of the electors, were recommended to them

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by the university of Oxford. He loved retirement, and
entirely devoted himself to the pursuit of knowledge. And
though he never published any thing himself, yet he was
very communicative, and ready to impart what he knew,
to others, either in conversation or in writing. His retired
situation at Gresham-college being agreeable, it did not
appear that he had any other views, but continued there
the remainder of his life, which was terminated by a fever
the 4th of November 1613, at 48 years of age, in the
midst of his pursuits, and before he had taken proper
care to collect and digest his learned labours ; which how-
ever were not lost; being reduced to order, and published
after his death. These were little or nothing mathemati-
cal, being of a miscellaneous nature, on the several sub-
jects of Weights, Money, Languages, Religion, Logic, the
Sabbath, Meteors, the Eye, Ethics, &c.

BRIDGE, a w^ork of carpentry or masonry, built over
.a river, canal, or the like, for the convenience of passing
from one side to the other ; and may be considered as a
road over water, supported by one or more arches, and
these again supported by proper piers or butments. Be-
sides these essential parts, may be added the paving at
top, the banquet, or raised footway, on each side, leaving
a sufficient breadth in the middle for horses and carriages,
also the parapet wall either with or without a balustrade,
or other ornamental and useful parts. The breadth of a
bridge for a great city should be such, as to allow an easy
passage for three carriages and two horsemen abreast in
the middle way, and for 3 foot passengers in the same
manner on each banquet : but for other smaller bridges,
a less breadth may suffice.

Bridges'are commonly very difficult to execute, on ac-
count of the inconvenience of laying foundations and wall-
ing under water. The earliest rules and instructions for
building of bridges are given by Alberti, in his Archit. 1,
8. Other rules were afterwards laid down by Palladio,
Serlio, and Scamozzi, which are collected by Blondel, in
his Cours d'Archit. pa. 6'29, &c. The best of these rules
were also given by Goldman, Baukhurst, and in Hawkes-
moor's History of London Bridge. M. Gautier has a con-
siderable volume expressly on bridges, ancient and mo-
dern. See also Riou's Short Principles for the Architec-
ture of Stone Bridges ; as also Emerson, Muller, Labelye,
and my own Principles of Bridges, in the 1st vol. of my
Tracts, besides neat accounts in the various new Ency-
clopaedias, as well as several French authors.

The conditions required in a bridge are, That it be well
designed, commodious, durable, and suitably decorated.
It should be of such a height as to be quite convenient for
the passage over it, and yet easily admitting through its
arches the vessels that navigate under it, and also the wa-
ter, even at high tides and floods; the neglect of this pre-
cept has caused the destruction of many bridges. Bridges
are commonly continued in a straight direction perpen-
dicular to the stream; though some think they should be
made rather convex towards the stream, the better to resist
floods, &c. And bridges of this sort have been executed
in some places, as Pont St. Esprit near Lyons. Again, a
bridge should,not be made in too narrow a part of a navi-
gable river, or one that is subject to tides or floods: because
the breadth being still more contracted by the piers, this
will increase the depth, velocity, and fall of the water under
the arches, and endanger the whole bridge and navigation.
There ought to be an uneven number of arches, or an even
number of piers; both that the middle of the stream or

chief current may flow freely without the interruption of
a pier ; and that the two halves of the bridge, by gradually
rising from the ends to the middle, may there meet in tht;
highest and largest arch ; and also, that by being open in
the middle, the eye in viewing it may have a better com-
mand and look directly through there. When the middle
and ends are of different heights, their difference however
ought not to be great in proportion to the length, that the
ascent and descent may be easy; and in that case also it
is more beautiful to make the top in one continued curve
than two straight lines forming an angle in the middle.
Bridges should rather be of few and large arches, than of
many smaller ones, if the height and situation will possi-
bly admit of it ; for this will leave passage for the water
and navigation, and be a great saving in materials and la-
bour, as there will be fewer piers and cejitres, and the
arches &c will require less materials ; a remarkable in-
stance of which appears in the difference between the
bridges of Westminster and Blackfriars, the expense of the
former being more than double the latter.

For the proper execution of a bridge, and making an
estimate of the expense &c, it is necessary to have three
plans, three sections, and an elevation. The three plans
are so many horizontal sections, viz, 1st a plan of the
foundation under the piers, with the particular circum-
stances attending it, whether of gratings, planks, pile&,
&c ; the 2d is the plan of the piers and arches ; and the
3d is the plan of the superstructure, with the paved road
and banquet. The three sections are vertical ones; the
1st of them a longitudinal section from end to end of the
bridge, and through the middle of the breadth ; the 2d a
transverse one, or across it, and through the summit of an
arch ; and the 3d also across, but taken upon a pier. The
elevation is an orthographic projection of one side or face
of the bridge, or its appearance as viewed at a distance,
showing the exterior aspect of the materials, with the
manner in which they are disposed, &c.

For the figure of the arches, some prefer the semicircle,
though perhaps without any just reason; others the el-
liptical form, as having many advantages over the semi-
circular; and some talk of the catenarian arch, though
its pretended advantages are only chimerical ; but the arch
of equilibration is the only perfect one, so as to be equally
strong in every part : see my Principles of Bridges. The
piers are of divers thickness, according to the figure, spun,
and height of the arches; as may be seen in the work
above mentioned.

With the Romans, the repairing and building of bridges
were committed to the priests, thence named Pontifices ;
next to the censors, or curators of the roads; but at lust
the emperors took the care of the bridges to themselves.
Thus the Pons Janiculensis was built of marble by Anto-
ninus Pius ; the PonsCestius was restored by Gordian ;
and Arian built a new one, which was called after his own
name. In the middle age, bridge-building was reckoned
among the acts of religion ; and, towards the end of the
12th century, St. Benezet founded a regular order of hos-
pitallers, under the name Pontifices, or bridge-builders,
whose office was to assist travellers, by making bridges,
settling ferries, and receiving strangers into hospitals, or
houses, built on the banks of rivers. We read of an hos-
pital of this kind at Avignon, where the hospitallers re-
sided under the direction of their first superior St. Benezet :
and the Jesuit Raynaldus has a treatise on St. John the
bridge-builder.

2 K 2

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Among the bridges of antiquity, that built by Trajau
over the Danube is the most magnificent. It was de-
molished by his next successor Adrian, the ruins of which
are still to be seen in the middle of the Danube, near the
city Warhel in Hungary. It had 20 piers, of square stone,
each of which was 150 feet high above the foundation, 60
feet in breadth, and 170 feet distant from one another,
which is the span or width of the arches ; so that the
whole length of the bridge was more than 1530 yards, or
one mile nearly.

In France, the Pont de Garde is a very bold structure;
the piers being only 13 feet thick, yet serving to support
an immense weight of a triplicate arcade, and joining two
mountains. It consists of three bridges, one over another;
the uppermost of which is an aqueduct, which carries
water to Nismes.

The aqueduct bridge of Alcantara, near the city of Lis-
bon, which was finished in 1732, is also a very magnifi-
cent piece of workmanship ; it consists of 35 arches, the
largest of which is 227 feet high, and 10SJ feet wide.

The bridge of Avignon, which was finished in the year
1188, consists of IS arches, and measures 1340 paces, or
about 1000 yards in length.

Over the several canals at Venice are laid nearly 500
bridges of different sizes ; the greater number of them are
of stone. The chief of these, called the Rialto, is cele-
brated as a master-piece of art : it consists of one flat and
bold arch, nearly 100 feet span, and only 23 feet high
above the water; it was built in 1591 from a design of
Michael Angelo. The breadth of the bridge, which is 43
feet,. is divided by two rows of shops into three narrow
streets, that in the middle being the widest ; and there is
in the centre an open arch-way, by which the three streets
communicate with one another. At each end of the Ri-
alto is an ascent of 56 steps; the view from its summit is
very' lively and magnificent. The whole interior of the
shops and the bridge is of marble. The foundation ex-
tends 90 feet, and rests upon 12000 elm piles. This
structure cost the republic 250,000 ducats. Poulet also
mentions a bridge of a single arch, in the city of Munster
in Bothnia, much nobler than that of the Rialto at Venice.
Yet these are nothing to a bridge in China, built, from one
mountain to another, consisting only of a single arch, 400
cubits long, and 500 cubits high, whence it is called the
flyingbridge; a figure of which is given in thePhilos. Trans.
Kircher also speaks of a bridge in the same country 360
• perches long without any arch, but supported by 300
pillars.

There are many bridges of considerable note in our own
country. The triangular bridge at Crowland in Lincoln-
shire, is esteemed the most ancient Gothic structure re-
maining entire in the kingdom; and was erected about the
year 860.

London bridge is on the old Gothic structure, with 20
small locks or arches, each of about 20 feet wide ; but
there are now only IS open, two having lately been thrown
into one in the centre, and another on one side is con-
cealed or covered up. It is 900 feet long, 6*0 high, and
74 wide ; the piers are from 25 to 34 feet broad, with star-
lings projecting at the ends : so that the greatest water-
way, when the tide is above the starlings, was 450 feet,
scarce half the breadth of the river; and below the star-
lings, the water-way was reduced to ly4feet, before the
late opening of the centre. London bridge was first built
with timber between the years 993 and 1016 ; and it was

repaired, or rather rebuilt wi'th timber in 11 63. The stone-
bridge was begun in 1176, and finished in 1209- It im-
probable there were no houses on this bridge for upwards
of 200 years ; since we re<id of a tilt and tournament held
on it in 1395. Houses it appears were erected on it after-
wards ; but being found of great inconvenience and nui-
sance, they were removed in 1758, and the avenues to it
also enlarged, and the whole made more commodious ;
the two middle arches were then thrown into one, by re-
moving the pier from between them ; the whole amounting
to above S0,000/.

There were other bridges in England built in the old
manner of London bridge ; as the bridge at Rochester,
which is 550 feet long, and has 11 arches; also the late
bridge at Newcastle-upon-Tyne, which was broken down
by a great flood in the year 1771, for want of a sufficient
quantity of water-way through the arches.

The longest bridge in England is that over the Trent at
Burton, built in the 12th century, .of squared free-stone,
and is strong and lofty ; it contains 34 arches, and the
whole length is 1545 feet. But this falls far short of the
wooden bridge over the Drave, which according to Dr.
Brown is at least 5 miles long.

But one of the most singular bridges in Europe is that
built over the Taaf in Glamorganshire, by William Ed-
wards, a poor country mason, in the year 1756. This
remarkable bridge consists of but one stupendous arch,,
which, though only S feet broad, and 35 feet high, is no
less than 140 span, being part of a circle of 175 feet dir
ameter.

There is also a remarkable bridge of one airh, built at
Colebrook Dale in 1779, o( cast-iron : and another slill
larger of the same metal, raised over the river "Wear, at
Sunderland, the arch being of 240 feet span. And since
that time several other iron bridges have been erected in
various places, accounts and drawings of which may be
seen in my Tracts before noticed.

Of modern bridges, perhaps the two finest in Europe.
are those of Westminster and Blackfriars over the river
Thames at London. The former is 1220 feet long, and
44 feet wide, having a commodious broad foot-path on
each side for passengers. It consists of 13 large and
two small arches, all semicircular, with 14 intermediate
piers. The arches all spring from about 2 feet above low-
water mark; the middle arch is 76 feet wideband the
others on each side decrease regularly by 4 feet at a time.
The two middle piers are each 17 feet thick at the spring-
ing of the arches ; and the others decrease equally on each
side by one foot at a time ; every pier terminating with a
saliant right angle against either stream. This bridge is
built of the best materials, and in a neat and elegant taste,
but the arches are too small for the quantity ol masonry
contained in it. This bridge was begun in 1738, and open-
ed in 1750; the whole sum of money granted and paid
for the erection of it, with the purchase of houses to take
down, and widening the avenues, lVc, amounted to38.<),500/.
Blackfriars bridge, nearly opposite the centre of the city
of London, was begun in 1700, and completed in 10 years
and three quarters; it js an exceeding light and elegant
structure, but the materials unfortunately do not seem to
be of the best sort, as many of the arch stones are decay-
ing. It consists of 9 huge, elegant, elliptical arches; the
centre arch being 100 led wide, and those on each side
decreasing in a regular gradation to the smallest, at each
extremity, which is 70 feet wide. The breadth of the

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bridge is 42 feet, and the length from wharf to wharf 995.
The upper surface is a portion of a very large circle, which
forms an elegant figure, and is of convenient passage over
it. The whole expense was 150,840/.

Wooden Bridges. The simplest case of these edifices,
is that in which the road-way is laid over beams placed
horizontally, and supported at each end by'piers or posts.
This method however is deficient in strength and widjth of
opening: it is therefore necessary, in ail works of any mag-
nitude, to apply the principles of trussing, as used in roofs
and arches. Wooden bridges of this kind are stiff frames
of carpentry, in which, by a proper disposition, beams are
put so as to. stand instead of solid bodies, as large as the
spaces which the beams enclose ;. and thus two or three
more of these are set in abutment with each other, like
mighty arch stones.

Palladio has given several very elegant designs of wooden
bridges ; one of which is that over the river Cismoine.
The river where this bridge is erected, is 100 feet wide;
this width is divided into six equal parts; and at the end
of each part, excepting at the banks, which are strength-
ened with pilasters of stone, the beams are- placed that
form the breadth of the bridge, upon which, a little space
being left at their ends, were placed other beams length-
wise, which form the sides. Over these, directly upon
the first, the colonelli (king-posts) were disposed on each
side; these king-posts are connected to the beams which
form the breadth of the bridge, by means of irons passing
through the projecting ends of the beams, and bolted and
pinned through both.

Mr. Coxe, in the first volume of his Travels, has slightly
described a very singular bridge at Wittingen in Switzer-
land ; the construction of which is quite simple. The span
is 230 feet, and it rises only 5. The arches approach to
a catenarian form, built up of 7 courses of solid logs of
oak, in lengths of 12 or 14 feet, and \6 inches or more in
thickness. These are all picked of a natural shape, suited
to the intended curve; so that the wood is no where cut
across the grain to trim it into shape. These logs are laid
above each other, so that their abutting joints are alternate
like those of a brick wall ; or, in the language of the work-
men, they break joint. It is indeed a wooden structure
simply built up by laying the pieces upon each other, ta-
king care to make the. abutting joints as close as possible.
They are not fastened together by pins or bolts, but held
united by iron straps, which surround them at the distance
of 5 feet from each other, where they are fastened by bolts
and keys. The two arches being erected and well butted
against the rock on each side, were freed from their sup-
ports, and allowed to settle. They are so placed that the
intended road intersects them about the middle of their
height. The road-way is supported by cross-joints which
rest on a long horizontal summer-beam ; and this is con-
nected with the arches on each side by uprights bolted into
thcin. The whole is covered with a roof which projects
over the arches on each side to defend them from the wea-
ther. Three of the spaces between these uprights have
struts or braces, which give the upper work a sort of truss-
ing in that part. This bridge is of a strength much more
than ade«matc to support any load that can be laid upon
it; though it is manifest, by the attempts to truss the ends,
that it was the contrivance of a person ignorant of prin-
ciple. It was the work of one Uliich Grubenhamm, of
Tuffcn, in the canton of Appcnzel, a carpenter without

education, but celebrated for several works of the same-
kind.

At Schaffhausen, in Switzerland, where the Rhine flows
with great rapidity, several stone bridges had been destroy-
ed, when in 1754 Grubenhamm offeree! to throw a wooden
bridge of a single arch across the river, which is. nearly
390 feet wide. The magistrates, however, required that it
should consist of two arches, and that he should for that
purpose employ the middle pier of the last stone bridge,
which would divide the new one into two unequal arches,
of 172 and 193 feet span. The carpenter did so, but con-
trived to leave it a matter of doubt whether the bridge is
at all supported by the middle pier. It was erected on a
plan nearly similar to the Wittingen bridge, at the expense
df about 8000/ sterling. Travellers inform us, that it shook
if a man passed over it; yet waggons heavily laden also
went over it without danger. We are sorry to add that this
curious bridge was burnt by the French when they evacu-
ated Schaffhausen in April 1799-

Iron Bridges, are those in which iron is the chief ma-
terial used. Iron bridges are the exclusive invention of.
British artists. The first that has been erected on a large
scale, is that over the river Severn at Colebrook Dale, in
Shropshire. This bridge is composed of 5 ribs, and each
rib of 3 concentric arcs, connected together by radiating
pieces. The interior arc forms a complete semicircle, but
the others ex-tend only to the cills under the road-way.
These arcs pass through an upright frame of iron at each
end, which serves as a guide; and the small space in the
haunches between the frames and the outer arc is fitted in
with a ring of about 7 feet diameter. Upon the top of
the ribs are laid cast-iron plates, which sustain the road-
way. The arch of this bridge is 100 feet 6 inches in
span; the interior ring is cast in two pieces, each piece
being about 70 feet in length. It was constructed in the
year 1779, by i\lr. Abraham Darby, iron-master at Cole-
brook Daie, and must be considered as a very bold effort
in the first instance of adopting a new material. The total
weight of the metal is 378-2- tons.

The second iron bridge of which the particulars have-
come to our knowledge was that designed by Mr. Thomas
Paine, author of many political works. It was construct-
ed by Messrs. Walker at Rotherham, and was brought to.
London and set up in a bowling-green at Paddington, where
it was exhibited for the first time. After which it was in-
tended to have been sent to America; but Mr. Paine not
being able to defray the expense, the manufacturers took
it back, and- the malleable iron was afterwards worked up
in the construction of the bridge at Wcarmouth.

The third iron bridge of importance erected in Great
Britain, was that over the river Wear at. Bishop Wear-
mouth, near Sunderland, the chief projector of which was
Rowland Burton, esq. M. p. It consists of a single arch,,
whose span is 236 feet; and as the springing, stones at
each side project 2 feet, the whole opening is 240 feet.
The arch is the segment of a circle of about 444. feet di-
ameter, its versed sine is 34 feet, and the whole height from
low water about 100 fit, admitting vessels of from 2 to-
300 tons burthen to pa^s under it without stnking their
masts. A series of 105 blocks form a. rib, and 6 of these
ribs compose the breadth of the bridge.. The spandrels,,
or the spaces between the-arch and the road-way, are filled
up with cast-iron circles which touch the outer circum-
ference of the arch, and at the same time support the

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road-way, thus gradually diminishing from the abutments
towards the centre of the bridge.. There are also diagonal
iron bars, which are laid on the tops of the ribs and extend-
ed to the abutments, to keep the ribs from twisting. The
superstructure is a strong frame of timber planked over to
support the carriage-road, which is composed of marl,
lime-stone, and gravel, with a cement of tar and chalk im-
mediately upon the planks, to preserve them. The whole
width of the bridge is 32feet. The abutments are masses
of almost solid masonry, 24 feet in thickness, 42 in
breadth at bottom, and 37 at top. The south pier is
founded on the solid rock, and rises from about 22 feet
above the bed of the river. On the north side the ground
was not so favourable, so that it was necessary to carry
the foundation 10 feet below the bed. The weight of the
iron in this extraordinary fabric amounts to 260 tons;
46 of these are malleable, and 214 cast. The entire ex-
pense was 27,000/.

Many other iron bridges have since been constructed
or proposed b}7 different persons, and in various situations,
which are exhibited and described in the first volume of
my Tracts.

There are various other kinds of bridges, besides these
already mentioned, such as hanging or pendant bridges,
draw-bridges, flying bridges, &c, &c, and even natural
bridges, of which kind a most wonderful one is described
by Mr. Jefferson, in his State of Virginia; and another
described by Don Ulloa, in the province of Angariz, in
South America. See also Arch.

BRIGGS (Henry), one of the greatest mathematicians
in the l6th and 17th centuries, was born at Warleywood,
near Halifax in Yorkshire, in 1556. From a grammar-
school in that country he was sent to St. John's-college,
Cambridge, in 1579; where after taking both the degrees in
arts, he was chosen fellow of his college in 1588. He ap-
plied himself chiefly to the study of mathematics, in which
he so greatly excelled, that in 1592 he was made examiner
and lecturer in that faculty ; and soon after, reader of the
physical lecture, founded by Dr. Linacer.

On the settlement of Gresham-college, in London, he
was chosen the first professor of geometry there, in 1596.
Soon after this, he constructed a table, for finding the
latitude, from the variation of the magnetic needle being
given. In the year 1609 he contracted an acquaintance
with the learned Mr. James Usher, afterwards archbishop
of Armagh, which was continued many years after by let-
ters, two of Mr. Briggs being still extant in the collection
of Usher's letters that were published : in the former of
these, dated August 1601, he writes, among other things,
that he was engaged in the subject of eclipses ; and in the
latter, dated the 10th of March l6l5, that he was wholly
taken up and employed about the noble invention of lo-
garithms, which had come out the year before, and in the
improvement of which he had afterwards so great a con-
cern. For Briggs immediatelyapplied himself to the study
and improvement of them; expounding them also to his
auditors in his lectures at Gresham-college. In these lec-
tures he proposed the alteration of the scale of logarithms,
from the hyperbolic form which Lord Napier had given
them, to that in which 1 should be the logarithm of the
ratio of 10 to 1 ; and soon after he wrote to Lord Napier
to make the same proposal to himself. In the year 1616
Briggs made a visit to Napier at Edinburgh, to confer
with him mi this change; and the next year he did the

same also. In these conferences, the alterations were agreed,
upon accordingly, and on Briggs's return from his second
visit, in 1617, he published the first chiliad, or 1000 of his
logarithms. See the Introduction to my Logarithms, for
an ample history of these transactions.

In 1619 he was made the first Savilian professor of
geometry at Oxford ; and resigned the professorship of
Gresham-college the 25th of July 1620 At Oxford he
settled himself at Merton-college, where he continued a
most laborious and studious life, employed partly in the
duties of his office as geometry lecturer, and partly in the
computation of the logarithms, and in other useful works.
In the year l622 he published a small tract on the
" North-west passage to the South Seas, through the con-
tinent of Virginia and Hudson's Bay ;" the reason of which
was probably, that he was then a member of the com-
pany trading to Virginia His next performance was his
great and elaborate work, the Arithmetica Logarithmica
in folio, printed at London in 1624; a stupendous work
for so short a time ! containing the logarithms of 30
thousand natural numbers, to 14 places of figures besides
the index. Briggs lived also to complete a table of loga-
rithmic sines and tangents for the 100th part of every de-
gree to 14 places of figures, besides the index ; with a ta-
ble of natural sines for the same 100th parts to 15 places,
and the tangents and secants for the same to ten places ;
with the description and construction of the whole. These
tables were printed at Gouda in 1631, under the care of
Adrian Vlacq, and published in 1633, under the title of
Trigcmometria Britannica. In the construction of these
two works, on the logarithms of numbers, and of sines
and tangents, our author, besides extreme labour and ap-
plication, manifests the highest power of. genius and in-
vention; as we here, for the first time, meet with several
of the most important discoveries in the mathematics, and
what have hitherto been considered as of much later in-
vention ; such as the Binomial Theorem ; the Differential
Method and Construction of Tables by Differences; the
Interpolation by Differences; with Angular Sections, and
several other ingenious compositions ; a particular account
of which may be seen in the Introduction to my Mathe-
matical Tables.

This truly great man terminated his useful life the 26th
of January 1630, and was buried in the choir of the cha-
pel of Merton-college. As to his character, he was not
less esteemed for his great sincerity and other eminent
virtues, than for his excellent skill in mathematics. Doc-
tor Smith gives him the character of a man of great pro-
bity; easy of access to all; free from arrogance, morose*
ness, envy, ambition, and avarice ; a contemner of riches,
and contented in his own situation; preferring a studious
retirement to all the splendid circumstances of life. The
learned Mr. Thomas Gataker, who attended his lectures
when he was reader of mathematics at Cambridge, repre-
sents him as highly esteemed by all persons skilled in ma-
thematics, both at home and abroad ; and says, that dcr
siring him once to give his judgment concerning judicial
astrology, his answer was, " that he conceived it to be a
mere system of groundless conceits." Ollghtred calls him
the mirror of the Bg*, for his excellent skill in geometry.
And one of his successors at Gresham-college, the learned
Dr. Isaac Barrow, in his oration there upon his admis-
sion, has drawn his character more fully ; celebrating his
great abilities, skill, and industry, particularly in perfect-

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ing the invention of logarithms, which, without his care
and pains, might have continued an imperfect and useless
design.

His writings were more important than numerous:
some of them were published by other persons : the list
of the principal part of them is as follows:

1. A Table to find the Height of the Pole; the. Magne-
tical Declination being given. This was published in Mr.
Thomas Blundeville's Theoriques of the Seven Planets;
London lu02, 4to.

2. Tables for the Improvement of Navigation. These
consist of, A table of declination to every minute of the
ecliptic, in degrees, minutes, and second's : A table of the
sun's prosthaphaereses: A table of equations of the sun's
ephemerides: A table of the sun's declination : Tables to
find the height of the pole in any latitude, from the height
of the pole star. These tables are printed in the 2d edi-
tion of Edward Wright's treatise, entitled, Certain Errors
in Navigation detected and corrected; London lGlO, 4to.

3. A Description of an Instrumental Table to rind the
Part Proportional, devised by Mr. Edward Wrjght. This
is subjoined to. Napier's table of logarithms, translated
into English by M r. Wright, and after his death published
by Briggs, with a preface of his own ; London, l6l6 and
1618, 12mo.

4. Logarithmorum Chilias Prima. Lond. l6l7, 8vo.

5. Lucubrationes et Annotationes in Opera Posthuma
J. Neperi ; Edinb. l6l9, 4to.

6. Euclidis Elementornm vi libri priores &c. Lond.
l620, folio. This was printed without his name to it.

7. A treatise of the North-west Passage to the South
Sea &c. By H. B., Lond. ]6'22, 4to. This was reprinted
in Purchas's Pilgrims, vol. 3, p. 852.

8. Arithmetica Logarithmica, &c. Lond.l6"24, folio.

9. Trigonometria Britannica, &c. Goudie 1(>33, folio.

10. Two letters to Archbishop Usher.

11. Mathematica ab Antiquis minus Cognita. — This is
a summary account of the most observable invention of
modern mathematicians, communicated by Mr. Briggs to
Dr. George llakewill, and published by him in his Apo-
logie; Lond. folio.

Besides these publications, Briggs wrote some other
pieces, that have not been printed : as,

(1.) Commentaries on the Geometry of Peter Ramus.

(2.) Duaj Epistolae ad celeberrimum virura Chr. Sever.

Longomontanum. One of these letters contained some

remarks on a treatise of Longomontanus, about squaring

the circle; and the other a defence of arithmetical gco-

. metry.

(3.) Animadverses Geometries : 4to.

(4.) De eodem Argumento: 4 to. — These two were in
the possession of the late; Mr. Jones. They both contain
a great variety of geometrical propositions, concerning the
properties of many figures, with several arithmetical com-
putations, relating to the circle, angular sections, &c. —
The two following were also in possession of Mr. Jones:

(5.) A treatise of Common Arithmetic; folio.

(6.) A letter to Mr. Clarke of Gravesend, dated 25 Feb.
1606 ; with which he sends him the description of a ruler
called Bedwell's rider, witb directions how to use it.

BRIGGS (William), an eminent physician in the lat-
ter part of the 17th century, was born at Norwich, for
which town his father was four times member of parlia-
ment. He studied at the university of Cambridge; and
afterwards travelled into France, where he attended the

lectures of the famous anatomist Vieussens, at Monlpe-
lier. On his return he published his Ophthalmographia,
in 1676. The year following he was made doctor of me-
dicine at Cambridge, and soon after fellow of the College
of Physicians at London. In lo'S2 he resigned his fellow-
ship to his brother: and the same year his Theory of Vi-
sion was published by Hooke. The enduing year he sent
to the Royal Society a continuation of that discourse,
which was published in their Transactions ; and the same
year he was appointed physician to St. Thomas's Hospi-
tal. In 16S4 he communicated to the Royal Society two
remarkable cases relating to vision, which were likewise
printed in their Transactions; and in J 685 he published
a Latin version of his Theory of Vision, at the desire of
Mr. Newton, afterwards Sir Isaac, then professor of ma-
thematics at Cambridge, with a recommendatory epistle
from him prefixed to it. He was afterwards made physi-
cian in ordinary to King William, and continued in great
esteem for his skill in his profession till his death, which
happened the 4th of September 1704.

Briggs's Logarithms, that species of them in which I
is the logarithm of the ratio of 10 to 1, or the logarithm
of 10. See Logarithms.

BROKEN Number, the same as Fraction; which see.
Broken Ray, or Ray of Refraction, in Dioptrics, is the
line into which an incident ray is refracted or broken, in
crossing the second medium.

BROUNCKER, or BROUNKER (William), Lord
Viscount, of Castle Lyons in Ireland, son of Sir William
Brounker, afterwards made viscount in 16"45, was born
about the year 16'20. He very early discovered a genius
for mathematics, in which he afterwards became very
eminent. He was made doctor of physic at Oxford, June
23, l6'46. In 1657 and 16"5S, he was engaged in a cor-
respondence by letters on mathematical subjects with Dr.
John Wallis, who published them in his Comniercium
Epistolicum, printed l6'58, at Oxford. Brounker was
one of the nobility and gentry who signed the remarkable
declaration concerning king Charles 11, published in April
1660. , ^

After the Restoration, Lord Brounker was made chancel-
lor and keeper of the great seal to the queen, also one of
the commissioners of the navy, and master of St. Kathe-
rine's hospital near the Tower of London. He was one of
those great men who first formed the Royal Society, of
which he was by the charter appointed the first president
in l662: which office he held, with great advantage to
the society, and honour to himself, till the anniversary
election, Nov. 30, 1677, when he was succeeded by Sir
Joseph Williamson. He died at his house in St. James's-
street, Westminster, the 5th of April 1684; and was suc-
ceeded in his title by his younger brother Marry, who
died in January 1687.

Lord Brounker had several papers inserted in the Phi-
losophical Transactions, the chief of which were, ]. Ex-
periments concerning the Recoiling of Guns. — 2. A Series
for the Quadrature of the Hyperbola; which was the first
series of the kind on that subject. — 3. Several of his let-
ters to Archbishop Usher were also printed in Usher's Let-
ters; as well as some to Dr.' Wallis, in his Commercium
Epistolicum, above mentioned ; particularly an approxi-
mation to the circular area by means of continued frac-
tions, of which he was the first inventor.

BROWN (Sir William), a noted physician and mis-
cellaneous writer of the 18th century. He first settled

BUL

[ 256 ]

BUR

at Lynn in Norfolk, where he published a translation of of the world, according to Philolaus, an ancient philo-
Dr. Gregory's Elements of Catoptrics and Dioptrics ; to sopher and astronomer. Afterward, in the year 1645, he

which he added, 1. A Method for finding the Foci of all
Specula and Lenses universally ; as also Magnifying or
Lessening a given object by a given Speculum or Lens, in
any assigned proportion. — 2. A Solution of those Pro-
blems which Dr. Gregory has left undemonstrated. — 3. A
particular account of Microscopes and Telescopes, from
Mr. Huygens ; with the discoveries made by Catoptrics
and Dioptrics. . " •

Having acquired a competence by his profession, he
removed" to Queen's-square, Ormond-street, London,
where he resided till his death, in 1774, at 82 years of
a»e ■ leaving by his will two prize-medals to be annually
contended for by the Cambridge poets.

Sir William Brown it seems was a very facetious man ;
and a great number of his lively essays, both in prose and
verse, were printed and circulated among his friends. The
active part taken by him in the contest with the licen-
tiates in 1768, occasioned his being introduced by Mr.
Foote in his Devil upon Two Sticks. — Upon Foote's exact
representation of him with his identical wig and coat, tall
figure, and glass stiffly applied to his eye, he senthim a com-
plimentary card for having so happily represented him; but
as he had forgot the muff, he sent him his own. — This good-
natured way of resenting disarmed Foote completely. —
He used to frequent the annual ball at the ladies' board-
ing-school, Queen-square, merely as a neighbour, a good-
natured man, and a lover of youthful company. A dig-
nitary of the church being there one day to see his
daughter dance, and finding this upright figure stationed
in the room, told him he believed he was Hermippus Re-
divivus who lived anhclitu puellarum. — When he lived at
Lynn, a pamphlet was written against him ; which he
nailed up against his house-door. — At the age of 80, on
St. Luke's day 1771, he came to Batson's coffee-house in
his laced coat and band, and fringed white gloves, to in-
troduce himself to Mr. Crosby, then lord mayor. A gen-
tleman present observing that he looked very well, he re-
plied, he had neither wife nor debts.

BRUIN (John de), a respectable mathematician and
philosopher, was born at Gorcum in 1620; and having
pursued a course of philosophy at Leyden, and prose-
cuted his studies at Bois-le-Duc and Utrecht, he removed
to Leyden, where he taught mathematics. He afterwards
became professor at Utrecht, and besides his official du-
ties made dissertations in private, and read lectures on
Grotius's treatise De Jure Belli et Pacis. He was di-
stinguished by his attachment to experimental philosophy,
and°by his observations in astronomy. He published dis-
sertations De vi altrice; De corporum gravitate et levi-
tate; De cognitione Dei naturali ; De kicis causa et ori-
oine &c. He had a dispute with Isaac Vossius, to whom
he wrote a letter printed at Amsterdam in l6'63, wherein
he criticises Vossius's book De natura et proprietate lu-
cis; and strenuously maintains the hypothesis of Des-
cartes. He died in 16'75, after he had been professor 23
years.

BULLIALD (Ismael), an eminent astronomer and
mathematician, was born at Laon in the Isle of France in
l6'05. He travelled in his youth, for the sake of im-
provement, and gave very early proofs of his astronomical
ocnius. Riccioli styled him, Astronomus profunda! inda-
ginis. He first, published lii.s dissertation entitled, Philo-
laus, sive de vero Systcmate Mundi ; or his true system

set forth his Astronomia Philolaica, grounded upon the
hypothesis of the earth's motion, and the elliptical orbit
described by the planet's motion about a cone. To which
he added tables entitled, Tabula; Philolaica;: a work
which Riccioli says ought to be attentively read by all
students of astronomy. — He considered also the hypothe-
sis or approximation of Bishop Ward, and found it not to
agree with the planet Mars ; and he showed, in his de-
fence of the Philolaic astronomy against the bishop, that
from fouT observations made by Tycho on the planet
Mars, that planet in the first and third quarters of the
mean anomaly, was more forward than it ought to be ac-
cording to Ward's hypothesis; but in the 2d and 4th qua-
drant of the same, the planet was not so far advanced as
that hypothesis required. He therefore set about a cor-
rection of the bishop's hypothesis, and made it answer
more exactly to the orbits of the planets, which were most
excentric, and introduced what is called, by Street in his
Caroline Tables, the Variation : these tables were calcu-
lated from this correction of Bulliald's, and exceeded all
others in exactness that had before been published. This
correction is, in the judgment of Dr. Gregory, a very-
happy one, if it be not set above its due place; and be ac-
counted no more than a correction of an approximation
to the true system : for by this means we are enabled to
gather the coequate anomaly a priori arid that directly
from the means, and the observations are well enough an-
swered at the same time; which, in Mercator's opinion,
no one had effected before. — It is remarkable that the el-
lipsis which he has chosen for a planet's motion, is such a
one as, if cut out of a cone, will have the axis of the cone
passing through one of its foci, viz, that next the aphelion.
In 1657 was published his treatise De Lineis Spiralibus,
Exerc. Geom. et Astron. Paris, 4to. In 1 682 came out at
Paris, in folio, his large work entitled, Opus novum ad
Arithmeticam Infinitorum ; a work which is a diffuse am-
plification of Dr. Wallis's Arithmetic of Infinites, and
which Wallis treats of particularly in the SOth chapter of
his Historical Treatise of Algebra. He wrote also two
Admonitions to Astronomers ; the first, concerning a new
star in the neck of the Whale, appearing at some times,
and disappearing at others; the second, concerning a ne-
bulous star in the northern part of Andromeda's girdle,
which had not been noticed by any of the ancients. This
star also appeared and disappeared by turns. And as
these phenomena were new and surprising, he strongly re-
commended the observing them to all that might be curi-
ous in astronomy.

BURGO (Lucas Pacioli de), a celebrated author
on arithmetic and algebra, who flourished in the 15th cen-
tury; and whose works on those subjects were the first that
we have in print. See an account of them in the article
Algebra.

BURNING, the action of fire on some pabulum or fuel
by which its minute parts are put into a violent motion,
and some of them, assuming the nature of fire, fly off in
orbem, while the rest are dissipated in vapour, or reduced
to ashes.

BuitNiis'C-G/rtss, or Burning-Mirror, a machine by
which the sun's rays are collected into a point; and by that
means their force and effect are extremely heightened, so
as to burn objects placed in it.

Burning-glasses arc of two kinds, convex and concave.

BUR

[ 257 ]

BUR

The convex ones are lenses, which, acting according to the Piece of Pompey's Pillar vitrifies, the black part in 50 sec.
laws of refraction, incline the rays of light towards the the white part in 54

axis, and unite them in a point or focus. The concave Copper ore in - 8

ones are mirrors or reflectors, whether made of polished Bone calcines in 4, and vitrifies in - - - 33

metal or silvered glass, and which acting by the laws of An emerald melts into a substance like a torquois stone ;

reflection, throw the rays back into a point or focus before a diamond weighing 4 grains loses J of its weight; the as-

the glass. bestos vitrifies, as all other bodies will do if kept long

The use of burning-glasses, it appears, is very ancient, enough in the focus; but when once vitrified, the mirror

many of the old authors recording some effects of them. ca^g° no further with them. PhiLos. Trans, vol. 4, p. 10,8

Diodorus Siculus, Lucian, Dion, Zpnaras, Galen, Anthe
mius, Eustatius, Tzetzes, and others, relate that by means
of them Archimedes set fire to the Roman fleet at the siege
of Syracuse. Tzetzes is so particular in his account of this
matter, that his description suggested to Kircher the me-
thod by which it was probably accomplished. That au-
thor says that " Archimedes set fire to Marcellus's navy
by means of a burning-glass composed of small square
mirrors, moving every way upon hinges; which, when
placed in the sun's rays, directed them upon the R.oman
fleet, so as to reduce it to ashes at the distance of a bow-
shot." And the burning power of reflectors is mentioned
in Euclid's Optics, theor. 31. Again, Aristophanes, in his
comedy of The Clouds, introduces Socrates as examining
Strepsiades about a method he had discovered of getting
clear of his debts. He replies, that " he thought of ma-
king use of a burning-glass which he had hitherto used in
kindling his fire; for should they bring a writ against me,
I'll immediately place my glass in the sun at some little dis-
tance from it, and set it on fire." Pliny and Lactantius have
also spoken of glasses that burn by refraction. The former

Tschirnhausen's reflecting mirrors produced equally-
surprising effects; as they may be seen described in the
Acta Erudit. for 1687, p. 52. And other persons have
made very good ones of wood, straw, paper, ice, and other
substances capable of taking a proper form and polish.

Every lens, whether convex, plano-convex, or convexo-
convex, collects the sun's rays, dispersed over its convex-
ity into a point by refraction; and it is therefore a burn-
ing-glass. The most considerable of this kind is that made
by Tschirnhausen, and described in the same Acta Erudit.
The diameters of his lenses are from 3 to 4 feet, having the
focus at the distance of 12 feet, and its diameters an inch
and a half. To make the focus more vivid, the rays are
collected a second time, by a second lens parallel to the
first, and placed at such a distance that the diameter of
the cone of rays formed by the first lens is equal to the
diameter of the second; so that it receives them all, and
the focus is reduced from an inch and a half to half the
quantity, and consequently its force is quadrupled. This
glass vitrifies tiles, slates, pumice-stones, &c, in a moment.
It melts sulphur, pitch, and all rosins, under water; the

calls them balls or globes of crystal or glass, which being ashes of vegetables, woods, and other matters, are trans

exposed to the sun, transmit a heat sufficient to set fire to
cloth, or corrode the dead flesh of those patients who
stand in need of caustics ; and the latter, after Clemens
Alexandrinus, observes that fire may be kindled by inter-
posing glasses filled with water between the sun and the
object, so as to transmit the rays to it.

Among the ancients, the most celebrated burning-mir-
rors were those of Archimedes and Proclus; by the former
was burnt the fleet of Marcellus, as above mentioned ; and

by the latter, the navy of Vitellius, besieging Byzantium, "IV."/^1, i"-" i"" V'

,„„„ ,• . r, 3 . , ' v . about 11| inches diameter: six of them are placed con-

according to £onaras was burnt to ashes. ,. * , , ., _t, c ■ . • *, c ,

tiguous to, and round the 7th, forming a kind ot sphen-

muted into glass, and every thing applied to its focus is
either melted, changed into a calx, or into fumes. The
author observes that it succeeds best when the matter ap-
plied is laid on a hard charcoal well burnt. But though the
force of the solar rays be thus found so surprising, yet the
rays of the full moon, collected by the same burning-glass,
do not show the least increase of heat.

Sir Isaac Newton presented a burning-glass to the Roy-
al Society, consisting of 7 concave glasses, so placed, that
all their foci join in one physical point. Each glass is

Among the moderns, the most remarkable burning-
glasses, are those of Magine, of 20 inches diameter; of Se-
patala of Milan, near 42 inches diameter, and which burnt
at the distance of 15 feet; of Settalaof Villette; ofTschirn-
hausen, of Buffon, of Trudaine, and of Parker.

Villette, a French artist at Lyons, made a large mirror,
which was bought by Tavernier, and presented to the king
of Prussia ; a second, bought by the king of Denmark; a

cal segment, whose subtense is about 34| inches: the com-
mon focus is about 22| inches distant, and about an inch
in diameter. This glass vitrifies brick or tile in 1 second,
and melts gold in 30 seconds.

M. Buffon also made a variety of very powerful burn-
ing-glasses, both as mirrors and as lenses ; but at length
concluded with one which is probably of the same nature
with that of Archimedes, and consisted of 400 mirrors re-

third, presented to the Royal Academy by the king of fleeting their rays all to one point, and with which he

France; and a fourth came to England, and was publicly
shown. This mirror is 47 inches wide, being a segment
of a sphere of 76 inches radius; so that its focus is about
•38 inches from the vertex ; and its substance is a compo-
sition of tin, copper, and tin-glass. Some of its effects

were as follow :

A silver sixpence melted in

A George the lst's halfpenny in -

and runs with a hole in
Tin melts in -
Oast iron in -
Slate in -

A fossil shell calcines in -
liut. I.

sec.

71
16
34

3
16

3

7

could melt lead and tin at the distance of 140 feet; and
with others he consumed substances at the distance of 210
feet. SeePhilos. Trans, vol.44; Buffon's Hist. Nat. Suppl.
vol. 1 ; or Montucla's Hist, des Math. vol. 1, p. 246.

It appears there is no solid substance capable of resisting
the efficacy of modern burning-glasses; though water, &c,
are not affected by them at all. Thils, Messrs. Macquer
and Baume have succeeded in melting small portions of
platina by means of a concave glass 22 inches diameter,
and 28 inches focus, though this metal is not fusible by
the strongest fires that can be excited in furnaces, or sus-
tained by any chemical apparatus. Yet it was long since
observed, by the Academicians del Cimento, that spirit-of-
2 L

BUR

[ 258 ]

BUT

wine could not be fired by any burning-glass which they
used; and notwithstanding the great improvements these
instruments have since received, M. Nollet has not been
able, by the most powerful burning-mirrors, to set fire to
any inflammable liquors whatever.

Howevey, a large burning-lens, for fusing and vitrifying
such substances as resist the fires of furnaces, and espe-
cially for the application of heat in vacuo, and in certain
other circumstances in which heat cannot be applied by
other means, has long been a desideratum with persons
concerned in philosophical experiments: and this it ap-
pears is now in a great measure accomplished by Mr. Par-
ker, an ingenious glass-manufacturer in Fleet-street, Lon-
don. His lens is made of flint-glass, and is 3 feet in dia-
meter ; but when fixed in its frame, exposes a surface of
32 inches in the clear ; the length of the focus is 6 feet 8
inches, and its diameter one inch. The rays from this large
lens are received and transmitted through a smaller, of 13
inches diameter in the clear within the frame, its focal
length 29 inches, and diameter of its focus 3-8ths of an
inch: so that this second lens increases the power of the
former more than 7 times, or as the square of 8 to the
square of 3.

From a great number of experiments made with this
lens, the following are selected to serve as specimens of its
powers :

Substances Fused.

Scoria of wrought iron
Common slate -
Silver, pure . -
Platina, pure -

Nickell

Cast iron, a cube -

Kearsh -

Gold, pure n

Crystal pebble - - - -

Cank, or terra ponderos$ -

Lava r

Asbestos -

Bar iron, a cube -

Steel, a cube - - - -

Garnet -

Copper, pure -

Onyx -

Zeolites -

Pumice-stone -

Oriental emerald -

Jasper -

White agate -

Flint, oriental - - - -

Topaz, or chrysolite -

Common limestone -

White rhomboidal spar

Volcanic clay - - - -

Cornish moorstone -

Rough cornelian -

Rotten-stone -

Time

Wgt.

in Sec.

n Grs.

2

12

2

10

3

20

3

10

3

16

3

10

3

10

4

20

6

7

7

10

7

10

10

10

12

10

12

10

17

10

20

33

20

10

23

10

24

10

25

2

25

10

30

10

30

10

45

3

55

10

60

10

60

10

60

10

75

10

SO

10

Burning Zone, or Torrid Zone, the space within 23|
decrees of the equator, both north and south.

BURROWS (William), a writer on the subject of
navigation in the l6th century. His " Discourse of the
Variation of the Compass or Magnetic Needle," is sub-
joined to Norman's " Newe Attractive." Burrows had
been a famous navigator, having used the sea from 1 5 years
of age, and for his merit was promoted to be controller of
the navy by Queen Elizabeth. He shows how to determine
the variation several ways, setting down many observations
of it, made by an azimuth compass of Norman's invention,
but improved by himself. He demonstrates the falsehood
of the rules used to find the latitudes by the guard-stars.
He particularizes many errors in the then sea-charts, oc-
casioned by the neglect of the variation ; adding, " But of
these coasts (towards the north) and of the inwarde partes
of the countries of Russia, Muscovia, &c, I have made a
perfect plat and description, by myne owne experience in
sundrie voiages and travailes, bothe by sea and lande, to
and fro in these partes, which I gave to her majestie in
anno 157S." And lastly, he justly blames Coignet's in-
strument, called a Nautical Hemisphere ; but speaks too
severely against the writers on navigation.

BUSHEL, a measure of capacity for dry goods; as-
grain, pulse, fruits, &c, containing 4 pecks, or 8 gallons,
or £ of a quarter. By act of parliament, made in 1697, it
was ordained that " Every round bushel with a plain and
even bottom, being made 18| inches wide throughout, and
8 inches deep, shall be esteemed a legal Winchester bush-
el, according to the standard in his majesty's exchequer."
Now a bushel being thus made will contain 2150'42 cu-
bic inches, and consequently the corn gallon contains only
268* cubic inches. — The weight of a bushel of good corn,
or pulse, &c, is Commonly accounted to be as follows ;
viz, of

Wheat - - 60 lb. avoirdupois.
Rye 56

Barley - - 50
Beer, or Big - 44
Oats 40

Pease - - 60

Beans - - 60

Potatoes (heaped) 56
But some experienced persons say that the bushel of good
wheat weighs and produces as below ; viz,

Bushel of wheat weighs - - 63 lb.
Yields of meal 62

Loss in grinding ... 1

Bran in the meal - 6

BUTMENTS, are those supporters, or props, by which
the feet of arches or the extremities of bridges arc sup-
ported; and should be made very strong and firm.

BUTTRESS, is an arch, or a mass of masonry, serving
to support the sides of a building, wall, or the like, on the
outside. See Arch, and Arch-Boutant.

[ 259 ]

c.

C A L

/"^AISSON, in Architecture, a kind of chest or flat-bot-
^■^ tomed vessel, in which the pier of a bridge is built,
then sunk to the bottom of the water, and the sides loos-
ened and taken off from the bottom, by a contrivance for
that purpose; the bottom of the caisson being left under
the pier as a foundation to it. The caisson is kept afloat
till the pier is built above the height of low-water mark;
and for that purpose, its sides are either made of more than
that height at first, or else gradually raised to it as it sinks
by the weight of the work, so as always to keep its top
above water. Mr. Labelye tells us, that the caissons in
which he built some of the piers of Westminster bridge,
contained 150 load of fir timber, of 40 cubic feet each,
and that it was of more tonnage or capacity than a 40-gun
ship of war.

Caisson, in Military Affairs, is sometimes used for a
chest ; and in particular for a bomb or shell chest, and is
used as a superficial mine, or fourneau. This is done by
filling a chest either with gunpowder and loaded shells, or
else with shells alone, and burying it in a spot where an
enemy, besieging a place, is expected to come, and then
firing it by a train to destroy the men.

CALAMUS Pastoralis, in Music, the Shepherd's
Pipe; an ancient musical instrument, the origin of the flute,
at first made of a straw or reed.

CALCULATION, the act of computing several sums,
by adding, subtracting, multiplying, dividing, &c. From
calculus, in allusion to the practice of the ancients, who
used calculi, or little stones, in making computations, in
taking suffrages, and in keeping accounts, &c; as we now
use counters, figures, &c. Calculation is more particu-
larly used to signify the computations in astronomy, tri-
gonometry, &c, for making tables of astronomy, of loga-
rithms, ephemcrides, finding the times of eclipses, and such
like.

CALCULATOR, a person who makes or performs cal-
culations.— It is also the name of a machine constructed
by Mr. Ferguson, in the shape of an orrery, for exhibiting
the motions of the earth and moon, and resolving a varie-
ty of astronomical problems.' See his Astron. 4to, p. 265,
or 8vo, p. 393.

CALCULATORES were anciently accountants who
reckoned their sums by calculi, or little stones, or coun-
ters.— In ancient canons too we find a sort of diviners or
enchanters, censured under the denomination of Calcula-
tores, probably so called from their calculating nativities.

CALCULUS denotes primarily a small stone, pebble,
or counter, used by the ancients in making calculations or
computations, taking of suffrages, playing at tables, and
the like.

Calculus denotes now a certain way of performing
mathematical investigations and resolutions. Thus, we
say the Arithmetical or Numeral Calculus, the Algebrai-
cal Calculus, the Differential Calculus, the Exponential
Calculus, the Fluxional Calculus, the Integral Calculus,
the Literal or Symbolical Calculus, &c; for which, see
each respective word.

Arithmetical or Numeral Calculus, is the method of

C AL

performing arithmetical computations by numbers. See
Arithmetic, and Number.

Algebraical, Literal, or Symbolical Calculus, is the me-
thod of performing algebraical calculations by letters or
other symbols. See Algebra.

Differential Calculus, is the arithmetic of the indefi-
nitely small differences of variable quantities; a mode of
computation much used by foreign mathematicians, and
introduced by Leibnitz, as similar to Newton's method of
Fluxions. See Differential &c.

Exponential Calculus, is the applying the fluxional
or differential methods to exponential quantities ; such as

0 , or x , or ay , &c. See Exponential.

Fluxional Calculus, is the method of fluxions, invent-
ed by Newton. See Fluxions.

Integral Calculus, or Summatorius, is a method of in-
tegrating, or summing up differential quantities ; and is si-
milar to the finding of fluents. See Integral and
Fluent.

Calculus Literalis, or Literal Calculus, is the same with
algebra, or specious arithmetic, so called from its using
the letters of the alphabet ; in contradistinction from nume-
ral arithmetic, in which figures are used.

CALENDAR, or Kalendar, a distribution of time
as accommodated to the uses of life ; or an almanac, or
table, containing the order of days, weeks, months, feasts,
&c, occurring in the course of the year : it is so called
from the word Calendaj, which among the Romans denoted
the first days of every month, and anciently was written
in large characters at the head of each month. See Al-
manac, Calends, Month, Time, Year, &c.

In calendars, the days were originally divided into oc-
toades, or eights; but afterwards, in imitation of the Jews
and Orientals, they were divided into hebdomades, or
sevens, for what we now call a week : which custom, Sca-
liger observes, was not in use among the Romans till after
the time of Theodosius.

Divers calendars are established in different countries,
according to the different forms of the year, and distribu-
tions of time: as the Persian, the Roman, the Jewish, the
Julian, the Gregorian, &c, calendars. — The ancient Ro-
man calendar is given by Ricciolus, Struvius, Danet, and
others ; in which we perceive the order and number of the
Roman holy-days and work-days. — The Jewish calendar
was fixed by Rabbi Hillel, abouttheyear 360 ; from which
time the days of their year may be reduced to those of the
Julian calendar. — The three Christian calendars are given
by Wolfius in his Elements of Chronology ; as also the
Jewish and Mohamedan calendars. Other writers on the
calendars are Victa, Clavius, Scaliger, Dlondel, &c.

The Roman Calendar was first formed by Romulus,
who distributed time into several periods for the use of his
followers and people under his command. He divided the
year into 10 months, of 304 days; beginning on the first
of March, and ending with December.

Numa reformed the calendar of Romulus. He, added
the months of January and February, making it to com-
2 L 2

C A L

[ 260 ]

C A L

mence on the first of January, and to consist of 355 days.
But as this was evidently deficient of the true year, he or-
dered an intercalation of 45 days to be made every 4 years,
in this manner, viz, Every 2 years an additional month of
22 days, between February and March ; and at the end of
each two years more, another month of 23 days ; the
month thus interposed, being called Marcedonius, or the
intercalary February.

Julius Caesar, with the aid of Sosigenes, a celebrated as-
tronomer of those times, further reformed the Roman ca-
lendar, whence arose the Julian calendar, and the Julian
or old stvle. Finding that the sun performed his annual
course in 365 days and a quarter nearly, he divided the
year into 365 days, but every 4th year 366 days, adding a
day to the 23d of February, which being the 6th of the
calends, and thus reckoued twice, gave occasion to the
name Bissextile, or what we also call leap-year.

This calendar was further reformed by order of the pope,
Gregory xin; from whence arose the term Gregorian ca-
lendar and style, or what is now called the new style,
which is now observed by almost all nations in Europe.
The year of Julius was too long by nearly 11 minutes,
which amounts to about 3 days in 400 years; the pope
therefore, by the advice of Clavius andCiaconius, ordained
that there should be omitted a day in every 3 centuries
out of 4 ; so that every century, which would otherwise
be a bissextile year, is made to be only a common year,
excepting only such centuries as are exactly divisible by
4, which happens once in 4 centuries. See Bissextile.
This reformation of the calendar, or the new style, as we
call it, commenced in the countries under the popish in-
fluence, on the 4th of October 1582, when 10 days were
omitted at once, which had been over-run since the time
of the council of Nice, in the year 325, by the surplus of
11 minutes each year. But in England it only commenced
in 1752, when 11 days were omitted at once, the 3d of
September being accounted the 14th that year ; as the sur-
plus minutes had then amounted to 11 days. And now,
since the year 1800, another day has been added on ac-
count of the odd minutes; so that now our new style dif-
fers from the old bv 12 days.

Julian Christian Calendar, is-that in which the days
of the week are determined by the letters a,b, c, r>,E,F, g,
by means of the solar cycle; and the new and full moons,
particularly the paschal full moon, with the feast of Easter,
and the other moveable feasts depending on it, by means
of golden numbers, orlunarcycles, rightly disposed through
the Julian year. See Cycle, and Golden Number.

In this calendar, it is supposed that the vernal equinox
is fixed to the 21st day of March,; and that the golden
numbers, or cycles of 19 years, constantly indicate the
places of the new and full moons; though both are erro-
neous ; and from hence arose a great irregularity in the
time of Easter.

GrejotvVm Calendar, is that which, bymcansof epacts,
rightly disposed through the several months, determines
the new and full moons, with the time of Easter, and the
moveable feasts depending on it, in the Gregorian year.
This differs therefore from the Julian calendar, both in
the form of the year, and in as much as epacts are substi-
tuted instead of golden numbers. See Epact.

Though the Gregorian calendar be more accurate than
the Julian, yet it is not without imperfections, as Scaliger
and Calvisius have fully shown; nor does it appear pos-

sible to devise any one that shall be quite perfect. Yet
the Reformed Calendar, and that which is ordered to be
observed in England, by act of parliament made the 24th
of George ii, come very near to the point of accuracy :
For, by that act it is ordered that "Easter-day, en which
the rest depend, is always the first Sunday after the full
moon, which happens upon, or next after the 21st day of
March ; and if the full-moon happens on a Sunday, Easter-
day is the Sunday after."

Reformed, or Corrected, Calendar is that which, re-
jecting all the apparatus of golden numbers, epacts, and
dominical letters, determines the equinox, and the paschal
full moon, with the moveable feasts depending on it, by
computation from astronomical tables. This calenda\-
was introduced among the protestant states of Germany
in the year 1700, when 11 days were omitted in the month
of February, to make the corrected style agree with the
Gregorian. This alteration in the form of the year, they
admitted for a time ; in expectation that, the true quantity
of the tropical year being at length more accurately de-
termined by observation, the Romanists wduld agree with
them on some more convenient intercalation.

French New Calendar, is quite of a different form of
calendar from all others. It commenced in France on the
22d of September 179-'j soon after the change of govern-
ment. At the time of printing this (viz, in July 1794), it
does not certainly appear whether this new calendar will
be made permanent or not; but merely as a curiosity in
the science of chronology, a very brief notice of it may
here be added, as follows.

The year, in this calendar, commences at midnight the
beginning of that day in which fall the true autumnal equi-
nox for the observatory of Paris. The year is divided into
12 equal months, of 30 days each; after which 5 supple-
mentary days are added, to complete the 305 days ot the
ordinary year : these 5 days do not belong to any month.
Each month is divided into three decades of 10 days each ;
distinguished by 1st, 2d, and 3d decade. All these are
named according to the order of the natural numbers, viz,
the 1st, 2d, 3d, Ike, month, or day of the decade, or of the
supplementary days. The years which receive an inter-
calary day, when the position of the equinox requires it,
which we call embolismic or bissextile, they call olympic ;
and the period of four, years, ending with an olympic year,
is called an olympiade; the intercalary day being placed
after the ordinary (we supplementary days, and making
the last day of the olympic year. Each day, from mid-
night to midnight, is divided into 10 parts, each part into
10 others, and so on to the last mcasui able portion of time.

In this calendar the months and days have new names.
The first three months of the year, of which the autumn is
composed, take their etymology, the first from the vintage
which takes place from September to October, and is called
Vendemaire; the second, Brumaire, from the mists and
low fogs, which show as it were the transudation of nature
from October to November: the third, I rimaire, from the
various colds which are generally felt from November to
December. The three winter months take their etymo-
logy, the first, Nivose, from the snow which whitens the
earth from December to January; the second, Pluviose,
from the rains which usually fall in greater abundance
from January to February; the third, V'entose, from the
wind which dries the earth from February to March. The
three spring months take their etymology, the first, Ger-

CAL

I M\ ]

C A L

minal, from the fermentation and developement of the sap
from March to April ; the second, Floreal, from the blow-
ing of the flowers from April to May ; the third. Prairial,
from the smiling fecundity of the meadow crops from May
to June. Lastly, the three summer months take their ety-
mology, the first, Messidor, from the appearance of the wa-
ving ears of corn and the golden harvests which cover the
fields from June to July; the second, Thermidor, from
the heat, at once solar and terrestrial, which inflames the
air from July to August; the third, Fructidor, from the
fruits gilt and ripened by the sun from August to Septem-
ber. Thus, the whole 12 months are,

Autumn. Spring.

Vendemaire Germinal

Brumaire Floreal

Frimaire. Prairial.

Winter. Summer.

Nivose Messidor

Pluviose Thermidor

Ventose. Fructidor.

From these denominations it follows, that by the mere
pronunciation of the name of the month, every one readily
perceives three things and all their relations, viz, the kind
of season, the temperature, and the state of vegetation :
for instance, in the word germinal, his imagination will
easily conceive, by the termination of the word, that the
spring commences ; by the construction of the word, that
the elementary agents are busied ; and by the signification
of the word, that the buds unfold themselves.

As to the names of the days of the week, or decade of
10 days each, which they have adopted instead of seven,
as these bear the stamp of judicial astrology and heathen
mythology, they are simply called from the first ten num-
bers; thus,

Primidi Sextidi

Duodi Septidi

Tridi Octidi

Quartidi Nonidi

Quintidi Decndi.

In the almanac, or annual calendar, instead of the mul-
titude of saints, one for each day in the year, as in the
popish calendars, they annex to every day the name of
some animal, or utensil, or work, or fruit, or flower, or ve- .
yetable, &c, appropriate and most proper to the times.
This calendar however was changed for the old one again,
with the change of the form of government. For the ca-
lendars of the Greeks, Egyptians, Jews, Syrians, Persians,
Mahomedans, Ethiopians, &c, see the article Year.

Calendar Mont/is, the solar months as they stand in
the calendar, viz, January 31 days, &c. The number of
days in each month may be suggested to the memory by
the following canon :

"Thirty days hath September,
April, June, and November;
February has twenty-eight alone,
And all the rest have thirty-one."
Astronomical Calendar, an instrument engraven upon
copper-plates, printed on paper, and pasted on board, with
a brass slider which carries a hair, and shows by inspec-
tion the sun's meridian altitude, right ascension, declina-
tion, rising, setting, amplitude, &c, to a greater exactness
than can be shown by the common globes.

CALENDS, Cakndce, in the Roman Chronology, de-
noted the first days of each month ; being so named from
KtzXew, (caleo) 1 call, or proclaim ; because that, before the

publication of the Roman Fasti, and counting their mouths
by the motion of the moon, a priest was appointed to ob-
serve the first appearance of the new moon ; who, having
seen her, gave notice to the president of the sacrifices lo
offer one; and calling the people together, he proclaimed
unto them how they should reckon the days until the
nones; pronouncing the word Caleo 5 times if the nones
should happen on the 5th da3', or seven times if they hap-
pened on the 7th day of the month.

The calends were reckoned backwards, or in a retro-
grade order: thus, for example, the first of May being
the calends of May, the last or 30th day of April was the
pridie calendarum, or2d of the calends of May; the 29th
of April the 3d of the calends, or before the calends: and
so back to the 1 3th, where the ides commence ; which are
likewise numbered backwards to the 5th, where the nones
begin ; and also reckoned after the same manner to the first
day of the month, which is the calends of April.

Hence conies this rule to find the day of the calends
answering to any day of the month, viz, Consider how
many days of the month are yet remaining after the day
proposed, and to that number add 2, for the number of
or from the calends. For example, suppose it were the
23d day of April, it would then be the 9th of the calends
of May : for April containing 30 days, from which 23
being taken, there remains 7 ; to which 2 being added,
makes the sum 9. And the reason for this addition of the
constant number 2, is because the last day of the month
is called the 2d of the calends of the month following.

CALIBER, or Caliper, is the thickness or diameter
of a round body, particularly the bore or width of a piece
of ordnance, or that of its ball.

Caliber Compasses, or Caliper Compasses, or simple
Calipers, a sort of compasses made with bowed or
arched legs, the better to take the diameter of any round
body ; as the diameters of balls, or the bores of guns; or
the diameter, and even the length of casks, and such like.
The best sort of calipers usually contain the following ar-
ticles, viz, 1st, the measure of convex diameters in inches
&c; 2d, of concave diameters; 3d, the weight of iron
shot of given diameters ; 4th, the weight of iron shot for
given gun bores ; 5th, the degrees of a semicircle; Oth, the
proportion of troy and avoirdupois weight; 7th, the pro-
portion of English and French; feet and pounds weight;
8th, factors used in circular and spherical figures; 9>h,
tables of the specific gravities and weights of bodies ; 10th,
tables of the quantity of powder necessary for the proof
and service of brass and iron guns ; 11th, rules for com-
puting the number of shot or shells in a complete pile;
12th, rules for the fall or descent of heavy bodies; 13th,.
rules for the raising of water ; 14th, rules for firing artih-
lery and mortars; 1 5th, a line of inches ; lfith, logarith-
mic scales of numbers, sines, versed sines, and tangents;
17th, a sectoral line of equal parts, or the line of lines;,
18th, a sectoral line of planes and superficies; and lf»th,
a sectoral line of solids.

Calippic, Period, in Chronology, a period of 76 years,
continually recurring; at every repetition of which, it
was supposed, by its inventor Calippus, an Athenian as-
tronomer, that the mean new and full moons would al-
ways return to the same day and hour.

About a century before, the golden number or cycle of.
1£) years had been invented by Meton, which Calippus
finding to contain 19 of Nabonassar's year, 4 days and
£|£, to avoid fractions he quadrupled it, and so produced)

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c

]

C A L

his period of 76' years, or 4 times 19; after which he
supposed all the lunations &c would regularly return to
the same hour. But neither is this exact, as it brings
them too late by a whole day in 225 years.

CALORIC, in the new Chemical Nomenclature, is that
supposed material substance which possesses the power of
exciting the feeling of heat in the animal body; or it is
the matter and the cause of heat.

The general properties of caloric are the following : If
it be considered as material, it must be a substance of ex-
treme subtlety and tenuity, entirely imponderable by hu-
man instruments, derived immediately from the sun toge-
ther with light, and like light obeying certain and similar
laws of reflection and refraction, and also elicited by com-
bustion and by various other instances of chemical action,
percussion or collision, friction, the electric spark, galva-
nism, &c: when uncombined, moving with immense velo-
city in every direction, and constantly tending to an equi-
librium ; penetrating all bodies in nature with more or less
facility, and thereby enlarging their dimensions, without
increasing their weight : when combined, probably enter-
ing into chemical union with other bodies, and thereby
constituting the different states of liquid, vapour, or gas :
and lastly, when applied suddenly to the animal body, oc-
casioning the sensations of warmth, heat, or, in greater in-
tensity, burning. Sec the articles Light and Heat.

Caloric, then, is a word used to denote that substance
by which the phenomena of heat are produced ; and the
same as was formerly termed igneous fluid, matter of heat,
and other analogous denominations. There are perhaps
few subjects respecting which a more remarkable versati-
lity of general opinion has been evinced, than with regard
to the existence or non-existence of this principle. Are
the physical effects of heat produced by the operation of
a material fluid, sui generis ? or is heat merely an affec-
tion, consisting in internal vibrations and collisions of its
particles, or in some other mode of corpuscular action, of
which we are ignorant, and is there consequently no such
thing as caloric ? These are the questions that have di-
vided the opinions of many of the ablest philosophers of
both ancient and modern times ; but though different opi-
nions still prevail on this subject, yet the former of these
hypotheses seems to be most generally received. Among
the ancients who supposed the materiality of heat, may
be reckoned Zeno and Cicero : the latter, in his book De
Natura Deorum, says expressly, that heat is combined
with water, as its liquefaction sufficiently proves; and that
it cannot freeze nor congeal into snow or hoar-frost, with-
out suffering that heat to escape, &c, &c: which is very
nearly the idea that we have of it at the present day ; and
indeed this appears to have been the most prevailing opi-
nion till the time of Lord Bacon, who, in his work De
Forma Calidi, considers heat as the effect of an intestine
motion, or mutual collision of the particles of the body
heated ; an expansive uiuluhitory motion in the minute
particles of the body, by which they tend with some rapi-
dity towards the circumference, and at the same time in-
cline a little upwards; which idea was also with some
modifications adopted by Descartes, Newton, Boyle, and
the other mechanical philosophers of the succeeding age.
The general opinion however among chemists, was still in
favour of the materiality of heat; and as chemistry be-
came more generally known, so this opinion gained
strength, till Count Rumford by some experiments seemed
to show, that heat was imponderable, and capable of being

produced, ad infinitum, from a finite quantity of matter,
which again threw some doubt on this question ; and
though these results appear, in some degree, to have been
confuted by subsequent experiments, yet is the matter at
present undecided, and will probably always be involved
in some obscurity. We cannot here enter into the merits
of the particular arguments that have been advanced on
both sides of this intricate and interesting inquiry ; but
we shall enumerate a few of the principal experiments
that have been made on this subject, and shall then leave
the reader to draw his own conclusion, as to the materia-
lity or immateriality of heat, or the cause of heat ; or, in
other words, of the existence or non-existence of that
substance which is defined by the term Caloric.

It is known, that when water freezes, a portion of heat is
given out by it during the congelation, which is so consi-
derable, that if we were to conceive it to be transmitted
to, and imbibed by an equal quantity of water at the tem-
perature of 32° of Fahrenheit's thermometer, the latter
would be heated to no less than 140°, or to 172°; if
therefore heat were a ponderable substance, it might be
imagined that a given quantity of water would become
lighter when frozen in a vessel hermetically sealed. This
circumstance therefore induced some philosophers to as-
certain, if possible, the ponderability of caloric, and thus
to decide the question on which so many opinions had
been formed. Of the experiments made for this purpose,
that of Dr. Fordyce seems to have been conducted with
the greatest caution. He took a glass globe, three inches
in diameter, with a short neck, and weighing 451 grains ;
poured into it 1700 grains of water from the New River,
London, and then sealed it hermetically. The whole of
which then weighed 2150-|4 grains, at the temperature of
32°: this was then put for 20 minutes into a freezing
mixture of snow and salt, till some of it was frozen, and
was then taken out, and wiped perfectly dry with a linen
cloth, and next with a clean dry piece of washed leather;
and being then immediately weighed, it was found to be
■~sth of a grain heavier than before : this was repeated five
times, at each time more of the water was frozen, and
more weight gained ; and when the whole was frozen, it
was found to have gained -rg-ths of a grain; but the tem-
■ perature of the globe, and contained ice, by having been
kept in the frozen mixture, had been reduced to 12° of
Fahrenheit's thermometer: and on allowing it to rise to
32°, it was found to weigh -^th of a grain more than
when the water was fluid. The beam in this experiment
was so adjusted that it would mark -^-^ of a grain.

By this result therefore it appears that water by freez-
ing, an operation in which it evolves caloric, becomes
heavier, and that of course the addition of caloric to a
body renders it absolutely lighter ; which if admitted goes
to prove, that caloric counteracts the cll'ect of gravitation,
as it does that of contiguous attraction or affinity. This
experiment was repeated by Morvcau and Chaussier, and
with the same result, the water being heavier after it had
been frozen in glass vessels hermetically sealed, than it
was before when in its fluid state.

This subject soon attracted the attention of Lavoisier;
and his experiments, which were published in the Me-
moirs of the French Academy, led him to conclude that
the weightof bodies is mil at all altered by healing or cool-
ing them, and Consequently that caloric produces no sen-
sible change in the weigh! ot bodies. Count Rumford also
made similar experiments, and was at first led to the same

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conclusion as Dr. Fordyce ; but afterwards, by varying
the experiments, be had reason to suppose that he had
been mistaken in the first instance, and he finally came to
the same conclusion as Lavoisier. Hence then we may
conclude from these experiments, and others of a simi-
lar nature which have been made on this subject, that if
caloric be possessed of weight, it exists at least in such a
state of rarity, that its gravity cannot be ascertained by
experiment. But Count Rumford went still further, and
endeavoured to prove that an inexhaustible supply of
heat may be produced from a finite quantity of matter ;
and that therefore heat must be an effect arising from
some species of corpuscular action among the constitu-
ent particles of the body, agreeably to the notions of
Bacon. This experiment was conducted in the following
manner. The Count caused a cylinder of brass to be
turned 7f inches in diameter, and P'S inches long, which
was bored like a cannon with a caliber 3P7 inches in dia-
meter, and 7'i inches deep; so that the bottom was 2"6
inches in thickness. The hollow cylinder contained 385^
cubic iiiches of brass, and weighed 113M3lbs avoirdupois.
By means of the engine used for boring cannon in the ar-
senal of Munich, a blunt borer, or flat piece of hardened
steel, 4 inches long, 0'6~3 inch thick, and 3-i inches wide,
was kept with one of its extremities, whose area was about
2y square inches, pressed against the bottom of this hol-
low cylinder on the inside, with a force of about 10,000lbs
avoirdupois, while the latter was turned about its axis,
with a velocity of 32 revolutions in a minute. In one
experiment, the cylinder was covered on the outside with
a coating of thick flannel, to prevent the access of heat
from the atmosphere ; and in another, the borer was made
to work through a collar of leathers, so as to prevent ac-
cess of air also to the interior of the bore ; and in a third,
the whole cylinder was immersed in water, the borer still
working through a collar of leathers, so as to prevent the
access of the water to the interior of the bore; again, in
a fourth, the collar of leathers was removed, and the wa-
ter had access to the bottom of the interior of the bore
where the friction took place. The result was, that in all
these cases heat was generated by the friction in suffi-
cient quantity to cause about 26-4-lbs of ice-cold water to
boil in two hours and a half; or at about the same rate
as that at which it would have been produced by 9 large
wax candles. The capacity of brass for heat, or its power
of producing it by friction, did not appear to be diminish-
ed, and it seemed that this generation of heat would have
gone on for ever, if the friction had been continued, or
that the source of it was inexhaustible. Now any
thing which an insulated body, or a system of bodies, can
continue to supply without limitation, cannot possibly be
a material substance; and hence the conclusion was, that
heat must be an effect arising from some species of cor-
puscularaction among the constituent particles of the body.

Some other experiments of M. Pictet, Mr. Davy, and
other celebrated chemists, seem to be in favour of the con-
clusions drawn by Count Rumford from those above
stated ; but there are others, that appear to contradict
this principle, some of which were made by the Count
himself.

If heat arise only from the corpuscular motion of the
particles of the body, it would follow that no heat could
be produced, but in those cases where motion could be
communicated; whence again heat could not be derived
through a vacuum. But Count Rumford, by a very satis-

factory experiment, has shown that heat may be commu-
nicated through a Torricellian vacuum, in which there
could be nothing to communicate or propagate motion ;
for though such a vacuum may be not quite complete, yet
the matter of it must be in such a state of rarity, as to
prevent us from supposing for a moment, that it is capable
of exciting motion in the particles of the body placed in
it; and consequently, that the conclusion drawn from the-
boring experiment was inaccurate : and a recent discovery
made by Dr. Herschel, has at last nearly put an end to-
the dispute; by demonstrating that caloric is not a pro-
perty of matter, but a peculiar substance; or at least that
we have the same reason for considering it as a substance,
as we have for believing light to be material.

Dr. Herschel had been employed in making observa-
tions on the sun by means of telescopes ; and to prevent
the inconvenience arising from the heat, he used coloured
glasses; but these glasses, when they were deep enough co-
loured to intercept the light, very soon cracked and broke
to pieces; which circumstance induced him to examine
the heating power of the different coloured rays. He there-
fore made each of them in its turn fall upon the bulb of a
thermometer, near which two other thermometers were
placed to serve as a standard ; and the number of degrees
that the thermometer exposed to the coloured ray rose
above the other two thermometers, indicated the heating
power of that ray. Pie thus found, that the most refran-
gible rays have the least heating power; this power gra-
dually increasing as the refrangibility di-
minishes. The heating power of the Violet = 16
green, violet, and red rays, as determined Green = 22'4
by these experiments are expressed by Red = 55
Dr. Herschel, by the annexed numbers.

It struck Dr. Herschel as very remarkable, that the il-
luminating power and the heating power of the solar rays
should follow such different laws : the first existing in
greatest perfection in the middle of the spectrum, and di-
minishing as we approach either extremity ; but the second
constantly increasing from the violet end, and being greatest
at the red end. This led him to suspect, that probably
the heating power did not stop at the termination of the
visible spectrum, but is continued beyond it; he therefore
placed the thermometer completely without the boundary
of the red ray, but still in the line of the spectrum; and
it rose still higher than when exposed to the red ray : and
on shifting the thermometer still furtherit continued to rise;
the rise not reaching its maximum till the thermometer
was half an inch beyond the visible spectrum : when shift-
ed still further, it sank a little, but the power of heating
did not cease at the distance of an inch and a half from
the real ray.

These experiments were afterwards repeated and fully
confirmed by Sir Henry Englerield, in the presence of
some very good judges. The apparatus was very different
from that of Dr. Herschel, being contrived on purpose to
obviate certain objections which had been made, to the
conclusion drawn by that celebrated philosopher ; and the
bulbs of the thermometers were mostly blackened. The
following are the results of these experiments.

Thermometer in the blue ray rose in 3™ from 55° to 56°

green - - 3

-

54

- 58

yellow - - 3

-

56

- 62

full red - 11

-

56

- 72

confines of red 2i

-

58

- 73]

beyond the visible light 24

-

61

- 79

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The thermometer with its bulb blackened, rose muA
more when placed in the same circumstances, than the
-thermometers whose bulb was either naked, or whitened
with paint ; as is shown in the following table :

R , f black thermometer in 3™ from 5S° to 6l°

' \ white thermometer 3 - 55 - 58
p. , l~ black thermometer 3 - 59 - 6-i

\ white thermometer 3 - 58 - 5S-£
Confines of ( black thermometer 3 - 59-71

red \ white thermometer 3 - 57i 60i

Both Dr. Herschel and Sir H. Englefield take notice
of a faint blush of red, of a semi-oval form, visible when
the rays beyond the end of the spectrum were collected
by a lens.

From these experiments it follows, that there are rays
emitted from the sun which produce heat, but have not
the power of illuminating; and that these are the rays
that produce the greatest quantity of heat: and conse-
quently caloric is emitted from the sun in rays, and the
rays of caloric are not the same with the rays of light.

We have thus enumerated many of the principal expe-
riments that have been made, in order to prove the mate-
riality or immateriality of heat ; and without insisting fur-
ther on ihis subject, on which prubably the present state
of knowledge is insufficient for forming a decided opinion.
It seems however that, under all the circumstances attend-
ing this inquiry, we ought rather to consider it as a mate-
rial substance ; because of the two theories, that which
supposes it to be so is by far the most intelligible, the
most agreeable to the analogy of nature, and the least ex-
ceptionable ; and we shall accordingly regard it as an
elastic fluid, sui generis, capable of pervading, with vari-
ous degrees of facility, all the solid bodies with which we
are acquainted, and of being imbibed and retained by
them in different proportions, according to their respective
degrees of specific attraction, or capacity for it.

It will easily be conceivced, that from the elasticity and
power of pervading other substances, which we have attri-
buted to this fluid, it must necessarily follow that when a
body is by any means charged with a larger quantity of it
than is proportional to its mass and capacity, when com-
pared with those of other bodies in its vicinity ; the sur-
plus will be communicated to those other bodies, until
the density of the fluid in every body in the system be-
comes equal ; in like manner as a quantity of air thrown
into one of a number of vessels communicating with cich
other, will pass from it to the others, until it becomes dis-
tributed among them all, in proportion to their respecti\e
capacities; or till it becomes of equal density in them
all : and this state of density or compression of the calo-
ric is called its temperature.

The celerity and facility with which (his fluid pervades
different substances, are however extremely various, as
may be shown by the following experiment. Take a small
cylindrical earthen or wooden vessel, between three and
four inches in diameter, and insert into it, through holes
drilled in its bottom for that purpose, several wires about
•yth of an inch in diameter, and 6 or 7 inches long, of
equal size in every respect, but of different metals, as
gold, silver, copper, iron, brass, and zinc; and also two
small rods, one of glass and the other of wood, of equal
dimensions with the wire ; so that they may each of them
project about ^ an inch on the inside of the vessel ; the
nlher parts being on the outside, and forming a kind of
btand for i(. Dip a portion of the lower extremities of

these wires into melted wax, so that they may become
equally coated with it ; and when the wax has cooled, fill
the vessel with a heated fluid, as boiling water, or melted
lead, and observe the intervals between the time of filling
it, and the melting of the wax ; which intervals, will in-
dicate the relative conducting power of the substances.

We shall now conclude this article by a slight investi-
gation of the effects arising from the condensation or ra-
refaction of this fluid in bodies : by the former their tem-
perature is elevated, or they are heated ; and by the latter
it is depressed, or they become cooled.

In our investigation of the most universal operations in
nature, we generally find that the more immediate instru-
ments employed for the production of the effects which we
perceive, are two counteracting powers, by whose co-ope-
ration, and mutual counterpoise, the necessary equilibrium
is preserved, and the purposes of their creation effected,
with greater regularity than would probably result from
the employment of a single agent. Thus, gravity is, as we
know, counteracted by inertia; and the effects of cohesion,
a no less universal principle than gravity, are in like man-
ner restrained and modified by the agency of caloric: and
as without inertia, all the celestial bodies would be drawn
together into some one part of extended space; so without
heat, all the matter in the universe would become a con-
gealed and concrete mass; and fluidity and organization,
vegetation, and life, would have no existence. Solids be-
come first expanded, and then liquified, by its influence,
and liquids assume the aeriform state. It is essential to
the idea of a solid body, that it should in a certain degree
resist both compression and dilatation; that is, that its
particles should repel each other when brought nearer to-
gether, and attract each other when further separated,
within certain limits; and it is incontrovertible that they
can only remain in a quiescent state, by the equilibrium
of these opposite forces. The latter of these consti-
tutes cohesion, and the existence of the former is probably
attributable to the substance of which we are now speak-
ing. But be this as it may, it is at least a fact, that the
repulsive energies of these corpuscles are increased, and
the sphere of action enlarged by its influence, for the
body becomes expanded ; that is, this equilibrium no
longer takes place, unless its molecular are removed fur-
ther from each other than when it was at a lower tempe-
rature. Elevate its temperature still further, and the body
fuses ; that is, the attractive power of its particles becomes
annihilated, or wholly counteiacted, and they are only kept
together, like a quantity of shot, by the action of gravity,
operating independently on each. Increase still the action
of heat on the body, and the sphere of repulsion of its parti-
cles, or the distance to which this power operates, again
becomes much further extended, perhaps infinitely so, and
is therefore sufficient to overpower the action of gravity,
and the body becomes resolved into an elastic fluid. We.
know of no power that is capable of resisting this univer-
sal solvent: gold itself may be expanded into vapour by
the action of heat; and as these ellects are always produ-
cible by the same cause, so is it at least probable, that the
converse of; this is also true; that whenever a substance
is formed in a stale of fluidity, it is attributable solely to
this cause, and that the abstraction of the caloric, by which
it' is held in solution, would again reduce it ton solid.
Mercury is, we know, capable, of congelation; and the at-
mosphere itself would probably become a concrete body,
if we could deprive it of all, or the greater part of its heat .

CAM

[

Hence then we arrive at a very distinctive and appro-
priate definition of caloric ; that it is that substance, or
affection of matter, which is the cause of fluidity, in all
or the greater part of the bodies that we meet with in
that state.

For a fuller and more detailed account of caloric and
its various properties, see Murray's or Thomson's Chemis-
try; and for the ancient opinionson this subjectsee Heat.

CAMBER-BEAM, a piece of timber cut arch-wise, or
with an obtuse angle in the middle. They are commonly
used in platforms, as for church-roofs, and other occasions
where long timbers are wanted to lie at a small slope. A
camber-beam is much stronger than another of the same
dimensions; for being laid with the hollow side down-
wards, and having good butments at the ends, they serve
for a kind of arch.

CAMELEON, one of the constellations of the southern
hemisphere, near the south pole, and invisible in our lati-
tude. There are ten stars marked in this constellation in
Sharpe's catalogue.

CAMELOPARDALUS, a new constellation of the
northern hemisphere, formed by Hevelius, consisting of 32
stars first observed by him. It is situated between Cepheus,
Cassiopeia, Perseus, the Two Bears, and Draco ; and it
contains 58 stars in the British catalogue.

CAMERA Molia, a name given by Kircher to a con-
trivance for blowing the fire, for the fusion of ores, with-
out bellows. This is effected by means of water falling
through a funnel into a close vessel, which sends from it
so much air or vapour, as continually blows the fire.. See
Hooke's Philos. Coll. No. 3, pa. 80.

Camera Lucida, a contrivance of Dr. Hooke to make
the image of any thing appear on a wall in a light room,
either by day or night. See Philos. Trans. No. 38, pa. 741.
Another camera lucida has lately been introduced by
William Hyde Wollaston, M. D. secretary of the Royal
Society.

Camera Obscura, or Dark Chamber, an optical ma-
chine or apparatus, representing an artificial eye, by which
the images of external objects, received through a double
convex glass, are shown distinctly, and in their native co-
lours, on a white ground placed within the machine, in
the focus of the glass. The first invention of the camera
obscura is ascribed to John Baptista Porta. See his Ma-
gia Naturalis, lib. 17, cap. .6, where he largely describes
the effects of it. See also the end of Gravesande's Per-
spective, and other optical writers, for the construction
and uses of various sorts of camera obscuras.

This machine serves for many useful and entertaining
purposes. For example, it is very useful in explaining
the nature of vision, representing a kind of artificial eye:
it exhibits very diverting spectacles; showing images per-
fectly like their objects, clothed in their natural colours,
but more intense and vivid, and at the same time accom-
panied with all their motions ; an advantage which no art
can imitate: and by this instrument, a person unacquaint-
ed with painting, or drawing, may delineate objects with
the greatest accuracy of drawing or colouring.

Theory of the Camera Obscura. The theory and prin-
ciple of this instrument may be thus explained. If any
object ab radiate through a small aperture L, upon a
white ground opposite to it, within a darkened room, or
box, &c ; the image of the object will be painted on that
ground in an inverted position. For, by the smallness of
the aperture, the rays from the object will cross each other
Vol. I.

there, the image of the point a being at a, and that of b
at b ; so that the whole object ab will appear inverted,
as at ab. And as the corresponding rays make eq^ual an-
gles on both sides of the aperture, if the ground be pa-
rallel to the object, their heights will be to each other di-
rectly as their distances from the aperture.

Construction of a Camera Obscura, in which the images
of external objects shall be represented distinctly, and in
their genuine colours, either in an inverted or an erect si-
tuation. 1st, Darken a chamber that has one of its win-
dows looking towards a place containing various objects
to be viewed, leaving only a small aperture open in .one
shutter. 2d, In this aperture fit a proper lens, either pla-
no-convex, or convex on both sides; the convexity form-
ing a small portion of a large sphere. But note, that if
the aperture be made very small, as of the size of a pea,
the objects will be represented even without any lens at
all. 3d, At a proper distance, to be determined by trials,
stretch a paper or white cloth, unless there be a white
wall at that distance, to receive the images of the objects:
or the best way is to have some plaster of Paris cast on a
convex mould, so as to form a smooth concave surface,
and of a curvature and size adapted to the lens, to be
placed occasionally at the proper distance. 4th, If it be
desired to have the objects appear erect, instead of in-
verted, this may be done either by placing a concave lens
between the centre and the. focus of the first lens; or by
reflecting the image from a plane speculum inclined to the
horizon in an angle of 45 degrees ; or by having two lenses
included in a draw-tube, instead of one.

That the images be clear and distinct, it is necessary
that- the objects be illuminated by the sun's light shining
upon them from the opposite quarter : so that, in a west-
ern prospect, the images will be best seen in a forenoon,-
an eastern prospect the afternoon, and a northern prospect
about noon ; a southern aspect is the least eligible of any.
But the best way of any is, if the lens be fixed in a pro-
per frame, on the top of a building, and made to move ea-
sily round in all directions, by a handle extended to the
person who manages the instrument ; the images being
then thrown down into a dark fbom immediately below it,
on a horizontal plaster of Paris ground a little concave:
for thus a view of all the objects quite around may easily
be taken in the space of a few minutes; as in the case of
the excellent camera obscura placed on the top of the
Royal Observatory at Greenwich.

The objects will be seen brighter, if the spectator first
wait a few minutes in the dark. Care should also be
taken, that no light escape through any chinks ; and that
the ground be not too much illuminated. It may further
be observed, that the greater distance there is between the
aperture and the ground, the larger the images will be;
but then at the same time the brightness is weakened more
and more with the increase of distance.
2 M

CAM

[ 266 ]

C A M

To construct a Portable Camera Obscura. 1st, Provide
a small box or chest of dry wood, about 10 inches broad,
and 2 feet long or more, according to the size of the lenses.
2d, In one side of it, as bd, fit a sliding tube ef with two
lenses; or, to have the image at a less distance from the
tube, with three lenses, convex on both sides; the diame-
ter of the two outer ones to be about 7 inches, but that of
the inner to be less, as 4-J or 5 inches. 3d, At a proper
distance, within the box, set up perpendicularly an oiled
paper gh, so that images thrown upon it may appear on
the other side. 4th, In the opposite side, ati, make a round
hole, for a person to look conveniently through with both
eyes. Then if the tube be turned towards the objects,
and the lenses be placed by trials at the proper distance,
by sliding the tube in and out, the objects will be seen de-
lineated on the paper, erect as before.

The machine may be better accommodated for draw-
ing, by placing a mirror to pass from g to c; for this will
reflect the image upon a rough glass plane, or an oiled pa-
per, placed horizontally at ab; and a copy of it may
there be sketched with a black-lead pencil.

Another Portable Camera Obscura is thus made. 1st, On
the top of a box or chest raise a little turret hi, open to-
wards the object ab. 2d, Behind the aperture, incline a
small mirror ab at an angle of 4-5 degrees, to reflect the
rays a« and u& upon a lens g convex on both sides, and
included in a tube gl. Or the lens may be fixed in the
aperture. 3d, At the distance of the focus of this lens
place a table, or board ef, covered with a white paper, to
receive the image ab. Lastly, In mn make an oblong
aperture to look through ; arid an opening may also be
made in the side of the box, for the convenience of drawing.

This sort of camera is easily changed into a show-box,
for viewing prints, &c: placing the print at the bottom of
the box, with its upper part inwards, where it is enlight-
ened through the front, left open for this purpose, either

by day or candle-light; and the print may be viewed
through the aperture in hi. — A variety of contrivances for
this purpose may be seen described in Harris's Optics,
b. ii. sect. 4. — Mr. Storer has also procured a patent for
an instrument of this kind, which he calls a delineator;
being formed of two double convex lenses and a plane mir-
ror, fitted into a proper box. One lens is placed close to
the mirror, making with it an angle of 45 decrees; the
othef being placed at right angles to the former, and fixed
in a moveable tube. If the moveable lens be directed to-
wards the object which is to be viewed or copied, and mo-
ved nearer to or farther from the mirror, till the imaoe is
distinctly formed on a greyed glass, laid upon that surface
of the upper lens which is next the eye, it will be found
m'ore sharp and vivid than those formed in the common in-
struments; because the image, is received so near the up-
per lens. And by increasing the diameter and curvature
of the lenses, the effect will be much heightened.

CAMPANI (Matthew), an eminent Italian optician,
born in the diocese of Spoletto, in the 17th century, but
resided chiefly at Rome, where he was very eminent for
his mechanical contrivances of optical and philosophical
instruments, and for several books on such subjects; par-
ticularly relating to pendulums, or timepieces, for the lon-
gitude; also to spectacles, and to telescopes, of which it
is said he made the very best and largest of his time. It is
said it was by his telescopes that the first Cassini made his
best observations and chief discoveries in the heavens; and
they still show at Paris, among the astronomical antiqui-
ties, several of Campani's object-glasses, of 8, 10, and 12
inches diameter. He was curate at Rome, and much re-
spected.

CAMPANI (Joseph), younger brother and pupil to
Matthew", above mentioned, was also very skilful in mecha-
nics and physic, and executed his brother's plans with great
ingenuity. They were both living in ]6"78.

CAMUS (Chakles-Stephen-Lewis), a celebrated
French mathematician, examiner of the Royal Schools of
Artillery and Engineers, secretary and professor of the
Royal Academy of Architecture, honorary member of that
of the Marine, and fellow of the Royal Society of London,
was born at Cressy en Brie, the 25th of August l6"99. His
early ingenuity in mechanics and his own entreaties, in-
duced his parents to send him to study at a college in Pa-
ris, at 10 years of age; where in the space of two years his
progress was so great, that he was able to give lessons in
mathematics, arid thus to defray his own expenses at the
college, without any further charge to his parents. By the
assistance of the celebrated Varignon, young Camus soon
ran through the course of the higher mathematics, and ac-
quired a name among the learned. He made himself more
particularly known to the Academy of Sciences in 1727,
by his' memoir on the subject of the prize which they had
proposed for that year, viz, " To determine the most ad-
vantageous way of masting ships;" in consequence of which
he was named that year Adjoint-Mechanician to the Aca-
demy; and in 1730 he was appointed professor of archi-
tecture. In less than three years after, he was honoured
with the secretaryship of the same; and the ISth of April
1733, he obtained the degree of Associate in the Academy,
where he distinguished himseli greatly by his memoirs on
living forces, or bodies in motion acted on I)}' forces, on
the figure of the teeth of wheels and pinions, on pump
work, and several other ingenious memoirs.

In 1736 he was sent, in company with Messieurs Clai-

CAN

[

]

CAN

raut, Maupertuis, and Monnier, on the celebrated expedi-
tion to measure a degree at the north polar circle; in which
he rendered himself highly useful, not only as a mathema-
tician, but also as a mechanician and aa artist, branches
for which he had a remarkable talent.

In 1741 Camus had the honour to be appointed Pen-
sioner-Geometrician in the Academy ; and the same year
he invented a gauging-rod and sliding-rule, proper at once
jo gauge all sorts of casks, and to calculate their contents.
About the year 17*7 he was named Examiner of the
Schools of Artillery and Engineers: and, in 1756, one of
the eight mathematicians appointed to examine by a new
measurement, the base which had formerly been measured
by Picard, between Villejuifve and Juvisi ; an operation
in which his ingenuity and exactness were of great utility.
In 1765 M. Camus was elected a fellow of the Royal So-
ciety of London; and died the 4th of May 17<5S, in the
69th year of his age; being succeeded by the celebrated
d'Alembert in his office of Geometrician in the French Aca-
demy; and leaving behind him a great number of manu-
script treatises on various branches of the mathematics.

The works published by M. Camus are :

1. Course of Mathematics for the use of the Engineers,
4 vols, in Svo.

2. Elements of Mechanics.

3. Elements of Arithmetic.

And his memoirs in the volumes of the Academy are:

1. Of Accelerated Motions by living forces: vol. for 1728.

2. Solution of a Geomet. Prob. of M. Cramer: 1732.

3. On the figure of the teeth and pinion's in clocks: 1733.

4. On the action of a musket-ball, piercing a pretty
thick piece of wood, without communicating any consi-
derable velocity to it: 1738.

5. On the best manner of employing buckets for raising
water: 1739.

6. A Problem in Statics: 1740.

7- On an Instrument for gauging of vessels: 1741.

8. On the Standard of the Ell Measure: 174fj.

9. On the tangents of Points common to several branches
of the same curve: 1747.

10. On the operations in measuring the distance between
the centres of the pyramids of Villejuifve and Juvisi, to
discover the best measure of the degree about Paris: 1754.

1 1. On the Masting of Ships : Prize torn. 2.

12. The Manner of working Oars: Mach. torn. 2.

13. A Machine for moving many Colters at once: Mach.
torn. 2.

CANAL, in general, denotes a long, round, hollow in-
strument, through which a fluid matter may be conveyed.
In which sense, it amounts to the same as what is other-
wise called a pipe, tube, channel, &c. Thus the canal of
an aqueduct, is the part through which the water passes;
which, in the ancient works of this kind, is lined with a
coat of mastic of a peculiar composition.

Canal more particularly denotes a kind of artificial
river, often furnished with locks and sluices, and sustained
by banks or mounds. They are contrived for divers pur-
poses: some for forming a communication between one
place and another; as the canals between Bruges and
Ghent, or between Brussels and Antwerp: others for the
decoration of a garden, or house of pleasure ; as the canals
of Versailles, I-'ontainbleau, St. James's Park, &c : and
others are made for draining wet and marshy lands; which
last, however, are more properly called water-gangs, drains,
ditches, &c.

It is needless to enumerate the many advantages arising
from canals and artificip.l navigations. Their utility is now
so apparent, that most nations in Europe give the highest
encouragement to undertakings of this kind, wherever they
are practicable. Nor did their advantages escape the ob-
servation of the ancients ; for in the earliest accounts of
society we read of attempts to cut through large isthmuses,
to make communications by water, either between one sea
and another, or between different nations, or distant parts
of the same nation, where land-carriage was tedious and
expensive.

Egypt is full of canals, dug to-receive and distribute the
waters of the Nile, at the time of its inundation. They
are dry the rest of the year, except the Canal of Joseph,
and four or five others, which may be ranked as consi-
derable rivers. There were also subterraneous canals, or
tunnels, dug by an ancient king of Egypt, by which those
lakes, formed by the inundations of the Nile, were con-
veyed into the Mediterranean Sea.

A new canal for conveying the waters of the Nile from
Ethiopia into the Red Sea without passing into Egypt, was
projected by Albuquerque, viceroy of India for the Por-
tuguese, to render Egypt barren and unprofitable to the
Turks. And M. Gaildereau attributes the frequency of
the plague in Egypt, of late days, to the decay or stopping
up of the canals; which happened upon the Turks be-
coming masters of the country.

Herodotus relates, that the Cnidians, a people of Coria,
in Asia Minor, designed to cut through the isthmus which
joins that peninsula to the continent; but were supersti-
tious enough to give up the undertaking, because it was
interdicted by an oracle.

Several kings of Egypt attempted to join the Red Sea
to the Mediterranean ; a project which Cleopatra was very
fond of. This canal was begun, according to Herodotus,
by Necus son of Psammeticus, who desisted from the at-
tempt on an answer from the oracle, after having lost 120
thousand men in the enterprise. It was resumed and com-
pleted by Darius son of Hystaspes, or, according to Dio-
dorus and Strabo, by Ptolemy Philadelphus ; who relate
that Darius relinquished the work on a representation
made to him by unskilful engineers, that the Red Sea, being
higher than the land of Egypt, would overflow and drown
the whole country. It was wide enough for two galleys to
pass abreast, and its length was four days sailing. Dio-
dorus adds, that it was also called Ptolemy's river; that
this prince built a city at its mouth on the Red Sea, which
he called Arsinoe, from the name of his favourite sister;
and that the canal might be cither opened or shut, as oc-
casion required. Diod. Sic. lib. 1 ; Strabo, Geog. lib. 17;
Herod, lib. 2. Soliman the 2d, emperor of the Turks, em-
ployed 50 thousand men in this great work; which was
completed under the caliphate of Omar, about the year
635 ; but it was afterwards allowed to fall into neglect and
disrepair ; so that it is now difficult to discover any traces
of it. Hist. Acad. Scienc. ann. 1703, pa. 110.

Both the Greeks and Romans intended to make a canal
across the Isthmus of Corinth, which joins the Morea and
Achaia, for a navigable passage by the Ionian Sea into the
Archipelago. Demetrius, Julius Caisar, Caligula, and
Nero, made several unsuccessful efforts to open this pas-
sage. But as the ancients were entirely ignorant of the
use of water-locks, their whole attention was employed in
making level cuts, which is probably the chief reason why
they so often failed in their attempts. Charlemagne formed
2 M 2

CAN [26

a design of joining the Rhine and the Danube, to make a
communication between the Ocean and the Black Sea, by
a canal from the river Almutz which discharges itself into
the Danube, to the Reditz, which falls into the Maine,
which last falls into the Rhine near Mayence or Mentz :
for this purpose he employed a prodigious number of work-
men; but he met with so many obstacles from different
quarters, that he was obliged to relinquish the attempt.

In China there is scarce a town or village without the
advantage either of an arm of the sea, a navigable river,
or a canal, by which means navigation is rendered so com-
mon, that there are almost as many people on the water
as the land. The great canal of China is one of the won-
ders of art, extending from north to south quite across the
empire, from Pekin to Canton, a distance of 825 miles, and
was made upwards of 900 years ago. Its breadth and
depth are sufficient to carry barks of considerable burthen,
which are managed by sails and masts, as well as towed
by hand. On this canal it seems the emperor employs
near ten thousand ships. It passes through, or by, 41
large cities; there are in it 75 vast locks and sluices, to
keep up the water, and pass the ships where the ground
will not admit of sufficient depth of channel, besides se-
veral thousand draw- and other bridges. Indeed, F. Ma-
gaillane assures us, there are passages by canals from one
end of China to the other, for the length of 6Y)0 French
leagues : and that a person may go from one end of the
empire to the other, either by canals or rivers, except a
single day's journey by land in crossing a mountain.

The French at present have many fine canals. That of
Briere, otherwise called the Canal of Burgundy, was begun
under Henry iv, and finished under the direction of Car-
dinal Richelieu in the reign of Louis xm.- This canal
makes a communication between the Loire and the Seine,
and so to Paris. It extends 1 1 French great leagues from
Briere to Montargis, and has 42 locks upon it.

The Canal of Orleans was begun in l6'75, for establish-
ing a communication also between the Seine and the Loire.
It is considerably shorter than that of Briere, and has only
20 sluices.

The Canal of Bourbon was but lately undertaken: its
design is to make a communication from the river Oise to
Paris.

But the greatest and most useful work of this kind, is
the junction of the Ocean with the Mediterranean by the
Canal of Languedoc, called also the Canal of the Two Seas.
It was proposed in the reigns of Francis i, and Henry iv,
and was begun and finished under Louis xi v ; having been
planned by Francis Riquet in I he year l6"6'6\ and finished
before his death, which happened in 1680. It begins with
a large reservoir 4000 paces in circumference, and 24 feet
deep, which receives many springs from the mountain
Noire. The canal is about 200 miles in length, extending
from Narbonne loTholouse, being supplied by a number
of rivulets in the way, and furnished with 104 locks or
sluices, of abdut 8 feet rise each. In some places it is
carried over bridges and aqueducts of vast height, which
give passage underneath to other rivers; and in some places
it is cut through solid rocks for a mile together.

The new canal of the lake Ladoga, cut from Volhowa
to the Neva, by which a communication is made between
the Baltic, or rather Ocean, and the Caspian Sea, was
begun by the czar Peter the 1st, in 1 7 J i> : by means of
which the English and Dutch merchandise is easily con-

]

CAN

veyed into Persia, without being obliged to double the
Cape of Good Hope. There was a former canal of com-
munication between the Ladoga lake and the river Wolga,
by which timber and other goods had been brought from
Persia to Petersburg; but the navigation of it was so dan-
gerous, that a new one was undertaken.

The Spaniards have several times had in view the dig-
ging a canal through the Isthmus of Darien, between North
and South America, from Panama to Nombre de Dios, to
make a ready communication between the Atlantic and the
South Sea, and thus afford a straight passage to China and
the East Indies.

In the Dutch, Austrian, and French Netherlands, there
is a great number of canals: that from Bruges to Ostend
carries vessels of 200 tons. But it would be an endless
task to describe the numberless canals in Holland, Ger-
many, Russia, &c. We will therefore only take a vie\y of
those in our own country.

In England, that ancient canal from the river Nyne, a
little below Peterborough, to the river Witham, three miles
below Lincoln, called by the modern inhabitants Caerdike,
may be ranked among the monuments of the Roman gran-
deur, though it is now most of it filled up. Morton will
have it made under the emperor Domitian. Urns and
medals have been discovered on the banks of this canal,
which appear to confirm that opinion. Yet some authors
take it to be a Danish work. It was 40 miles in length ;
and, so far as appears from the ruins, must have been very
broad and deep. Notwithstanding that early beginning,
it is but very lately that canals have been revived in this
country. They are now however become very numerous,
particularly in the counties of York, Lincoln, and Cheshire.
Most of the counties between the mouth of the Thames and
the Bristol Channel are connected, either by natural or ar-
tificial navigations; those upon the Thames and Isis reach-
ing within 20 miles of those upon the Severn.

The canal for supplying London with water by means
of the New River, was projected and begun by Mr. Ed-
ward Wright, author of the celebrated treatise on Navi-
gation, about the yeai\l6'0S ; but finished by Mr. (after-
wards Sir Hugh) Middleton, five years after. This canal
commences near Ware, in Hertfordshire, and takes a
course of 60 miles before it reaches the cistern at Islington,
which supplies several water-pipes that convey it to the
city and parts adjacent. In some places it is 30 feet deep,
and in others it is conveyed over a valley between two
hills, by means of a trough supported by wooden arches,
and rising above 23 feet in height.

The Dukeof Bridgewater'sCanal, projected and executed
under the direction of Mr. Brindley, was begun about the
year J 759. It was first designed only for conveying coals
to Manchester, from a mine in the duke's estate; but lias
since been applied to many other useful purposes of in-
land navigation. This canal begins at a place called
Worsley Mill, 7 miles from Manchester, where a bason is
made capable of holding all the boats, and a great body of
water which serves as a reservoir or head to the naviga-
tion. The canal runs through a hill by a subterraneous
passage, large enough for admitting long Sat- bottomed
boats, which are towed by a rail on each hand, mar three
quarters of a mile, to the coal-works. There the passage
divides into two channels, one of which goes off 300 yards
to the right, and the other as many to the left; and both
may be continued at pleasure. The passage is in some

CAN

[ 269 J

CAN

places cut through the solid 'rock, and ill others arched
over with brick ; and air-funnels, some of which are near
37 yards perpendicular, are cut, at certain distances,
through the rock to the top of the hill. The arch at its
entrance is about 6' feet wide, and about 5 feet .high from
the surface of the water ; but widens within, so that in some
places the boats may pass one another, and at the pits it
is 10 feet wide. When the boats are loaded and brought
out of the bason, five or six of them are linked together,
and drawn along the cabal by a single horse, and thus
reaching Manchester in a course of 9 miles. It is broad
enough for two barges to pass or go abreast; and on one
side there is a good road for the passage of the people, and
the horses or mules employed in the work. Tlie canal is
raised over public roads by means of arches; and-it passes
over the navigable river Irwell near 50 feet above it; so
that large vessels in full sail pass under the canal, while
the duke's barges are at the same time passing over them.
This canal joins that which passes from the river Mersey
towards the Trent, taking in the whole a course of 34 miles.

The Lancaster Canal begins near Kendal, and terminates
near Eccleston, comprehending the distance of 72| miles.

The canal from Liverpool to Leeds is 108-j miles in
length: that from Leeds to Selby, 23J miles; from Chi-
chester to Middlewich, 265 miles; from the Trent to the
Mersey, 88 miles ; from the Trent to the Severn, 4-6| miles.
The Birmingham Canal joins this near Wolverhampton,
and is 24f miles: the Droitwich Canal is b\ miles: the
Coventry Canal, commencing near Lichfield, and joining
that of the Trent, is 36'J miles : the Oxford Canal breaks
off from this, and is 82 miles: the Chesterfield Canal joins
the Trent near Gainsborough, and is 41- miles.

A communication is now formed, by means of this in-
land navigation, between Kendal and London, by way of
Oxford ; between Liverpool and Hull, by the way of Leeds;
and between the Bristol Channel and the Humber, by the
junction formed between the Trent and the Severn. Other
schemes have been projected, which the present spirit of
improvement will probably soon carry into execution, of
opening a communication between the German and Irish
seas, so as to reduce a hazardous navigation of more than
800 miles by sea, into a little more than 1 50 miles by land,
or inland navigation; and also of joining the lsis with the
Severn ; besides many other canals in various parts of
England.

In Scotland, a navigable canal between the Forth and
Clyde, which divides that country into two paits, was me-
ditated upon mote than a century since, for transports and
small ships of war. It was again projected in the ye;ir
1722, and a survey made; but nothing more was done tiil
176"l, when the then Lord Napier, at his own expense, had
a survey, plan, and estimate, made on a small scale. In
1764, the trustees for fisheries, &c, in Scotland, procured
another survey, plan, and estimate, of a canal 5 feet deep,
which was to cost 79,000 pounds. In 1766, a subscrip-
tion was obtained by a number of the most respectable
merchants in Ghisgow, for making a canal 4 feet deep and
24 feet in breadth; but when the bill was nearly obtained
in parliament, it was given up on account of the smallness
of the scale, and a new subscription set on foot for a canal
7 feet deep, estimated at 150,000 pounds. This obtained
the sanction of parliament; and the work was begun in
1768, by Mr. Smeaton the engineer. The extreme length
of the canal from the Forth to the Clyde is 35 miles, be-
gi lining at the. mouth of the Carron, and eliding at Dalmure

Burnfoot on the Clyde, 6 miles below Glasgow, rising and
falling 160 feet by means of 39 locks, 20 on the east side
of the summit, and 19 on the west, as the tide does not
ebb so low in the Clyde as in the Forth by 9 feet; and it
was deepened to upwards of 8 feet. This canal was finish-
ed a few years since, after having experienced some inter-
ruptions and delays, for want of resources, and is esteemed
the greatest work of the kind in this island. Vessels draw-
ing S feet water, with 19 feet in the beam and 73 feet in
length, pass with ease; and the whole enterprise displays
the art of man in a high degree. To supply the canal with
water was of itself a very great work. There is one reser-
voir of 50 acres 24 feet deep, and another of 70 acres 22
feet deep, in which many rivers and springs terminate,
which it is expected will afford sufficient supply of water
at all times. '

The Practice of Canal Digging and Inland Navigations.

The particular operations necessary for making artificial
navigations depend upon a variety of circumstances. The
situation of the ground; the vicinity or connexion of it
with rivers; the ease or difficulty with which a proper
quantity of water can be obtained: these and many other
circumstances necessarily produce great variation in the
structure of artificial navigations, and augment or diminish
the labour and expense of- executing them. When the
ground is naturally level, and unconnected with rivers, the
execution is easy, and the navigation is not liable to be dis-
turbed by floods: but when the ground risijs and falls, and
cannot be reduced to a level, artificial methods of raising
and lowering vessels must be employed; which also vary
according to circumstances.

Sometimes a kind of temporary sluices are employed,
to raise boats over falls or shoals in rivers, by a very sim-
ple operation. Two pillars of mason-work, with grooves,
are fixed, one on each bank of the river, at some distance
below the shoal. The boat having passed these pillars,
strong planks are let down across the river by pulleys into
the grooves; by which means the water is dammed up to
a proper height for allowing the boat to pass up the rivev
over the shoal.

The Dutch and Flemings at this day sometimes, when
obstructed by cascades, form an inclined plane or rolling-
bridge upon dryland, along which their vessels are drawn
from the river below the cascade, into the river above it.
This, it is said, was the only method employed by the an-
cients, and is still sometimes used by the Chinese. These
rolling-bridges consist of a number of cylindrical rollers
which turn easily on pivots. And a mill is commonly
built near; so that the same machinery may serve the dou-
ble purpose of working the mill and drawing up vessels.

But in the present improved state of inland navigation,
the^e falls and shoals are commonly surmounted by means
of what are called locks or sluices. A lock is a bason
placed lengthwise in a river or canal, lined with walls of
masonry on each side, and terminated by two gates placed
across the canal, where there is a cascade or natural fall
of the country; and- so constructed, that the bason being
filled with water by an upper sluice to the level of the
waters above, a vessel may ascend through the upper gate ;.
or the water in the lock being reduced to the level of the
water at the bottom of the cascade, the vessel may descend
through the lower gate : for when the waters are brought
to a level on either side, the gate on that side may be easily
opened.

But as the lower gate is strained in proportion to the

CAN

C 270 ]

CAN

depth of water it supports, when the perpendicular height
of the water exceeds 12 or 13 feet, it becomes necessary to
have more locks than one. Thus, if the fall be l6 feet,
two locks are required, each of 8 feet fall ; and if the fall
be 25 feet, three locks are necessary, each having 8 feet 4
inches fall. — It is evident that the side-walls of locks
should be made very strong : and where the natural foun-
dation is bad, they should be founded on piles and plat-
forms of wood. They should likewise slope outwards, in
order to resist the pressure of the earth from behind.

To illustrate this by representations : Plate 6, fig. 1, is
a perspective view of part of a canal, with several locks,
&c; the vessel L being wilhin the lock AC. — Fig. 2 is an
elevation or upright section along the canal ; the vessel l
about to enter. — Fig. 3, a like section of a lock full of
water; the vessel l being raised to a level with the water
in the superior canal. — And fig. 4 is the plan or ground
section of a lock : where l is a vessel in the inferior canal ;
C, the under gate ; a, the upper gate ; gh, a subterraneous
passage for letting water from the superior canal run into
the lock ; and kf, a subterraneous passage for water from
the lock to the interior canal.

x and t (fig. 1) are the two flood-gates, each of which
consists of two leaves, pressing against one another, so as
to form an obtuse angle, the better to resist the pressure
of the water. The first (x) prevents the water of the su-
perior canal from falling into the lock; and the second (y)
dams up and sustains the water in the lock. These flood-
gates ought to be very strong, and to turn freely on their
hinges. They should also be made very tight and close,
that as little water as possible may be lost. And, to make
them open and shut with ease, each leaf is furnished with
a long lever a 6, a6; cb, cb.

By the subterraneous passage gh (fig. 2, 3, 4) which
descends obliquely, by opening the sluice g, the water is
let down from the superior canal d, into the lock, where
it is stopped and retained by the gate c when shut, till the
water in the lock be on a level with the water in the su-
perior canal d ; as represented in fig. 3. When, on the
other hand, the water contained by the lock is to be let
out, the passage gh must be shut, by letting down the
sluice g ; the gate a must also be shut, and the passage
kf opened by raising the sluice K. A free passage being
thus given to the water, it descends through kf, into the
inferior canal, until the water in the lock be on a level with
the water in the inferior canal b; as represented in fig. 2.

Now suppose it be required to raise the vessel l (fig. 2)
from the inferior canal b to the superior one D. If the
lock be full of water, the sluice c. must be shut, as also
the gate a, and the sluice k opened, so that the water in
the lock may run out till it becomes level with the water
in the inferior canal b. When the water in the lock is
on a level with the water at b, the leaves of the gate c are
opened by the levers cb, which is easily performed, the
water on each side of the gate being in equilibrio ; the
vessel then sails into the luck. After this, the gate c and
the sluice k are shut, and the sluice g opened, in order
to fill the lock, till the water in the lock, and consequently
the vessel, be upon a level with the water in the superior
canal d; as is represented in fig. 3. The gate a is then
opened, and the vessel passes into the canal d.

Again, let it be required to make a vessel descend from
the canal d into [he inferior canal B. If the lock be
empty as in lig. 2, the gate c and sluice k must be shut,
and the upper sluice g opened, so that the water in the

lock may rise to a level with the water in the upper canal
d. Then, opening the gate a, the vessel will pass through
into the lock. This done, shut the gate a and the sluice
G ; then open the sluice k, till the water in the lock be
on a level with the water in the inferior canal ; this done,
the gate c is opened, and the vessel proceeds into the ca-
nal b, as was required.

Other contrivances have lately been made for raising and
lowering the vessels between the under and upper canal.

CANCER, the Crab, one of the twelve signs of the zo-
diac, usually drawn on the globe in the form of a crab, and
in books of astronomy denoted by a character resembling
the number sixty-nine, turned sideways, thus ss.

The number of stars in the sign Cancer, Ptolemy makes

13, Tycho 15, Bayer and Hevelius 29, and Flamsteed S3.

Tropic of Cancer, a little circle of the sphere parallel

to the equinoctial, and passing through the beginning of

the sign Cancer.

CANDLEMAS, or the Purification, a feast of the
church, held on the 2d of February, in memory of the pu-
rification of the Virgin; taking its name of Candlemas,
either from the number of lighted candles used by the Ro-
mish church in the processions of this day, or because that
the church then consecrated candles for the whole year.

CANES Venatici, the Hounds, or the Greyhounds, one of
the new constellations of the northern hemisphere, which
Hevelius has formed out of the unformed stars of the old
catalogues. These two dogs are further distinguished by
the names of Asterion and Chara. They contain 23 stars
according to Hevelius, but 25 in the British catalogue.

CANICULAR, a name given, by many of the earlier
astronomers to the constellation which we call the Lesser
Dog, and Canis Minor, but some Procyon and Antecanis.
See Canis Minor.

It is also used for one of the stars of the constellation
Canis Major; called also simply the Dog-star ; and by the
Greeks 'S.sigio;, Sirius. It is situated in the mouth of the
constellation, and is the largest and brightest of all the
stars in the heavens. From the heliacal rising of this star,
that is, its emersion from the sun's rays, which now hap-
pens with us about the 11th of August, the ancients rec-
koned their DiesCaniculares, or Dog-days.

The Egyptians and Ethiopians began their year at the
heliacal rising of Canicula; reckoning to its rise again the
next year, which is called the Annus Canarius.

CANICULAR Days, or Dog-days, denote a certain
number of days, before and after the heliacal rising of ca-
nicula, or the dog-star, in the morning. The ancients
imagined that this rising of the star occasioned the sultry
weather usually felt in the latter part of the summer, or
dog-days ; with all the distempers of that sickly season :
Homer's II. lib. 5, v. 10, and Virgil's^n. lib. 10, v. 270.
Some authors say, from Hippocrates ami Pliny, that the
day this star first rises in the morning, the sea boils, wine
turns sour, dogs begin to grow mad, the bile increases
and irritates, and all animals grow languid ; also that the
diseases it usually occasions in men, are burning fevers,
dysenteries, and phrensies. The Romans sacrificed a
brown dog every year to Canicula at his rising, to ap-
pease its rage. All this however arose from the ground-
less idea that the dog-star, so rising, was the occasion of
the. extreme heat and the diseases of that season : for the
rising of the star not only varies, in any one year, as the
latitude varies, but is always later and later every year in
all latitudes ; so that in time the star may, by the same

C A N

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C A N

rule, be charged with bringing frost and snow, when he
rises in winter.

The dog-days were commonly reckoned about 40, viz,
20 days before and 20 days after the heliacal rising ; and
almanac-makers have usually set down the dog-days in
their almanacs to the changing^ time of the star's rising,
by which means they had at length fallen considerably
after the hottest time of the year, till of late we have ob-
served an alteration of them in the almanacs,'and very
properly, from July 3 to August 11. For, by the dog-
days, the ancients meant to express the hottest time of the
year, which is commonly during the month of July, about
which time the dog-star rose heliacally in the time of the
most ancient astronomers that we know of: but the pre-
cession of the equinoxes has carried this heliacal rising in-
to a much later and cooler part of the year ; and because
Hesiod tells us that the hot time of the year ends on the
50th day after the summer solstice, which brings us to
about August 10 or M', the above alteration appears to
be \cry proper.

Canicular Year, denotes the Egyptian natural year,
which was computed from one heliacal rising of Canicula
to the next. This year was also called annus canarius,
and annus cynicus ; and by the Egyptians themselves the
Sethicyear, from Seth, by which name they called Sirius.
Some call it also the heliacal year. This year consisted
ordinarily of 365 days, and every 4th year of 366 ; by
which means it was accommodated to the civil year, like
the Julian account. And the reason why they chose this
star, in preference to others, to compute their time by,
was not only the superior brightness of that star, but be-
cause in Egypt its heliacal rising was a time of very sin-
gular note, as coinciding with the greatest augmentation
of the Nile, the reputed father of Egypt. Ephestion adds,
that from the aspect of canicula, its colour, &c, the Egyp-
tians drew prognostics concerning the rise of the Nile;
and according to Florus, predicted the future state of the
year. So it is no wonder that the first rising of this star
was observed with great attention. Bainbrigge, Canicul.
cap. 4, p. 26.

CANIS Major, the Great Dog, a constellation of the
southern hemisphere, below the feet of Orion, and one of
the old 4S constellations. The Greeks, as usual, have
many fables of their own about the exaltation of the dog
into the skies; but the origin of this constellation, as well
as its other name Sirius, lies more probably anion" the
Egyptians, who carefully watched the rising of this°star,
and by it judged of the swelling of the Nile, calling the
star the sentinel and watch of the year; and hence, ac-
cording to their manner of hieroglyphic writing, repre-
sented it under the figure of a dog. They also called the
Nile Siris; and hence their Osiris.

The stars in thisconstellation, Ptolemy makes29; Tycho
however observed only 13, and Hevelius 21 ; but in Flam-
steed's catalogue they are 31.

Camis Minor, a constellation of the northern hemi-
sphere, just below Gemini, and is one of the 48 old con-
stellations. The Greeks fabled that this is one of Orion's
hounds; but' the Egyptians were most probably the in-
ventors of this constellation, and they may have given it
this figure to express a little dog, or watchful creature, go-
ing before, as conductor of the larger, or rising before it :
and hence the Latins have called it Antecanis, the star
before the dog.

The stars in this constellation are, in Ptolemy's cata-

logue 2, the principal of which is the star Procyon; in
Tycho's, 5; in Hevelius's, 13; and in Elamsteed's, 14.

CANNON, in Military Affairs, a long round hollow
engine, made of iron or brass, &c, for throwing balls, &c,
by means of gunpowder. The length is distinguished into
three parts ; the first reinforce, the second reinforce, and
the chase: the inside hollow where the charge is lodged,
being also called the chase, or bore. But for the several
parts and members of a cannon, see Astragal, Base-
king, Bore, Breech, Cascabel, Chase, Muzzle,
Ogee, Rein force-ring, Trunnions, &c. See also
Gun, and Gunnery. \

Cannon were first made of several bars of iron adapted
to each other lengthways, and hooped together with strong
iron rings. They were employed in throwing stones and
metal of several hundred weight. Others were made of
thin sheets of iron rolled up, and hooped : and on emer-
gencies they have been made of leather, with plates of iron
or copper. They are now constituted of cast-iron or brass;
being cast solid, and the tube bored out of the middle by
means of a very strong machine for that purpose. When
or by whom cannon were first invented is uncertain.

Larrey makes brass cannon the invention of J. Owen;
and asserts that the first known in England, were in 1535 ;.
and further that iron cannon were first cast here in 1547.
He acknowledged that cannon were known before; and re-
marks that at the battle of Cressy, in 1346, there were 5
pieces of cannon in the English army, which were the first
ever seen in France. Mezeray also observes, that King
Edward struck terror into theFrench army, by 5 or6pieces
of cannon ; it being the first time they had met such thun-
dering machines.

In the list of aids raised for the redemption of King John
of France, in 1308, mention is made of an officer in the
French army called Master of the King's Cannon, and of
his providing 4 large cannon for the garrison of Harfleur.
But Father Daniel, in his Life of Philip of Valois, produces
a proof from the records of the chamber of accounts at
Paris, that cannon and gunpowder were used in the year
1338. And D(u-Cange even finds mention of the same
engines in Froissart, and other French historians, sometime
earlier.

The Germans carry the invention of cannon farther
back, and ascribe it to Albertus Magnus, a Dominican,
monk, about the year 1250. But Isaac Vossius finds
cannon in China upwards of 1700 years ago; being used,
by the emperor Kitey, in the year of Christ 85. The an-
cients too, of Europe and Asia, had their tier)' tubes, or
Cannae, which being loaded with pitch, stones, and iron;
balls, were exploded with a vehement noise,, smoke, arid*
great effect.

Cannon were formerly made of a very great length,.which
rendered them exceedingly heavy, and their use very
troublesome and Confined. But it has lately been found
by experiment, that there is very little added to the force of
the ball by a great length of the cannon; and therefore they
have very properly been much reduced both in their length
and weight, and rendered easily manageable on all occa-
sions. They were formerly distinguished by many hardi
and terrible names, but are now only named from the
weight of their ball;, as a 6 pounder, a 12 pounder, a 24
pounder, or a 36 pounder, which is the largest size now.
used by the English for battering.

CANON, in Algebra, Arithmetic, Geometry, &c, is a
general rule for resolving all cases of a like nature with the:

CAN

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CAN

present inquiry. Thus the last step of every equation is
a canon, and if turned into words, becomes a rule to re-
solve all cases or questions of the same kind with that pro-
posed:

Tables of sines, tangents, &c, whether natural or arti-
ficial, are also called canons.

Canon, in Ancient Music, is a method of determining
the intervals of musical notes. Ptolemy, rejecting the Ari-
stoxenian way of measuring the intervals in music by the
magnitude of a tone, formed by the difference between a
diapente and a diatessaron, thought that they should be
distinguished by the ratios which the sounds terminating
those intervals bear to one another, when considered ac-
cording to their degree of acuteness or gravity ; which, be-
fore Anstoxenus, was the old Pythagorean method." He
therefore made the diapason consist in a double ratio ; the
diapenfe consist in a sesquialterate ; the diatessaron, in a
sesquitertian ; and the tone itself, in a sesquioctave; and
all the other intervals, according to the proportion of the
sounds that terminate them : therefore, taking the canon,
for a determinate line of any length, he shows in what man-
ner it must be cut, to represent the respective intervals :
and this method answers exactly to experiments in the dif-
ferent lengths of musical chords. From this canon, Ptolemy
and his followers have been called Canonici ; as those of
Aristoxenus were called Musici.

Canon, in Modern Music, is a short composition of
two or more parts, in which one leads and the other fol-
lows; and is a fugue so bound up, or restrained, that the
following part or parts must precisely repeat the same
notes, with the same degrees rising or falling, which were
expressed by the, leading part; it is therefore tied to so
strict a rule, that it is called canon.

Pascal Canon, a table of the moveable feasts, showing
the day of Easter, and the other feasts depending on. it,
for a cycle or period of 19 years. It is said that the Pascal
Canon was the calculation of Euscbius of Cassarea, and
that it was made by order of the council of Nice.

CANOPUS, a name given by some of the old astrono-
mers to a star under the 2d bend of Eridanus. These
writers say that the river in the heavens is not the Eridanus,
but the Nile, and that this star commemorates an island
made by that river, which was called by the same name.

Canopus is also the name of a bright star of the first
magnitude in the rudder of Argo, one of the southern
constellations. Its situation, as given by several authors,
at different times, is as follows:

Authors

Dates

Longit.

Lat.

Rt. Aseen.

Declin. So.

F. Thomas

Jan. 1682

8° 05 52'

75° 15'

93° 32' 20"

S2° 31' 33"

F. Noel

1697

93 54

52 29

for. Halley

1700

10 52

72 49

F. Feuille

Mar. 1709

52 30 4

CANTALIVERS, in Architecture, arc the same with
modillions, except that the formerare plain, and the latter
carved. They are both a kind of cartouses, set at equal
distances under the corona of the cornice of a building.

CANTON (John), an ingenious natural philosopher,
was born at Stroud, in Gloucestershire, in 1718; and was
placed, when young, under the care of Mr. Davis, an able
mathematician of that place, with whom he learnt the
rudiments of arithmetic, &c. He next proceeded to higher
parts of the mathematics, and particularly to algebra and
astronomy, in which he had made a considerable progress,
when his father took him from school and set him to learn
his own business, which was that of a broad-cloth weaver.

This circumstance did not check his zeal for acquiring
knowledge. All his leisure time was devoted to the assi-
duous cultivation of science; and he had now acquired a
sufficient knowledge of astronomy to calculate lunar
eclipses and other phenomena, and to construct various
kinds of sun-dials, which he did at times when he ought
to have slept, and without the knowledge of his father,- who
feared that the prosecution of such studies might injure his
health. It was during this discouragement, and at these
hours, that he computed, and cut upon stone, with no
better an instrument than'a common knife, the lines of a
large upright sun-dial, on which, beside, the hour of the
day, were shown the sun's rising, his place in the ecliptic,
and some other particulars. When this was finished and
made known to his father, he permitted it to be placed
against the front of his house, where it excited the admi-
ration of several neighbouring gentlemen, and introduced
young Canton to their acquaintance, which was followed
by the offer of the use of their libraries. In the library of
one of these gentlemen he found Martin's Philosophical
Grammar, which was the first book that gave him a taste
for natural philosophy. In the possession of another gen-
tleman he first saw a pair of globes ; a circumstance that
afforded him great pleasure, from the great ease with which
he could resolve those problems he had hitherto been ac-
customed to compute.

Among other persons with whom our author became
acquainted in early life, was Dr. Henry Miles of Tooting;
who perceiving that he possessed abilities too promising to
be confined within the narrow limits of a country town,
prevailed on his father to permit him to come to London.
Accordingly he arrived at the metropolis the 4th of March
1737, and resided with Dr. Miles at Tooting till the 6th
of May following; when he articled himself, for the term
of 5 years, as an assistant to Mr. Samuel Watkins, master
of an academy in Spital Square. In this situation his in-
genuity, diligence, and prudence, were so distinguished,
that on the expiration of his servitude in Ma}' 1742, he
was taken into partnership with Mr. Watkins for 3 years ;
which gentleman he afterwards succeeded in the school,
and there continued during his whole life.

Towards the end of 1745, electricity received a great
improvement by the discovery of the famous Leyden phial.
This event turned the thoughts of most of the philoso-
phers of Europe to that branch of natural philosophy; and
Canton, who was one of the first to repeat and to pursue
the experiment, found his endeavours rewarded by many
discoveries. — Towards the end of 174£), he was engaged
with his friend, the late ingenious Benjamin Robins, in
making experiment? to determine the height to which
rockets may be made to ascend, and at what distance their
light may be seen. — In 1750 was read at the Royal So-
ciety Mr. Canton's " Method of making Artificial Mag-
nets, without the use of, and yet far superior to, any natu-
ral ones." This paper procured him the honour of being
elected a member of the Society, and the present of their
gold medal. The same year he was complimented with
the degree of M. A. by the university of Aberdeen. And
in 1751 he was chosen one of the council of the Royal
Society.

In J75'2, our philosopher was so fortunate as to be the
first person in England who, by attracting the electric fire
from the clouds during a thunder-storm, verified Dr.
Franklin's hypothesis of the similarity of lightning and
electricity. Next year his paper entitled " Electrical Ex-

CAN

[ 273 ]

CAP

periments, with an attempt to account for their several
phenomena," was read at the Royal Society; in which
Mr. Canton mentions his having discovered, by many ex-
periments, that some clouds were in a positive, and some
in a negative state of electricity: a discovery which was
also made by Dr. Franklin in America much about the
same time. This circumstance, together with our author's
constant defence of the doctor's, hypothesis, induced that
excellent philosopher, on his arrival in England, to pay
Mr. Canton a visit, and gave rise to a friendship which
ever after continued between them. — In the Ladies' Diary
for 1756, our author answered the prize query that had
been proposed in.the preceding year, concerning the meteor
called shooting-stars. The solution, though only signed
a. m. was so satisfactory to his friend, the excellent ma-
thematician Mr. Thomas Simpson, who then conducted
that ingenious and useful little work, that he sent Mr.
Canton the prize, accompanied with a note, in which he
said he was sure that he was not mistaken in the author
of it, as no correspondent besides him, could have given
that answer. — Our philosopher's next communication to
the public was a letter in the Gentleman's Magazine for
September 1759, on the electrical properties of the tour-
malin, in which the laws of that wonderful stone are laid
down in a very concise and elegant manner. On the 13th
of December in the same year, was read at the Royal So-
ciety, An attempt to account for the Regular Diurnal
Variation of the Horizontal Magnetic Needle ; and also for
its Irregular Variation at the time of an Aurora Borealis.
A complete year's observations of the diurnal variations
of the needle are annexed to the paper. — Nov. 5, 176l,
our author communicated to the Royal Society an ac-
count of the Transit of Venus of the 6th of June that year,
observed in Spital-square; and in 1762 a letter addressed
by him to Dr. Franklin was read, containing some remarks
on Mr. Delaval's electrical experiments. On the l6th of
Dec. the same year, another curious addition was made
by him to philosophical knowledge, in a paper entitled
Experiments to prove that Water is not Incompressible.
And on Nov. 8, the year following, were read before the
Society, his farther Experiments and Observations on
the Compressibility of Water, and some other Fluids.
These experiments are a complete refutation of the famous
Florentine experiment, which so many philosophers have
mentioned as a proof of the incompressibility of water. For
this communication he received the Society's gold medal.

In 1768 our author communicated to the Society, An
easy method of making a phosphorus that will imbibe and
emit light like iheBolognian Stone ; with experiments and
observations. When he first showed to Dr. Franklin the
instantaneous light acquired by some of this phosphorus
from the near discharge of an electrified bottle, the doctor
immediately exclaimed, " And God said, Let there be light,
and there was light."

The Dean and Chapter of St. Paul's having, in a letter,
dated March 6, 1769, requested the opinion of the Royal
Society relative to the best method of fixing electrical
conductors to preserve that cathedral from injury by light-
ning, Mr. Canton was one of the committee appointed to
take the letter into consideration, and to report their opi-
nion thereon. The gentlemen who joined Injn in this bu-
siness were, Mr. Delaval, Dr. Franklin, Dr. Watson, and
Mr. Wilson. Their report was made on the 8th of .June
following; and the mode roeonimendeif by them has been

Vol.1.

carried into execution. Our author's last communication
to' the Royal Society, was a paper read Dec. 21, 170Q,
containing Experiments to prove that the Luminousness
of the Sea arises from the Putrefaction of its animal Sub-
stances.

Besides the papers above mentioned, Mr. Canton wrote
a number of others, both in the earlier and the latter parts
of his life, which appeared in several publications, and
particularly in the Gentleman's Magazine. — He died of a
dropsy, the 22d of March, 1772, in the 54th year of his
age

CAPACITY, is the solid content of any body. Also
our hollow measures for corn, beer, wine, &c, are called
measures of capacity.

Capacity, in the Modern Doctrine of Heat, signifies
the proportional capability of a given quantity of any sub-
stance to absorb and retain caloric; or that disposition or
property by which various bodies respectively require more
or less of this fluid to superinduce any given temperature
in a given mass. See Caloric and Temperature.

CAPE, or Promontory, is any high land runnin-j out
with a point into the sea; as Cape Verde, Cape Horn, the
Cape of Good Hope, &c.

CAPELLA, a bright star of the first magnitude, in the
left shoulder of Auriga.

CAPILLARY Tubes, in Physics, are very minute pipes,
whose canals are exceedingly narrow; being so called from
their resemblance to a hair in smallness. Their usual dia-
meter may be from -^ to 3?^ of an inch : though Dr. Hooke
assures us that he drew tubes in the flame of a lamp much
smaller, and resembling a spider's thread.

The Ascent of Water, fyc, in capillary tubes, is a noted
phenomenon in philosophy. Take several small glass tubes,
of different diameters, and open at both ends; immerse
them a little way in water, and the fluid will be seen to
stand higher in the tubes than the surface of the water
without, and higher as the tube is smaller, almost in the
reciprocal ratio of the diameter of the.tube; and that both
in open air and in vacuo. The greatest height to which
Dr. Hooke ever observed the water to stand, in the smallest
tubes, was 21 inches above the surface in the vessel.

From this property, of a higher ascent in smaller tubes,
a neat experiment exhibits a pleasing appearance in this
manner: Select a number of these glass tubes all different
in their diameter, from the very smallest to the largest size;
fix these against a small piece of board, all parallel and
near to each other, with their lower ends all on a level,
and so as that the tubes may succeed each other in a re-
gular gradation, from the smallest to the largest; then dip
the whole machine into water with the lower ends equally
immersed, and it will be seen that, by the fluid ascending
to different heights in the tubes, the tops of the columns
will appear to form a regular curve line, being highest in
the smallest tube, and lowest in the largest; and if the
water be a little tinged with cochineal, or ink, 6ec, it will
render the effect more visible and pleasing. — A more pleas-
ing variety of this experiment js made in the following
manner: Take two equal square pieces of perfectly flat
glass, of any convenient size, 6' or 8 inches, for instance, on
each side; connect two of their equal edges close together
by any means, but separate their opposite edges a little
from each other, by fixing a small .bit of chip, &c, be-
tween them, so as to give the two the appearance of a very
thin wedge, which will represent, and have the effect of,
2 N

CAP

an infinite number of contiguous tubes as disposed paral-
lel to the said closed 'and opened edges; for then, on dip-
ping the bottoms in the colon it-.: water, this will be seen
_to ascend up between the two planes, highest on the side
next their closed edges, but gradually lower towards the
opposite sides, the top of the fluid forming a regular hy-
perbolic curve line, to which the closed edges and the bot-
tom are seen as the asymptotes.

This ascent and suspension of the water in the tube, is
by Dr. Jurin, Mi. Hauksbee, and other philosophers, as-
cribed to the attraction of the periphery of the concave
surface of the tube, to which the uppersurface of the water
is contiguous and adheres.

This does not however happen uniformly the same in all
fluids, some standing higher than others; and in quicksil-
ver the contrary takes place, as that fluid stands lower
within the tube than its surface in the vessel, and the lower
as the tube is smaller. See Philos. Trans. No. 355, &c, or
Cote's Hydr. and Pneum. Lect. p. 265. But these effects
suppose that the. tubes are used in the clean state in which
they are naturally presented ; for, by certain precautions,
mercury may be shown to have a like elevation above the
level. And if the interior of these tubes be coated over
with greasy matter, such as oil or tallow, the usual effect
will not take place, as the fluid will retain its level.

Ann her phenomenon of these tubes is, that such of them
as would only naturally discharge water by drops, when
electrified, yield a contrnued and accelerated stream ; and
the acceleration is proportional to the smallness of the
tube: indeed, the effect of electricity is so considerable,
that it produces q continued stream from a very small
tube, out of which the water had not before been able to
drop. Priestley's Hist. Electr. 8vo, vol. 1, p. 171, ed.
3d.

The celebrated philosopher, Laplace, has written pretty
fully on the theory of this capillary attraction, confirming
the opinion of Jurin and Hauksbee, &c, above mentioned,
and explaining it by mathematical calculations, in the
Supplement to the 10th book of his Celestial Mechanics,
on Capillary Attraction. On this subject, see also Dr.
Young's paper on the cohesion of fluids, in the Philos.
Trans, for 1805, or in his Natural Philos. vol. 2; or Gre-
gory's Mechanics, vol. l,p. 393; or, translation of Haiiy's
Nat. Philos. vol. 1, p. 158; Nicholson's Journal, Nos. 74,
76, N. S.; or Retrospect of Philosophical Discoveries, No.
10.

Among the various methods of determining the diame-
ter of a uniform capillary tube, the following appears the
best and most accurate. Pour into the tube a certain
quantity of mercury, whose weight in troy gKiins is iv, and
measure carefully the length, /, of the tube which it oc-
cupies, then is the diameter, d = -0192523 „/-. For the

specific gravity of mercury beingl3568, a cubic inch weighs
3435-16' grains: hence, 1 : d^l x '785398 :: 3435'l6
grains :w: . Whence, multiplying extremes and means, &c,

we get d = </ - , = '0192523 J%.

v a.vlb-K, x 78539S X / J v I

CAPITAL, in Architecture, the uppermost part of a
column or pilaster, serving as a head or crowning to it,
feeing placed immediately over the shaft, and under the
entablature. It is made differently in the different orders,
and is that indeed which chiefly distinguishes the orders
themselves.

[ 274 ] ' C A H

Capita l of a Basiion, is an imaginary line dividing any
work into two equal and similar parts; or a line drawn
from the angle of the polygon to the point of the bastion,
or from the point of the bastion to the middle of the
gorge:

CAPONiFRE, or Caponniere, in Fortification, is a
passage made from one work to another, of 10 or 12 feet
wide, and about five feet deep, covered on each side by a
parapet, terminating in a glacis or slope: sometimes it is
covered with planks und earth.

CAPRA, or the Ske-goat, a name given to the star Ca^
pella, on the left shoulder of Auriga; and sometimes to the
constellation Capricorn. Some again represent Capra as
a constellation in the northern hemisphere, consisting of 3
stars, comprised between the 45th and 55th degree of la-
titude. The poets fable her to be Amalthea's goat, which
suckled Jupiter in his infancy.

CAPRICORN, the Goat, a southern constellation, and
the 10th sign of the zodiac, as also one of the 48 original
constellations received by the Greeks from the Egyptians.
The figure of this sign is drawn as having the fore part of ;.
goat,- but the hinder part of a fish; and sometimes simply
under the form of a goat. In writing, it is denoted by a
character representing the crooked horns of a goat's head,
thus Vf .

The stars in this constellation, in Ptolemy's and Tycho's
catalogue, are 28; in that of Hevelius, 29; though it is to
be remarked that one of those in the tail, of the 6th mag-
nitude, marked the 27 th in Tycho's 'book, was lost in He-
velius's time. Flamsteed gives 51 stars to this sign.

Tropic ofCAPRicoRN, a small circle of the-sphere, pa-
rallel to the equator, passing through the beginning of Ca-
pricorn, or the winter solstice, or the point of the sun's
greatest south declination.

CAPSTAN, a large massy column shaped like a trun-
cated cone; being set upright on the deck of a ship, and
turned by levers or bars, passing through holes in its upper
extremity. The capstan is a kind of perpetual lever, or
an axis-in-peritrochio, which, by means of a strong rope
or cable passed round serves to raise very great weights;
such as to hoist sails, to weigh the anchor's, to draw the
vessels on shore, and hoist them up to be refitted, &c.

CAPUT Draconis, or dragon's head, a name given by
some to a fixed star of the first magnitude, in the head of
the constellation Draco.

CARACT, or Carat, a name given to the weight which
expresses the degree of goodness or fineness of gold. The
whole quantil of metal is considered as consisting of 24
parts, which arc the carats; so that the carat is the
24th part of the whole: this carat is divided into 4 equal
parts, called grains ot a carat, and the grains into halves
and quarters.

Winn gold is purified to the utmost degree possible, so
that it loses' no more by further trials, it is considered as
quite pure, and said to be 24 carats fine; if it lose 1 carat,
or l-24th in purifying, it was of 23 carats fine; and if it
lose 2 carats, it was 22 carats line; and so on.

CARC \SS, is a hollow case formed of ribs of iron, and
covered over with pitched cloth, &C, about the size of
bomb shells ; or sometimes made all of iron except two or
three holes for the lire to blaze through. These were filled
with various combustibles, to fire houses, when thrown out
of mortars into besieged places.

CARCAV1 (PETEKDE),wasbornat Lyons, but in what

CAR

C 2?5 ]

CAR

year is not known. He was counsellor to the Parliament
of Toulouse, afterwards counsellor to (he grand council,
and keeper of the king's library. He was appointed geo-
metrician to the French Academy of Sciences in 1666, and
died at Paris in 1()S4. There are extant some letters of
his, printed among ihose of Pes Carles.

CARDAN (Hieronymus, or JEROM),one of the most
extraordinary geniuses of his. age, was born' at Pavia, in
Italy, Sept. 24, 1501. At 4 years old he was carried to
Milan, his father being an advocate and physician in that
city: at the age of 20 he entered the university of the same
city, and prosecuted his studies with great success. Inl524
he went to Padua, and the same year was admitted to the
degree of master of arts, and the year following, that of
doctor of physic. He married about the year 1531 ; and
became professor of mathematics at Milan in J 533. In
1539 he was admitted a member of the college of physi-
cians at Milan; in 1543 he read public lectures inmedi-
cine there, and the same at Pavia the year following; but
in consequence of non-payment of his salary, he discon-
tinued them, and returned to Milan.

Inl552 he went into Scotland, having been sent for by the
archbishop of St. Andrew's, to cure him of an afflicting
asthma, after trying the physicians, of the king of France
and of the emperor of Germany, without receiving any
benefit. He began to recover from the day that Cardan
prescribed for him : our author returned at the end of 6
weeks and 3 days, leaving him prescriptions which in two
years wrought a complete cure. On this visit Cardan
passed through London, and calculated King Edward's
nativity; being famous for his pretensions in astrology, as
well as his skill in mathematics and medicine. Returning
to Milan, after 4 months absence, he remained there till
the beginning of Oct. 1552; and then went to Pavia, from
Whence he was invited to Bologna in 1562. He taught in
this last city till the year 1570; at which time he was
thrown into prison; but some months after he was sent
home to his own house. He quitted Bologna in 1571;
and went to Rome, where he lived for some time without
any public employment. He was however admitted a
member of the college of physicians, and received a pen-
sion from the Pope, till the time of his death, which hap-
pened at Rome on the 21st of September 1575.

Cardan, at the same time chat he was one of the greatest
geniuses and most learned men of his age, in all the sci-
ences, was one of the most eccentric and fickle in conductof
all men that ever lived; despising every good principle and
opinion, and without one friend in the world. The same
capriciousness that was remarkable in his outward con-
duct, is also observable in the composition of his numerous
and elaborate works. In many of his treatises the reader
is stopped almost every moment by the obscurity of his text,
or by digressions from the point in hand. In his arithme-
tical writings there are several discourses on the motions
of the planets, on the creation, on the Tower of Babel, and
such like. And the apology which he made for these fre-
quent digressions is, that he mightby that means enlarge and
fill up his book, his bargain with the bookseller being at so
much per sheet; and that he worked as much for his daily
support as for fame. The Lyons edition of his works, printed
in 1663, consists of no less than 10 volumes folio.

In fact, when we consider the transcendent qualities of
Cardan's mind,- it cannot be denied that he cultivated it
with every species of knowledge, and made a greater pro-
gress in philosophy, in the medical art, in astronomy, in

mathematics, and the other sciences, than the most part of
his contemporaries who had applied themselves but to one
of those sciences. In particular, he was perhaps the best
algebraist of his time, a science in which he made great
improvements; and his' labours in cubic equations espe-
cially have rendered his name immortal, the rules for re-
solving them having ever since borne his name, and are
likely to do so as long as this science exists, although he
received the first knowledge of them from another person;
the account of which, and his disputes with Tartalea, have
been given at large under the article Algebra.

Scaliger affirms, that Cardan, having by astrology pre-
dicted and fixed the time of his death, abstained from all
food, in order that his prediction might be verified, and
that his continuance' to live might not discredit his'art.
It is further remarkable, that Cardan's father also died in
this manner, in the year 1524, having abstained from sus-
tenance for nine days. Our author, too, informs us that
his father had white eyes, and could see in the night-time. s

CARDINAL Points, in Geography, are the cast, west,
north, and south points of the horizon.

' Cardinal Points of the Heavens, or of a Nativity, are
the rising and setting of the sun, the zenith and nadir.

Cardinal Signs, are those at the four quarters, or the
equinoxes and solstices, viz, the signs Aries, Libra, Cancer,
and Capricorn.

Cardinal Winds, are those that blow from the four
cardinal points, viz, the east, west, north, and south winds.

CARDIOIDE, the name of a curve so called by C»s-
tilliani. — But it was first treated of by Koersma, and b.y
Carre. See Philos. Trans. 1741, and Memoires de l'Acad.
1705.

Tfts Cardioide is thus ge-
nerated, apb isacircle,and
ab its diameter. Through
one extremity a of the
diameter draw any num-
ber of lines apq, cutting
the circle in p ; upon these
set off always pq equal to
the diameter ab; so shall
the points q be always in
the curve of the cardioide.

From this generation of the curve, its chief properties
are evident, viz, that,

everywhere pq = ab,

cq, or qq is = a« or 2ab,

AQ = AB± AP,

p always bisects qq.

The cardioide is an algebraical curve, and the equation
expressing its nature is th,is : putting

fl = AB the diameter,

x = od perp. to eiB,

y = dq perp. to an ; then is
y — 6ay3 -+- 2^y — 6ax"-y ■+■ x*

-+- 12a-y" — 8a'y -+• Sarx*
which is the equation of the curve.

Many properties of the cardioide may be seen in the-
places above cited.

CARRE' (Lewis), an ingenious mathematician and
philosopher, was born in the year l663, in the province of
Brie in France. His father, a substantial farmer, intended
him for the church. But, after going through the usual
course of education for that purpose, and having an utter
aversion to it, he refused toenter upon that function; by
3N2 '

}=c

CAR

[ 276 ]

C A 11

which he incurred his father's displeasure. His resources
being thus cut off, he was obliged to quit the university,
and look into the world for some employment. In this
exigency he had the good-fortune to be engaged as an
amanuensis by the celebrated Father Malebranche ; by
which he found himself at once transported from the mazes
of scholastic darkness, to the source of the most brilliant
and enlightened philosophy. Under this great master he
studied mathematics and the most sublime metaphysics.
After seven years spent in this excellent school, M. Cane
found it necessary, in order to procure himself some less
precarious establishment, to teach mathematics and philo-
sophy in Paris; but especially that philosophy which, on
account of its tendency to improve our morals, he valued
more than all the mathematics'in the world. And ac-
cording^' his greatest care was to make geometry serve as
an introduction to his well-beloved metaphysics.

Most of Carre's pupils were of the fair sex. The first
of these, who soon perceived that his language was rather
the reverse of elegant and correct, told him pleasantly
that, as an acknowledgment for the pains he took to teach
her philosophy, she would teach him French; and he ever
after owned that her lessons were of great service to him.
In general he seemed to set more value on the genius of
women than that of men. ••

M. Carre, although he gave the preference to the meta-
physics, did not neglect mathematics; and while he taught
both, he took care to make himself acquainted with all
the new discoveries in the latter. This was all that his
constant attendance on his pupils would allow him to. do,
till the year 1697, when Mi Varignon, so remarkable for
his extreme scrupulousness in the choice of his eleves,
took M. Carre to him in that station. Soon after, viz, in
the year 1700, our author, thinking himself bound to do
something that might render him worthy of that title, pub-
lished the first complete work on the Integral Calculus,
under the title of " A method of measuring Surfaces and
. Solids, and finding their Centres of Gravity, Percussion,
and Oscillation." He afterwards discovered some errors
in the work, and was candid enough to own and correct
them in a subsequent edition.

In a short time M. Carre became associate, and at
length one of the pensioners of the Academy. And as
this was a sufficient establishment for one, who knew so
well how to keep his desires within just bounds, he gave
himself up entirely to study ; and as he enjoyed the ap-
pointment of Mechanician, he applied himself more par-
ticularly to mechanics. He took also a survey of every
branch relating to music; such as the doctrine of sounds,
the description of musical instruments; though he de-
spised the practice of music, as a mere sensual pleasure.
Some sketches of his ingenuity and industry in this way
may be seen in the Memoirs of the French Academy of
Sciences. He also composed some treatises on other
branches of natural 'philosophy, and some on mathema-
tical subjects; all which he bequeathed to that illustrious
body; though it does not appear that any of them have
yet been published. It is not unlikely that he was hin-
dered from putting the last hand to them by a train of dis-
orders proceeding from a bad digestion, which, after ha-
rassing him during the course of five or six years, at
length brought him to the grave in 1711, at 48 years of
age<

His,memoir8 printed in the volumes of the Academy,
•vilh the years of the volume;, are as below.

1. Rectification of Curve Lines by Tangents : 1701. .

2. Solution of a Prob. proposed to Geometricians, &c.
1701.

3. Reflections on the Table of Equations : 1701.

4. On'the Cause of the Refraction of Light: 1702.

5. Why the Tides are always augmenting from Brest to
St. Malo, and diminishing along the coasts of Normandy :
1702.

6. Number and Names of Musical Instruments : 1702.

7. On the Vinegar which causes small stones to roll
upon an inclined plane: 1703.

S. On the Rectification &c, of the Caustics by reflec-
tion: 1703.

9. Method for the Rectification of Curves : 1704.

10. Observations on the Production of Sound : 1704.

1 1. On a Curve formed from a Circle : 1705.

12. On the Refraction of Musket-balls in water, and on
the Resistance of that fluid : 1705.

13. Experiments on Capillary Tubes: 1705.

14. On the Proportion of Pipes to have a determinate
quantity of water : 1705.

15. On the Laws of Motion : 1706.

10. On the Properties of Pendulums ; with some new
properties of the Parabola : 1707.

17- On the Proportion of Cylinders that their sounds,
may form the musical chords : 170.9.

18- On the Elasticity of the Air : 1710.

19. On Catoptrics: 1710.

20. On the Monochord : in the Machines, torn. 1, with
some other pieces, not. mathematical.

CARRIAGE, of a Cannon, is the machine upon which
it is mounted ; serving to poiuj or direct it for firing, and
to convey it from place to place.

Wheel Carriage, one that is mounted and moved
about upon wheels. Horses draw in general, to most ad-
vantage, when the direction of their draft is parallel to
the ground, or rather a little upwards. A carriage also
goes easiest when the centre of gravity is placed very high ;
since, when once put in motion, it continues it with very
little labour to the horses.

CARTES (Rene des), one of the most eminent phi-
losophers and mathematicians of the 17lh century, or in-
deed of any age whatever. He was descended of an an-
cient noble family in Touraine in France, b\?ing a younger
son of a counsellor in the parliament of Rennes, and was
born March 31, 1596. His,fathcr gave him a liberal edu-
cation, and the more so as he observed in him the appear-
ance of -a promising genius, using to call him the Philoso-
pher, on account of his insatiable curiosity in asking the
reasons of every thing that lie did not understand.

He soon made a remarkable progress in his studies, and
from his infancy showed a decided taste for natural and
philosophical knowledge. Disgusted with the jargon of
an absurd philosophy, lie found only in mathematics the
certainty with which he was charmed. He gave himself
up chiefly to this science, and thence he derived the most
solid and indisputcd part of his renown. '\ el this philo-
sopher, who taught others to think, who bloke the yoke of
antiquity, and re-established season on her throne, this very
Des Cartes, was himself led astray by his imagination. Let
us respect his errors; lor never did an ordinary man fall

into the like. His edifice of Vortices, like the philosophy
ol Aristotle, has been demolished by the elloi Is of his suc-
cessors J though his system concerning tire nature of ani-
mals, in which nc saw no principle, superior to mecha-

CAR

[ 277 ]

C A R

nism, still divides the literary world. But if we do not
always find truth in the works of Des Cartes, we are at no
loss to trace evident marks of genius. That philosopher
was a profound thinker, and spent his life in solitude. In
vain did Cardinal Richelieu, in the name of the king, offer
him important posts: he preferred his retirement to the
slavery of honours. Yet, yielding to the urgent solicita-
tions of Queen Christina of Sweden, who wished to see and
converse with him, he repaired to Stockholm. But the
hours of their interviews were not regulated by the habits
of Des Cartes, nor congenial to his nature. That great
man, born with a weak constitution, which was rendered
still more delicate by his custom of meditating in bed,
could not long sustain the task of rising every morning at
five o'clock, in the winter rigours of a northern climate,
to repair to the library of Christina. A defluction on the
lungs terminated his life on the 11th of February, 16.50.
In 1667, his corpse was conveyed to France, and is now
deposited in the Pantheon at Paris.

"Nature," says Voltaire, "had favoured Des Cartes with
a strong and clear imagination, whence he became a very
singular person, both in private life and in his manner of
reasoning. This imagination could not be concealed even
in his philosophical writings, which are every where .
adorned with very brilliant ingenious metaphors. Nature
had almost made him a poet ; and indeed he wrote a piece
of poetry for the entertainment of Christina, queen of Swe-
den, which however was suppressed in honour of his me-
mory. He extended the limits of geometry as^ far beyond
the place where he found them, as Newton did after him ;
and first taught the method of expressing curves by equa-
tions. Heapplicd this geometrical and inventive genius to
dioptrics, which when treated by him became a new art ;
and if he was mistaken in some thingsi the reason is, that a
man who discovers a new tract of land, cannot at once
know all the properties of the soil. Those who come after
him, and make these lands fruitful, are at least obliged to
him for the discovery." Voltaire acknowledges, that there
are innumerable errors in the rest of Des Cartes's works :
but adds, that geometry was a guide which he himself had
in some measure formed, and which would have safely
conducted him through the several paths of natural phi-
losophy: yet he had at last abandoned this guide, and
gave entirely into the humour of framing hypotheses ; and
then philosophy was no more than an ingenious romance,
fit only to amuse the ignorant.

It has been pretty generally acknowledged, that he bor-
rowed his improvements in algebra from Harriot's Artis
Analytical Praxis; which is highly probable, as he was in
England about the time when Harriot's book was publish-
ed, and follows the manner of Harriot/, except in the me-
thod of noting the powers. On this head the following anec-
dote is related by Dr. Pell, in Wallis's Algebra, pa. 198.
Sir Charles Cavendish, then resident in Paris, discoursing
there with M. Roberval, concerning Des Cartes' Geome-
try, then lately published: I admire, said Roberval, that
method in Des Cartes, of placing all the terms of the equa-
tion on one side, making the whole equal to nothing, and
how he fell upon it. 1 he reason why you admire it, said
Sir Charles, is because you arc a Frenchman; for if you
were an Englishman, you would not admire it. Why so r
asked Roberval. Because,' replied Sir Charles, we in Eng-
land know whence he had it; namely, from Harriot's Al-
gebra. What book is that? says Roberval ; I never saw
.'1. Next time you come to my chamber, said Sir Charles,

I will show it to you. Which a while after he did : and
on perusal of it, Roberval exclaimed with admiration, II
Va vu .' il I'a vu ! He had seen it ! he had seen it ! rinding
all that in Harriot which he had before admired in Des
Cartes, and not doubting but that Des Cartes had it from
thence. See also Montucla's History of Mathematics.

The real improvement of Des Cartes in algebra and geo-
metry, I have particularly treated of under the article Al-
gebra ; and his philosophical doctrines are displayed in
the article Cartesian Philosophy, here following. He
was never married, but had one natural daughter, who
died when she Was five years old. There have been seve-
ral editions of his works, and commentaries upon them ;
particularly those of Schooten on his Geometry.

CARTESIAN Philosophy, or Cartesianism, the system
of philosophy advancer) by Des Cartes, and maintained by
his followers, the Cartesians.

The Cartesian philosophy is founded on two great prin>-
ciples, the one metaphysical, the other physical. The
metaphysical one is this : I think, therefore I am, or I
exist: the physical principle is, that nothing exists but
substances. Substance he makes of two kinds; the one
a substance that thinks^ the other a substance extended :
so that actual thought and actual extension make the es-
sence of substance.

The essence of matter being thus fixed in extension,
Des Cartes concludes that there is no vacuum, nor any
possibility of it in nature; but that the universe is abso-
lutely full : by this principle, mere space is quite ex-
cluded ; for extension being implied in the idea of space,,
matter is so too.

Des Cartes defines motion to be the translation of a
bodv from the neighbourhood of others that are in con-
tact with it, and considered as at rest, to the neighbour-
hood of other bodies : by which he destroys the distinc-
tion between motion that is absolute or real, and that
which is relative or apparent. He maintains that the
same quantity of motion is always preserved in the uni-
verse, because God must be supposed to act in the most
constant and immutable manner. And hence also he de-
duces his three laws of motion. See Motion.

On these principles, Des Cartes explains mechanically
how the world was formed, and how the present phenome-
na of nature came to arise. He supposes that God created
matter of an indefinite extension, which he separated into
small cubic portions or masses, full of angles : that he
impressed two motions on this matter; the one, by which
each part revolved about its own centre; and another, by
Which an assemblage, or system of them, turned round a
common centre. From whence arose as many different
vortices, or eddies, as there were different masses of- mat--
ter, thus moving about common centres.

The consequence of these motions in each vortex, ac-
cording to Des Cartes, is as follows : The parts of matter
could not thus move and revolve among one another, with-
out having their angles gradually broken ; and; this conti-
nual friction of parts and angles must produce three ele-
ments: the first of these, an infinitely fine, dust, formed of
the angles broken off; the second, the spheres remaining,
after all the angular parts are thus removed ; and those
particles not yet rendered smooth and spherical, but still
retaining some of their angles, and hamous parts, form the
third element.

Now the first orsubtilest element, according to the laws
of motion, must occupy the centre of each system, or vorr-

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tcx, by reason of the small ness of its parts : and this is the
matter which constitutes the sun, with the fixed stars
above, and the fire below. The second element, made up
of spheres, forms the atmosphere, and all the matter be-
tween the earth and the fixed stars ; in such sort, that the
largest spheres are always next the circumference of the
vortex, and the smallest next its centre. The third ele-
ment, formed of the irregular particles, is the matter that
composes the earth, and ;:11 terrestrial bodies, together
with comets, spots in the sun, &c.

He accounts for the gravity of terrestrial bodies from
the centrifugal force of the ajther revolving- round the
earth: and on the same general principles he pretends to
explain the phenomena of the magnet, and to account for
all the other operations in nature.

The Cartesian system has received many alterations and
amendments; ingenious men, for full a century, employing
their talents in reforming one part, and new-modelling
another; but it is now generally acknowledged that the
foundation was faulty, and the superstructure erroneous ;
so that the fabric, though allowed to be. a work of genius,
is abandoned to neglect and ruin, and is pointed at as a
memorial of philosophical presumption.

CARTOUCHE, in the Military Art, a case of wood,
about 3 inches thick at the bottom, girt with marlin, hold-
ing about 400 musket balls, besides 6 or 8 balls of iron of
a pound weight, to be fired from howitzers for the defence
of a pass, &c.

CARTRIDGE, in the Military Art, the charge or load
of a fire-arm, wrapped up in a thick paper, pasteboard, or
parchment, to be the more readily charged, or conveyed
into the piece. Cartridges of cannon and mortars are usually
in cases of pasteboard, or tin, sometimes of wood half a
foot long ; taking up the place of the bullet in the piece to
whose caliber the diameter is proportioned. Those of
muskets, pistols, and small-arms, only contain the charge
of powder, with a ball wrapped up in thick paper.

Flannel cartridges are now preferred, because they do
not retain the fire, and are therefore less liable to acci-
dents in loading. A parchment cap is made to cover
them, in order to prevent the dust of powder from passing
through them, which is taken off before they are put into
the piece.

Cartridge-box, a case of wood or turned iron, co-
vered with leather, holding a dozen musket-cartridges. It
is worn upon a belt, and hangs a little lower than the right
pocket-hole.

CARY (Robert), a learned English chronologer and
divine, was born at Cockington, in the county of Devon,
about the year 1615. He took his degrees in arts, and
ll. d. in Oxford. After returning from his travels he was
presented to the rectory of Portlemouth, near Kingsbridge
in Devonshire: but not long after he was drawn over by
the presbyterian ministers to their part)', and chosen mo-
derator in that part of the second division of the county of
Devon, which was appointed to meet at Kingsbridge. And
yet, on the restoration of Charles the 2d, he was one of the
first to congratulate that prince on bis return, and soon
after was preferred to the archdeaconry of Exelcr; but
from which he was however some lime afterward ejected.
The remainder of his days he spent at his rectory at Portle-
mouth, and died in 16'88, at 73 years of age. — He pub-
lished Palcelogia Chronica, a chronological account of an-
cient time, in three, parts: l, Didactical; 2, Apodidacti-
cal; 3, Canonical: in 1677.

CASATI (Paul), a learned Jesuit, bom at Placentia
in l6l7- He entered early among the Jesuits; and after
having taught mathematics and divinity at Rome, he was
sent into Sweden to Queen Christina, whom he prevailed
on to embrace the popish religion. His writings are as
follow :

1. Vacuum Proscriptum. — 2. Terra Machinis mota. —
3. Mechanicorum, libri octo. — 4. De Igne Dissertationes,
— 5. De Angelis Disputatio Theolog. — 6. Hydrostaticae
Dissertationes. — 7- Optica? Disputationes. Itisremarkable
that he wrote this treatise on optics at 88 years of age, and
after he was blind. He was also author of several books
in the Italian language.

CASCABEL, the knob or button of metal behind the
' breech of a cannon, serving as a kind of handle by which
to elevate and direct the piece; to which someadd the fillet
and ogees as far as the base- ring.

CASEMATE, or Cazemate, in Fortification, a kind
of vault or arch, of stone-work, in that part of the flank
of a bastion next the curtain; serving as a battery, to defend
the face of the opposite bastion, and the moat or ditch.

It is now seldom used, because the batteries of the enemy
are apt to bury the artillery of the casemate in the ruins of
the vault : beside, the great smoke made by the discharge
of the cannon, renders it intolerable to the men. So that,
instead of the ancient covered casemates, later engineers
have contrived open ones, only guarded by a parapet, &c.

Casemate, or Casement, in Architecture, a hollow
moulding, which some architects make ■§. of a circle, and
others is Casement is also used for a small moveable win-
dow, usually within a larger, being made open to turn on
hinges.

CASERNS, orCAZERNS,in Fortification, small rooms,
or huts, erected between the ramparts and the houses of
fortified towns, or even on the ramparts themselves ; to
serve as lodgings for the soldiers on immediate duty, to
ease the garrison.

CASE-Shot, or Cannister-Siiot, are a number of
small balls put into a round tin cannister, and so shot out
of great guns. These have superseded, and been sub-
stituted instead of, the grape-shot, which have been laid
aside. >

CASSINI (John-Dominic), an eminent astronomer,
was born of noble parents, at a town in Piedmont in Italy,
June 8, 1625. After laying a proper foundation in his
i studies at home, he was sent to continue them in a college
of Jesuits at Genoa. He had an uncommon turn for
Latin poetry, which he exercised so very earl)', that some
of his poems were published when he was but 11 year..
old. At length he met with books of astronomy, which
he read with great eagerness. Pursuing the bent of his in -
clinations in this way, he soon acquired -a very extensive
knowledge of thesubject, and in HioO thesenate of Bologna,
invited him to be their public mathematical professor,
Cassini was but 2 i years of age when he went'to Bologna,
where he taught mathematics, and made celestial obser-
vations with great ease and assiduity. — In l6'S2 a comet •
appeared, which he observed with great accuracy ; and he
discovered that counts were not bodies accidentally gene-
rated in the atmosphere, as had been supposed, but of the
same nature, and probably governed by the same laws, as
the planets. The same year he resolved an astronomical
problem, which Kepler and Bulliald had given up as in-
solvable ; viz, to determine geometrically the apogee and
excentricity of a planet, from its true and mean place. —

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In 1653, when a church in Bologna was repaired and en-
larged, he obtained leave of the senate to correct and settle
a meridian line, which had been drawn by an astronomer
in 1575 — in 1657 he attended, as an assistant, a noble-
man, who was sent to Rome to compose some differences,
which had arisen between Bologna and Ferrara, from the
inundations of the Po ; and he showed so much skill and
judgment in the management of the affair, that in l66'3
the pope's brother appointed him inspector general of the
fortifications of the castle of Urbino: and he had after-
wards committed to him the care of all the rivers in the
ecclesiastical state.

Meanwhile he did not neglect his astronomical studies,
but cultivated them with great care. He made several
discoveries relative to the planets Mars and Venus, parti-
cularly the revolution of Mars about his axis: but the
point he had chiefly in view, was to settle an accurate
theory of Jupiter's satellites; which, after much labour
and observation, he happily effected, and published it at
Rome, among other astronomical pieces, in 1 666.

Picard, the French astronomer, getting Cassini's tables
of Jupiter's satellites, found them so very exact, that he
conceived the highest opinion of his skill ; and from that
time his fame increased so fast in France, that the governs
ment desired to have him a member of the Academy. Cas-
sini however could not quit his station without leave of
his superiors ; and therefore the king, Lewis the 14th, re-
quested of the pope and the senate of Bologna, that Cassini
might be permitted to come into France. Leave was
granted for 6 years; and he came to Paris in the begin-
ning of 1669, where he was immediately made the king's
astronomer. When this term of 6 years was nearly ex-
pired, the pope and the senate of Bologna insisted on his
return, on pain of forfeiting his revenues and emoluments,
which had hitherto been faithfudy remitted to him: but
the minister Colbert notwithstandiv.tr prevailed on him to
stay, and he was naturalized in 1&73; the year in which
he was married.

The Royal Observatory of Paris had been finished some
time. The occasion of its being built was this: In 1638,
the celebrated Mersenne was the chief institutor and pro-
moter of a society, where several ingenious and learned
men met together to talk upon physical and astronomical
subjects; amopg whom were Gassendi, Descartes, Mon-
mort, Thevenot, Bulliald, our countryman Hobbes, &c:
and this society was kept up by a succession of learned
men for many years. At length the government consider-
ing that a number of such men, acting in a body, would
succeed much better in the promotion of science, than if
they acted separately, each in his particular art or pro-
vince, established under the direction of Colbert, in 1666,
the Royal Academy of Sciences: and for the advancement
of astronomy in particular, erected the Royal Observatory
at Paris, and furnished it with all kinds of instruments
that were necessary to make observations. The founda-
tion of this noble pile was laid in l667, and the building
completed in l6'70. Of this observatory, Cassini was ap-
pointed the first inhabiter; which he took possession of in
Sept. 1671., In 1672 he endeavoured to determine the
parallax oCMars and the sun : and in l677 he proved that
the diurnal rotation of Jupiter about his axis was per-
formed in .9 hours 58 minutes, from the motion of a spot
in one of his larger belts : also in l684 he discovered four
satellites of Saturn, besides that which Huygcns had ob-
served. In 1693 he published a new edition of his Tables

of Jupiter's Satellites, corrected by later observations.
In 1695 he took a journey to Bologna, to examine the
meridian line, which he had fixed there in lfJ55 ; and he
showed, in the presence o.f eminent mathematicians, that
it had riot varied in the least, during that 40 years. In
1700 he continued the meridian line through France,
which Picard had begun, to the very southern limits of
that country.

Our author resided at the royal observatory for more
than 40 years, making many excellent and useful disco-
veries. The titles of his publications occupy nearly 5
pages in the closely printed index of Rozier, and conse-
quently far exceed our bounds even to enumerate in this
place. He had die misfortune in his latter years to be de-
prived of his sight; and he ended a useful and honourable
life September 14th 1712, at the age of 87 years.

CASSINI (James), a celebrated French astronomer,
and member of the several Academies of Sciences of France,
England, Prussia, and Bologna, was born at Pan.-, Feb. 18,
1677, being the younger son of John-Dominic Cassini,
above mentioned, whom he succeeded as astronomer at
the royal observatory, the elder son having lost his life at
the battle of La Hogue.

After his first studies in his father's house, in which it is
not to be supposed that mathematics and astronomy were
neglected, lie was sent to study philosophy at the Maza-
rine college, where the celebrated Varignon was then pro-
fessor of mathematics; from whose assistance young Cas-
sini profited so well, that at 15 years of age he supported
a mathematical thesis with great honour. At the age of
17 he was admitted a member of the Academy ot Sciences;
and the same year he accompanied his father in his jour-
ney to Italy, where he assisted him in the verification of
the meridian at Bologna, and other measurements. On
his return he made other similar operations in a journey
into Holland, where he discovered some errors in the mea-
sure of the earth by Snell, the result of which was com-
municated to the Academy in 1702. He also visited
England in 1696, where he was made a member of the
Royal Society. — In 1712 he succeeded his father as astro-
nomer royal at the observatory. — In 1717 he gave to the
Academy his researches on the distance of the fixed stars,
in which he showed that the whole annual orbit, of near
200 millions of miles diameter, is but as a point in com-
parison of this distance. The same year he communicated
also his discoveries concerning the inclination of the orbits
of the satellites in general, and especially of those of Sa-
turn's satellites and ring. — In 1725 he undertook to deter-
mine the cause ot the moon's libration.

In 1732 an important question in astronomy exercised
the ingenuity of our author. His father had determined,
by his observations, that the planet Venus made one com-
plete revolution about her a:sis in 23 hours : and M. Bian-
chini had published a work in 1729, in which he settled
the period of the same revolution at 24 days 8 hours.
From an examination of Bianchini's observations, which
were upon the spots in Venus, he discovered that he had
intermitted his observations for the duration of 3 hours,
from which cause he had probably mistaken new spots for the
old ones, and so had been led into the mistake. He soon
afterwards determined the nature and quantity of the ac-
celeration of the motion of Jupiter, at half a second per
year, and of that of the retardation of Saturn at two mi-
nutes per year ; that these quantities would go on increas-
ing for 2000 years, and then would decrease again. — In

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1740 he published his Astronomical Tables ; and his Ele-
ments of Astronomy; both of which are very extensive
and accurate works.

Though astronomy was the principal object ot our au-
thor's consideration, he did not confine himself absolutely
to that branch, but made occasional excursions into other
fields. We owe also to him, for example, Experiments on
Electricity, or the light produced by friction ; Experi-
ments on the recoil of fire-arms ; Researches on the rise of
the mercury in the barometer at different heights^ above ,
the level of the sea ; Reflections on the perfecting of burn-
ing-glasses; and other memoirs.

The French Academy had properly judged that one of
its most important objects, was the measurement of. the
earth. In 1669 Picard measured a little more than a de-
gree of latitude to the north pf Paris; but as that extent
appeared too small from which to conclude the whole cir-
cumference with sufficient accuracy, it was resolved to
continue that measurement on the meridian of Paris to the
north and south, through the whole extent of the country.
Accordingly, in 16S3, M. Lahire continued that on
the northside of Paris, and the older Cassini that on the
south side. The latter was assisted in 1700 in the con-
tinuation of this operation by his son James. The same
work was farther continued by the same Academicians;
and finally the part left unfinished by Lahire in the north,
was finished in 171S by our author, with the late Maraldi,
and Lahire the younger.

These operations produced a considerable degree of pre-
cision. It appeared also, from this measured extent of 6
degrees, that the degrees were of different lengths in diffe-
rent parts of the meridian ; and in such sort that our
author concluded, in the volume published for 171S,
that they decreased more and more towards the pole,
and that therefore the figure of the earth was that of an
oblong spheroid, having its axis longer than the equa-
toriafdiameter. He also measured the perpendicular to
the sarrfe meridian, and compared the measured distance
with the differences of longitude, as before determined by
the eclipses of Jupiter's satellites ; whence he inferred, that
the length of the degrees of longitude was smaller than it
'would be on a sphere, and that therefore again the figure
of the earth was an oblong speroid ; contrary to the deter-
mination of Newton by the theory of gravity. In conse-
quence of these assertions of our author, the French go-
vernment sent two different classes of mathematicians, the
one to measure a degree at the equator, and the other at
the polar circle ; and the comparison of the whole deter-
mined the figure to be an oblate spheroid, contrary to
Cassini's determination.

After a long and laborious life, our author James Cas-
sini lost his life by a fall in April 1756', in the 80th year
of his age, and was succeeded in the Academy and Obser-
vatory by his second eon Cesar-Francois de Tliury. He
published A Treatise on the Magnitude and Figure of the
Earth ; as also The Elements or Theory of the Planets,
with tables ; besides a great number of papers ill the Me-
moirs of the Academy, from the year 16.99 to '755.

CASSINI de Tiiuiiy (Cesar-Fuancois), a celebra-
ted French astronomer, director of the observatory, and
member of most of the learned societies of Europe, was
born at Paris June 17, 1 7 14, being the second son of James
Cassini, whose occupations and talents our author inhe-
rited and supported with great honour. lie received his
first lessons in astronomy and mathematics from MM. Ma-

raldi and Camus; and his progress was so great, that at 10
years of age he was able to calculate the phases of the total
eclipse of the sun of 1727- At the age of 18 he accom-
panied his father in his two journeys undertaken for draw-
ing the perpendicular to the observatory meridian, from
Strasbourg to Brest. From'that time a general chart of
France was devised ; for which purpose it was necessary
to traverse the country by several lines, parallel and per-
pendicular to the meridian of Paris, and our author was
intrusted with this business. He did not content himself
with the measure of a degree by Picard ; suspecting even
that the measures which had been taken by his father and
grandfather were not exempt from some errors the imper-
fections of their instruments at least would be liable to, he
a<?ain undertook to measure the meridian at Paris, by meant
of a new series of triangles, of a smaller number, and more
advantageously disposed. This great work was published
in 1740, with a, chart showing the new meridian of Paris,
by two different series of triangles, passing along the sea-
coasts to Bayonne, traversing the frontiers of Spain to the
Mediterranean and Antibes, and thence along the eastern
limits of France to Dunkirk, with parallel and perpendi-
cular lines described at the distance of 6000 toises from
one another, from side to side of the country. — In 1735
he had been received into the academy as adjoint supernu-
merary at 21 years of age.

A tour which our author made in Flanders, in company
with the king, about 1741, gave rise to the particular chart
of France, at the instance of the king. Cassini published
different works relative to these charts, and a great number
of the sheets of the charts themselves.

In I76I Cassini undertook an expedition into Germany,
for the purpose of continuing to Vienna the perpendicular
of the Paris meridian; to unite the triangles of the chart
of France with the points taken in Germany; to prepare
the means of extending into this country the same plan as
in France; and thus to establish successively for all Eu-
rope a most useful uniformity. Our author was at Vienna
the 6th of June 176l, the day of the transit of Venus over
the sun, of which he observed as much as the state of the
weather would permit him to do, and published the ac-
count thereof in his Voyage en Allcmagne.

Finally, M. Cassini, always meditating the perfection of
his grand design, proposed at the peace between England
and France in 1 7S3, the joining of certain points taken upon
the English coast with those which had been determined
on the coast of France, and thus to connect the general
chart of the latter with that of the British isles, like as he
had before united it with those of Flanders and Germany.
The proposal was favourably received by the Knglish go-
vernment, and presently carried into effect, under the di-
rection of the Royal Society, the execution being com-
mitted to the late General Roy; after whose death the bu-
siness was for some time suspended; but it was afterwards
revived under the auspices of the Duke ol Richmond, Ma-
ster-General of the Ordnance, and the execution commit-
ted to the care of Col. Edward Williams and Captain (now
Col.) William Mlldge, both respectable officers of the Ar-
tillery, and Mr. Isaac Dalby, who had before accompanied
and assisted General Boy; from whose united skill and
zeal the happiest prosecution of this business was to be ex-
pected ; as it has since been most ably completed by Col.
Mudge, now Lieut. Governor of the Royal Military Aca-
demy; where as a cadet he received his mathematical edu-
cation under the direction of the author oi this Dictionary.

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The volumes of the Memoirs of the French Academy
for the years between 1735 and 1770 contain a prodigious
number of papers by Gassini. They consist chiefly of
astronomicalobservationsand questions; amongwhichare,
Researches concerning the parallax of the sun, the moon,
Mars, and Venus ; On astronomical refractions, and the
effect caused in their quantity and laws by the weather ;
Numerous observations on the obliquity of the ecliptic,
and on the law of its variations. In short, he cultivated
astronomy for 50 years, a period the most important for
that science that ever elapsed, for the magnitude and va-
riety of objects, in which he commonly sustained a consi-
derable share.

M. Cassini was of a very strong and vigorous constitu-
tion, which carried him through the many laborious ope-
rations in geography and astronomy which he conducted.
An habitual retention of urine however rendered the last
12 years of his life very painful and distressing, till it was
at length terminated by the small-pox the 4th of September
1784, in the 71st year of his age; being succeeded in the
academy, and as director of the observatory, by his only
son Count John-Dominic Cassini; who is the 4th in order
by direct; descent in that honourable station.

CASSIOPEIA, one of the 48 old constellations, placed
near Cepheus, not far from the north pole. The Greeks
probably received this figure, as they did the rest, from the
Egyptians, and in their fables added it to the family in the
neighbouring part of the heavens, making her the wife of
Cepheus, and mother of Andromeda. They pretend that
she was placed in this situation, to behold the destruction
of her favourite daughter Andromeda, who is chained just
by her on the shore, to be devoured ; and this as a punish-
ment for her pride and vanity in presuming to stand the
comparison of beauty with the Nereids.

In the year 1572 there burst out all at once in this con-
stellation a new star, which at first surpassed Jupiter him-
self in magnitude and brightness; but it diminished by de-
grees, till it quite disappeared at the end of 18 months.
This star alarmed all the astronomers of that age, many of
whom wrote dissertations upon it; among the rest Tycho
Brahe, Kepler, Maurolycus, Lycetus, Gramineus, and
others. Beza, the Landgrave of Hesse, Rosa, and some
more, wrote to prove it a comet, and the same that appear-
ed to the Magi at the birth of Christ, and that it came to
declare his second coming: these were answered by Tycho.

The stars in the constellation of Cassiopeia, are in Pto-
lemy's Catalogue 13, in Hevelius's 37, in Tycho's 16', and
in Flamsteed's 55.

CASTELLI (Benedict Abb ate), an eminent mathe-
matician, was born of an ancient and noble family at Brescia,
in the year 1577. In 1595 he entered into a monastery
of the order of St. Benedict, in his native city. He was
for some time a disciple of Galileo at Florence, and assisted
him in his astronomical observations, and afterwards main-
tained a regular correspondence with him. Under his
name, the apology of Galileo against the censures of Leu-
dovico delle Colombo and Vincent di Grazia appeared,
though it was principally written by Galileo himself. From
l6'l5 to 1625 he occupied the mathematical chair at Pisa.
In 1625 Caslclli was invited to Rome by Pope Urban viii,
and made mathematical professor in the college clella Sa-
pienza. The subject of his particular attention, and in the
investigation of which he chiefly excelled, was the motion
of water; and in 1628 he published two works upon it,

Vol. I.

which gained much reputation ; viz, The Mensuration of
running waters, and Geometrical Demonstrations of the
mensuration of running waters. These have been lately
inserted in the collection of the author's works on similar
topics, printed at Florence, with other treatises on the la-
guna of Venice, on the improvement of the Pontine, Bo-
lognese, Ferrarese, and Romagnese marshes, &c. Gugliel-
mini, though in other things he impugns Castelli, allows
him the honour of having first applied geometry to the
motion of water; and Montucla (Hist. Math. t. 2, p. 201)
calls him " The creator of a new part of hydraulics." He
died at Rome in 1644.

CASTOR, a moiety of the constellation Gemini ; called
also Apollo. Also a star in this constellation, whose lati-
tude, for the year 1700, according, to Hevelius, was
10° 4' 20" north ; and its longitude 53 16° 4' 14".

CASTOR and Pollux. See Gemini.

CASTOR and Pollux, in Meteorology, is a fiery me-
teor, which at sea appears sometimes adhering to a part of
the ship, in the form of a ball, or even several balls. When
one is seen alone, it is properly called Helena; but two are
called Castor and Pollux, and sometimes Tyndarida?.

By the Spaniards, Castor and Pollux are called San
Elmo; by the French, St. Elme, St. Nicholas, St. Clare,
St. Helene; by the Italians, Hermo; and by the Dutch,
Vree Vuuren.

The meteor Castor and Pollux, it is commonly thought,
denotes a cessation of the storm, and a future calm ; as it
is rarely seen till the tempest is nearly spent. But Helena
alone portends ill weather, and denotes the severest part
of the storm yet behind.

When the meteor adheres to the masts, yards, &c, it is
concluded, from the air not having motion enough to dis-
sipate this flame, that a profound calm is at hand ; but if
it flutter about, that it denotes a storm.

CASTRAMETATION, the art, or act, of encamping
an army.

CATACAUSTICS, or Catacaustic Curves, in the Higher
Geometry, and in Optics, are the species of caustic curves
formed by reflection. These curves are generated after the
following manner: If there bean jt_
infinite number of rays ab, ac,
ad, &c, proceeding from the ra-
diating point a, and reflected at
any given curve bcdh, so that
the angles of incidence be still
equal to the angles of reflection ;
then the curve beg, to which the
reflected rays bi, ce, dt,&c, are
everywhere tangents, as at the
points 1, e, f, &c, is the cata-
caustic, or caustic-by-reflection.

Or it is the same thing as saying, that a caustic curve is
that formed by joining the points of concurrence of the
several reflected rays.

Some properties of these curves are as follow. If the
reflected ray ib be produced to K, so that ab = bk, and
the curve kl be the evolute of the caustic beg, beginning
at the point k; then the portion of the caustic be is
= (ac — ab) -1- (ce — bi), that is, the difference of the
two incident rays added to the difference of the two re-
flected rays.

When the given curve BCD is a geometrical one, the
caustic will be so too, and will always be rectifiable. The
20

CAT [ 2i

caustic of the circle is a cycloid, or epicycloid, formed' by
the revolution of a circle upon a circle.

Thus, abd being a semi-
circle, exposed to parallel
rays; then those rays which
fall near the axis cb will be
reflected to r, the middle
point of bc ; and those which
fall at a, as they touch the
curve only, will not be re-

fleeted at all; but any intermediate ray Hi will be reflect-
ed to a point K, somewhere between a and F, And since
every different incident ray will have a different focal
point, therefore those various focal points will form a curve
line aef in one quadrant, and fgd in the other, being the
cycloid above mentioned. The caustic of the common cy-
cloid, when the rays are parallel to its axis, is also a com-
mon cycloid, described by the revolution of a circle on the
same base. The caustic of the logarithmic spiral, is the
same curve.

The catacaustic may be beautifully exhibited experi-
mentally by exposing the inside of a smooth bowl, or glass,
to the sun-beams, or strong candle-light ; for then this curve
aei'&d will appear plainly delineated on any white surface
placed horizontally within the same, as on the surface of
milk contained in the bowl.

The principal writers on the caustics are, l'Hospital,
Carre, &c. See Memoires de l'Acad. an. 1666 & 1703.

CATACOUSTICS, or Cataphonics, is the science of re-
flected sounds; or that part of acoustics which treats of
the properties of echoes.

Catacoustics, in Military Language, are ecoutes or
small galleries, from distance to distance, in front of the
glacis of a fortified place, all of which communicate with
a gallery that is carried parallel to the covert-way. The
besieged make use of them in going to meet the enemy's
miners, and interrupt their progress.

CATADIOPTRICAL Telescope, the same as Reflecting
Telescope ; which see.

CATALOGUE of the Stars, is a list of the fixed stars,
disposed according to some order, in their several con-
stellations ; with the longitudes, latitudes, right ascensions,
&c, of each. ,

Catalogues of the stars have usually been disposed,
either as collected into certain figures called constellations,
or according to their right ascensions, that is the order of
their passing over the meridian. All the catalogues, from
the most ancient down to Flamsteed's inclusively, were of
the first of these forms, or in constellations ; but most of
the others that have since appeared are of the latter form,
as being much more convenient for most purposes. In-
deed one has lately been constructed in which the stars are
disposed in classes according to zones or degrees of polar
distance.

Hipparchus of Rhodes first undertook to make a cata-
logue of the stars, about 128 years before Christ ; in which
he made use of the observations of Timocharis and Ari-
styllus, for about HOyears beforebim. Ptolemy retained
Hipparchus's catalogue, containing 1026 fixed stars in 48
constellations, though he himself made abundance ot ob-
servations, with a view to a new catalogue, an. Dom. 140
Albategni, a Syrian, brought the same down to his own
time, viz, about the year of Christ 880. Anno 1437, Ulugh
Beigh, or Beg, king of Parthia and India, formed a new

]

CAT

catalogue of 1022 fixed stars, or according to some 10l6»
since translated out of Persian into Latin by Dr. Hyde, in
1665. The third person who made a catalogue of stars
from his own observations was Tycho Brahe, who deter-
mined the places of 777 stars, for the end of the year 16OO;
which Kepler, from other observations of Tycho, after-
wards increased to the number of 1000 in theRudolphine
tables ; adding those of Ptolemy and other authors, omit-
ted by Tycho ; so that his catalogue amounts to above
ll60. About the same time, William landgrave of Hesse,
with his mathematicians Byrgius and Rothman, deter-
mined the places of 400 stars from new observations, rec-
tifying them for the year 1593 ; which Hevelius prefers to
those of Tycho. Ricciolus, in his Astronomia Reformata,
determined the places of 101 stars for the year 1700, from
his own observations : for the rest, he followed Tycho's
catalogue; altering it where he thought proper. Anno
1667 Dr. Halley, in the island of St. Helena, observed
350 of the southern stars, not visible in our horizon. The
same labour was repeated by Father Noel in 1710, who
published a new catalogue of the same stars constructed
for the year 1687. Also De la Caille, at the Cape of
Good Hope, made accurate observations of about 10
thousand stars near the south pole, in the years 1751 and
1752 ; the catalogue of which was published in the Me-
moirs of the French Academy of Sciences for the year
1752, and in some of his own works, as more particularly
noticed below.

Bayer, in his Uranometria, published in ]fj03 a cata-
logue of ll60 stars, compiled chiefly from Ptolemy and
Tycho, in which every star is marked with some letter of
the Greek alphabet ; the brightest or principal star in any
constellation being denoted by the first letter of the al-
phabet, the next star in order by the second letter; and so
on; and when the number of stars exceeds the Greek al-
phabet, the remaining stars are marked by the letters of
theRoman alphabet ; which letters are preserved by Flam-
steed in his catalogue, and also by Senex on his globes, and
indeed by most astronomers since that time.

In 1673, the celebrated John Hevelius, of Dantzic, pub-
lished, in his Machina Ccelcstis, a catalogue of 188S stars,
of which 1553 were observed by himself; and their places
set down for the end of the year 1660. But this catalogue,
as it stands in Flamsteed's Historia Ccelestis of 1725, con-
tains only 1520 stars.

The. most complete catalogue ever given from the la-
bours of one man, was the Britannic catalogue, compiled
from the observations of the accurate and indefatigable
Mr. Flamsteed, the first Royal Astronomer at Greenwich ;
who for a long series of years devoted himself wholly to
that employment. As there was nothing wanting, either
in the observer or apparatus, this may be considered as a
perfect work, so far as it goes. It is however to be re-
setted that the edition did not pass through his own
hands : that now extant was published by authority, but
without the author's consent, and contains 2734 stars.
Another edition was published in 1725, pursuant to his
testament, containing 3000 Stars, with their places adapted
to the beginning of the year 1(>8<); to which is added Mr.
Sharp's catalogue of the southern stars not visible in our
hemisphere, set down for the year 1720'. See vol. 3 of his
Historia Ccelestis, in which are printed the catalogues of
l'(()lemy,UlughBeigh,Tyeho. the Prince of Hesse, and Heve-
lius; with an account of each of them in the Prolegomena.

CAT

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CAT

The first catalogue, we believe, that was printed in the
new or second form, according to the order of the right
ascensions, is that of Lacaille,. given in his Ephemerides
for the 10 years between 1755 and 1765, and printed in
1755. It contains the right ascensions and declinations
of 307 stars,which are rectified to the beginning of the year
1750. — In 1757, Lacaille published his Astronomise Fun-
damenta, containing a catalogue of the right ascensions
and declinations of 398 stars, likewise adapted to the be-
ginning of 1750. — And in 1763, the year after his death,
was published the Caelum Australe Stelliferum of the
same author ; containing a catalogue of the places of 1942
stars, all situated to the southward of the tropic of Ca-
pricorn, and observed by him while he was at the Cape of
Good Hope, in 1751 and 1752; their places being also
suited to the beginning of 1750. — In the same year was
published his Epltemerides for the 10 years between 1765
and 1775 ; in the introduction to which are given the
places of 515 zodiacal stars, all deduced from the obser-
vations of the same author; and adjusted to the beginning
of the year 1765.

In the Nautical Almanac for 1773, is given a catalogue
of 387stars, in right ascension, declination, longitude, and
latitude, derived from the observations of the late cele-
brated Dr. Bradley, and adapted to the beginning of the
year 1760. This small catalogue, and the results of
about 1200 observations of the moon, are all that the pub-
lic have yet seen of the multiplied labours of this most ac-
curate and indefatigable observer, though he has now (1814)
been dead upwards of 52 years. ,

In 1775 was published a thin volume entitled Opera
Inedita, containing several papers of the lateTobias Mayer,
and among them a catalogue of the right ascensions and
declinations of 998 stars, which may be occulted by the
moon and planets ; the places being adjusted to the be-
ginning of the year 1756.

At the end of the first volume of Astronomical Obser-
vations made at the Royal Observatory at Greenwich,
published in 1776, Dr. Maskelyne, the late Astronomer
Royal, gave a catalogue of the places of 34 principal
stars, in right ascension and north-polar distance, adapted
to the beginning of the year 1770. These being the result
of several years' repeated observations, made with the ut-
most care and the best instruments, may be presumed to
be exceedingly accurate.

In 1782, M. Bode, of Berlin, published a very exten-
sive catalogue of 5058 of the fixed stars, collected from
the observations of Flamsteed, Bradley, Hevelius, Mayer,
Lacaille, Messier, Monnier, D'Arquier, and other astro-
nomers; all rectified to the beginning of the year 1780 ;
and accompanied with a Celestial Atlas, or sets of maps
of the constellations, engraved in a most delicate and
beautiful manner.

To these may be added, Dr. Herschel's catalogue of
double stars, printed in the Philos. Trans, for 1782 and
1783; Messier's nebulse and clusters of stars, published
in the Connoissance des Temps for 1784; and Herschel's
catalogue of the same kind, given in the Philos. Trans, for
1786, and other years.

In 1789, the Rev. Francis Wollaston published A Spe-
cimen of a General Astronomical Catalogue, in Zones
of North-polar Distance, and adapted to Jan. 1, 179°-
These stars are collected from all the catalogues before-
mentioned, from that of Hevelius downwards. This work
contains five distinct catalogues; viz, Dr. Maskclyne's

new catalogue of 36 principal stars ; a general catalo<me
of all the stars, in zones of north-polar distance ; an in-
dex to the general catalogue ; a catalogue of all the stars,
in the order in which they pass the meridian ; and a cata-
logue of zodiacal stars, in longitude and latitude.

In 1792, Dr. Zach published at Gotha, Tabula? Mo-
tuum Solis, to which is annexed a new catalogue of the
principal fixed stars, from his own observations made in
the years 1787, 1788, 1789, 1790. This catalogue con-
tains the right ascensions and declinations of 3S1 principal
stars, adapted to the beginning of the year 1800.

The catalogues of Dr. Bradley, Lacaille, Zach, and
Mayer, are published by Professor Vince, in the second
volume of his Astronomy. M. Lalande has published a
new catalogue of more than 12,000 stars, in the volumes
of the Connoissance des Temps, from the year 7 (1799)
to the year 12. Almost all these are stars which had
not been before observed. M. C. Vidal has lately com-
municated to the Lyceum of Toulouse a catalogue of
888 austral stars, from the 5th to the 7th magnitude
inclusively : every star has been observed three times,
and all are reduced to a mean position, regard being
had to the effect of refraction, the aberration of light,
and the rotation of the earth's axis. The mean position
of all these stars has been calculated to a common period,
viz. Jan. 31, 1798 ; the equation and precession of the
equinoxes being previously allowed for. The place of
M. Vidal's observations was Mirepoix, a situation ad-
mirably suited to his purpose, by the serenity of its at-
mosphere, and the excellence of the instruments with
which the observatory isfurnished,and commanding nearly
six degrees of the heavens southwards more than Paris.
On this account C. Lalande and his nephew C. Francois
Lalande, in their grand work of completing a catalogue
of 48,000 stars, engaged M. Vidal to form a catalogue of
the austral stars, which he has executed with great suc-
cess and admirable precision. From the history of astro-
nomy for 1S00 by Jerome de Lalande, it appears that
M. F. Lalande has terminated the labour commenced
August 5, 1789, and determined the places of 50,000
stars, from the pole to 2 or 3 degrees below the tropic of
Capricorn. In 1S00, S. Piazzi, of Palermo, published his
large catalogue, of 5500 stars.

CATAPULT, Catapulta, a military engine, much used
by the ancients for throwing huge stones, and sometimes
large darts and javelins, 12, 15, or even 18 feet long, on
the enemy. It is sometimes confounded with the Balista,
which is more peculiarly adapted for throwing stones ;
some authors making them the same, and others dif-
ferent.

The catapulta, which it is said was invented by the
Syrians, consisted of two large timbers, like masts of ships,
placed against each other, and bent by an engine for this
purpose ; these being suddenly unbent again by the stroke
of a hammer, threw the javelins with prodigious force.
Its structure and the manner of working it are described
by Vitruvius; and a figure of it is also given by Perrault.
M. Folard asserts that the catapult made infinitely more
disorder in the ranks, than our cannon charged with iron
balls. See Vitruv. Archit. lib. x. cap. 15 and 18; and
Perr. notes on the same ; also Rivius, p. 597-

Josephus takes notice of the surprising effects of these

engines, and says, that the stones thrown out of them, of

a hundred weight or more, beat down the battlements,

knocked the angles off the towers, and would level a whole

2 0 2

CAT

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CAT

file of men from one end to the other, if the phalanx were
ever so deep.

See plate V, fig. 3 and 4, for two forms of the catapult,
the one for throwing darts- and javelins, the other for stones.

CATENARY, a curve line which a chain, cord, or
such like, forms itself into, by hanging freely from two
points of suspension, whether these be in the same hori-
zontal line or not; as the curve bac, formed by a heavy
flexible line suspended by any two points c and b. It is
otherwise called the elastic curve.

acts in the perpendicular direction op ; that is, the three
forces which retain the curve ap in its position, act in the
directions of the sides of the elementary triangle opv ; but,
by the principles of mechanics, any three forces, keeping a
body in equilibrio, are proportional to the three sides of a
triangle drawn in the directions in which those forces act;
therefore it follows, that the forces keeping ap in its po-
sition, viz, the tension at a, the tension at p, and the weight
of ap, are respectively as op,p~e, and op,
that is, as y, z, and x.

But the tension at a is the constant quantity a, and the
weight of the uniform curve ap may be expounded by its
length s; therefore it follows that x : j ; ; z : a; which
was to be proved. /

Also from this last proportion, by proper analogy, or
similar combinations of the terms, there arises this other
property, x : </(xx -+- yj) or z : : s : ^/'(aa -+- zi), or zi. =

The nature of this curve was sought after by Galileo,
who thought it was the same with the parabola; but
though Jungius detected this mistake, its true nature was
not discovered till the year l6'Ql, in consequence of M.
John Bernoulli having published it as a problem in the
Acta Eruditorum, to the mathematicians of Europe. In
1697 Dr. David Gregory published an investigation of the
properties before discovered by Bernoulli and Leibnitz;
in which he asserts that an inverted catenary is the best
figure for an arch of a bridge, &c. See Philos. Trans, abr.
vol. 4; also Bernoulli Opera, vol. 1. pa. 48, and vol. 3,
pa. 491 ; and Cotes's Harmon. Merfsur. pa. 108.

The catenary is a curve of the mechanical kind, and can-
not be expressed by a finite algebraical equation, in simple
terms of its absciss and ordinate; but is easily expressed
by means of fluxions : thus, if aq be its axis perpendicu-
lar to the horizon, and pq an ordinate parallel to the same,
or perp. to aq ; also pq another ordinate indefinitely near
the former, and po parallel to aq : then, a being some gi-
ven or constant quantity, the fundamental property of the
curve is this, viz, to : op :: at? : a, or x : j> :: z : a, that
is, the fluxion of the axis, is to the fluxion of the ordinate,
as the length of the curve, is to the given quantity a;
where x = aq, y = PQ, and 2 = ap. This, and the other
properties of the curve, will plainly appear from the fol-
lowing considerations : First, supposing the curve hung up
by its two points b and c, against a perpendicular or up-
right wall : then, every lower part of the curve being kept
in its position by the tension of that which is immediately
above it, the lower parts of the curve will retain the same
position unvaried, by whatever points it is suspended above;
thus, if it were fixed to the wall by the point f, or G, &c,
the whole curve cab will remain just as it was ; for the
tensions at f and o have the same effect on the other
parts of the curve, as when it is fixed by those points:
and hence it follows that the tension of the curve at the
point a, in the horizontal direction, is a constant quantity,
whether the two legs or branches of the curve, on both
sides of it, be longer or shorter; which constant tension
at a let be denoted by the quantity fit.

Now, because any portion of the curve, as ap, is kept
in its position by three forces, viz, the tensions at its ex-
tremities a and p, and its own weight, of which the ten-
sion at a acts in the direction ah or po, and the tension
at p acts in the direction of vp, and the weight of the line

xt/{aa -+- 22), or x = \/f~
give x = \/(a2 -+- s2). But

•; and the fluents of these

at the vertex of the curve,
where x = 0, and 2 = 0, this becomes 0 = 1/ {a1 -+- 0) =a;
and therefore, by correction, the true equation of the flu-
ents is j: = \/(«" -+- s2) — a, or a -+- x = \,/(a2 + z~) : and
hence also 2 = \/[{a + x)- — a~~\ = ^/('2ux -+■ x~), and

a = - : either of which is the equation of the curve,

in terms of the arc and its absciss; from this it appears
that a -+- x is the hypothenuse of a right-angled triangle,
whose two legs are a and 2. So that, if in qa and ha pro-
duced, there be taken ad = a, and ae = the curve z or
ap; then will the hypothenuse de be = a ■+- .r or dq,
And hence, any two of these three, a, x, s, being given, the
third is given also.

Again, from the first simple property, viz, *-.y : : z: a,
or ax = zj, by substituting the value of s above found, it

becomes ax = i*/(2ax -+- x'2), or»= —. -r-: and the

fluent of this equation is y = 2s x hyp. log. of [^/x -+-
\/{2a -+- x)]. But at the vertex of the curve, where x = 0
and y = 0, this becomes 0 = 2« x hyp. log. of *J1a;
therefore the correct fluent is y = 2« x hyp. log. of

— y — ; an equation to the curve also, in terms of

x and y, but not in simple algebraic terms. This last
equation however may be brought to much simpler terms
in different ways; as first, by squaring the logarithmic
quantity and dividing its coefficient by 2, then y = a x
^/(2ax + .r3)

hyp. log. of

■ = a x hyp. log. :

and 2dly, by multiplying both numerator and denomina-
tor by \/(2a ■+■ x) — */x, then squaring the product, and
dividing the coefficient by 2, which gives y = a x hyp. log.

: + x , , (: x .t)3 . . : + .1

— ; — - = a x hyp. 105. -, 5 = a x hyp. I02. .

</{z "— x-) •" b s — or -" s -s— x

The catenary is the cvolute to the equitangential curve.

CATHETUS, in Geometry, a name by which the per-
pendicular leg of a right-angled triangle is sometimes call-
ed ; or it is in general any line or radius falling perpendi-
cularly on another line or surface.

CatHETTJS of Incidence, in Catoptrics, is a right line
drawn from a radiant point, or point of incidence, perpen-
dicular to the reflecting line, or plane of the speculum.

Cathetus of Reflection, or of the Eye, a right line
drawn from the eye, perpendicular to the plane of re-
flection.

Cathetus, in Architecture, denotes the axis of a cp-

CAT

[ -285 ]

CAT

lumn, &c. In the Ionic capital, it signifies a line pass-
ing perpendicularly through the eye or centre of the vo-
lute.

CATOPTRICS, the science of reflex vision ; or that part
of optics which explains the laws and properties of light
reflected from mirrors, or specula.

.The first treatise extant on catoptrics, is that which was
composed by Euclid : this was published in Latin in l6"04
by John Pena ; it is also contained in Herigon's Course of
Mathematics, and in Gregory's edition of Euclid's works ;
though it is suspected by some that this piece, was not
the work of that great geometrician, notwithstanding it is
ascribed to him by Proclus in lib. 2, and by Marinus in
his Preface to Euclid's Data. Alhazen, an Arabian author,
composed a large volume of optics about the year 1100,
in which he treats pretty fully of catoptrics: and after him
Vitello, a Polish writer, composed another about the year
1270. Tacquet, in his Optics, has very well demonstrated
the chief propositions of plane and spherical speculums.
And the same is very ably done by Dr. Barrow in his Op-
tical Lectures. There are also Trabe's Catoptrics, David
Gregory's Elements of Catoptrics, Wolfius's Elements of
Catoptrics, and those of Dr. Smith, contained in his learned
and very elaborate Treatise on Optics, and many others of
less note.

As this subject is treated under the general term Optics,
the less need be said of it here. The whole doctrine of
catoptrics depends on this simple principle, that the
angle of incidence is equal to the angle of reflection ; that
is, that the angle in which a ray of light falls upon any
surface, called the angle of incidence, is equal to the angle
in which it quits it, when reflected from it, called the an-
gle of reflection; though it is sometimes defined that the
angles of incidence and reflection, are those which the in-
cident and reflected rays make, not with the reflecting sur-
face itself, but with a perpendicular to that surface, at
the point of incidence, which are the complements to the
others: but it matters not by what name these angles are
called, as to the truth and principles of the science ; since,
if the angles are equal, their complements are also equal.
This principle of the equality between the angles of inci-
dence and reflection, is mere matter of experience, being a
phenomenon that has always been observed to take place,
in every case that has fallen under observation, as near at
least as mechanical measurements can ascertain; and
hence it is inferred that it is a universal law ofnature, and
to be considered as matter A o B

of fact in all cases. Thus,
let AC be an incident ray,
falling on the reflectingsur-
fece DE,and cb the reflect-
ed ray, also co perpendicu-
lar to de; then is the an-
gle acd = BCE.or the angle aco = bco.

Of this law in nature, viz, the equality between the an-
gles cf incidence and reflection, it is remarkable, that in
this way, the length or route ac -+- cb, in a ray passing
from any point a to another given point b, by being re-
flected from any surface de, is the shortest possible, namely
ac -t-CB is shorter than the sum ag -t- cb, of any other
two lines inflected at the line de; and hence also the pas-
sage of the ray from a to B is performed in a shorter time
than if it had passed by any other way.

From this simple principle, and the common properties

of linear geometry, the chief phenomena of catoptrics are
easily deduced, and are as follows:

1. Rays of light reflected from a plane surface, have the
same inclination to
each other after re-
flection as they had
before it. Thus, the
rays ac, ai, ak, is-
suing from the radi-
ant point a, and re-
flected by the sur-
face de into the lines
CF, il, km ; these
latter lines will have the same inclination to each other as
the former ac, ai, ak, have. For draw abg perpendicu-
lar to de, and produce fc, li, mk backwards to meet
this perpendicular, so shall they all meet in the same point
G, and ab will in every case be equal to bd : for the inci-
dent iACB is equal to the reflected Z. fce, which is equal
to the opposite angle bcg; so that the two triangles abc,
gbc, have the angles at C equal, as also the. right angles at
b equal, and consequently the 3d angles at a and g equal;
and having also the side bc common, they are equal in all
respects, and ab=bg; and the same for the other rays:
consequently the angles bgc, bgi, bgk, are respectively
equal to the angles bag1, bai, bak; that is, the reflected
rays have the same inclinations as the incident ones have.

2. Hence it is that the image of an object, seen by re-
flection from a plain mirror, appears to proceed from a
place g as far beyond, or on the other side of the reflect-
ing plane de, as the object a itself is before the plane.
This is when the incident rays diverge from some point
as a.

But if the case be reversed, and fc, li, mk, be consi-
dered as incident rays, issuing from points, f, l, m, and
converging to some point g beyond the reflecting plane ;
then ca, ia, ka will become the reflected rays, and they
will converge to the point a as far before the plane as the
point g is beyond it.

So that universally, when the incident rays diverge from
a point a, the reflected rays will also diverge from a point
G ; and when the incident rays converge towards a point g,
the reflected ones will also converge to a point a; and in
both cases these two points are at equal distances on the
opposite sides of the reflecting plane de.

3. Parallel rays reflected from a concave spherical sur-
face, converge after re-
flection. For, let af, cd,
eb be three parallel rays
falling on the concave
surface fb, whose cen-
tre is c. To the centre
draw the perpendiculars
fc and bc; also draw
fm making the re-
flected angle, cfm, e-
qual to the incident
angle cfa; and in like
manner bm to make the
angle cbm == the angle
cbe; so shall the rays at
and eb bc reflected in-
to the converging rays
fm and bm. As to the

CAT

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CAT

ray cd, being perpendicular to the surface, it is reflected
back again in the same line DC.

4. Converging rays falling on the concave surface are
made to converge more. Thus, let gb and nr be the in-
cident rays : then because tse incident angle hfc is larger
than the angle AFC, therefore the equal reflected angle
nfc is greater than the reflected angle mfc, and so the
point n is below the point M-, or the line FN below the
line fm ; in like manner bn is below bm ; that is, the re-
flected rays FN and bn are more converging in this case,
than fm and bm in the other.

5. The focus to which all parallel rays, falling near the
vertex d, are reflected, is in the middle of the radius m.
For, because the /.MFC = Z.AFC, which is = the al-.
tern'ate Zfcm, therefore the sides opposite these angles
are also equal,, namely the side fm = cm ; consequently,
when the point f is very near the vertex d, then the sum
CM -+- mf is nearly = CD, and consequently cm nearly =
MD,orthe focus of the parallel rays is nearly in the middle
of the radius. But the focus of other reflected rays is either
above or below that of the parallel rays ; namely, below
when the incident rays are converging, and above when
they are diverging ; as is evident by inspection ; thus, n
the reflected focus of the converging rays gb and hf, is
below m; i that of the diverging rays yb and yf, is
above m.

6". Incident and reflected rays are reciprocal, or so that
if the reflected rays be returned back, or considered as
incident ones, they will be reflected back into what were
before their incident rays. And hence it follows that di-
verging rays, after reflection from a concave spherical sur-
face, become either parallel or less diverging than before.
Thus, the incident rays mf and mb are reflected into the
parallel rays fa and be, and the rays nf and nb are re-
flected into fh and bg, which are less diverging; also the
rays if and IB are reflected into fk and bl, which con-
verge,— And hence all the phenomena of concave mirrors
will be evident.

7. Rays reflected from a convex speculum, become
quite contrary to those reflected fron a concave one; so
that the parallel rays become diverging, and the diverging
rays become still more diverging ; also converging rays
will become either diverging, or parallel, or else less con-
verging. Thus bdf be-
ing a spherical surface,
whose centre is c, pro-
duce the radii cbv and
cft which are perpen-
dicular to the surface;
then it is evident that
the parallel rays af and
f.b will be reflected into
the diverging ones fk
and bl; and the di-
verging rays yb and
yf become BO and fp
which are more diverg-
ing; also the converging
rays hf and gb become Fit and bs which diverge, or else
kf and lb become fa and he which arc parallel, or else
lastly pf and ob become fy and by which are converging.

8. Hence, as in the concave speculum, so also in the
convex one, of parallel incident rays af and eb, the ima-
ginary focus m of their reflected rays fk and bl, is in

the middle of the radius when the speculum is a small
segment of a sphere : but the reflected imaginary focus of
other rays is either above or below the middle point m,
viz n being that of the converging rays gb and hf, below
M ; but i, that of the diverging rays yb and yf, above m.

9. When the speculum is the small segment of a sphere,
either convex or concave, and the incident rays either
converging or diverging, the distances of the foci, or points
of concurrence, of the incident rays, and of the reflected
rays, from the vertex of the speculum, are directly pro-
portional to the distances of the same from the centre of it;
that is, yd : id : : yc : ic,
and qd : nd : : qc : no

For because the radius cf, or the same produced, bisects
the. angle yfi in the concave speculum, or the external
angle yfp in the convex one, therefore yf : if : : yc : ic;
but when v is very near to d, then yf and if become
nearly yd and id ; consequently yd : id : : yc : ic.

In like manner, because cf bisects the angle qfn in the
convex, or its external angle nfh in the concave specu-
lum, therefore qf : fn : : qc : cn ; but when f is very
near to d, then qf and fn become nearly qd and nd;
and therefore qd : nd : : qc : uc.

For example, suppose it were required to find the focal
distance of diverging rays incident on a convex surface,
the radius of the sphere being 5 inches, and the distance
of the radiant point from the surface 20 inches. Here
then are given yd = 20, and cd = 5, to find id : then
the theorem yd : id : : yc : ic,

in numbers is 20 : id : : 25 : 5 — id,

or by permutation 20 : 25 : : id : 5 — id,
and by composition 4-5 : 20 : : 5 : id = ^ = %° =
2| the focal distance sought.

And if it should happen in any case, that the value of
id in the calculation comes out a negative quantity, the
focal distance must then be taken on the contrary side of
the surface.

From the foregoing principles may be deduced and col-
lected the following practical maxims, for plane and sphe-
rical mirrors, viz,

I. In a Plane Mirror.

(1.) The image will appear as far behind the mirror, as
the object is before it.

(2.) The image will appear of the same size, and in the
same position as the object.

(3.) Any plain mirror will reflect the image of an ob-
ject of twice its own length and breadth.

II, In a Spherical Convex Mirror.

(1.) The image will always appear behind the mirror,
or within the sphere.

(2.) The image will be in the same position, but less
than the object.

(3.) The image will be curved, but not spherical, like
the mirror.

(4.) Parallel rays falling on this mirror, will have the
image at half the distance of the centre from the mirror.

(5.) In converging rays, the distance of the object must
be equal to half the distance of the centre, to make the
image appear behind the mirror.

(6\) Diverging rays will have their image at less than
half the distance of the centre.

III. In a Spherical Concave Mirror.

(1.) Parallel rays have their focus, or the image, at half
the distance of the centre.

CAT

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C A V

(J.) In the centre of the sphere the image appears of
the same dimensions as the object.

(3.) Converging rays form an image before the mirror.

(4.) In diverging rays, if the object be at less than half
the distance of the centre, the image will be behind the
mirror, erect, curved, and magnified ; but if the distance
of the object be greater, the image will be before the mir-
ror, inverted and diminished.

(5.) The solar rays, being parallel, will be collected in
a focus at half the distance of its centre, where their heat
will be augmented in proportion as the surface of the mir-
ror exceeds that of the focal spot.

(6.) If a luminous body be placed in the focus of a
concave mirror, its rays, being reflected in parallel lines,
■will strongly enlighten a space of the same dimensions with
the mirror, at a great distance. If the luminous object
be placed nearer than the focus, its rays will diverge, and
so enlighten a larger space, but not so strongly. And on
this principle it is that reverberators are constructed.

Catoptric Dial, a dial that exhibits objects by re-
flected rays. See Reflecting Dial.

Catoptric Telescope, a telescope that exhibits objects
by reflection. See Reflecting Telescope,

Catoptric Cistula, a machine, or apparatus, by which
small bodies are represented extremely large, and near ones
extremely wide, and diffused through a vast space ; with
other very pleasing phenomena, by means of mirrors, dis-
posed by the laws of catoptrics in the concavity of a kind
of chest.

There are various kinds of these machines, accommo-
dated to the various intentions of the artificer : some mul-
tiply the objects, some magnify, some deform them, &c.
The structure of one or two of them will suffice to show
how many more may be made.
To make a Catoptric Cistula to represent several different

scenes of objects, when viewed through different holes.
D E

Provide a polygonal cistula, or box, like the mult-an-
gular prism abcdef, and divide its cavity by diagonal
planes ad, be, cf, intersecting in the centre, into as
many triangular cells as the chest has sides. Line those
diagonal partitions with plain mirrors; and in the sides of
the box make round holes, through which the eye may
look within the cells of it. These holes are to be covered
with plain glasses, ground withinsidc, but not polished, to
prevent the objects in the cells from appearing too di-
stinctly. In each cell are to be placed the different ob-
jects whose images are to be exhibited ; then covering up
the top of the chest with a thin transparent membrane,
or parchment, to admit the light, the machine is complete.

For, from the laws of reflection, it follows, that the
images of objects, placed within the angles of mirrors, are

multiplied, and appear some more remote than others;
by which the objects in one cell will appear to take up
more room than is contained in the whole box. There-
fore by looking through one hole only, the objects in one
cell will be seen, but those multiplied, and diffused through
a space much larger than the whole box. Thus every
hole will afford a new scene; and according to the dif-
ferent angles the mirrors make with each other, the repre-
sentations will be different : if they be at an angle greater
than a right one, the images will be monstrous, &c.

To make a Catoptric Cistula to represent the objects within
it prodigiously midtiplied, and diffused through a vast
space.

Make a polygonous cistula or box, as before, but with-
out dividing the inner cavity into any apartments, or cells;
line the insides cbhi, bhla, almf, &c, with plane mir-
rors, and at the holes pare off the tin and quicksilver,
that the eye may look through ; place any object in the
bottom mi, as a bird in a cage, &c.

Now by looking through the aperture hi, each object
placed at the bottom will be seen vastly multiplied, and
the images removed at equal distances from one another,
like a great multitude of birds, or a large aviary.

CAVALIER, in Fortification, a mount of earth raised
in -a fortress higher than the other works, on which to
place cannon. &c for scouring the field, or opposing a
commanding work. Cavaliers are of different shapes;
and are bordered with a parapet, to cover the cannon
mounted upon them; their situation is also various, either
in the curtain, the bastion, or the gorge. The cavalier is
sometimes called a double bastion, and its use is to over-
look the enemy's batteries, and to scour their trenches.

CAVALIERI (Buna ventura), a celebrated Italian
mathematician of the 17th century. He was born at
Milan in 1598, and became a friar of the order of the Je-
suati of St. Jerome. Cavalieri was a disciple of Galileo,
and the friend of Torricelli. He was professor of mathe-
matics at Bologna : where several of his books were pub-
lished, and where he died in the year l647. His works
that have been published, as far as I can discover, are as
follow :

1. Dircctorium Generate Uranometricum ; 4to, Bono-
nia?, 1632. — In this work the author treats of Trigonome-
try ; and Logarithms, their construction, uses, and appli-
cations. The work includes also tables of logarithms of
common numbers ; with trigonometrical tables of natural
sines, and logarithmic sines, tangents,, secants and versed
sines.

2. Lo Spechio Ustorio overo Trattato delle Settioni
Coniche : 4to, Bologna, 1632. — An ingenious treatise of
conic sections.

3. Geometria Indivisibilibus continuorumnovaquadam
ratione promota: 4 to, Bononiee, 1635 ; and a 2d edition
in 1653. — This is a curious original work in geometry, in
which the author conceives the geometrical figures as re-
solved into their very small elements, or as made up of an
infinite number of infinitely small parts, and on account
of which he passes in Italy for the inventor of the infinite-
simal calculus.

4. Trigonomctria Plana et Sphajrica, Linearis, et Loga-
rithmica: 4to, Bononia:, 1643. — A very neatand ingenious
treatise on Trigonometry ; with the tables of sines, tan-
gents, and secants, both natural and logarithmic.

5. ExercitationesGcometricaj Sex; 4lo, Bunoniu;, 1(>47.
This work contains Exercises on the method of Indivisi-

C A V

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C A V

bles; Answers to the objections of Guldini; The use of In-
divisibles in cossic powers or algebra, and in considera-
tions concerning gravity; with a miscellaneous collection
of problems.

CAVALLO (Tiberius), f. r. s. was born in the city
of Naples the 30th of March 1749. His father, who was
an eminent physician, died when Tiberius was scarcely 11
years of age. His relations continued to give him a libe-
ral education; and as he was intended for a commercial
life, with that view tliey sent him to England. He arrived
in London in 1771, and continued in this country during
the rest of his life. Here his inclination led him to pur-
sue scientific studies, the rudiments of which he had re-
ceived at the university of Naples, and he attained to a
considerable rank in several of the more fashionable and
popular physical sciences. In March 1799, he was elected
member of the Royal Academy of Sciences of Naples ;
and in December the same year he was elected home mem-
ber of the Royal Society of London ; where he died the
26th of December 1809, at 60 years of age.

Besides several ingenious papers inserted in the Philos.
Trans., Mr. C. published separate and very interesting
volumes on various popular and useful subjects: as, A
Complete Treatise of Electricity in Theory and Practice,
1 vol. 8vo. 1777 (enlarged to 3 vols, in 1793). — An Essay
on the Theory and Practice of .Medical Electricity, 1 vol.
8vo. 17S0. — A Treatise on the Nature and Properties of
Air, and other Permanently Elastic Fluids, with an Intro-
duction to Chemistry, 1 vol. 4to, 17S1. — The History and
Practice of Aerostation ; 1 vol. Svo, 1785. — Mineralogical
Tables; folio, (accompanied with an 8vo explanatory
pamphlet) 1785. — A Treatise on Magnetism, in Theory
and Practice; I vol. Svo, 1787- — Description and Use of
the Telescopical mother-of-pearl Micrometer; Svo, 1793.
—An Essay on the Medicinal Properties of Factitious
Airs, with an Appendix on the nature of Blood ; 1 vol.
8vo, 1798, many of which have gone through several edi-
tions.

CAUDA Capricorni, a fixed star of the -1th magnitude,
in the tail of Capricorn; called also by the Arabs, Dineb
Algedi ; and marked y by Bayer.

Cauda Ceti, a fixed star of the 3d magnitude; called
also by the Arabs, Dineb Kaetos ; marked /3 by Bayer.

Cauda Cygni, a fixed star of the 2d magnitude in the
Swan's tail ; called by the Arabs, Dineb,Adigege, or Elde-
giagich ; and marked a. by Bayer.

Cauda Delphini, a fixed star of the 3d magnitude, in
the tail of the Dolphin ; marked £ by Bayer.

Cauda Draconis, or Dragon's tail, the moon's southern
or descending node.

Cauda Leonis, a fixed star of the first magnitude in the
Lion's tail ; called also by the Arabs, Dineb Eleccd ; and
marked /3 by Bayer. It is called also Lucida Cauda.

Cauda Ursce Mujoris, a fixed starof the"3d magnitude,
in the tip of the Great Bear's tail ; called also by the
Arabs, Alalioth, and Benenath ; and marked jj by Bayer.

Cauda Una Blinoris, a fixed star of the 3d magnitude,
at the end of the Lesser Bear's tail; called also the Pole
Star, and by the Arabs, Aliukabah ; and marked a byTBayer.

CAVENDISH (Henry), f. r.s. son of the late Lord

Charles Cavendish, was bom Oct. 10, 1731, and died Feb.
24, 1810, consequently in his 79th year. Mr. C lived
and died a bachelor ', devoting the whole hours and atten-
tions of his long life mosl assiduously in cultivating the
sciences, mathematical, physical, chemical, &c, in all

which he excelled in a ver}' eminent degree. His genius
and labours were rewarded with the most brilliant disco-
veries and improvements in the various branches of science;
the records of which are chiefly to be found in the volumes
of the Philos. Trans. He had studied, and rendered him-
self familiarly conversant with, every part of Newton's
philosophy; the principles of which he applied, 40 years
ago, to an investigation of the laws on which the pheno-
mena of electricity depend. Pursuing the same research,
on the occasion of the first experiments with the torpedo,
he gave a satisfactory explanation of the remarkable
powers of electrical fishes; pointing out that distinction
between common and animal electricity, which has since
been amply confirmed by the brilliant discoveries in Gal-
vanism. Having very early turned his attention to pneu-
matic-chemistry, in 1766 he ascertained the extreme levity
of inflammable air, now called hydrogen gas; a discovery
on which are founded many curious experiments, particu-
larly that of aerial navigation. In the same line of science,
he made the important discovery of the composition of
water, by the union of two airs; which laid the founda-
tion of the modern system of chemistry, which rests prin-
cipally on this fact, and that of the decomposition of water,
announced soon afterwards by M. Lavoisier.

As the purity of atmospherical air had been a subject
of controversy, Mr. C. contrived essential improvements
in the method of performing experiments with the eudio-
meter; by means of which he was the first who showed,
that the proportion of pure air, in the atmosphere, is nearly
the same in all open places. The other, and much larger
portion of our atmosphere, he sagaciously conjectured to
be the basis of the acid of nitre ; an opinion which he soon
brought to the test, by an ingenious and laborious experi-
ment, completely proving its truth; whence this air has
now very generally obtained the name of nitrogen. So
many and great discoveries spread his fame throughout
Europe, and he was universally considered as one of the
greatest philosophers of the age.

Among the labours of his latter days, was the nice and
difficult experiment, by a mode and instrument contrived
by the late Rev. John Michell, by which he determined
the mean density of the earth, in a result not widely dif-
ferent from what was obtained in the more perfect way by
the attraction of mountains. This density forms an ele-
ment of consequence in delicate calculations of astronomy,
as well as in geological inquiries. Even in the last year of
his life, he proposed and described improvements in the
manner of dividing large astronomical instruments, which
promise very great advantages.

These pursuits, together with reading, of various kinds,
by which he acquired a deep insight into almost every
topic of general knowledge, formed the whole occupation
of his life, and indeed constituted his sole amusement.
The love of truth alone was sufficient lo till his mind.
From his attachment to such occupations, and the con-
stant resource he found in them, together with a shyness
and diffidence natural to his disposition, his habits had,
from early life, been secluded, lit- manners were mild —
his mind firm — his nature benevoleni and complacent. He
was liberal, without being profuse: and charitable, with-
out ostentation.

CAVETTO, a hollow member or moulding, containing
a quadrant of a circle, and having an effect just contrary
to that of a. quarter round. It is used as an ornament 111
cornices.

CEN [2

CAULICOLES, Cauliculi, in Architecture, are those 8
lesser branches or stalks in the Corinthian capital, which
spring out from the four greater principal caules or stalks.

CAUSE, Physical, that which produces. a sensible cor-
poreal effect. Thus, the sun is a physical cause of light.
Others define a physical cause to be that which produces
its effects by a physical virtue, or natural influence.

CAUSTIC CjUitVE, in the higher geometry, is a curve
forlned by the concourse or coincidence of the rays of
light, when either reflected, or refracted, from some other
curve.

Every curve therefore has its twofold caustic: accord-
ingly, caustics are divided into catacaustics, and diacaus-
ties; the one formed by reflection, the angle of reflection
being equal to that of incidence ; the other by refraction.
See the two articles Catacaustics and Diacaustics.

To pursue the doctrine of caustics, as it relates to .op-
tics, consult Haiiy's Philosophy, vol. 2 ; and as a branch
of the doctrine of curves, consult the writers on fluxions,
as Hays, or Hodgson, or Stone, &c.

CAZEMATE. See Casemate.

CAZERN. See Casern.

CEGlNUS,a fixed star of the 3d magnitude, in the left
shoulder of Bootes; and marked y by Bayer.

CELERITY, is the velocity or swiftness of a body in
motion ; or that affection of a body in motion by which it
can pass over a certain space in a certain time.

CELESTIAL Globe, &c. See Globe, &c.

CELLAR1US (Christopher), a learned geographer
and historiographer of the 17th century. He was born
in 1638, at Smalcalde in Franconia, where his father was
minister. Our author was successively rector of the col-
leges at Weymar, Zeits, and Mersbourg, and professor of
eloquence and history in the university founded by the king,
of Prussia at Hall in 1 693, where he composed the greatest
part of his works.

His great application to study hastened the infirmities of
old age ; for it has been said, he would spend whole days
and nights together over his books, without any attention
to his health, or even the calls of nature. He died in 1707,
at 69 years of age.

Cellarius was author of a great number of books, on
various subjects ; but those for which he has a place here,
are his geographical works, which are as follow:

1. Notitia Orbis Antrqui, 2 vols, in 4to; and is esteemed
the best work extant on the ancient geography.

2. Atlas Ccelestis; in folio.

3. Historia Antiqua, 2 vols, in 12mo ; being an abridge-
ment of universal history.

CENTAURUS, the Centaur, one of the 48 old constel-
lations, being a southern one, and is in form half man and
half horse. It is fabled by the Greeks that it was Chiron
the Centaur, who was the tutor of Achilles and Esculapius.
The stars of this constellation are, in Ptolemy's catalogue
37, in Tycho's 4, and in the Britannic catalogue, with
Sharpe's appendix, 35.

CENTER, and Centering. See Centre, and Cen-
tring.

CENTESM, the 100th part of any thing.

CENTIGRADE, the division into grades or degress by
hundredth parts, called also Centesimals, contracted into
Ccntemeg, on still more contracted Cents.

The Nort"n'-American states have their money (dollars),
divided into hundredth parts or cents. And the French
nation have lately divided all their money, weights, and

Vol. I.

]

C E N

measures, decimally ; which is allowed to be the base
way. They have even divided the circular quadrant into
cents, for trigonometrical uses, on the centigrade way,
having 100 in the quadrant, while we have 90 : there-
fore the magnitude of their degree, is to that of ours, as
9 to 10; but the number of their degrees to the number
of ours, as 10 to 9 : hence, to change French degrees into
English, diminish them by one-tenth, or multiply them by
9, and cut off one figure for a decimal: and to find the
number of French degrees answering to a given number
of English, annex a cipher to the latter, then divide them
by 9-

The French have also their centigrade thermometer,
&c. In that instrument they divide the distance between
the freezing and boiling points into 100, which we divide
into 180; therefore the length of their thermomelrical de-
gree, is to the length of ours, as 180 to .100, or as 18 to 10,
or 9 to 5 ; and the number of theirs to the number of ours,
as 10 to IS, or 5 to 9; therefore, to change French ther-
mometer degrees into English, multiply the former by 9*
and divide by 5 : so, to find the English degrees answering
to 15 French, the 5th of 15 is 3, then 3 x 9 = 27 are the
English degrees answering to 15 French : but as the French.
freezing point is at 0, and ours at 32, therefore 27 + 32 =
59 is the point or degree on the English thermometer, an-
swering to that of 1 5 on the French.

CENTRAL, something relating to a centre. Thus we
say central eclipse, central forces, central rule, &c.

Central Eclipse, is when the centres of the luminaries
exactly coincide, and fall directly in a line with the eye.

Central Forces, are forces having a tendency directly
towards or from some point or centre; or forces which
cause a moving body to lend towards, or recede from, the
centre of motion. And accordingly they are divided into
two kinds, in respect to their different relations to the
centre, and hence are called centripetal, and centrifugal.

The doctrine of central forces constitutes a consider-
able branch of the Newtonian philosophy, and has been
greatly cultivated by mathematicians, on account of its
extensive use in the theory of gravity, and other branches
of physico-mathematical science.

In this doctrine, it is supposed that matter is equally
indifferent to motion or rest, or rather that it always re-
sists any attempt to change its state; or that a body at
rest never moves itself, and that a body in motion never of
itself changes either the velocity or the direction of its mo-
tion; but that every motion would continue uniformly,
and its direction rectilinear, unless some external force 01
resistance affected it or should act upon it. Hence, whe*;
a body at rest always tends to move, or when the velocity
of any rectilinear motion is continually accelerated or re:
tarded, or when the direction of a motion is continually
changed, and a curve line is thereby described, it is sup
posed that these circumstances proceed from the influence
of some power that acts incessantly ; which power may
be measured, in the first case, by the pressure of the quies-
cent body against the obstacle which prerents it from
moving, or by the velocity gained or lost in the second
case, or by the flexure of the curve described in the 3d
case ; due regard being had to the time in which these
effects are produced, and other circumstances, according
to the principles of mechanics. Now the power or force
of gravity produces effects of each of these kinds, which
fall under our constant observation near the surface of the
earth ; for the same power which render* bodies heavy,

2P

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CEN

while they are at rest, accelerates their motion when they
descend perpendicularly; or retards them when they as-
cend; and bends the track of their motion into acurveline,
when they are projected in a direction oblique to that of
their gravity. But we can judge of the forces or powers
that act on the celestial bodies by effects of the last kind
only. And hence it is, that the doctrine of central forces
is of so much use in the theory of the planetary motions.

Sir I. Newton has treated of central forces in lib. 1, sec
2, of his Principia, and has demonstrated this fundamental
theorem of central forces, viz, that the areas which re-
volving bodies describe by radii drawn to an immoveable
centre, lie in the same immoveable planes, and are pro-
portional to the times in which they are described. Prop. 1.
It is remarked by a late eminent mathematician, that
this law, which was originally observed by Kepler, is the
onlygeneral principle in the doctrine of centripetal forces;
but since this law, as Newton himself has proved, cannot
hold in cases where a body has a tendency to any other
than one and the same point, there seems to be wanting
some law that may serve to explain the motions of the
moon and satellites which gravitate towards two different
centres: the law he lays down for this purpose is, that
when a body is urged by two forces tending constantly to
two fixed points, it will describe, by lines drawn from the
two fixed points, equal solids in equal times, about the
line joining those fixed points. See Machin, on the Laws
of the Moon's Motion, in the Postscript. This short trea-
tise is published at the end of the English translation of
Sir Isaac Newton's Principia. See also a demonstration
of this law by Mr. William Jones, in the Philos. Trans,
vol. 5Q. Very learned tracts have also been since given,
when the motion respects, not two only, but several cen-
tres, by many ingenious authors, and practical rules de-
duced from them for computing the places &c of planets

and satellites; as by Lagrange, Laplace, Waring, &c, &c.

See Berlin Memoirs ; those of the Academy of Sciences at

Paris ; and the Philos. Trans, of London.

M. Demoivre gave elegant general theorems relating

to central forces, in the

Philos. Trans, and in his

Miscel. Analyt. 231. — Let

mpq be any given curve, in

which a body moves : let P

be the place of the body at

any time; s the centre of

force, or the point to which

the central force acting on

the body is always directed; pg the radius of curvature

at the point p; and st perpendicular to the tangent pt ;

then will the centripetal force be every where proportional

to the quantity • —3. Vid. ut supra.

M. Varignon has also given two general theorems on
this subject in the Memoirs of the Acad. an. 1700, 1701 ;
and has shown their application to the motions of the
planets. See also the same Memoirs, an. 1706, 1710.

Mr. Maclaurin has also treated the subject of central
forces very ably and fully, in his Treatise on Fluxions,
art. 416 to 493 ; where he gives a great variety of expres-
sions for these- forces, and several elegant methods of in-
vestigating thern.

Laws of Centra i. Forces.

1.1 be followingis a very clear and comprehensive rule,
fc>r which wc arc obliged to the Marquis de I'Hopital :

Suppose a body of any determinate weight to revolve uni-
formly about a centre, with any given velocity ; find from
what height it must have fallen, by the force of gravity, to
acquire that velocity ; then, as the radius of the circle it
describes, is to double that height, so is its weight to its
centrifugal force. So that, if b be the body, or its weight
or quantity of matter, v its velocity, and r the radius of
the circle described, also g = 16^ feet ; then, by the

laws of falling bodies we have4g2 : z?2:: g : — theheightdue

to the velocity v; and by rule r :— : : b : — = f the cen-

trifugal force. Hence, if the centrifugal force be equal to
the gravity, the velocity is equal to that acquired by fall-
ing through half the radius.

2. The central force of a body moving in the periphery
of a circle, is as the versed -p A D (t
sine am of the indefinitely
small arc ae; or it is as the
square of that arc ae direct-
ly, and as the diameter ae
inversely. For am is the
space through which the
body is drawn from the tan-
gent in the given time, and
2am is the proper measure
of the intensity of the central
force. But, ae being very
small, and therefore nearly equal to its chord, we shall
have by the nature of the circle

AE3

ab : ae : : ae : am = — .

AB

3. If two bodies revolve uniformly in different circles;
their central forces are in the duplicate ratio of their velo-
cities directly, and the diameters or radii of the circles in-

■ ' .. . - V3 J)3 V3 t'3

versely; that is, f :/:: — :—::—:—.
J d d n r

For the force, by the last article, is as — or — ; and the ve-

J AB D

locity v is as the space ae uniformly described.

4. And hence, if the radii or diameters be reciprocally
in the duplicate ratio of the velocities, the central forces
will be reciprocally in the duplicate ratio of the radii, or
directly as the 4th power of the velocities; that is, if
V2 : »2 : : r : r, then f :/::»•-: r'2 : : v4 : »\

5. The central forces are as the diameters of the circles
directly, and squares of the periodic times inversely. For
if c be the circumference described in the time t, with the

velocity v; then the space c = tv, or v = - ; hence, using

this value of v in the 3d rule, it becomes

F : / : : — „ : —= : : — r : - : : — r : ii since the diameter

DT df TJ I' T t

is as the circumference.

6. If two bodies, revolving in different circles, be acted
on by the same central force; the periodic times are in
the subduplicate ratio of the diameters or radii of the cir-
cles; for when p =/, then — j = -., and d : d : : i3 : 1°,

T* t~

or t : t : : ^/d : ^/d : : v'r : v'r.

7. If the velocities be reciprocally as the distances from
the centre, the central forces will be reciprocally as the
cubes of the same distances, or directly as the. cubes of the
velocities. That is, if v : v : : r : r, then is F :/: : r3 :

3 3 3 7- r v'J nJ r'1 n" 13

rj : : vJ : tr. lot E ; f : : — : - : : - : — ; ; r; : rj : :

C E N

C 201 ]

C E N

8. If the velocities be reciprocally in the subduplicate
ratio of the central distances, the squares of the times will
be as the cubes of the distances : thus, if v2 : »2 : : r : r,

then is t2 : /2 : : r3 : r3. For by 3d and 5th — : -::—:-,

and

(by this) = :,, and hence

t2 : t- : : r3 : r3.

9- Therefore, if the forces be reciprocally as the squares
of the central distances, the squares of the periodic times
will be as the cubes of the distances ; or when f : f : : r2 : k~,

then is t2 : t2 : : r3 : r3. For by 5th, f : / : : -j- : -, and by

this f :/ : : r* : r2, theref. — : - : : r2 : r% and hence
r f

Tz : tz : : R3 : r3.

Exam. From this, and some of the foregoing theorems,

may be deduced the velocity and periodic time of a body

revolving in a circle, at any given distance from the earth's

centre, by means of its own gravity. Put g = 1 6^ feet,

the space described by gravity, at the surface, in the first

second of time, viz, = am in the foregoing fig. and by rule

2 ; then putting r = the radius ac; it is ae = \/(ab x

am) = y"2gr the velocity in a circle at its surface, in one

second of time; and hence, putting c = S'lilSy &c, the

circumference of the earth being 2cr = 25,000 miles, or

132,000,000 feet, it will be </2gr : 2cr : : l" : cj— =

5078 seconds nearly, or lh 24m 38', the periodic time at
the circumference : Also the velocity there, or \/2gr is =
26*000 feet per second nearly. Then, since the force of
gravity varies in the inverse duplicate ratio of the distance,
by rules 8 and g, it is ^/r : yV : : v or 25000 : 26000

y"- = v the velocity of a body revolving about the earth

at the distance r ; and t/r3 : ^/r3 : : t or 5078" : 5078

\/-t = t the time of revolution in the same. So if, for

instance, it be the moon revolving about the earth at the
distance of 60 semidiameters; then r = 60r, and the above
expressions become v = 26000^/-^ == 3357 feet per se-
cond, or 38f miles per minute, for the velocity of the moon

in her orbit; and t = 5078 y^ = 2360051 seconds, or

27-re days nearly, for the periodic time of the moon in her
orbit at that distance.

Thus also the ratio of the forces of gravitation of the
moon towards the sun and earth may be estimated. For,
1 year or 36"5J days being the periodic time of the earth
and moon about the sun, and 27-;% days the periodic time
of the moon about the earth, also 60 being the distance of
the moon from the earth, and 23920 the distance from
the sun, in semidiameters of the earth, by art. 5 it is

SO 23920 - 23902 27'32 . ...

»T^J : ^ 5 : ;/or 1 : — — x — -. = 2£; that is, the

27'33 365-3S3 60 305-25 9

proportion of the moon's gravitation towards the sun, is to
that towards the earth, as 2-| to 1 nearly.

Again, we may hence compute the centrifugal force of
a body at the equator, arising from the earth's rotation.
For, the. periodic time when the centrifugal force is
rqual to the force of gravity, it has been shown above, is
5078 seconds, and 23 hours 56 minutes, or 86"l60 se-
conds, is the period of the earth's rotation on its axis:
therefore, by art. 5, as 8616O2 : 5078' : : 1 : T|¥, the cen-
trifugal force required, which therefore is the 28<)th part of
gravity at the earth's surface. Simpson's Flux. pa. 240, &c.

By similar means we may determine the centrifugal force
of bodies in different latitudes ; for as the earth turns
round its axis, it is evident that those bodies on the surface
of it, which lie nearer to the axis, or, which is the same
thing, lie nearer to the poles, describe circles smaller than
those which lie nearer to the equator, though they are all
performed in! the same interval of time, viz, 23h 56"m.
Hence the periodical times being equal, the central forces
are as the radii of the circles (5) ; and since, in different
latitudes, we may use the following proportion; viz, as the
radius is to the co-sine of a given latitude, so is the cen-
trifugal force of bodies situated at the equator, to the cen-
trifugal force of bodies at thnt given latitude. Now as the
co-sines decrease in length the nearer they approach the
poles, so the tendency of bodies to fly off from the surface
of the earth is greatest at the equator, but diminishes in
approaching towards the poles ; and hence we perceive why
the earth has been found, by means of unquestionable mea-
surements and other observations, to be an oblate spheroid,
whose polar diameter is the shortest. This circumstance
furnishes a convincing evidence of the earth's daily rota-
tion about its axis.

For another example, suppose a to be a ball of 1 ounce,
which is whirled about the centre c, so as to describe the
circle aeb, each revolution being made in half a second ;
and the length of the cord ac equal to 2 feet. Here t = \,

r = 2, and it having been found above that c*/ — = t is

the periodic time at the circumference of the earth when
the centrifugal force is equal to gravity ; hence then, by

art. 5, as 4 : r- : : f or 1 : /, which proportion becomes

£. ■ I • • 1 ■ 2S1 = 1££ = iLHHil' = 9.819 = the cen-
ic i3 gtf g 16-^j

trifugal force, or that by which the string is stretched, viz,
nearly 10 ounces, or 10 times the weight of the ball

Lastly, suppose the string and
ball to be suspended from a point
d, and describes in its motion a
conical surface ad b; then put-
ting dc = a, AC=r, and ad
= h; and putting f=1 the
force of gravity as before; then
will the body a be affected by
three forces, viz, gravity acting -^-t
parallel to dc, a centrifugal force
in the direction ca, and the ten-
sion of the string, or force by which it is stretched, in the
direction da : hence these three powers will be as the three
sides of the triangle adc respectively, and therefore as CD

or a : ad or A : : 1 : - the tension of the string as com-

a

pared with the weight of the body. Also DC or a : ac or

r : : 1 : — the general expression for the centrifugal force

above found ; hence, gt7, = lac1 and consequently

t = c^/— = 1'108^/a = the periodic time.

10. When the force by which a body is urged towards
a point is not always the same, but is either increased or
decreased as some power of the distance ; several curves
will then arise according to that power. If the force de-
crease as the squares of the distances increase, the body
will describe an ellipsis, and the force is directed towards
one of its foci ; so that in every revolution the body once
2 P 2

C E N

approaches towards it, and once recedes from it: also the
excentricity of the ellipse is greater or less, according to
the projectile force; and the curve may sometimes become
a circle, when the excentricity is nothing; the body may
also describe the other two conic sections, the parabola
and hyperbola, which do not return into themselves, by
supposing the velocity greater in certain proportions. Also
if the force increase in the simple ratio as the distance in-
creases, the body will still describe an ellipse; but the force
will in this case be directed to the centre of the ellipse;
and the body, in each revolution, will twice approach to-
wards it, and again twice recede from that point. For the
investigations of these and other particulars, consult Simp-
son's Fluxions, vol. 1, sect. 12, and Gregory's Median.
vol. 1, chap. 3, bk. 2.

The general doctrine of central forces is called most into
application in investigating the orbits of the planets as. ur-
ged towards the eentral body the sun: and, considered in
that viejv, the following deduction of the principal laws is
offered.

(1). When a body, urged by a centripetal force, moves
in any curve; then, in every point of the curve, the torcc
is in a ratio, compounded of the direct ratio of the body's
distance from the centre of force, and the inverse ratio of
the cube of the perpendicular on the tangent to the same
point of the curve, drawn into the radius of curvature of
the same point.

(2). Hence, if the body move in a circle, by a force
tending to a point in the circumference, the force will be
everywhere in tloe inverse ratie of the 5th power of the
body's distance from that point.

(3). When a body moves in a conic section, the force
urging it towards the focus is reciprocally proportional to
the square of the distance.

(4). Hence, if a body move in an ellipsis, by a centri-
petal force tending to its centre, that force will be directly
as the distance.

(5). Hence'results an easy construction, for determi-
ning the radius of curvature in any conic
section. Thus, from the given point a,
draw ab through the focus F, and ar
perpendicular to the curve at a, cutting
the axis vc in C; at c erect cb perpen-
dicular to ac, and draw br perpendicular
ab; then ra is the radius of curvature
at the point a.

In the parabola, the construction becomes a little sim-
pler. For, since af = fc, and acb a right angle, there-
fore f is the centre of a circle passing through a, c, b, and
af = fb. Hence, if af be produced till fb = af, and
br be drawn perpendicular to ab, meeting the perpendi-
cular ar; then r will be the centre of curvation of the
parabola.

Hence, in all the conic sections, ac : ab : : ab : ar
the radius of curvature; which therefore is a 3d propor-
tional to AC and ab. And when the point a coincides
with the vertex v, then AC : ab becomes the ratio of the
two semiaxes; that is, the radius of curvature at the ver-
tex, or extremity of the axis, is equal to a 3d proportional
to the. two semiaxes, and is therefore equal to the semipa-
rameter of the axis.

Further, the radii of curvature at the extremities of^ the
transverse and conjugate axes, are to each other recipro-
cally as the cubes of the axes. For, let a denote the fpr-

[ 292 ] C E N

mer semiaxis, and c the latter: then the radius of curva-
ture to the former is -, and to the latter is - ; and

(6). When a body moves in a parabola, by a centripetal
force tending to the focus ; then the velocity of the body
in the parabola, is to the velocity of a body describing a
circle at the same distance, everywhere as ,y/2 to 1. But
the velocity of a body in an ellipsis, is to the velocity of a
body moving in a circle at the same distance, in a less ra-
tio than that of \/Z to 1. And the velocity in an hyper-
bola, is to the velocity in a circle, in a greater ratio than
that of a/2 to 1. — If the body move in the nautical spiral,
its velocity is everywhere equal to the velocity of a body
describing a circle at the same distance.

(7). Supposing a body is projected in a given direction,
with a given velocity, and that it is urged by a force which
is always reciprocally as the square of the distance ; re-
quired to determine the curve in which the body will
move. — It is plain, in the first place, that the path of the
body will be one of the conic sections. Let the body be
projected in the right line ad, with the given velocity c :
and since the absolute quantity of the centripetal force is
known, there will thence be given the velocity with which
a body can describe a circle at the distance af, by the
same force: for it is equal to that acquired by a body in
falling through ijiv, while urged constantly by the same
force. Let that velocity be c. Erect ac perpendicular to

ad, in which take ar a 4th proportional to e1, c1, — ,

where fp is a perpendicular on ad ; then a r will be the ra-
dius of curvature at a. Draw hb perpendicular fo af,
and bc perpendicular to ar; then drawing FC, it will give
the position of the axis. Make the angle eac equal to the
angle fac Then, if ae be parallel to re, the figure in
which the body moves will be a parabola. But if it meet
the axis fc in e; and if the points f and e fall on the same
side of the point c, the figure will be an hyperbola; but if
f and e fall on contrary sides, the figure will be an ellip-
sis. Hence, with the foci e and f, and the axis af j; ae,
the section may be described in which the body will move.

For further particulars on central forces, the reader may
consult Newton's Principia, Varignon in the Memoirs of
the French Academy, 1700,1701; Maclaurin's Fluxions;
Simpson's Fluxions; Gregory's Mechanics, vol. 1 ; Daw-
son's Essay on the inverse problem of central forces, in
the Manchester Memoirs, or in Leybourn's Mathemati-
cal Repository, Nos. 4 and 5, it. s.; and Wa ring's paper
on central forces, Philos. Trans', vol. 7S, or my Abridg-
ment, v. If), p. 384.

Central Rule, is a rule or method discovered by
Mr. Thomas Bak^er, rector of Nympton in Devonshire,
which he published in his Geometrical Key, in the year
l6S4, for determining tjie centre of a circle which shall cut
a given parabola in as many points as a given equation, to
bc constructed, has real roots ; which he has applied with
good success in the construction of all equations as far as
the 4th power inclusive.

The Central Rule is chiefly founded on this property of
the parabola ; that if a line be inscribed in the curve per-
pendicular to any diamtger, the rectangle of the segments
of this line, is equal to the rectangle of the intercepted
part of the diameter and the parameter of the axis.

C E N

[ 293 ]

C E N

The Central Rule has the advantage over the methods of
constructing equations by Descartes and Delatteres, which
are liable to the trouble of preparing the equations by ta-
king away the second terra ; whereas Baker's method ef-
fects the same thing without any previous preparation
whatever. See Philos. Trans. N° 157.

CENTIIE, or Center, in a general sense, signifies a
point equally remote from the extremes of a line, plane, or
solid ; or a middle point dividing them so that some cer-
tain effects are equal on all sides of it.

Centre of Attraction, or Gravitation, is the point to
which bodies tend by gravity ; or that point to which a
revolving planet or comet is impelled or attracted, by the
force or impetus of gravity.

Centre, in Architecture. This term is tised to denote
a frame of timber, constructed for the purpose of sup-
porting the stones or bricks forming an arch or vault,
during the erection. Thus the centre serves as 'a founda-
tion for the arch to be built upon, which at the completion
of the work is struck ,pr taken down, and then the arch
will support itself by the reciprocal pressure of its parts.

Centre of a Bastion, isL a point in the middle of the
gorge, where the capital line commences, and which is
usually at the angle of the inner polygon of the figure. Or
it- is the point where the two adjacent curtains produced
intersect each other.

Centre of a Circle, is the point in the middle of a cir-
cle, or circular figure, from which all lines drawn to the
circumference are equal.

Centre of a Conic Saction, is the middle point of any
diameter, or the point in which all the diameters intersect
and bisect one another.

In the ellipse the centre is,within the figure ; but in the
hyperbola it is without, or between the. conjugate hyper-
bolas; and in the parabola it is at an infinite distance
from the vertex.

Centre of Conversion, in Mechanics, a. term first used
by M. Parent, and may be thus conceived: Suppose a
stick laid on stagnant water, and then drawn by a thread
fastened to it, so that the thread always makes the same
angle with the stick, either a right angle or any other;
then it will be found that the stick will turn on one point
of it, which will be immoveable; and this point is termed
the centre of conversion.

This effect arises from the resistance of the fluid to the
stick partly immersed in it. And if, instead of the body
thus floating on a fluid, the same be conceived to be laid
on the surface of another body ; then the resistance of this
plane to the stick will still have the same effect, and will
determine the same centre of conversion, And this resist-
ance is precisely what is called friction, so prejudicial to
the effects of machines.

M. Parent has determined this centre in some certain
cases, with much laborious calculation. When the thread
is fastened to the extremity of the stick, he found that the
distance of the centre from this extremity would be nearly
■J4 °f the whole length. But when it is a surface or a so-
lid, there will be some change in the place of this centre,
according to the nature of the figure. See Mem. of the
Acad, of Sciences, vol. 1, p. 191.

C en t he of a Carve, of the higher kind, is the point
where two diameters meet. — When all the diameters meet
in the same point, it is called, by Sir Isaac Newton, the
scnera I centre.

Centre of a Dial, is the point where its gnomon or

stile, which is placed parallel to the axis of the earth,
meets the plane of the dial ; and from this all the hour-
lines are drawn, in such dials as have centres, viz, all ex-
cept that whose plane is parallel to the axis of the earth
where there can be no centre; in which qase all the hour-
lines will be parallel to the stile, and to one another, the
centre being as it were at an infinite distance.

Centre of Equilibrium, is the same with respect to bo-
dies immersed in a fluid, as the centre of gravity is to bo-
dies in free space; being a certain point, upon which if
the bod}- or bodies be suspended, they will rest in any po-
sition. To determine this centre, see Emerson's Mechan.
prop. 92, p. 134.

Centre of Friction, is that point in the base of a body
on which it revolves, into which if the whole surface of the
base, and the mass of the body were collected, and made
to revolve about the centre of the base of the given body,
the angular velocity destroyed by its friction would be
equal to the angular velocity destroyed in the given body
by its friction in the same time. — See Vince on the Moffon
of Bodies affected by friction, in the Philos. Trans. 1785.

Centre of Gravity, is that point about which all the
parts of a body do in any situation exactly balance each
other. Hence, by means of this property, if the body be
supported or suspended by this point, the body will rest in
any position into which it is put; as also, that if a plane pass
through the same point, the segments on each side will equi-
ponderate, neither of them being able to move the other.

The centre of gravity of a body is not always within the
body itself : thus the centre of gravity of a ring is not in
the substance of a ring, but in the axis of its circumscribing
cylinder: and the centre of gravity of a hollow staff, or of
a bone, is not in the matter of which it is constituted, but
somewhere in its imaginary axis. Every body however
has a centre of gravity, and so has every system of bodies.

If a heavy body be sustained by two or more forces,
their directions must meet either at the centre of gravity
of that body, or in the vertical line which passes through it.

When a body stands upon a plane, if a vertical line
passing through the centre of gravity fall within the base
on which the body stands, it will not fall over; but if
that vertical line passes withqut the base, the body will
fall, unless it be prevented by a prop or cord. Whore the
vertical line falls upon the extremity of the base, the body
may stand, but the equilibrium may be disturbed by a
very trifling force; and the nearer this line passes to any
edge of the base, the more easily the body is thrown over;
the nearer it falls to the middle of the base, the more
firmly the body stands.

The whole gravity, or the whole matter, of a body, may
be conceived united in the centre of gravity ; and in de-
monstrations it is usual to conceive all the matters as re-
ally collected in that point.

Through the centre of gravity passes a right line, called
the diameter of gravity ; and therefore the intersection of
two such diameters determines the centre of gravity. Also
the plane upon which the centre of gravity is placed, is
called the plane of gravity; so that the common intersec-
tion of two such planes determines the diameter of gravity.

In homogencal bodies, which may be. divided length-
ways into similar and equal parts, the centre of gravity is
the same with the centre of magnitude. Hence therefore
the centre of gravity of a line is in the middle point of it,
or that point which bisects the line. Also the centre of
gravity of a parallelogram, or cylinder, or any prism

C E N

[ 29* 3

CEN

whatever, is in the middle point of the axis. And the
centre of gravity of a circle or any regular figure, is the
same as the centre of magnitude.

Also, if a line can be so drawn as to divide a plane into
equal and similar parts, that line will be a diameter of
gravity, or will pass through the centre of gravity ; and
ft is the same as the axis of the plane. Thus the line
drawn from the vertex and perpendicular to the base of
an isosceles triangle, is a diameter of gravity ; and thus
also the axis of an ellipse, cr a parabola, &c, is a diameter
of gravity. The centre of gravity of a segment or arc of
a circle, is in the radius or line perpendicularly bisecting
its chord or base.

Likewise, if a plane divide a solid in the same manner,
making the parts on both sides of it perfectly equal and
similar in all respects, it will be a plane of gravity, or will
pass through the centre of gravity. Thus, as the inter-
section of two such planes determines the diameter of gra-
vity, the centre of gravity of a right cone, or spherical seg-
ment, or conoid, &c, will be in the axis of the same.

The centre of gravity of a ship, is always before the
point, which is the middle of her absolute length : for the
fore-part, being of greater capacity than the after-part,
must consequently have greater weight ; and therefore it
carries the centre of gravity forwards in proportion to its
greater weight (which in large ships is from 50 to 80 tons),
and also to'the interval between every centre of gravity of
each particular part both forward and aft. When a ship
is at sea and loaded, the centre of gravity may well be
supposed not to change, unless the cargo be moved. But
experience shows that the fore or after-part of the bottom
of a ship plunges and labours more and more, in propor-
tion as the wind acts with more or less force on the sails;
because ships are generally not masted according to the
point velique; so that a ship, which has the centre of the
effort of her sails ill-placed, draws always more water for-
ward or aft when the impulse of the wind upon her sails
is very powerful, than when she is at ease under her bur-
then. From the centre of gravity of the floating line of a
ship let a perpendicular be raised, and continued till it be
intersected by the direction of the impulse of the water on
the bows, in sailing directly before the wind ; and where
these two lines cut each other, that point is the point ve-
lique, and where the centre of effort of all the sails should
be placed.

Common Centre of Gravity of two or more bodies, or the
different parts of the same body, is such a point as that,
if it be suspended or supported, the system of bodies will
•equiponderate, and rest in any position. Thus, the point
of suspension in a common balance beam, or steelyard, is
the centre of gravity of the same.

When any number of bodies move in right lines with
uniform motions, their common centre of gravity moves
also in a right line with a uniform motion ; and the sum
of their motions, estimated in any given direction, is pre-
cisely the same as if all the bodies in one mass were car-
ried on with the direction and motion of their common
centre of gravity. Nor is the centre of gravity of any
number of bodies affected by their collision or actions on
each other. For further particulars on the motion of the
centre of gravity of any number of bodies, see Emerson's
Mcchan. Props. 48, 49, &c,

Laws and Determination of the Centre of Gravity.
1. The centre of gravity of two equal bodies or masses
!>, equally distant from their respective centres. For these

are as two equal weights suspended at equal distances irpna
the point of suspension; in which case they will equi-
ponderate, and rest in auy position.

© - Gb

2. If the centres of gravity of two bodies, a and b, be
connected by the right line ae, the distances ac and bc,
from the common centre of gravity c, are reciprocally as
the weights or bodies a and B ; that is, ac : bc : : b : a.

See this demonstrated under the article Lever.

Hence, if the weights of the bodies a and b be equal,
their common centre of gravity c will be in the middle of
the right line ab, as in the foregoing article. Also since
a : B : : bc : ac, therefore a x ac = b x bc; whence
it appears that the powers of equiponderating bodies arc
to be estimated by the product of the mass multiplied by
the distance from the centre of gravity; which product is
usually called the momentum of the weights.

Further, from the foregoing proportion, by composition
it will be a -T- b : a : : ab : bc, c* a -+- b : B : : ab : ac.
So that the common centre of gravity c of two bodies will
be found, if the product of one weight by the whole di-
stance between the two, be divided by the sum of the two
weights. Suppose, for example, that a = 12 pounds,
b == 4lb, and ab = 36" inches : then lfj : 12 : : 36 : 27 =
bc, and consequently AC = 9, the two distances from the
common centre of gravity.

v ff v % s ©~Q—^ '

3. The common Centre of Gravity of three or more given
bodies or points a, b, c, d, &c, will be thus determined. —
If the given bodies lie all in the same straight line ad ;
by the last article, find p the centre of gravity of the two
a and b, and q the centre of gravity of c and d ; then,
considering p as the place of a body equal to the sum of
a and B, and q as the place of another body equal to both
c and d, find s the common centre of gravity of these two
sums, viz, a -+- b collected in P, and c -+- d united in q ;
so shall s be the common centre of gravity of all the four
bodies, a, b, c, d. And the. same for any other number of
bodies, always consideringthesum of any number of them
as united or placed in their common centre of gravity,
when found.

Otherwise, thus. Take the distances of the given bodies
from some fixed point as v, calling the distance va = a,
vb = b, vc = c, vd = d, and the distance of the centre
of gravity vs = x; then sa = x — a, sb = x — b, sc =
c — x, kd = d — x, and by the nature of the lever
a . (x — a) -+■ b . (x — b) = c (c — x) -+- d (d — x) ;
hence ax -+- bx -i- cx -+- dx = \a -+■ b6 -+- cc ■+• vd, and

x = = vs the distance sought; which

A+t+C+D

therefore is equal to the sum of all the momenta, divided
by the sum of all the weights or bodies.

Or thus. When the bodies arc not in the same straight
lino, connect them with the lines ah, cd ; then, as before,
find v the common centre of a and B, and <} the common
centre of c and d; then, conceiving a and b united in p,

CEN

[ 295 ]

CEN

and c and d united in q, find s the common centre of p
and Q, which will be the centre of gravity of the whole.

Or the bodies may be all reduced to any line vhb, &c,
drawn in -any direction whatever, by perpendiculars bb,
cc, &c, let fall thereon from the centres of gravity of
those bodies; and then the common centre 5 in this line,
as before, will be at the same distance from v as the true
centre s is; and consequently the perpendicular from s
will pass through s the real centre.

4. From the foregoing general expression, viz,

ha + b6 + cc + &c , c . c

x = , for the centre ot gravity 01 any

A + B + C -f- &C ' ° J J

system of bodies, may be derived a general method for
finding that centre in any body; for a, b, c, &c, may be
considered as the elementary parts of any body, whose
sum or mass is 11 = a + b + c &c, and ah, sb, cc, &c,
are the several momenta of all these parts, viz, the product
of each part multiplied by its distance from the fixed
point v. Hence then, in any body, find a general ex-
pression for the sum of the momenta, and divide it by the
content of the body, so shall the quotient be the distance
of the centre of gravity from the vertex, or from any other
fixed point, from which the momenta are estimated.

A C B

5. Thus, in a right line ab, all the particles which com-
pose it may be considered as so many very small weights,
each equal to x, which is therefore the fluxion of the
weights, or of the line denoted by x. So that the small
weight x multiplied by its distance from a, viz x, is xx the
momentum of that weight x; that is, xx is the fluxion of
all the momenta in the line ab or x; and therefore its flu-
ent f.r2 is the sum of all those momenta; which being di-
vided by x the sum of all the weights, gives \x or £ab for
the distance of the centre of gravity c from the point a ;
that is, the centre c is in the middle of the line ab.

6. Also in the parallelogram, whose axis or length

dT>

ab = x, and its breadth de = b;
drawing de parallel and indefi-
nitely near de, the areola (/dec = •
bx will be the fluxion of all the A
weights, which multiplied by its
distance x from the point a, gives

bxx for the fluxion of all the mo- <? E

menta, and consequently the fluent lix2, is the sum of all
those momenta themselves; which being divided by bx
the sum of all the weights, give fr = xAb for the distance
of the centre c from the extremity at a, and is therefore in
the middle of the axis, as is known from other principles.

And the process and conclusion will be exactly the
same for a cylinder, or any prism whatever, making b to
denote the area of the end or of a transverse section of the
body.

7. In a triangle abc; the line ad
drawn from one angle to bisect the op-
posite side, will be a diameter of gravity,
or will pass through the centre of gravity :
for if that line be supported, or conceiv-
ed to belaid upon the edge of something.
the two halves of the triangle on both
sides of that line will just balance one $
another, since all the parallels EP &c to the base will be
bisected, as well as the base itself, and so the two halves of
«ach line will just balance each other. Therefore, putting
the base bc = b, and the axis or bisecting line ad = a,

the variable part as =r; then, by similar triangles ad :
bc :■ : as : ef, that is a : b : ,fx : — = ef; which, as a

weight, multiplied by x, gives — for the fluxion of the

weights; and this again multiplied by x = as, the distance

bx?x
from a, gives - — for the fluxion of the momenta; the

fluent of which, or — divided by — the fluent for the
3a J la

weights, gives \x = §as for the distance of the centre of
gravity from the vertex a in the triangle aef; and^when
x = ad, then -f-AD is the distance of the centre of gravity
of the triangle abc.

The same otherwise, without Fluxions. Since a line
drawn from any angle to the middle of the opposite side
passes through the centre of gravity, therefore the inter-
section of aiiy two of such lines, will be that centre : thus
then the centre of gravity is in the line ad ; and it is also
in the line cg bisecting ab ; it is therefore in their inter-
section s. Now to determine the distance of s from any
angle, as a, produce cg to meet bh parallel to as in h ;
then the two triangles ags, bgh are mutually equal and
similar; for the opposite angles at g are equal, as are the
alternate angles at 11 and s, and at a and b, also the side
ag = bg; therefore the other sides bh, as are equal.
But the triangles cds, cbh are similar, and the side
CB = 2cd, therefore bh or its equal as = 2ds, that is
as = f-AD, the same as was found before. And in like
manner cs = §cg.

8. In a trapezium. Divide the ^

figure into two triangles by the
diagonal ac, and find the centres
of gravity e and F of these trian-
gles; join ef, and find the common
centre G of these two by this pro-
portion, abc : adc : : fg : eg, or
abcd : adc : : ef : eg.

In like manner, for any other
figure, whatever be the number of sides, divide it into se-
veral triangles, and find the centre of gravity of each ;
then connect two centres together, and find their common
centre as above ; do the same with this and the centre of
a third, and so on, always connecting the last found com-
mon centre to another centre, till the whole are included
in this process; so shall the last common centre be that
which is required.

g. In the Parabola bac Put ad=x, bb = y, and
the parameter = p. Then, by the nature of the figure,
px = yl, and Qy = I^Jpx; hence
2x^/px is the fluxion of the mass,
and Ixx^/px is the fluxion of the
momentum; then the fluent of the
latterdivided by that of the former,
or %x*^/px divided by |.vv/;;.r,
gives |x=}ad, for ag, the di-
stance of the centre of gravity g,
from the vertex a of the parabola.

1 0. In the Circular Arc a b d,
considered as a physical line
having gravity. It is mani-
fest that the centre of- gra-
vity G of the arc, will be
somewhere in the axis, or
middle radius bc, c being

&

C E N

c

3

C E N

the centre of the circle, which is considered as the point of
suspension. Suppose v indefinitely near to a, and fh parallel
toBC. Put the radius bc or ac = r, thesemiarc ab =z, and
thesemichord ae = t; then is ah = .jr,and a f = k the fluxion
of the mass; therefore ce x 2 is the fluxion of the mo-
mentum. But, by similar triangles, ac or r : ce : : af or
z : ah or x, therefore rx = ce x z, and so rx is also the
fluxion of the momentum ; the fluent of which is rx, and

thib divided by z the whole weight, gives — = — =

J E ° % AB

= cg the distance of the centre of gravity from the

JIBD ° J

centre c of the circle; being a 4th proportional to the
given arc, its chord, and the radius of the circle.

Hence, when the arc becomes the semicircle ibk,

the above expression becomes — or = =

r IB l'5708r T5708

•6366V, viz a third proportional to a quadrant and the radius.
11. In the Circular Sector abdc. Here also the centre
of gravity will be in the axis or middle radius bc. Now
with any smaller radius describe the concentric arc ms,
and put the radius ac or bc = r, the arc abd = a, its
chord aed = c, and the variable radius cl or cm = y ;

then as r : y : : a : — = the arc lmn, and r : y : :c : —

j r > j r

= the chord lon ; also, by the last article, the distance
of the centre of gravity of the arc lmn is

CM x aed cy , , ay

= — ; hence the arc lmn or

LUMJ

multiplied byi

gives -^ the fluxion of all the weights or mass, and this
multiplied by — the distance of the common centre of
gravity, gives — the fluxion of the momentum ; the

flilent of which, viz — divided by — , the fluent of the

ar J 2r'

mass, gives '-^ for the distance of the centre of gravity of
the sector clmn from the centre c; and when y = r, it
becomes — = cg for that of the sector cabd proposed ;

3a 1 r ;

being § of a 4th proportional to the arc of the sector, its
chord, and the radius of the circle.

Hence, when the sector becomes a semicircle, the last

expression becomes — = — or •£ of a 3d proportional

to a quadrantal arc and the radius. Or it is equal to

— = '4244r from the centre c ; where p = 3-14l6\ So
op *

that the distanced the centre of gravity of the sector, is

just equal to % of that of its arc.

12. In the Cone add, Putting a = dc, b = area of

tliebase aeb, and x = Dc'anyva-

riable altitude; then as a3:*2: : b:

bx

— =z area aeb ; hence the fluxion
of the mass is — r whose fluent,

a" '

or the solid, is — ; and the fluxion
of the momentum is — -, whose

fluent is — ; then this fluent di-

vided by the former fluent, the
whole weight or mass, gives |x or

'Idc for the distance of the centre of gravity of the cone
Vab, or f DC for that of the cone dab "below the vertex v.

And the tame is the distance in any other pyramid. So
that all pyramids of the same altitude, have the same cen-
tre of gravity.

13. General Scholium to the foregoing articles, for the
place of the centre of gravity in geometrical figures.

(1). The centre of gravity of every geometrical figure,
lies in its axis or diameter,
being that line which is
drawn from the middle of
the base of one end, to the
middle of the opposite end,
or to the opposite angle or
vertex; because this line
bisects all the ordinates or
sections parallel to the said
base or ends. And the distance of the centre of gravity,
as measured on that axis, either from the base or from
the vertex, is for the several figures as below.

(2). The centre of gravity is in the middle of that axis,
or length, or equally distant from both ends, or coinciding
with the centre of magnitude in all regular figures, or such
as have all the said parallel sections equal ; such as right
lines, all parallelograms and prisms, all regular plane po-
lygons and solid polyhedrons, also the circle and sphere.

(3). In a triangle and paraboloid, it is ■§■ of the axis
from the base, or f from the vertex.

(4). In a parabola it is -| from the base, or 4 from the
vertex.

(5). In a circular arc, it is from the centre of the cir-
cle, a 4th proportional to the arc, its chord, and the ra-
dius of the circle, viz, — .

a

(6). In a circular sector, the distance from the centre is,

or | of the said 4th proportional to the arc, chord,

and radius.

(7). In a semicircle the distance from the centre is — ,

or f of a third proportional to the arc and radius.

(8). In a circular segment, the distance from the cen-
tre is ' or -i- of the cube of the chord divided by

12 area xl *

the area of the segment, or T'T of a 4th proportional to
the said area, the square of the chord, and the same
chord.

(9). In any pyramid or cone, it is £ from the base, or
^ from the vertex.

(10). In the hemisphere, or hemispheroid, it is J from
the base, or T from the vertex.

(11). In the segment of a sphere, or of a spheroid, the

distance from the base is — ^—, h ; where h denotes the
6d— th

height of the segment, and d the whole diameter or axis.

(12). In the segment of an hyperbolic conoid, the di

stance from the base is — -.If, where % and d are the

6(1 + ill

samo as above, viz, the height of the segment and tln<
whole diameter or axis.

(13). In any figure, composed of two parts, or for any
two bodies: find the centre of gravity of each of the two
bodies, or two parts, and connect these two centres by a
light line, in which line will lie the common centre of
gravity of the compound body; and the distance of that
ccntro from the centre of each body, or each part, will be

CEN

regulated by this proportion, viz, that each distance is re-
ciprocally proportional to the mass or weight, or content
of each body ; or, as the sum of the contents of the two
bodies, or of the whole compound mass, is to the whole
distance of the two centres, so is each particular body or
part, to the distance from the centre of the other body or
part.

14. To determine the Centre of Gravity in any Body me-
chanically. Lay the body on the edge of any thing, as a
triangular prism, or such like, moving it backwards and
forwards till the parts on both sides are in equilibrio ; then
is that line just in, or under the centre of gravity. Balance
it again in another position, to find another line passing
through the centre of gravity ; then the intersection of
these two lines will be the centre required.

The same may be done by laying the body on an hori-
zontal table, as near the edge as possible without its fall-
ing, and that in two positions, as lengthwise and breadth-
wise : then the common intersection of the two lines con-
tiguous to the edge, will be its centre of gravity. Or it
may be done by placing the body on the point of a style,
&c, till it rest in equilibrio. It was by this method that
Borelli found that the centre of gravity in a human body
is between the nates and pubis ; so that the whole gravity
of the. body is collected into the place of the genitals; an
instance of the wisdom of the Creator, in placing the
membrum virile in the part which is the most convenient
for copulation.

The same otherwise: thus, Hang the body up by any
point; then a plumb-line hung over the same point, will
pass through the centre of gravity ; because that centre
will always descend to the lowest point when the body
comes to rest, which it cannot do except when it falls in
the plumb-line. Therefore, marking that line upon it,
and suspending the body by another point, with the plum-
met, to find another such line, the intersection of the two
will give the centre of gravity.

Or thus. Hang the body by two strings from the same
tack, but fixed to different points of the body ; then a
plummet, hung by the same tack, will fall on the centre
of gravity.

In the 4th volume of the New Acts of the Academy of
Petersburg, is the demonstration of a very general theo-
rem concerning centres of gravity, by M. Lhuilier; a
particular example only of the general proposition will
be as follows : Let a, b, c, be the centres of gravity of
three bodies; a, b, c their respective masses, and Q their
common centre of gravity. Let right lines qa, qb, qc, be
drawn from the common centre to that of each body, and
the latter be connected by right lines ab, AC, and bc; then

qa2 x a -h qb2 x b -t- qc2 x c =

ah ., nc j he.

AB- X ; -+- AC' X ■ -+- BC' X ;

a+b+c - a + b + c a + b+c

Uses of the Centre of Gravity. This point is of the
greatest use in mechanics, and many important concerns
in life; because the place of that centre is to be considered
as the place of the body itself, in computing mechanical
effects ; as in the oblique pressures of bodies, banks of
earth, arches of bridges, and such like.

The same centre is even useful in finding the superficial
andsolid contentsof bodies ; forit isagencral rule, that the
superficies or solid generated by the rotation of a line or
plane about any axis, is always equal to the product of
the said line or plane drawn into the circumference or
path described by the centre of gravity. For example, it

Vol. I.

[ 297 ] CEN

was found above at art. 1 1, that in a semicircle, the di-
stance of the centre of gravity from the centre of the cir-

and therefore the path of that centre, or the

3P

circumference described by it while the semicircle re-
volves about its diameter, is §r; also the area of the- semi-
circle is ipr2; hence the product of the two is fpi'3; which,
it is well known, is equal to the solidity of the sphere ge-
nerated by the revolution of the semicircle.

And hence also is obtained another method of finding
mathematically the centre of gravity of a line or plane,
from the contents of the superficies or solid generated by
it. For if the generated superficies or solid be divided by
the generating line or plane, the quotient will be the cir-
cumference described by the centre of gravity; and con-
sequently this divided by 2p gives the radius, or distance
of that centre from the axis of rotation. For example,
in the semicircle, whose area is ipr°-, and the content of
the sphere generated by it fpr3; here the latter divided by

the former is fr, and this divided by 2p gives — for the

distance of the centre of gravity from the axis, or from
the centre of the semicircle, the same as before. The pro-
perty last mentioned, relative to the relation between the
centre of gravity and the figure generated by the revolu-
tion of any line or plane, is mentioned by Pappus, in the
preface to his 7 th book ; and Father Guldini has more
fully demonstrated it in his 2d and 3d books on the Cen-
tre of Gravity.

The principal writers on the centre of gravity are Ar-
chimedes, Pappus, Guldini, Wallis, Casatus, Carre, Hays,
Wolfius, &c. See also ray Mensuration, p. 382, 4th edit.

Centre of Gyration, is that point in which if the
whole mass be collected, the same angular velocity will
be generated in the same time, by a given force acting at
any place, as in the body or system itself. This point dif-
fers from the centre of oscillation, inasmuch as in this
latter case the motion of the body is produced by the gra-
vity of its own particles, but in the case of the centre of
gyration the body is put in motion by some other force
acting at one place only.

To determine the Centre of Gyration, in any body, or
system of bodies composed of the parts a, b, c, &c, moving
about the point s, when urged by a force/ acting at any
point p. See the following figure. Let a be that centre ;
then, by mechanics, the angular velocity generated in the

f ■ sp , ,

system by the force/, is as-— r^— r^-^e!Tc'md'by

the same, the angular velocity of the matter placed all in
the noint r is ^-^ 3 5 then since these two are

1 (A -t- B ■+- c &c) X sir"

to be equal, their equation will give sr

= >/-

B2 ■+- C . SCB &C

for the distance of the centre

of gyration sought, below the axis of motion.

Now because the quantity a . sa2 -+- b . sb2 -+- &c is =
sg. so . b, where g is the centre of gravity, o the centre of
oscillation, and b the whole body or sum of a, b, c, &c ;
therefore it follows that sr2= sg . so; that is, the dis-
tance of the centre of gyration, is a mean proportional be-
tween those of gravity and oscillation.

And hence also, if p denote any particle of a body,
placed at the distance d from the axis of motion j then is

sum of all the pa? , , , ,

sr = : — : whence the point r may bc dctrr-

budy b

mined in bodies by means of Fluxions.
2Q

GEN

The distance of the centre of gyration from the point or
axis of motion, being a mean proportional between the dis-
tances of the centres of gravity and oscillation lrom the
sam« axis. Hence, when any two of these distances is
known, the third may readily be determined.

The distance srt of the centre of gyratrbn from the axis
of rotation in several bodies will be as follows :

1. In a right line, or thin cylinder, revolving about the
end, sr = length x ^/$.

2. The plane of a circle, or cylinder, revolving about the plane from every particle
axis, sr = radius x ^\. in the body; a suppo-

3. The periphery of a circle, about the diameter, sr = sition which will not al-
radius x ,^/\.

4. A wheel with a very thin rim, revolving about its
axle, sr = radius.

5. The plane of a circle, about the diameter, su = \
radius.

6. The surface of a sphere, about the diameter, sr =
radius xv/f. ,'.''•'

7. A globe, about the diameter, sr = radius x */%,
S. A cone, about the axis, sr = radius x y/-^.
Cent-re of Inertion. See Centre of Gravity

[ 298 ] C E N

of Fluxions, where it is introduced as one of the examples
of that method. See Haves, Carue, Woluus, Simp-
son, Fluxions, &c.

To determine thejCentrc of Oscillation, in any Compound
Mass or Body mn, or of any System of Bodies a,b, c, &c.

Let mn be the plane of vibration, to which plane con-
ceive all the matter to

be reduced by letting fall
perpendiculars to this

ter the vibration of the
body, because the parti-
cles are still at the same
distance from the axis of
motion. Let o be the
centre of oscillation, and
G the centre of gravity;
through the axis s draw

sgo, and the horizontal

line st ; then from every particle a, b, c, &c, let fall per-
Centre of Magnitude, is the point which is equally pendiculars A«and xp, b/> and Br/, cc and cr, &c, to these
distant from all the similar external parts of a body. This two lines ; and join sa, sb, sc ; also draw got and on per-
is the same as the centre of gravity in- homogeneal bodies pendicular to ST.

that can be cut into like and equal parts according to Now the forces of the weights a, b, c, to turn the body

their length, as in a cylinder or any other prism. about the axis, are a . sp, B . sr/, — c . sr ; and, by rfle-

Centre of Motion, is the point about which any body, chanics, the forces opposing or resisting that motion are

or system of bodies, moves, in a revolving motion. a . s a2, b . SB2, c . sc" ; therefore the angular velocity ge-

Centre of Motion oiaship, is the point upon whicha ves- , , . a . sp + b ■ so — c ■ sr ,.,

, ... i ,, ,■ •» '■■• ■■. r .. r nerated by the whole system is 3 —, — ■—,. Inlike

sel oscillates or rolls, when put in motion. J J a. sa' + b.sb -+- c. sc

Centre of Oscillation, is that point in the axis or line manner, the angular velocity which any bod)' or particle

of suspension of a vibrating body, or system of bodies, in p, situated in o, generates in the system, by its weight, is

which if the whole matter or weight be collected, the vi- p^m Qr sn or j^ because of the similar triangles sgot,

brations will still be performed in the same time, and with p. so so- so. so'

the same angular velocity, as before. Hence, in a com- son. But, by the conditions of the problem, the time and

angular velocity are alike in both these cases ; therefore

•these two expressions must be equal to each other, that is,

sm a . sa + b . so — c . sr , . ,

— , and consequently

pound pendulum, its distance from the point of suspension
is equal to the length of a simple pendulum whose oscilla-
tions are isochronal with those of the compound one.

Mr. Huygens, in his Horologium Oscillatorium, first
showed how to find the centre of oscillation. At the be-
ginning of his discourse on this subject, he says, that Mer-
sennus first proposed the problem to him while he was yet
very young, requiring him to resolve it in the cases of sec-
tors of circles suspended by their angles, and by the mid-
dle of their bases, both when they oscillate sideways and
flatways; as also for triangles and the segments of circles,
either suspended from their vertex or the middle of their
bases. But, continues he, not having immediately disco-
vered any thing that would open a passage into this busi

Again, by mechanics,
B . sr/ — c . sf is equal

sg a . sp + b . su — c . sr
the sum of the forces a . sp -t

(a -+- b ■+- c) sot the force of the same matter collect-
ed all into its centre of gravity g ; therefore

A . SA3 + B . SB3 + C . SC3 ... . , , . . ,

so = — - _ , _, — , which is the distance of the

(A -t-

•«0

centre of oscillation o below the axis of suspension.

Farther, because it was found under the article Centre
of Gravity, that (a ■+■ b -+- c) . so = a . sa ■+- b . sb ■+- c. sc,

. , A .SA3 -+- B.SB3 -|- C. SC3 . . ,.

ness, I was repulsed at first setting out, and stopped from therefore so = A . sa + B . s„ + c . sc is the same distance

a farther prosecution of the subject; till being farther in- of ,ne ccnlre 0f oscillation; where any of the products

cited to it in adjusting the motion of the pendulums of my A _ sa> B _ s^ &c are to De considered negative when the

clock, I surmounted all difficulties, going far beyond Dcs- pojnts a> ft; &c lie above the point s, or where the axis

cartes, Fabry, and others, who had done the thing in a passcs through

few of the most easy cases only, without any sufficient de-
monstration ; and solving not only the problems proposed
by Mersenne, but many others that were much more dif-
ficult, and showing a general way of determining this cen-
tre, in lines, superficies, and solids.

In the Leipsic Acts for 1691 and 1714, this doctrine is
handled by the two Bernoullis: and the same is also done
by Herman, in his treatise De Motu Corporum Solidorum
ct Kluidorum.

It may also be seen in treatises on the. Inverse Method

Again, because, by Fuel. II. 12 and 13,

sa2 = sg2 -t- ga'1 — 2sg . on,
and sb2 = sg* -1- gb2 -1- 2sg . cb,
and sc2 = so2 ■+- gc2 + 2sg . cc, &c ;
and since by Mechanics, the sum of the last terms is no-
thing, namely — 2sg . aa ■+■ 2sg . g4 ■+- 2sg.gc&c=0;
therefore the sum of the others, or a . sa2 •+■ B . SB1 &c

= ( A -1-11 &C) SG* + A . GA1 -Hl-CPZ + C. GC2 &.C, OT
= I) . SG2 + A . GA2 ■+- B . (ill" -I- C .GC2 &C ;

where b denotes the body, or sum A -+- B -t- C &c of all

C E N

[

b . SG

GO =

the parts: this value then being substituted in the nume-
rator of the 2d value of so above-found, it becomes

b . so3 ■+- a . ga3 -f- b , Grr &c
so = ; , or

b . SG
A . GA2 + B . GB3 &C

SO = SG H — - ;

b . SG

From which it appears that the centre of oscillation is
always below the centre of gravity, and that the difference

,. , , . A.GA3 + B.GB3 &.C

or distance between them is Go =

It follows also from hence, that sg

A . GA3 + B . GB3 &C , , . .

i — ; that is, the rectangle sg . go is always

" the body b ' ° J

the same constant quantity; wherever the point of suspen-
sion s is placed, since the point g and the bodies a, b, &c,
are constant. Or go is always reciprocally as sg, that is
go is less as sg is greater; and the points g and o coin-
cide when sg is infinite; but when s coincides with G, then
go is infinite, or o is at an infinite distance.

To find the Centre of Oscillation by means of Fluxions.
From the premises is derived this general method for the
centre of oscillation, viz, let a: be the abscissa of an oscilla-
ting body, and y its corresponding ordinate or section; then
will the distance so of the centre of oscillation below the
axis of suspension s, be equal to the fluent of^x2* divided
by the fluent of yxx. So that, if from the nature or equa-
tion of any given figure, the value of y be expressed in
terms of x, or otherwise, and substituted in these two
fluxions; then the fluents being taken, and the one divided
by the other, the quotient will be the distance to the cen-
tre of oscillation in terms of the absciss x.

But when the body is suspended by a very fine thread
of a given length a, then the fluent of (a + x)2. yx divided
by the fluent of (a ■+■ x)yx gives the distance of the same
centre of oscillation below the point of suspension.

Ex. For example, in a right line, or rectangle or cy-
linder or any other prism, whose constant section hy, or
the constant quantity u; then yr** is ax2x, whose fluent
is -j-ax3 ; also yxx is aix, whose fluent is lax'1 ; and the
quotient of the former -Jar3 divided by the latter fax2, is
f j: for the distance of the centre of oscillation below the
vertex in any such figure, namely having every where the
same breadth or section, that is, at two-thirds of its length.
. In like manner the centre of oscillation is found for
various figures, vibrating flatways, and are as they are ex-
pressed below, viz,

Nature of the Figure. When suspended by Vertex.

Isosceles triangle \ of its altitude

Common Parabola f of its altitude

Any Parabola x its altitude.

As to figures moved laterally or sideways, or edge-
ways, that is about an axis perpendicular to the plane of
the figure, the finding the centre of oscillation is some-
what difficult; because all the parts of the weight in the
same horizontal plane, on account of their unequal dis-
tances from the point of suspension, do not move with the
same velocity; as is shown by Huygens, in his Horol.
Oscil. He found, in this case, the distance of the centre
of oscillation below the axis, viz,

In a circle, - - £ of the diameter:

In a rectangle, susp. by one angle, -| of the diagonal :
In a parabola, suspi by its vertex, -f axis
The same snap, by mid. of base, $ axis

Ju a sector of a circle - - -

In a cone

C E N

radius base3

• - - - 4 axis h — :

3 5 axis

■Sr2
- - - - e h , where r is

r the rad. added to the length ol

\ param.
+• 4- param,

rndius

i -luii'd

In a sphere - - -

the radius, and g = a
the thread.

See also Simpson's Fluxions, art. 183 &c.

To find the Centre of Oscillation Mechanically or Experi-
mentally. Make the body oscillate about its point of sus-
pension ; and hang up also a simple pendulum of such a
length that it may vibrate or just keep time with the other
body : then the length of the simple pendulum is equal to
the distance of the centre of oscillation of the body below
the" point of suspension.

Or it will be still better found thus: Suspend the body
very freely by the given point, and make it vibrate in small
arcs, counting the vibrations it makes in any portion of
time, as a minute, by a good stop-watch ; and let that
number of oscillations made in a minute be called n : then
shall the distance of the centre of oscillation be

so = — inches. For, the length of the pendulum vi-
brating seconds, or 60 times in a minute, being 39f inches,
and the lengths of pendulums being reciprocally as the
square of the number of vibrations made in the same time,.

therefore re* : 60* : : 39r : > the length of the pendu-
lum which vibrates n times in a minute, or the distance
of the centre of oscillation below the axis of motion.

Centre of Percussion, in a moving body, is that point
where the percussion or stroke is the greatest, in which
the whole percutient force of the body is supposed to be
collected ; or about which the impetus of the parts is ba-
lanced on every side, so that it may be stopped by an im-
moveable obstacle at this point, and rest on it, without
acting on the centre of suspension.

1 . When the percutient body revolves about a fixed point,
the centre of percussion is the same with the centre of os-
cillation ; and is determined in the same manner, viz, by
considering the impetus of the parts as so many weights ap-
plied to an inflexible riiiht line void of gravity; namely, by
dividing the sum of the products of the forces of the parts
multiplied by their distances from the point of suspension,
by the sum of the forces. And therefore what has been
above shown of the centre of oscillation, will hold also of
the centre of percussion, when the body revolves about a
fixed point. For instance, that the centre of percussion
in a cylinder is at f of its length from the point of suspen-
sion, or that a stick of a cylindrical figure, supposing the
centre of motion at the hand, will strike the greatest blow
at a point about two-thirds of its length from the hand. —
See the investigation at large in my Course of Mathematics,
p. 44.

2. But when the body tnoves with a parallel motion, or
all its parts with the same celerity, then the centre of per-'
cussion is the same as the centre of gravity ; for the mo-
menta ara: the products of the weights and celerities ; and
to multiply equiponderating bodies by the same velocity,
is the same thing as to take equimultiples; but the equimul-
tiples of equiponderating bodies do also equiponderate;
therefore equivalent momenta are disposed about the cen-
tre of gravity, consequently in this case the two centres
coincide, and what is shown of the one will hold in the
other.

Centre of Percussion, in a fluid, is the same as out of il.
2 tv> 2

CEN

[ 300 ]

CEN

Centre of a Parallelogram, the point in which its dia-
gonals intersect.

Centre of Position. See Centre of Gravity.

Centre of Pressure, of a fluid against a plane, is that
point against which a force being applied, equal to the
whole pressure, but in a contrary direction, it will keep
the surface at rest. It is evident that, on a plane parallel
to the horizon, or on any plane where the pressure is uni-
form, the centre of pressure coincides with the centre of
- gravity of the plane. But for other cases, it will be found
by the following method.

To find the Centrcof Pressure of any Plane Surface.

Let abcd be the surface of the fluid, vw, the plane ;
which being pro-
duced, let CDbeits
intersection with
the surface, p the
centre of pressure,
and g the centre of
gravity; and con-
ceive the whole
plane to be divid-
ed into an indefi-
nite number of ex-
tremely small parts
of which one is x :
draw pq, Gg, xv,
perpendicular to the surface, and pa, an, xm perpendicu-
lar to cd; and join Qffi, gn, vm, then it is manifest that the
triangles VQa, Ggn, xvm are similar. Now, by the princi-
ples of hydrostatics, the pressure on x, perpendicular to
vw, is as a: x xv; and by mechanics, its effect to turn the
plane about cd, is as x x xv x xm; but an : Gg : : xm : xv

=a xm x — ; hence the effect of the pressure at x, to turn

en

the plane about cd, is as x x xm1 x — ; therefore the whole

1 ' Gil

effect is as the sum of all the x x xm* x — . But if a =

GI2

the area of vw, the whole quantity of pressure on vw = a
x Gg; therefore the effect of the pressure at p, to turn the
plane about cd, is as jxGgn pa. Hence a x Gg x pa =
sum of all the x x xm1 x — ; consequently

i of all the x x xm? x •

<3g-

Hence it appears, that p is at the same distance from cd
as the centre of percussion is, cd being the axis of suspen-
sion. They do not however in general coincide or lie in
the same line, that is, in the line «G ; for the efficacy of the
pressure at or, to turn the plane about kg, is as x x xv x
run (or since xv varies as xm), as x x xm x mn; but the sum
of all the * x xm x mn, is not generally = 0, therefore the
whole pressure will not necessarily balance itself upon the
line on. The situation of the line pa must be determined
by making thesum of all the x x xm x mn = 0, which in any
particular case may be done by a fluxional process. It is
not therefore true, in general, that the centres of pressure
and percussion are the same point.

Cent he of a Regular Polygon, or Regular Body, is the
same as that of the inscribed, or circumscribed circle or
ipherfe.

Centre of Rotation, is that point about which a body,
otherwise at liberty, revolves or tends to revolve when it is
acted upon unequally at different points, or by a force

the direction of which does not pass through its centre of
gravity. M. J. Bernoulli was the first who published any
thing on this subject; and he first found the point about
which a body at rest would begin to revolve when struck
by another body, and he called this point the centre of spon-
taneous rotation, to distinguish it from the centre of forced
rotation. He observed however that D. Bernoulli had
discovered the same; he has also mentioned the curve de-
scribed by that point in the progressive motion of the body,
and has described a method of inquiry by which the ve-
locity of the bodies may be found after the stroke. Mr.
Vince has treated this subject much at large, and with
great perspicuity, in the Philos. Trans, vol. 70, for 1780,
p. 540, &c.

Centre of a Sphere, is the same as that of its genera-
ting semicircle, or the middle point of the sphere, from
whence all right lines drawn to the superficies, are equal.

Centre of Spontaneous Rotation, is that point which re-
mains at rest the instant a body is struck, or about which
the body begins to move. .The term was first used by John
Bernoulli to distinguish this centre from the centre of forced
rotation, or that about which a body is compelled to re-
volve, such for instance as the point of suspension of a
pendulum. See Centre of Rotation.

Centring of an Optic Glass, the grinding it so as that
the thickest part be exactly in the middle.

Cassini the younger has a discourse expressly on the ne-
cessity of well centring" the object glass of a large tele-
scope, that is, of grinding it so as that the centre may fall
exactly in the axis of the telescope. Mem. Acad. 1710.

Indeed, one of the greatest difficulties in grinding large
optic glasses is, that in figures, so little convex, the least
difference will throw the centre 2 or 3 inches out of the
middle. And yet Dr. Hooke remarks, that though it were
better the thickest part of a long object glass were exactly
in the middle, yet it may be a very good one when it is
an inch or two out of it. Philos. Trans, No. 4.

CENTRING, or Centering, in Building, denotes the
frame of timber by which the brick or stone work of
arched vaulting is supported during its erection, and from
which it receives its form and curvature. For their de-
scription and properties, see the Principles of Bridges, in
my new Tracts, vol. 1, p. 105.

CENTRIFUGAL Machine, a curious machine invented
by a Mr. Erskine, for raising water by means of a centri-
fugal force, combined with the pressure of the atmosphere.
This machine consists of a large tube of copper, &c, in the
form of a cross, placed perpendicularly in the water, and
resting at the bottom on a pivot. At the upper part of the
tube is a horizontal toothed wheel, which touches the teeth
of another in a vertical position ; so that by the aid of a
double winch the whole machine is moved round with very
great velocity. Near the bottom of the perpendicular part
of the tube is a valve opening upwards; and near the two
extremities, but on the contrary sides of the arms, or cross
part of the tube, are two other valves opening outwards.
These two valves are kept shut by means of springs, till the
machine is put in motion, when the centrifugal velocity of
the water fbrccs them open, and discharges itself into a cis-
tern or reservoir placed there for that purpose. On the
upper part of the arm are two holes, which are closed by
pieces that screw into the metal of the tube. Before the
machine can work, these holes must be opened, and water
poured in through them, till the whole tube be full; by
these means all the air is forced out of the machine, and,

C E N

[ 301 J

C E R

the water supported in the tube by means of the valve at
the bottom. The tube being thus filled with water, and
the holes closed by their screw caps, it is turned round by
the winch, when the water in the arms of the tube acquires
a centrifugal force, opens the valves near the extremities of
the arms, and flies out with a velocity nearly equal to that
of the extremities of the said arms.

Centrifugal Force, is that by which a body revolving
about a centre, or about another body, endeavours to re-
cede from it. And

CENTRIPETAL Force, is that by which a moving
body is perpetually urged towards a centre, and made to
revolve in a curve instead of a right line.

Hence, when a body revolves in a circle, these two forces,
viz, the centrifugal and centripetal, are equal and contrary
to each other, since neither of them gains upon the other,
the body being in a manner equally balanced by them.
But when, in revolving, the body recedes farther from the
centre, then the centrifugal exceeds the centripetal force ;
as in a body revolving from the lower to the higher apsis,
in an ellipse, and respecting the focus as the centre. And
when the revolving body approaches nearer to the centre,
the centrifugal is less than the centripetal force; as while
the body moves from the farther to the nearer extremity
of the transverse axis of the ellipse.

It is one of the established laws of nature, that all motion
is of itself rectilinear, and that the moving body never re-
cedes from its first right line, till some new impulse be
superadded in a different direction; after that new impulse
the motion becomes compounded, but it is still rectilinear,
though not in the same line or direction as before. To
movcin a curve it must receive a new impulse in a diffe-
rent direction every moment, a curve not being reducible
to any number of finite right lines. If then a body, con-
tinually drawn towards a centre, be projected in a line
that does not pass through that centre, it will describe a
curve; in every point of which, as
a, it will endeavour to recede from
the curve, and proceed in the tan-
gent ad; and if nothing hindered, it
would actually proceed in it; so as
in the same time in which it de-
scribes the arch ae, it would re-
cede the length of the line de, perpendicular to ad, by its
centrifugal force: or, being projected in the direction ad,
but continually drawn out of its direction into a curve by
a centripetal force, so as to fall below the line of direction
by the perpendicular space de; then the centrifugal or
centripetal force is as this line of deviation de, supposing
the arch ae indefinitely small.

The doctrine of centrifugal forces was first mentioned by
Huygens, at the end of his HorologiumOscillatorium, pub-
lished in 1673, and demonstrated in the volume of his
Posthumous Works ; as also by Guido Grando, who gave
a few easy cases in bodies revolving in the circumference
of circles. But Newton, in his Principia, was the first vvho
fully prosecuted this doctrine; at least as far as regards the
conic sections. After him there have been several other
writers on thissubject; as Leibnitz, Varignon, in the Mem.
<le lAcad., Keil, in the Philos. Trans, and in his Physics,
Bernoulli, Herman, Cotes, in his Harmonia Mensurarum,
Maclaurin, in his Geometrica Organica, and in his Flux-
ions, and Euler, in his book de Motu, where he considers
the curves described by a body acted on by centripetal
forces tending to several fixed points.

See also tbe article Central Forces, where this doctrine is
more fully explained.

CENTROBARYCO, the same as centre of gravity.
Centrobaryc Method, is a method of determining the
quantity of a surface or solid, by means of the generating
line or plane, and its centre of gravity. The doctrine is
chiefly comprised in this theorem : Every figure, whether
superficial or solid, generated by the motion of a line or
plane, is equal to the product of the generating magnitude
and the path of its centre of gravity, or the line described
by its centre of gravity.

See more of this subject in the art. Centre of Gravity.
CENTRUM, in Geom. Mechanics, &c. See Centre.
Centrum Phonicum, in Acoustics, is the place where
the speaker stands in polysyllabical and articulate echoes.

Centrum Phonocampticum, is the place or object that
returns the voice in an echo.

CEPHEUS, a constellation of the northern hemisphere,
being one of the 48 old asterisms. The Greeks fable that
Cepheus was a king of Ethiopia, and the father of Andro-
meda, the princess who was delivered up to be destroyed
by a sea monster, from which she was rescued by Perseus.
The stars of this constellation, in Ptolemy's catalogue, are
13, in Tycho's 11, in Hevelius's 51, and in the Britannic
catalogue 35.

CERBERUS, one of the new constellations, formed by
Hevelius out of the unformed stars, and added to the 48
old asterisms. It contains only 4 star9, which are enume-
rated under Hercules in the Britannic catalogue.

CERES, or Piazzi, in Astronomy, a new primary
planet intermediate between the orbits of Mars and Jupi-
ter, was discovered on the 1st of January 1801, by M.
Piazzi, astronomer royal at Palermo in Sicily, who also
gave it this name.

M. Piazzi in a brief account he has published of the
discovery of this planet states, that having been engaged
for nine years in rectifying the positions of the stars as
Collected in the catalogues of various astronomers, he was
searching on the first of January 1801. among many
others, for the 87th in the catalogue of the zodiacal stars
of the Abbe de la Caille, when he observed that this star
was preceded by another, which according to his usual
custom he wished to observe also ; especially as it did not
interrupt the principal observation. Its light was some-
what faint, and its colour resembled that of Jupiter; but
like many others, which, in regard to their magnitude, are
usually placed in the 8th class. At that time no doubt
arose respecting the nature of it ; but on the evening of
the 2d, having repeated his observations, and rinding that
they did not correspond either in time or zenith distance,
he suspected that some error had been committed in his
observations on the preceding day. He then began to en-
tertain some idea that it might perhaps be a new planet.
In the evening of the 3d his conjecture was confirmed,
and he assured himself it was not a fixed star. However
before he would speak of it, he waited till the evening of
the 4th, in which he had the satisfaction of finding that
it had moved according to the same laws which it had ob-
served on the preceding days. At this time the motion
was retrograde ; but on the 10th of January it became di-
rect. He continued to observe the planet till the 13th of
February, when he was obliged by illness to discontinue
his observations. M. Piazzi then transmitted accounts of
his observations to several celebrated astronomers, in or-
der that they might calculate the orbit of the new plauct,

CEU

[ 309 ]

C H A

and trace out its progress in the heavens: but it eluded CHAIN, in Surveying, is a lineal measure, consisting

every search that was made for it, until December 7th, of a certain number of iron links, usually 100 ; serving

when it was re-discovered by the assiduous Dr. Zach of to take the dimensions of fields &c.

Saxe-Gotha; and soon after it was observed by Dr. Ol- At every 10th link is usually fastened a small brass

bers at Bremen ; by Mechain, at Paris ; by the royal as- plate, with a figure engraven upon it, or else cut into dif-

tronomerat Greenwich ; by Dr. Herschel, at Slough ; and ierent shapes, to show how many links it is from one end

by various other persons both in England and Scotland. of the chain.

Elements of its Theory according to Burckhardt.

Inclination of orbit

-

0s 10° 37'

Place of the ascending node

-

2 21 6

Epoch of 1801

T

2 17 19

Place of the aphelion

-

10 26 9

Passage of the aphelion Jan. 1,

1801

J 8 hours

Excentricity of orbit

-

0-0784

Mean distance

-

2-7677

Time of revolution

-

4-6o6 years.

From the observations of Dr. Herschel it is probable

Chains are of various kinds and lengths ; as

1. A chain of 100 feet long, each link one foot, is used
for measuring of large distances only, when no regard is
had to acres &c, in the superficial content.

2. A chain of one pole or l6 feet and a half in length ;
is particularly useful in measuring and laying out gardens
and orchards, or the like, by the pole or rod-measure.

3. A chain of 4 poles, or 66 feet, or 22 yards, in letigth,
called Gunter's chain, and is peculiarly adapted to the
business of land-surveying, because 10 square chains just

that the diameter of this planet, is much less than that of make an English acre of land ; so that the dimensions be-
the moon ; it appeared under a disc of less than 2". Yet, ing taken in these chains, they are readily reduced to acres
small as is this planet, M. Schroeter suspects it has two by dividing by 10, or barely cutting off the last figure from
satellites. It has been expected for nearly a century, that the square chains. But it is still better in practice to pro-
such a discovery as this of M. Piazzi, would be made by ceed thus, viz, count the dimensions, not in chains, but all
some diligent astronomer. Maclaurin, Lambert, Bode, in links; then the contents are in square links; and five
Zach, Capel Lofft, and others have supported the conjee- figures being cut off for decimals, the rest are acres ; that
ture. And six astronomers assembled at Lilienthal, in is four figures to bring the square links to square chains,
September 1S00, resolved to establish a society of 24 and one more to bring the square chains to acres,
practical astronomers, dispersed throughout different parts In this chain, the links are each 7 inches and ,%%, or
of Europe, for the express purpose of searching out this 7'92 inches in length, which is very nearly \ of a foot,
planet between Mars and Jupiter: they elected Schroeter And hence any number of chains or links are easily reduced
as their president, and Dr. Zach was chosen their secre- to feet or inches, or the contrary : the best way of doiq(

tary. Accordingly, the consequence has been, that no
less than four new planets have been discovered, all nearly
agreeing in size, period, and situation ; viz, the above one
by Piazzi, one by Harding of Lilienthal, and two by Ol-
bers of Bremen, all of which have received names of dif-
ferent heathen divinities : but it may be remarked on this
occasion, that it would be a more fit practice to call the
several new planets after the names of their respective dis-
coverers, as Herschel, Piazzi, Olbers, &c.

CETUS, the Whale, a southern constellation, and one

which is this : multiply the number of links by 66, then
cut off two figures for decimals, and the rest are feet : or
multiply links by 22 for yards, cutting off two figures.

Cha*n Rule, called also Rule of Exchange, Rule of Re-
duction, and Rule of Equation, is a mode of arbitration
very useful in the higher operations of exchange, as well
as in the arbitration of bullion, specie, and merchandise.
Foreignmerchants,itseems,areveryexpert in their applica-
tion of this rule to commercial computations; and that it is,
in a great measure, to this accomplishment that their ac-

o^ the 48 old asterisms. The Greeks pretend that it was knowledged superiority in the practice of exchange may

the sea monster sent by Neptune to devour Andromeda,
butwaskilled byPerseus. — In the neck of the whale is a re-
markable star, Collo Ceti, which appears and disappears pe-
riodically, or rather becomes brighter and fainter by turns,
arising it is supposed from the alternate turnings of its
bright and dark sides towards us, as it revolves upon its
axis, or else from the flatness of its form. The period of
its changes is about 312 days. Bulliald in Phil. Trans,
vol. 2, Hevelius, ibid. vol. 6, Herschel, ibid. vol. 70, Ma-
rald. in Mem. Acad. 1/ 19.

The stars in the constellation Cetus, in Ptolemy's cata-
logue, are 22, in Tycho's 21, in Hevelius's 45, and in the
Britannic catalogue *)7.

CEULEN, or Keui.en (Ludolpii Van), a noted
Dutch professor of mathematics, who flourished in the
latter part of the l6th century. Ludolph was sumamed
Van Ceulen', because he was from Cologne, which is called
Ceulen or Kculeri in Hutch. He was author of several
pieces, both in arithmetic and geometry, in the Dutch lan-
guage, some of which were translated into Latin by Wille-
brord Snell, as mentioned under his life. Ceulen was
chiefly remarkable for his approximation to the circum-
ference of the circle, to 36 places of figures, which he or-
dered to be engraved on his tombstone.

be attributed. See Arbitration, also Dr. Kelly's
Cambist.

CHALDER, or CHALDRON, of Coals, an English dry-
measure of capacity consisting of 36 bushels heaped up.

The chaldron of coals is accounted to weigh about 28
cwt. or 3136 pounds. — On ship-board, 21 chaldrons of
coals are allowed to the score.

By act of parliament, a Newcastle chaldron is to weigh
52| cwt, or 3 waggons of 17 £ cwt. or 6 carts of 8| cwt.
each, making 52-J cwt. to the chaldron. The statute Lon-
don chaldron is to consist of S6 bushels heaped up, each
bushel to contain a Winchester bushel and one quart, and
to be H)^ inches diameter externally. Now it has been
found by repeated trials that 15 London chaldrons are
equal to 8 Newcastle chaldrons, which, reckoning 52-£
cwt. to the latter, gives 28 cwt. to the former, or 3136'lbs
to the London chaldron.

This I find nearly confirmed by experiment. I weighed
one peck of coals, which amounted to 'Jljjlbs. Then 4
times this gives 87jb fqr the weight of the bushel ; and 36"
times the bushel gives 3132lbs for the chaldron ; to which
if the weight of the odd quart be' added, or 3lb nearby,,
it gives 31351b for the weight of the chaldron, which is
only 1 pound short of that by statute.

C 1-1 A

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C H A

CHAMBER of a Mortar, or Cannon, is a cell or cavity
at the bottom of the bore, to receive the charge of powder.
It is not found by experience that chambers have any
sensible effect on the velocity of the shot, unless in the
largest ordnance, as mortars or very large cannon. Nei-
ther is it found that the form of them is very material ; a
small cylinder is as good as any ; though mathematical
speculations may show a preference of one form over ano-
ther. But in practice, the chief point to be observed, is,
tc have the chamber of a size just to contain the charge
of powder, and no more, that the ball may lie close to the
charge ; and that its entrance may point exactly to the
centre of the ball.

CHAMBERS (Ephraim), author of the dictionary
of sciences called the Cyclopaedia. He was born at Mil-
ton, in the county of Westmoreland, where he received
the common education for qualifying a youth for trade
and commerce. When at a proper age, he was appren-
ticed to Mr. Senex the globe-maker, a business which is
connected with literature, especially with geography and
astronomy. It was during Mr. Chambers's residence with
this skilful artist, that he acquired that taste for literature
which accompanied him through life, and directed all his
pursuits. At this time, he formed the design of his grand
work, the Cyclopaedia; some of the first articles of which
were written behind the counter. To have leisure to pur-
sue this work, he quitted Mr. Senex, and took chambers
at Gray's Inn, where he chiefly resided during the rest of
his life. The first edition of the Cyclopaedia, which was
the result of many years intense application, appeared in
1728, in 2 vols folio. The reputation that Mr. Chambers
acquired by the execution of this work, procured him
the honour of being elected f. r.s. Nov. 6, 1729- In
less than ten years time, a second edition became neces-
sary; which accordingly was printed, with corrections and
additions, in 1738; and this was followed by a third edi-
tion the very next year.

Though the Cyclopaedia was the chief business of Mr.
Chambers's life, and may be regarded as almost the sole
foundation of his fame, his attention was not wholly con-
fined to this undertaking. He was concerned in a perio-
dical publication called The Literary Magazine, which was
begun in 1735. In this work he wrote a variety of articles;
particularly a review of Morgan's Moral Philosopher.
He was also concerned with Mr. John Martyn, professor
of botany at Cambridge, in preparing for the press a trans-
lation and abridgment of the Philosophical History and
Memoirs of the Royal Academy of Sciences at Paris;
which work was not published till 1742, some time after
our author's decease, in 5 volumes 8vo. Mr. Chambers
was also author of the translation of the Jesuit's Perspec-
tive, from the French, in 4to; which has gone through
several editions. He was alsojoint editor, with Dr. Shaw,
of a translation of Boerhaave's Institutiones Chemia;, un-
der the title of A New Method of Chemistry; including
the Theory and Practice of that Art, &c, published jointly
by P. Shaw, m. b. and E.Chambers, gent, with additional
notes and sculptures, Lond. 4to, 1727: and it afterwards
went through several editions, with augmentations. Mr.
C. was also the editor of the Abridgment of Lord Bacon's
Philosophical Works, in 3 vols 4to, 1739; and of those
of Mr. Boyle, 3 vols 4to, 1725. Besides which, he was
theauthor of 3 Chemical Lectures, read in London in 1731
and 1732, and Scarborough 1733, for the improvement of
arts, trades, and natui a) philosophy, 8vo, 2d edit. 1725.

Subsequent to which, he published Essays for the im-
provement of arts, manufactures, and commerce, by means
of chemistry, 2d edit, improved by P. Shaw, fellow of the
college of physicians, and physician in ordinary to his
majesty.

Mr. Chambers's close and unremitting attention to his
studies at length impaired his health, and obliged him oc-
casionally to take a country lodging, but without much
benefit ; he afterwards visited the south of France, but
still with little effect; he therefore returned to England,
where he soon after died, at Islington, May 15, 1740, and
was buried in Westminster Abbey.

After our author's death, two more editions _of his Cy-
clopaedia were published. The proprietors afterwards pro-
cured a supplement to be compiled, by Mr. Scott and Dr.
Hill, which extended to two volumes more; and the whole
has since been reduced into one alphabet in 4 volumes, by
Dr. Rees, forming a very valuable body of the sciences.
And the same is now extending in a new edition, in 4to,
under the title of Encyclopaedia.

CHAMBRANLE, the border, frame, or ornament of
stone or wood, surrounding the three sides of doors, win-
dows, and chimneys. This is different in the different or-
ders : when it is plain, and without mouldings, it is called
simply and properly, band, case, or frame. In an ordinary
door, it is commonly termed door-case; in a window, the
window-frame.

The chambranle consists of three parts; the two sides,
called ascendants ; and the top, called the traverse or su-
percilium.

CHAMFER, or Ch'amfehet, an ornament, in archi-
tecture, consisting of, half a scotia ; being a kind of small
furrow or gutter on a column.

CHAMFERING, is used for cutting the edge or the
end of any thing bevel, or aslope.

CHANCE, the Doctrine and Laws of, are the same as
those of Expectation, or Probability, &c ; which see.
Chances, in play, consist of the number of ways by which
events may happen. Thus, if a halfpenny, or other piece
of money, be tossed up, there are two events, or chances,
or sides that may turn up, namely, one chance for turning
up a head, and one for the contrary ; that is, it is an equal
chance to throw a head or not. And in throwing a com-
mon die, which has 6 faces, there are in all 6 chances,
that is one chance for throwing an ace or any other single
point, and 5 chances against it; or it is 5 to 1 that such
assigned point does not Come up.

The term chance is applied to events, to denote that they
happen without any necessary, foreknown, or intending
cause; or it is used to denote the bare possibility of an
event when nothing is known either to prevent or hinder
it : thus, when we say a thing happens by chance, we
really mean no more than that its cause is unknown to7
us ; not, as some vainly imagine, that chance itself can be
the cause of any thing.

Doctrine of Chances, History of. This subject,
which is no less useful than curious, does not appear to
have engaged the attention of mathematicians in former
times so much as its importance required. Until the be-
ginning of the last century little is to be found in any of
their writings concerning this subject. Of the few pro-
blems which they had been accustomed to investigate,
they withheld the solutions both from the public and from
each other ; and they seem to have considered the doc-
trine of chances rather as an exercise for their ingenuity,

C H A

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C H A

than as capable of being applied to any useful purpose, the world must be the effect of the wisdom and power of

Before Mr. Huygens published his book De Ratiociniis an intelligent cause ; and thus to confirm the argument

in Ludo Alea;, no person had treated the subject metho- taken from final causes for the existence of the Deity." —

dically ; and, with the exception of Messrs. Pascal and Rees.

Fermat, who had solved a few problems of no great im- For the doctrine itself consult Demoivre, T. Simpson,

portance or difficulty, he appears to have been the first and a work on the same subject lately published by the

who attempted either to give rules for the solutions of any celebrated Laplace. See also Credibility, Gaming,

question, or to lay down the principles from which those
solutions might be deduced. To this work succeeded a
small anonymous tract, On the Laws of Chance, which
was published in London in l6"2, and a French publica-
tion of not much larger size, entitled L' Analyse des Jeux
de Hazard, written by M. Monmort, and published in the
year 1708. In this latter work, the author having chiefly
insisted on the same mode of reasoning with Huygens, in
the solution of his problems, Mr. Demoivre (who consi-
dered such reasoning as neither genuine nor natural) was
induced in his celebrated work on the Doctrine of Chances
(which was first published in 1717), to adopt a plainer
and less exceptionable mode, in which he has proceeded
from the most simple to the most complicated cases ; so
that by the variety of his problems, as well as by the im-
provements and additions which he made in two subse-
quent editions, he had rendered his work one of the best
and most copious that has ever been written on the sub-
ject. In the year 1740, Mr. Thomas Simpson published
a small treatise on The Nature and Laws of Chance,
which, like his other publications, is not only clear and
concise, but contains some problems, whose solutions had
either never been attempted, or at least never before com-
municated to the public. Prior however to the two last-
mentioned publications, a posthumous work of Mr. James
Bernoulli was published in the year 1713, entitled De Arte
Conjectandi, containing an explanation of Huygens's tract,
and the solution of a great variety of other problems, de-
duced from the general principles of combination. The
second part of this valuable work has lately been trans-
lated into English by Mr. Baron Maseres, with notes and
commentaries. In the year 176*5, and at other times, M.
Dalembert, in his Opuscules &c, wrote different essays
on the doctrine of chances ; and not long ago M. Con-
dorcet published a small treatise on the same subject. In
addition to these, which are the principal publications on
the subject, may be noticed a small tract, De Mensura
Sortis, given by Demoivre in his Miscellanea Analytica,
and some other papers written by him, by Messrs. Ber-
noulli, Euler, and others, in the Acts of Leipsic, the Jour-
nal des Scavans, the Philosophical Transactions, &c,
among which may be particularly mentioned an Essay on
the Method of calculating the exact Probability of all
Conclusions founded on Induction, and a Supplement to
that Essay; the one preserved from the papers of the late
Rev. Mr. Bayes, and communicated with an appendix, by
Dr. Price, to the Royal Society in the year 176'2 ; the
other chiefly written by Dr. Price, and communicated in
the following year. These tracts contain the investigation
of a problem, the converse of which had formerly exer-
cised the ingenuity of Bernoulli, Demoivre, and Simpson.
Indeed both the problem and its converse may justly be
considered not only as the most difficult, but as the most
important that can be proposed on the subject ; having
(as Dr. Price well observes) "no less an object ill view
Ihan to show what reason we have for believing that there
are in the constitution of things, fixed laws, according to
which events happen; and that therefore, the frame of

Expectation, and Probability.

CHANDELIERS, in Fortification, a kind of wooden
parapet, consisting of upright timbers supporting others
laid across the tops of them, 6 feet high, and fortified with
fascines, &c. They are used to cover the workmen in ap-
proaches, galleries, and mines. And they differ from
blinds only in this, that the former serve to cover the men
before, and the latter over head.

CHANGES, the permutations or variations of any num-
ber of things, with regard to their position, order, &c ;
as how many changes may be rung on any number of
bells, or how many different ways any number of persons
may be placed, or how many several variations may be
made of any number of letters, or any other things pro-
posed to be varied.

To find out such numbers of changes, multiply continu-
ally together all the terms in a series of arithmetical pro^
gression, whose first term and common difference are each
unity or I, and the last term the number of things pro-
posed to be varied, thus 1x2x3x4x5 &c, till the
last number be the proposed number of things. For let
there be two things as a and 6. Then since they may be
placed either thus ab or ba, it is obvious that the number
of changes is 1 x 2 or 2. If three things, as a, b, c be
supposed ; they will admit of 6 variations as in a b c
the margin ; for each of the three may be com-
bined three different ways with each combina-
tion of the other two; thus a may be first
combined with be and then with cb, and hence
the number of changes arises, viz, 1 x 2 x

3 = 6. c b a
And if there be 4 things, each of them may be combined

4 ways with each combination of the other three, that is
4 times 6 ways, orl x2x3x4=24 ways.

In like manner, the combinations
of 5 things are 1x2x3x4x5 =120

of 6 things are 1x2x3x4x5 x 6 = 720
&c.

So that if it be proposed to assign how many different
ways a company of b persons may be placed, at table tor
instance, the answer will be 720 ways. Also the number
of changes that can be rung on 7 bells, is 1 x 2 x 3 x 4
x 5 x 6' x 7 or 720 x 7 = 5040,

CHAPITERS, the crowns or upper parts of a pillar or
column.

CHAPPE (Jean d'Auteroche), a French astrono-
mer, was born at Mauriac, in Auvergne, March 2, 1728.
A taste for drawing and mathematics appeared in him at
a very early period ; and he owed to Dom. Germain a
knowledge of the first elements of mathematics and astro-
nomy. M. Cassini, after assuring himself of the genius of
this young man, undertook the cultivation of it. He em-
ployed him on the map of France, and the translation of
Halley's tables, to which he made considerable additions.
The king chargod him in 1753 with drawing the plan of
(he county of Bitche, in Lorraine, all the elements of
which he determined geographically. He occupied him-

b

a

C

b

c

a

a

c

b

c

a

b

C H A

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C H A

self greatly with the two comets of 1760 ; and the fruit of
his labour was his Elementary treatise on the theory of
those comets, enriched with observations on the zodiacal
light, and on the aurora borealis. Soon after he went to
Tobolsk, in Siberia, to observe the transit of Venus over
the sun ; a journey which greatly impaired his health.
After two years absence he returned to France in 1762,
where he occupied himself for some time in arranging the
great number of observations he had then made. Mr.
Chappe also went to observe the next transit of Venus, viz,
that of 1769j at California, on the west side of North
America, where he died of an epidemic disease, the 1st of
August 1769. He had been named adjunct astronomer
to the Academy the 17th of January 1759.
The published works of M. Chappe, are,

1. The Astronomical Tables of Dr. Halley; with ob-
servations and additions: in 8vo, 1754.

2. Voyage to California to observe the transit of Venus
over the sun, June 3, 1769: in 4to, 1772.

3. He had a considerable number of papers inserted in
the Memoirs of the Academy, for the years 1760, 1761,
1764, 1765, 1766, 1767, and 1768; chiefly relating to
astronomical matters.

CHAPTREL, the same with Impost.
CHARACTERISTIC, of a Logarithm, the same as In-
dex, or Exponent. This term was first used by Briggs in
the 4th section of his Arithmetica Logarithmica, where he
treats particularly of it; meaning by it, the integral or
first part of a logarithm towards the left hand, which ex-
presses 1 less than the integer places or figures in the num-
ber answering to that logarithm, or how far the first figure
of this number is removed from the place of units ; namely,
that 0 is the characteristic of all numbers from 1 to 10 ;
and 1 the characteristic of all those from 10 to 100 ; and 2
the characteristic of all those from 100 to 1000 ; and so on
CHARACTERS, are certain marks used by astrono-
mers, mathematicians, &c, to denote certain things, whe-
ther for the sake of brevity, or perspicuity, in their ope-
rations.

I. Astronomical Characters.

The twelve Signs or Constella-
tions of the Zodiac.
V Aries, the Ram
« Taurus, the Bull
H Gemini, the Twins
25 Cancer, the Crab
SI Leo, the Lion
»E Virgo, the Maid
j± Libra, the Balance
iti Scorpio, the Scorpion
f Sagittary, the Archer
Vf Capricorn, the Goat
ZZ Aquarius, the Water-bearer
H Pisces, the Fishes.
The Aspects, Time, Motion, Sfc.
6 Conjunction ° Degrees

g Opposition ' Minutes or Primes

* Srxtile " Seconds, &c.

a Quartile a. m. Ante mcrid. or m. morn.

A Trine p. m. Post merid. or a. aftern.

h, m, s, Hours, rain. sec. .
2. Mathematical §c. Characters.
• Numerical Characters used by different Nations.
The most common numerical characters, arc those
Vol. I.

Planets, 8fc.

Q The Sun
J The Moon
© The Earth
$$ Mercury

Venus

Mars

Jupiter

Saturn

Herschel, or the

Georgian Planet
£$ Ascending Node
J5 Descending Node

called Arabic or Indian, viz, 1, 2, 3, 4, 5, 6, 7, 8, p,
with 0 for cipher or nothing.

The Roman numeral characters are seven, viz, 1 one,
v five, x ten, l fifty, c a hundred, d or id five hundred,
M or do or cio a thousand. Other combinations are as
in the following synopsis of the Roman Notation

1 = 1

2 = n: As often as any character is repeated,
' 3 == in so many times its value is repeated.

4 = in 1. or iv : A less character before a greater

5 = v diminishes its value.

6 = vi : A less character after a greater increases

7 = vii its value.

8 == viii

9 = ix
10 = x
50 — l

100 = c

500 = d or id :

For every o added, this becomes

10 times as many.

For every c and 0, set one at each

end, it becomes 10 times as much.

A line over any number increases

it 1000 fold.

1000 = m or cio :
2000 = MM
5000 = idd or v:
6000 = vi
10000 = x or cciod
50000 = iooo
60000 = lx or ccciooo
100000 = c or ccciooo
1000000 = m or ccccidooo
2000000 = mm, &c.

Greek Numerals.
The Greeks had three ways of expressing numbers.
First, For every single letter, according to its place in the
alphabet, was denoted a number from a 1 to ui 24; in
which manner the books of Homer's Ilias are distinguish-
ed. Secondly, Another way was by dividing the alphabet
into {1st) 8 units, a 1, jS 2, &c; {2nd) 8 tens, 1 10, % 20,
&c; {3d) 8 hundreds, p 100, tr 200, &c: and thousands,
they expressed by a point or accent under a letter, as a.
1000, /3 2000, &c. Thirdly, By six capital letters, thus,
1 {tot for fua) I, n {irsvrs) 5, A {Sana) 10, H {Hexarov)
100, X (%iAia) 1000, M (f/.utia) 10000 : and when the let-
ter n inclosed any of these, except I, it showed that the
inclosed letter was 5 times its own value, as | A| 50, |H| 500,
|X| 5000, |MJ 50000.

Hebrew Numerals.
The Hebrew alphabet was divided into nine units, as
« 1,12, &c; nine tens, as » 10, 3 20; nine hundreds, as
p 100, 1 200, &c, T 500, D 600, J 700, e\ 800, y 900.
Thousands were sometimes expressed by the units prefixed
to hundreds, as lb"in 1534, &c. and even to tens, as
J?N 1070, &c. But more commonly thousands were ex-
pressed by the word e^K 1000, o»B^N 2000; and cd'S^k

with the other numerals prefixed to signify the number of
thousands, as 0>B^>NJ 3000, &c.

Characters used in Arithmetic and Algebra.
The first letters of the alphabet, a, b, c, &c, denote
given quantities; and the last letters, z,y, x, &c, represent
such as arc unknown or sought. Stifelius first used the
capitals a, b, c, &c, for the unknown or required quan-
tities After that, Vieta employed the capital vowels a,
2 11

C H A

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C H A

B, I, o, u, y, tor the unknown or required quantities, and
the consonants b, c, d, &c, for known or given numbers.
Harriet changed Vieta's capitals into the small letters, viz,
a, e, i, 6, u, for unknown, and b, c, d, &c, for known quan-
tities. And Descartes changed Harriot's vowels for the
latter letters z, y, x, &c, and the consonants for the leading
letters, a, b, c, d, &c.

Newton denotes the several orders of fluxions of variable
quantities by as many points over the latter letters ; as, of
x, y, z; as x,y, sc are the 1st fluxions,

x, y, a are the 2d fluxions,

i, y, ~ are the 3d fluxions.
And Leibnitz denotes the differentials of the same quanti-
ties by prefixing d to each of them, thus dx, dy, dz.

PoweYs of quantities are denoted by placing the index
or exponent after them, towards the upper part; thus a2
is the 2d power, a3 the third power, and an the n power of

a. Diophantus marked the powers by their initials, thus
&*, xy, $8', $xy, kk", &c, for dynamis, cubus, dynamody-
namis, &c, or the 2d, 3d, 4th, &c, powers; and the same
method has been practised by several of the early writers,
since the introduction of algebra into Europe: but the
first of them, as Paciolus, Cardan, &c, used no mark for
powers, but the words themselves. Stifel, and others about
his time, used the initials or abbreviations ty, 5, $, 55'
&c, of res or coss, zenzus, cubus, zenzizenzus, &c, barba-
rous corruptions of the Italian cosa, census, cubo, censi-
census, &c. But he used also numeral exponents, both po-
sitive and negative, to the general characters or roots a,

b, c, &c. Bombelli made use of a half circle thus ^ as
a general character for the unknown or quantity required
to be found in any question, and the several powers <»f it
he denoted by figures set above it; thus *i-, •!,, sL, are
the 1st, 2d, 3d powers of w ; which powers he called dig-
nities. Stevinus used a whole circle for the same unknown
quantity, with the numeral index within it, and that both
integral and fractional ; thus ©, ®, ©, ©, are the 0, 1,
2, 3 powers of the general quantity O; also @, ©, ©,
he uses as the square root, cube root, 4th root of the same;
and ©, the cube root of the square,

and ©, the square root of the cube, and so on.
And these fractional exponents were adopted and farther
practised by his commentator Albert Girard. So that
Stevinus ought to be esteemed the first person who render-
ed general the notation of all powers and roots in the same
way, the former by integral, and the latter by fractional
exponents. Harriot denoted his powers by a repetition of
the letters; thus a, aa, aaa, &c. And Descartes, instead
of this, set the numeral index at the upper part of the let-
ters, as at present, thus a, a2, a3, &c ; though, I am in-
formed, by such as have seen Harriot's posthumous papers,
that he also there makes use of exponents.

The character ^/ is the sign of radicality, or of a root,
being derived from the initial r. or r, which was used at
first by Paciolus, Cardan, &c. This character y', I find
first used by Stifel, in 1544, and by Robert Recorde in
1557. The bare character y/ denotes the square root only ;
but at first they prefixed the initial of the name after it, to
denote the several roots: as ^ q the quadrate or square
r<»>t, .y/c the cube root, &c. But the numeral indices of
the root were prefixed by Albert Girard, exactly the same
as lh«y are used at present, viz, \/ , \/ , */, the 2d, 3d, or

4th root.

The character

represents addition, and a positive

quantity. At first the word itself was used, plus, piu. or
the initial p. by Paciolus, Cardan, Tartalea, &c. And the
character -<- for addition occurs in Stifelius.

— is the character of subtraction, and denotes also «f,
negative quantity; which also first occurs in the same au-
thor Stifelius. Before that, the word minus, mene, or the
initial m. was used. Other characters have also been
sometimes used by other authors, for addition and subtrac-
tion ; but they are now obsolete.

x is the character of multiplication, and was intro-
duced by Oughtred.

-f- representing division, was introduced by Dr. Pell.

Division is also denoted like a fraction, thus - or — = 2.

b a

= denotes equality, and was used by Robert Recorde.
Descartes employs x> for the same purpose.

The character : : for proportionality, or equality of ra-
tios, was introduced by Oughtred; as was also the mark
-— continued proportion.

> for greater, and < for less, were used by Harriot.
And -3 and -H were used by Oughtred for the same pur-
poses.

Dr. Pell used ©- for involution, and Lv for evolution.

00 denotes a general difference between any two quan-
tities, and was used by Dr. Wallis.

The Parenthesis ( ), as a vinculum, was invented by
Albert Girard, and used in such expressions as these,
\/(72 ■+■ v/5120), and b (bj -t- cq), both for universal
roots, and multiplication, &c.

The straight-lined vinculum, , was used by Vieta

for the same purpose ; thus a — b in b -f- c.

Characters in Geometry and Trigonometry.
D A Square /. An Angle

A A Triangle L_ A Right angle

□ A Rectangle _]_ Perpendicular

O or © A Circle U Parallel.

CHARGE, in Electricity, in a strict sense, imports the
accumulation of the electric matter on one surface of an
electric, as the| Leyden phial, a pane of glass, &r, while an
equal quantity passes off from the opposite surface. Or,
more generally, electrics are said to be charged, when the
equilibrium of the electric matter on the opposite surface
is destroyed, by communicating one kind of electricity on
one side, and the contrary kind to the opposite side: nor
can the equilibrium be restored till a communication be
made by means of conducting substances between the two
opposite surfaces : and when this is done, the electric is
said to be discharged. The charge properly refers to one
side, in contradistinction from the other; since the whole
quantity in the electric is the same before and after the
process of charging; and the operation cannot succeed,
unless what is gained on one side is lost by the other, by
means of conductors applied to it, and communicating ei-
ther with the earth, or with a sufficient number of non->
electrics. To facilitate the communication of electricity
to an electric plate, &c, the opposite surfaces are coated
with some conducting substance, usually with tin-foil,
within some distance of the edge; in consequence of which
the electricity communicated to one part of the coating is
readily diffused through all parts of the surface of the elec-
tric in contact with it; and a discharge is easily made by
forming a communication with any conductor from one
coating to the other. If the opposite coatings approach

C H A.

too near each other, the electric matter forces a passage
from one surface to the other before the charge is com-
plete. And some kinds of glass have the property of con-
ducting the electricity over the surface, so that they are
altogether unfit for the operation of charging and dis-
charging. If indeed the charge he too high, and the glass
plate or phial too thin, the attraction between the two op-
posite electricities forcesapassage through theglass, making
a spontaneous discharge, and the glass becomes unfit for
farther use. See Conductors, Electrics, Leyden
Phial, &c.

Charge, in Gunnery, the load of a piece of ordnance,
or the quantity of powder and ball, or shot, with which it
is prepared for execution.

The charge of powder, for proving guns, is equal to the
weight of the ball ; but for service, it is \ or \ the weight
oftheball, orstillless; and indeed in most cases of service,
the quantity of powder used is too great for the intended
execution. In the British navy, the allowance for 32-
pounders is but -^ of the weight of the ball. For it is pro-
bable that, if the powder in all ship guns was reduced to
| the weight of the ball, or even less, it would be a con-
siderable advantage, not only by saving ammunition,
but by keeping the guns cooler and quieter, and at the
same time more effectually injuring the vessels of the
enemy. With the present allowance of powder, the guns
are heated, and their tackle and furniture strained; and
all this only to render the ball less efficacious ; for a ball
which can but just pass through a piece of timber, and in
the passage loses almost all its motion, is found to rend
and fracture it much more, than when it passes through
with considerable velocity. See Robins's Tracts, vol. 1,
pa. 290, 291 ; and my own Tracts, vols. 2 and 3.

Again, the same author observes, that the charge is not
to be determined by the greatest velocity that may be pro-
duced; but that it should be such a quantity of powder as
will produce the least velocity necessary for the purpose
in view; and if the windage be moderate, no field-piece
should ever be loaded with more than £, or at the utmost
4- of the weisht of its ball in powder; nor should the charge
of any battering piece exceed -j of the weight of its bullet.
lb. pa. 266.

Different charges of powder, with the same weight of
ball, produce different velocities in the ball, which are in
the subduplicate ratio of the weights of powder ; and when
the weight of powder is the same, and the ball varied, the
velocity produced is in the reciprocal subduplicate ratio
of the weight of the ball : which is agreeable both to theory
and practice. See my paper on Gunpowder in the Philos.
Trans. 1 778, pa. 50 ; and my Tracts, vols. 2 and 3.

But this is on a supposition that the gun is of an inde-
finite length ; whereas, on account of the limited length of
guns, there is some variation from this law in practice, as
well as in theory ; in consequence of which it appears that
the velocity of the ball increases with the charge only to a
certain point, which is peculiar to each gun, where the
velocity is the greatest; and that by farther increasing the
charge, the velocity gradually liminishes, till the bore is
quite full of powder. By an easy fluxionary process it is
found that, calling the length of the bore of the gun b, the
length of the charge producing the greatest velocity, ought

to be , or about * of the length of the bore;

2'7182H1&28

where 2718281823 is the number whose hyp. log. is 1.
But, for several reasons, in practice the length of the charge

[ 307 ] C H A

producing the greatest velocity, falls short of that above
mentioned, and the more so as the gun is longer. From
many experiments I have found the length of the charge
producing the greatest velocity, in guns of various lengths
of bore, from 15 to 40 calibers, as follows:

Length of Bore Length of Charge

for greatest Veloc.

3
T5-

in Calibers
15
20
30
40

YE

3

-rs
3

CHARLES'S WAIN, a name by which some of the as-
tronomical writers, in our own language, have called Ur-
sa Major, or the great bear; though some writers say the
lesser bear. Indeed both of the two bears have been called
waggons or wains, and by the Latins, who have followed
the Arabians, two biers, Feretrum majus et minus.

CHART, or Sea-Chart, a hydrographical or sea map,
for the use of navigators ; being a projection of some part
of the sea in piano, showing the sea coasts, rocks, sands,
bearings, &c. Fournier ascribes the invention of sea-charts
to Henry son of John king of Portugal. These charts are
of various kinds, the Plain chart, Mercator's or Wright's
chart, the Globular chart, &c.

In the construction of charts, great care should be ta-
ken that the several parts of them preserve their position
to one another, iu, the same order as on the earth ; and
it is probable that the finding out of proper methods to
do this, gave rise to the various modes of projection.

There are many ways of constructing maps and charts ;
but they depend chiefly on two principles. First, by con-
sidering the earth as a large extended flat surface; and the
charts made ob this supposition are usually called Plain
Charts. Secondly, by considering the earth as a sphere;
and the charts constructed on this principle are sometimes
called Globular Charts, or Mercator's Charts, or Reduced
Charts, or Projected Charts.

Plain C/irtrts have the meridians, as well as the paral-
lels of latitude, drawn parallel to each other, and the de-
grees of longitude and latitude everywhere equal to those
at the equator. And therefore such charts must be de-
ficient in several respects. For, 1st, SinCe in reality all
the meridians meet in the poles, it is absurd to represent
them, especially in large charts, by parallel right lines.
2dly, As plain charts show the degrees of the several pa-
rallels as equal to those of the equator, therefore the di-
stance of places lying east and west, must be represented
much larger than they really are. And 3dly, In a plain
chart, while the same rhumb is kept, the vessel appears to
sail on a great circle, which is not the case. Yet plain
charts made for a small extent, as a few degrees in length
and breadth, may be tolerably exact, especially for any
part within the torrid zone; and even a plain chart made for
the whole of this zone will differ but little from the truth.

Mercator's Chart, like the plain charts, has the meri-
dians represented by parallel right lines, and the degrees
of the parallels or longitude, every where equal to those
at the equator; so that they are increased more and more,
above their natural size, as they approach towards the pole;
but then the degrees of the meridians, or of latitude, are
increased in the same proportion at the same part; so that
the same proportion is preserved between them as on the
globe itself. This chart has its name from that of the au-
thor, Girard Mercator, who first proposed it for use in the
year 1556, and made the first charts of this kind ; though
2R2

CHA

[ 308 ]

CHE

they were not altogether on true or exact principles, nor
does it appear that he perfectly understood them. Nei-
ther indeed was the thought originally his own, viz, of
lengthening the degrees of the meridian in some propor-
tion, for this was hinted by Ptolemy near two thousand
years ago. It was not perfected however till Mr. Wrightlirst
demonstrated the same about the year 15.90, and showed
a ready way of constructing this chart, by enlarging the
meridian line by the continual addition of the secants. See
his Correction of Errors in Navigation, published in 1599-

Globular Chart, is a projection so called from the con-
formity it bears to the globe itself; and was proposed by
Messrs. Senex, Wilson, and Harris. This is a meridional
projection, in which the parallels are equidistant circles,
having the pole for their common centre, and the meri-
dians curvilinear and inclined, so as all to meet in the pole,
or common centre .of the parallels. By which means the
several parts of the earth have their proper proportion of
magnitude, distance, and situation, nearly the same as on
the globe itself; which renders it a good method for geo-
graphical maps.

Selenographic Charts, are particular descriptions of the
appearances, spots, and maculae of the moon.

Topographic Charts, are draughts of some small parts
only of the earth, or of some particular place, without re-
gard to its relative situation, as London, York, &c.

For the Construction of Charts, see Geography,
Maps, Projection, &c.

CHASE, of a Gun, is its bore or cylinder.

CHAULNES (The Duke de), a peer of France, re-
markable as an astronomer and mathematician, was
born at Paris, Dec. 30, 1714. — At an early period he dis-
covered a singular taste and genius for the sciences ; and
in the tumults of armies and camps, he cultivated mathe-
matics, astronomy, mechanics, &c. He was named hono-
rary academician the 27th of February 1743, and few
members were more punctual in attending the meetings of
that body ; where he often brought different constructions
and corrections of instruments of astronomy, of dioptrics,
and achromatic telescopes. These researches were followed
with a new parallactic machine, more solid and convenient
than those that were in use ; together with many reflec-
tions on the manner of applying the micrometer to those
telescopes, and of measuring exactly the value of the parts
of that instrument. The Duke of Chaulnes proposed many
other works of the same kind, when death surprised him
the 23d Sept. 1769.

He had several papers published in the volumes of Me-
moirs of the Academy of Sciences, as follow :

1. Observations on some Experiments in the 4th part
of the 2d book of Newton's Optics: .an. 1755. — 2. Ob-
servations on the Platform fordividingmathematical instru-
ments: 1765. — 3. Determination of the distance of Arc-
turus from the sun's limh, at the summer solstice : 1765.
— 4. On some means of perfecting astronomical instru-
ments : 1765. — 5. On some experiments relative to diop-
trics : 1767. — 6. The art of dividing mathematical instru-
ments : 1768 — 7. Observations on the Transit of Venus,
.lune 3, 176£). — 8. New method of dividing mathematical
and astronomical instruments.

CHAUSE TRAPPES, or Caltrops, or Crowsfeet, are
iron instruments of spikes about 4 inches long, made like
a star, in such a manner that whichever way they fall, one
point stands always upwards, like a nail. They are usu-
ally thrown and scattered into moats and breaches, to

gall the horses' feet, and stop the hasty approaches of the
enemy.

CHAZELLES (John-Matthew), a French mathe-
matician and engineer, was born at Lyons in 1657, and
educated there in the college of Jesuits, from whence he
removed to Paris in 1675. He first became acquainted
with Du Hamel, secretary to the Academy of Sciences,
and through him with Cassini, who employed him with
himself at the Observatory, where Chazelles greatly im-
proved himself, and also assisted Cassini in the measure-
ment of the southern part of the meridian of France. Ha-
ving, in l6S4, instructed the Duke of Montemar in the ma-
thematical sciences, this nobleman procured him the ap-
pointment of hydrography-professor to the galleys of Mar-
seilles. In discharging the duties of this department, he
made numerous geometrical and astronomical observations,
from which he drew a new map of the coast of Provence.
— He also performed many other services in that depart-
ment, and as an engineer, along with the armies and naval
expeditions. To make observations in geography and
astronomy, he undertook also a voyage to the Levant, and
among other things he measured the pyramids of Egypt,
and found the four sides of the largest of them exactly to
face the four cardinal points of the compass. He made
a report of his voyage, on his return, to the Academy of
Sciences, upon which he was named a member of their
body in 1695, and had many papers inserted in the vo-
lumes of their Memoirs, from 1603 to 1708. Chazelles
died at Marseilles the lfjth of January 1710.

CHEMIN des Ronds, in Fortification, the way of the
rounds, or a space between the rampart and the low para-
pet under it, for the rounds to go about it.

CHEMISE, a wall that lines a bastion, or ditch, or the
like, for its greater support and strength.

CHEVAL de Frise, pi. Chevaux de Frise, or Friseland
Horse, so called because it was first used in that country.
It consists of a joist or piece of timber, about a foot in
diameter, and 10 or 12 long, pierced and transversed with
a great number of wooden spikes of 5 or 6 feet in length,
and armed or pointed with iron. It is also called turnpike,
or tourniquet, and is chiefly used to stop a breach, defend
a passage, or make a retrenchment to stop the cavalry.

CHEVRETTE, in Artillery, an engine to raise guns or
mortars into their carriage. It is formed of two pieces of
wood of about 4 feet long, standing upright upon a third,
which is square. The uprights are about a foot asunder,
and pierced with holes exactly opposite to one another,
to receive a bolt of iron, which is put in cither higher or
lower at pleasure, to serve as a support to a handspike by
which the gun is raised up.

CHEYNE (George), a British physician and mathe-
matician, was born in Scotland, l6'71, and educated at
Edinburgh under Dr. Pitcairn. He passed his youth in
close study and great abstemiousness; but on coming to
London, at 30 years of age, lie suddenly changed his whole
manner of living; which had such an effect on his consti-
tution, that his body grew to an immense size, weighing
it is said 448 pounds. From this load of oppression how-
ever he was afterwards in a great measure relieved, by
means of a milk and vegetable diet, which reduced his
weight to about one-third of what it had been, and restored
him to a good stale of health ; by which his life was pro-
longed to the 72d year of his age.

Dr. Cheync was author of various medical and other
tracts, and of a treatise on the Inverse method of Flux-

CHO

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C H. R

ions, under the title of Fluxionum Methodus Inversa; sive
quantitatum fluentiura leges generaliores : in 4to, 1703.
Upon this book Demoivre wrote some animadversions in
an 8vo vol. 1704; which were replied to byCheynein 1705.
Besides this work, he published Philosophical Principles
of Religion, natural and revealed. Also the English Ma-
lady, or a Treatise on Nervous Diseases.

CHILIAD, an assemblage of several things ranged by
thousands. It was particularly applied to tables of loga-
rithms, because they were at first divided into thousands.
Thus, in the year 1624, Mr. Briggs published a table of
logarithms for 20 chiliads of absolute numbers; afterward,
he published 10 chiliads more; and lastly, one more;
making in all 31 chiliads.

CHILIAEDRON, a solid figure having a thousand faces.

CHILIAGON, a regular plane figure of a thousand sides
and angles.

CHORD, a right line connecting the two extremes of
an arch ; so called from its resem-
blance to the chord or string of a
bow; as ab, which is common to
the two parts or arches adb, aeb
that make up the whole circle. The
chords have several properties:

1. The chord is bisected by a
perpendicular cf drawn to it from
the centre. — 2. Chords of equal
arcs, in the same or equal circles,

are equal to each other. — 3. Unequal chords have to one
another a less ratio than that of their arcs. — 4. The chord
of an arc, is a mean proportional between the diameter
and the versed sine of that arc.

Scale or Line of Chords. See Plane Scale.

Chord, or Cord, in Music, denotes the string or line
by whose vibrations the sensation of sound is excited; and
by whose divisions the several degrees of tune are deter-
mined. Chords of gold wire in harpsichords, yield a sound
almost twice as strong as those of brass ; while chords of
steel impart a feebler sound than those of brass.
To divide a Chord ab in the most simple manner, so as to
exhibit all the original concords.

Divide the given line into two equal parts at c; then sub-
divide the part cb into two equal parts at d, and again the
part CD into two equal parts at e. Here a c to a b is an
octave; AC to ad a fifth ; ad to ab a fourth ; ac to ae
a greater third, and ae to ad a. less third; ae to eb a
greater sixth, and ae to ab a less sixth. Malcolm's Trea-
tise of Music, ch. 6. sec. 3. See Monochord.

To find the number of V ibrations made by a Musical Chord
or Siring in a given time; having given its weight, length,
and tension. Let / be the length of the chord in feet, 1
its weight, or rather a small weight fixed to the middle
and equal to that of the whole chord, and w the tension,
or weight by which the chord is stretched. Then shall
the time of one vibration be expressed by

V V/^T7()»find consequently -^-v'-y- will be the number
of vibrations per second.

For example, suppose iv = 28800, or the tension equal
to 28800 times the weight of the chord, and the length of
it 3 feet; then the last theorem gives 354 nearly for the
number of vibrations made in each second of time.— But
if w were 14400, there would be- made but 250 vibrations

per second ; and if w were only 288, there would be no
more than 35i|- vibrations per second. See my Tracts,
vol. 3, prob. 21 & 22, pa. 240'.

On the supposition that chords or strings are perfectly
elastic, the following proportions have been deduced ma-
thematically. 1. If the length of the chord and the tend-
ing force be given, the inflecting force will be nearly as the
space through which the chord is bent. 2. If the length
of the string and the space through which it is bent be
given, the inflecting force is as the tension. 3. If the tend-
ing force, and the space through which the chord is bent
be given, the inflecting force is inversely as the length of
the chord. 4. If the diameter of the chord, and the ten-
sion be given, the times of vibration are as the length of
the chord : on which principle the monochord is founded.
5. If the tension and length of the chord be given, the time
of vibration is as the diameter of the chord. 6. If the di-
ameter, and length of the chord be given, the time of vi-
bration is inversely as the square root of the tension.

Chord, in Music, is also used for the union of two or
more sounds uttered at the same time, and forming toge-
ther a complete harmony.

Chords are divided into perfect and imperfect. In the
perfect chord we have the third, fifth, and octave of the
fundamental sound : they are also subdivided into major
and minor, according as the thirds which enter into their
composition are flat or sharp. Imperfect chords are those
in which the sixth, instead of the fifth, prevails ; and in
general all those whose lowest arc not their fundamental
sounds.

Chords are again divided into consonances and disso-
nances. The consonances are the perfect chord, and its
derivatives. Every other chord is a dissonance. A table
of both, according to the system of M. Rameau, may be
seen in Rousseau's Musical Dictionary, vol. 1, pa. 27-

CHROMATIC, a species of music which proceeds by
semitones and minor thirds. The word is derived from
the Greek ^wpa, which signifies colour, and perhaps the
shade or intermediate shades of colour, which mingle and
connect colours, like as the small intervals in this scale
easily slide or run into each other.

Boethius and Zarlin ascribe'the invention of the chro-
matic genus to Timotheus, a Milesian, in the time of Alex-
ander the Great. The Spartans banished k their city on
account of its softness. The character of this genus, ac-
cording to Aristides Quintillianus, was sweetness and
pathos.

CHROMATICS, is that part of optics which explains
the several properties of the colours of light, and of natu-
ral bodies.

Before the time of Newton, the notions concerning co-
lour were very vague and extravagant. The Pythagoreans,
called colour the superficies of bodies : Plato said that it
was a flame issuing from them : According to Zeno, it is
the first configuration of matter: And Aristotle said it was
that which made bodies actually transparent. Descartes-
accounted colour a modification of light, and he imagined
that the difference of colour proceeds from the prevalence
of the direct or rotatory motion of the particles of light.
Grimaldi, Dechales, and many others, conjectured that
the differences of colour depended on the quick or slow
vibrations of a certain elastic medium pervading the uni-
verse. Rohault conceived, that the different colours were
made by the rays of light entering the eye at different an-
gles with respect to the optic axis. And Dr. Hooke ima~

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C H R

gined that colour is caused by the sensation of the oblique
or uneven pulse of light ; which being capable of but two
varieties, he concluded there could be no more than two
primary colours.

Sir I. Newton, in the year 1 666, -began to investigate
this subject ; when finding that the coloured image of the
sun, formed by a glass prism, was of an oblong, and not
of a circular form, as, according to the laws of equal re-
fraction, it ought to be, he conjectured, that light is not ho-
mogcneal; but that it consists of rays of different colours,
and endued with divers degrees of refrangibility. And,
from a further prosecution of his experiments, he conclu-
ded that the different colours of bodies arise from their re-
flecting this or that kind of rays most copiously. This
method of accounting for the different colours of bodies
soon became generally adopted, and still continues to be
the most prevailing opinion. It is hence agreed that the
light of the sun, which to us appears white and perfectly
homogeneal, is composed of no less than seven different
colours, viz. red, orange, yellow, green, blue, purple, and
violet or indigo : that a body which appears of a red co-
lour, has the property of reflecting the red rays more pro-
fusely than the rest ; and so of the other colours, the orange,
yellow, green, &c: also that a body which appears black,
instead of reflecting, absorbs all or the most part of the
rays that fall upon it; while, on the contrary, a body
which appears white, reflects the greatest part of all the
rays indiscriminately, without separating them one from
another.

The foundation of a rational theory of colours being
thus laid, the next inquiry was, by what peculiar mecha-
nism, in the structure of each particular body, it was ren-
dered capable of reflecting one kind of rays more than ano-
ther ; and this is attributed, by Sir I. Newton, to the den-
sity of these bodies. Dr. Hooke had remarked, that thin
transparent substances, particularly soap-water blown into
bubbles, exhibited various colours, according to their thin-
ness ; and yet, when they have a considerable degree of
thickness, they appear colourless. And Sir Isaac himself
had also observed, on compressing two prisms hard toge-
ther, in order to make their sides (which happened to be
rather convex) to touch one another, that in the place of
contact they were both perfectly transparent, as if they
had been but one continued piece of glass : but round the
point of contact, where the glasses were a little separated
from each other, rings of different colours appeared. And
when he afterwards, further to elucidate this matter, cm-
ployed two convex glasses of telescopes, pressing their con-
vex sides upon each other, he observed several series of
circles or rings of such colours, different, and of various
intensities, according to their distance from the common
central pellucid point of contact.

As the colours were thus found to vary according to
the different distances between the glass plates, Sir Isaac
conceived that they proceeded from the different thickness
of the plate of air intercepted between the glasses; this
plate of air being, by the mere circumstance of thinness
or thickness, disposed to reflect or transmit the rays of this
or that particular colour. Hence he concluded, that the
colours of all natural bodies depend on their density, or
the magnitude of their component particles : and hence
also he constructed a table, in which the thickness of a
plate necessary to reflect any particular colour, was ex-
pressed in millionth parts of an inch.

From a great variety of such experiments, and observa-

tions upon them, our author deduced his theory of co-
lours. And from which it appears that every substance
in nature is transparent, provided it be made sufficiently
thin ; as gold, the densest substance we know of, when re-
duced into thin leaves, transmits a blueish-green light
through it. If we suppose any body therefore, as gold for
instance, to be divided into a vast number of plates, so
thin as to be almost perfectly transparent ; it is evident
that all, or the greatest part of the rays, will pass through
the upper plates, and when they lose their force will be
reflected from the under ones. They will then have the
same number of plates to pass through which they had
before penetrated; and thus, according to the number of
those plates through which they are obliged to pass, the
object appears of this or that colour, just as the rin^s of
colours appeared different in this experiment of the two
plates, according to their distance from one another, or
the thickness of the plate of air between them.

This theory of colours has been illustrated and con-
firmed by various experiments, made by other philoso-
phers. Mr.E. II. Delaval produced similar effects by the
infusion of flowers of different colours, and also by the in-
timate mixture of the various metals with glass, when they
are reduced to very fine parts ; the denser metals impart-
ing to the glass the less refrangible colours, and the lighter
ones those colours that are more easily refrangible. Dr.
Priestley and Mr. Canton also, by laying very thin leaves
or slips of the metals upon glass, ivory, wood, or metal,
and -passing an electrical stroke through them, found that
the same effect was produced, viz, that the substrated was
tinged with different colours, according to the distance
from the point of explosion.

However, the Abbe Mazeas and M. du Tour contend-
ed, that the colours between the glasses are not to be as-
cribed to the thin stratum of air, since they equally pro-
duced them by rubbing and pressing together two flat
glasses, which cohered so strongly that it required the
greatest force to move or slide them over each other. Sec
Priestley's History of Vision.

Of Newton's 8t/i Ejper. in the Id Hook of Optics.

The event of this experiment, which has been contra-
dicted by repetitions of the same by other philosophers,
having been the occasion of much controversy ; and rela-
ting to a material part of the doctrine of chromatics, it
will not be improper here to give an account of what has
passed concerning it. Newton found, that when light, by
contrary refractions through different mediums, is so cor-
rected, that it emerges in lines parallel to the incident rays,
it continues ever after t<6 be white. But that if the emer-
gent rays be inclined to the incident ones, the whiteness
of the emerging light will, by degrees, in passing on from
the place of emergence, become tinged at its edges with
colours. And these laws he inferred from experiments
made by refracting light with prisms of glass, placed with-
in a prismatic vessel of water.

By theorems deduced from this experiment he infers,
that the refraction of the rays of every sort, out of any
medium into air, are known by having the refraction of the
rays of any one sort : and also, that the refraction out of
one medium into another is found, whenever we have the
refractions out of them both, into any third medium.

Now the same experiment, having been since performed
by other persons, and turning out contrary to what is sta-
ted above, has occasioned rather free reflections to be
thrown on Newton concerning it; but which however ha\e

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been very satisfactorily obviated by Mr. Peter Dollond,
in a pamphlet on this subject; as we shall show below.

In the first place then, M. Klingenstierna, a Swedish
philosopher, having in the year 1755 considered the con-
troversy between Euler and Mr. John Dollond, relative to
the refraction of light, formed a theorem of his own, from
geometrical reasoning, by which he was induced to believe
that the result of Newton's experiment could not be as he
has represented ; except when the angles of the refracting
mediums are small. See the paper on this matter by Klin-
genstierna in the pamphlet above cited by Mr. Peter Dol-
lond.

This paper of Klingenstierna being communicated to
Mr. John Dollond by Mr. Mallet, to whom it was sent for
that purpose, made Dollond entertain doubts concerning
Newton's report, of the result of his experiment, and de-
termined him to have recourse to experiments of his own,
which he accordingly made in the year 1757< as follows.

Having cemented two glass planes together by their
edges, so as to form a, prismatic vessel, when closed at the
ends or bases; and the edge being turned downward, he
placed it in a glass prism with one of its edges upward,
filling up the vacancy with clear water; so that the refrac-
tion of the prism was contrived to be contrary to that of
the water, iu order that a ray of light, transmitted through
both these refracting mediums, might be affected by the
difference only between the two refractions. As he found
the water to refract more or less than the glass prism, he
diminished or augmented the angle between the glass plates,
till the two contrary refractions became equal, which he
discovered by viewing an object through this double prism.
For when it appeared neither raised nor depressed, he con-
cluded that the refractions were equal, and that the emer-
gent rays were parallel to the incident ones.

Now, according to the prevailing opinion, he observes,
that the object ought to have appeared through this
double prism in its natural colour ; for if the difference
of refrangibility had been in all respects equal, in the two
equal refractions, they would have rectified each other.
But this experiment fully proved the fallacy of the re-
ceived opinion, by showing that the divergency of the
light by the glass prism, was almost double of that by the
water; for the image of the object, though not at all re-
fracted, was yet as much infected with prismatic colours,
as if it had been seen through a glass wedge only, having
its angle near 30 degrees.

This experiment is the very same with that of New-
ton's above-mentioned, notwithstanding the result was so
strikingly different. Mr. Dollond plainly saw however,
that if the refracting angle of the water-vessel could have
admitted of a sufficient increase, the divergency of the
coloured rays would have been greatly diminished, or
entirely rectified ; and that there would have been a very
great refraction without colour, as he had already pro-
duced a great discolouring without refraction: but the
inconveniency of so large an angle as that of the prismatic
vessel must have been, to bring the light to an equal di-
vergency with that of the glass prism, whose angle was
about 60 degrees, rendered it necessary to try some expe-
riments of the same kind with smaller angles.

Accordingly he procured a wedge of plate-glass, whose
angle was only y degrees ; and, using it under the same
circumstances, he increased the angle of the water-wedge,
in which it was placed, till, the divergency of the light by

the water was equal to that by the glass; that is, till the
image of the object, though considerably refracted by the
excess of the refraction of the water, appeared neverthe-
less quite free from any colours proceeding from the dif-
ferent refrangibility of the light.

Many conjectures were made as to the cause of so great
a difference in the results of the same experiment; but
none that gave any great satisfaction; till it was after-
wards shown to be probably owing to the nature of the
glass then used by Newton. This conjecture is made by
Mr. Peter Dollond, son of John, the inventor of the achro-
matic telescope, in a pamphlet by him published in
defence of his father's invention, against the misrepre-
sentations of some persons who have unjustly attempted
to give the invention to other philosophers, who them-
selves never imagined that they had any just claim to it.
After a full and satisfactory vindication of his father, Mr.
P. Dollond then adds,

" I now come to a more agreeable part of this paper,
which is, to endeavour to reconcile the different results of
the Sth experiment of the 2d part of the 1st book of New-
ton's Optics, as related by himself, and as it was found by
Dollond, when he tried the same experiment, j Sn the year
1757. Newton observes, that light, as often \as by con-
trary refractions it is so corrected, that it emergeth in
lines parallel to the incident, continues ever after to be
white. Now Dollond says, when he tried the same ex-
periment, and made the mean refraction of the water equal
to that of the glass prism, so that the light emerged in
lines parallel to the incident, he found the. divergency of
the light by the glass prism to be netlrly double to what
it was by the water prism. The light appeared to be so
evidently coloured, that it was directly said by some per-
sons who were present, that if Newton had actually made
the experiment, he must have perceived it to have been so.
Yet who could for a moment doubt the veracity of New-
ton ? or feel indifferent in vindicating if possible the as-
sertion of such a character? Many conjectures were there-
fore made by different persons. Mr. Murdoch in par-
ticular gave a paper to the Royal Society in defence of
Newton ; but it was such as very little tended to clear
up the matter. Philos. Trans, vol. 53, p. 192. — Some
have supposed that Newton made use of water strongly im-
pregnated with saccharum saturni, because he mentions
sometimes using such water, to increase the refraction,
when he used water prisms instead of glass prisms. New-
ton's Opt. p. 62. — And others have thought that he tried
the experiment. with so strong a persuasion in his own
mind, that the divergency of the colours was always in the
same proportion to the mean refraction, in all sorts of
refracting mediums, that he did not attend so much to
that experiment as he ought to have done, or as he usually
did. None of these conjectures having appeared at all
satisfactory, I have therefore endeavoured to discover the
true cause of the difference, and thereby show, how the
experiment may be made to agree with Newton's de-
scription of it, and to get rid of those doubts, which have
hitherto remained to be cleared up.

" it is well known, that in Newton's time the English
were not the most famous for making optical instruments:
telescopes, opera-glasses, &c, were imported from Italy
in great numbers, and particularly from Venice; where
they manufactured a kind of glass which was much more
proper for optical purposes than any made in England at

C H R

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CUR

that time. The glass made at Venice was nearly of the
same refractive quality as our own crown-glass, but of a
much better colour, being sufficiently clear and transpa-
rent for the purpose of prisms. It is probable that New-
ton's prisms were made with this kind of glass ; and it
appears to be the more so, because he mentions the spe-
cific gravity of common glass to be to water as 2'58 to 1,
Newton's Opt. p. 247, which nearly answers to the specific
gravity we find the Venetian glass generally to have. Hav-
ing a very thick plate of this kind of glass, which was pre-
sented to me about 25 years ago by the late Professor
Allemand, of Ley den, and which he then informed me had
been made many years ; I cut a piece from this plate of
glass to form a prism, which I conceived would be similar
to those made use of by NevUon himself. I have tried
the Newtonian experiment with this prism, and find it
answers so nearly to what Newton relates, that the dif-
ference which remains may very easily be supposed to
arise from any little difference which may and does often
happen in the same kind of glass made at the same place
at different times. Now the glass prism made use of by
Dollond to try the same experiment, was made of English
flint-glass, the specific gravity of which I have never known
to be less than 322. This difference in the densities of
the prisms, used by Newton and Dollond, was sufficient to
cause all the difference which appeared to the two expe-
rimenters in trying the same experiment.

" From this it appears, that Newton was accurate in
this experiment as in all others, and that his not having
discovered that which was discovered by Dollond so many
years afterwards, wai owing entirely to accident ; for if his
prism had been made of glass of a greater or less density,
he would certainly have then made the discovery, and re-
fracting telescopes would not have remained so long in their
original imperfect state." See Achromatic, and Tele-
scope.

Mr. Delaval's experiment on the colours of opaque bodies. —
Besides the experiments of this gentleman, before-men-
tioned, on the colours of transparent bodies, he has also
published an account of some made upon the permanent
colours of opaque substances, the discovery of which must
be of the utmost consequence in the arts of colour-making
and dyeing.

The changes of colour in permanently coloured bodies,
our author observes, are produced by the same laws that
take place in transparent colourless substances ; and the
experiments by which they are investigated consist chiefly
of various methods of uniting the colouring particles into
larger masses, or dividing them into smaller ones. Sir
Isaac Newton made his experiments chiefly on transparent
substances ; and in the few places where he treats of others,
he acknowledges his want of experiments. He makes the
following remark however on those bodies which reflect
one kind of light and transmit another, viz, that if these
glasses or liquors were so thick and massy that no light
could get through them, he questioned whether they would
not, like other opaque bodies, appear of one and the same,
colour in all positions of the eye; though he could not
yet affirm it from experience. Indeed it was the opinion
of this great philosopher, that all coloured matter reflects
the rays of light, some reflecting the more refrangible rays
most copiously, and others those that are less so ; and that
tins is at once the true and only reason of these colours.
He was likewise of opinion that opaque bodies reflect the

light from their anterior surface, by some power of the
body evenly diffused over and external thereto. With re-
spect to transparent coloured bodies, he thus expresses
himself: "A transparent body which looks of any colour
by transmitted light, may also appear of the same colour
by reflected light; the light of that colour being reflected
by the farther surface of that body, or by the air beyond
it : and then the reflected colour will be diminished, and
perhaps cease, by making the body very thick, and pitch-
ing it on the back-side, to diminish the reflection of its
farther surface, so that the light reflected from the tingeing
particles may predominate. In such cases the colour of
the reflected light will be apt to vary from that of the
light transmitted."

To investigate the truth of these opinions, Mr. Delaval
entered upon a course of experiments with transparent
coloured liquors and glasses, as well as with opaque and
semitransparent bodies. And from these experiments he
discovered several remarkable properties of the colouring
matter ; particularly, that in transparent coloured sub-
stances it does not reflect any light; and when, by inter-
cepting the light which was transmitted, it is hindered
from passing through such substances, they do not vary
from their former colour to any other, but become en-
tirely black.

This incapacity of the colouring particles of transparent
bodies to reflect light, being deduced from very numerous
experiments, may therefore be taken as a general law.
It will appear the more extensive, if it be considered that,
for the most part, the tingeing particles of liquors, or other
transparent substances, are extracted from opaque bo-
dies; that these bodies owe their colours to those particles,
in the same manner as the transparent substances ; and
that by the loss of them they are also deprived of their
colours.

Notwithstanding these and many other experiments, the
theory of colours is not yet determined with certainty ; and
it must be acknowledged that very strong objections might
be brought against every hypothesis on this subject that has
been invented. The discoveries of Newton however are
sufficient to justify the following Aphorisms.

Aphorism 1. All the colours in nature arise from the
rays of light.

2. There are seven primary colours, namely red, orange,
yellow, green, blue, indigo, and violet.

3. Every ray of light may be separated into these seven
primary colours.

4. The rays of light, in passing through the same me-
dium, have different degrees of rcfrangibility.

5. The difference in the colours of light arises from its
different rcfrangibility: that which is the least refrangible
producing red; and that which is the most refrangible,
violet.

6. By compounding any two of the primary, as red and
yellow, or yellow and blue, the intermediate colour, orange
or green, may be produced.

7. The colours of bodies arise from their dispositions to
reflect one sort of rays, and to absorb the others: those
that reflect the least refrangible rays appearing red ; and
those that reflect the most refrangible, violet.

8. Such bodies as reflect two or more sorts of rays, ap-
pear of various colours.

9. The whiteness of bodies arises from their disposition
to reflect all the rays of light promiscuously.

C H R [3

10. The blackness of bodies proceeds from their inca-
pacity to reflect any of the rays of light. — And from theii
thus absorbing all the rays of light that are thrown upon
them, it arises, that black bodies when exposed to the sun,
become hot sooner than all others.

Of the Diatonic Scale of Colours. — Newton, in the course
of his investigations of the properties of light, discovered
that the lengths of the spaces occupied in the spectrum by
tha seven primary colours, exactly correspond to the
lengths of chords that sound the seven notes in the dia-
tonic scale of music : which is made evident by the fol-
lowing experiment.

B a b r d r S Q

Ai

;d

F g h ■ ■ i * I »' E

1 » 3 » 3. S3. l

2 T^ T T * t 9 - .±

On a paper, in a dark chamber, let a ray of light be
largely refracted into the spectrum abcdef, marking upon
it the precise boundaries of the several colours, as a, b, c,
&c; and across the spectrum draw the perpendicular
lines ag, bh, &c. Then it will be found that the spaces,
by which the several colours are bounded, viz, BagF con-
taining the red, abhg containing the orange, bcik contain-
ing the yellow, &c, will be in exact proportion to the di-
visions of a musical chord for the notes of an octave ; that
is, as the intervals of these numbers 1 , -| , |-, \ , f , | , T^., \.

CHRONOLOGICAL, belonging to chronology.

Chronological Characters, are characters by which
times are distinguished. Of these, some are natural, or
astronomical ; others are artificial or historical. The na-
tural characters are such as depend on the motions of the
stars or luminaries; as eclipses, solstices, equinoxes, the
different aspects of planets, &c. And the artificial cha-
racters are those that have been invented and established
by men ; as the solar cycle, the lunar cycle, &c. Histo-
rical Chronological Characters are those supported by the
testimonies of historians, when they fix the dates of cer-
tain events to certain periods.

CHRONOLOGY, the art of measuring and distinguish-
ing time; with the doctrine of dates, epochs, eras, &c.
The measurement of time in the most early periods, was
by means of the seasons, or the revolutions of the sun and
moon. The succession of Juno's priestesses at Argos served
Hellanicus for the regulation of his narrative ; while Epho-
rus reckoned his matters by generations. Even in the his-
tories of Herodotus and Thucydides, there are no regular
dates for the events recorded ; nor were there any endea-
vours to establish a fixed era until the time of Ptolemy
Philadelphus, who attempted it by compaiingand correct-
ing the dates of the olympiads, the kings of Sparta, and
the succession of the priestesses of Juno at Argos. Era-
tosthenes and Apollodorus digested the events related by
them, according to the succession of the olympiads and of
the Spartan kings.

The chronology of the Latins is still more uncertain.
The records of the Romans were destroyed by the Gauls;
and Fabius Pictor, the most ancient of their historians,
was obliged to borrow the chief part of his information
from the Greeks. In other European nations the chrono-
logy is still more imperfect, and of a later date : and even
in modern times a considerable degree of confusion and
inaccuracy has arisen, from the want of attention in the

Vol. I.

13 ] C H R

historians to ascertain the dates and epochs with preci-
sion.

Hence it is evident, how necessary a proper system of
chronology must be for the right understanding of history,
and also how difficult it must be to establish such a sys-
tem. For this purpose, however, several learned men have
spent much time, particularly Julius Africanus, Eusebius
ofCa?sarea, George Cyncelle, John of Antioch, Dennis,
Petau, Clavius, Calvisius, Scaliger, Vieta, Newton, Usher,
Simson, Marsham, Helvicus, Vossius, Strauchius, Blair,
Playfair, and Hales.

A system of this kind is founded, 1st, On Astronomical
observations, especially of the eclipses of the sun and moon,
combined with calculations of the years and eras of dif-
ferent nations. 2d, The testimonies of credible authors.
3d, Such epochs in history as are so well attested and de-
termined, that they have never been controverted. 4th,
Ancient medals, coins, monuments, and inscriptions.

The most obvious division of time, is derived from the
apparent or real revolutions of the luminaries, the sun
and moon. Thus, the apparent revolution of the sun, or
the real rotation of the earth on her axis causing the sun
to appear to rise and set, constitutes the vicissitudes of
day and night, which must be evident to the most' bar-
barous and ignorant nations. The moon, by her revolu-
tion about the earth, and her changes, as naturally and ob-
viously forms months ; while the great annual course of
the sun through the several constellations of the zodiac,
points out the larger division of the year.

The Day is-divided into hours, minutes, &c; while the
month is divided into weeks, and the year into months,
having particular names, and a certain number of days. —
See a particular account of each of these under the respec-
tive words.

Besides the natural divisions of time arising immedi-
ately from the revolutions of the heavenly bodies, there are
others which are formed from some of the less obvious
consequences of these revolutions, and are called Cycles,
or circles. The most remarkable of these are, 1, The
Solar Cycle, or cycle of the sun, a period or revolution of
28 years, in which time the days of the months return to
the same days of the week, the sun's place to the same
signs and degrees of the ecliptic on the same months and
days, and the leap-years begin the same course over again
with respect to the days of the week on which the days of
the months fall. 2, The Lunar Cycle, or cycle of the
moon, commonly called the Golden Number, is a revolu-
tion of 19 years; in which time the conjunctions, opposi-
tions, and other aspects of the moon, are on the same days
of the months as they were 19 years before,and within an
hour and a half of the time of the day.

The Indiction, or Roman lndiction, is a period of 15
years, used only by the Romans for indicating the times of
certain payments made by the subjects to the republic.

The Cycle of Easter, called also the Dionysian Period, is
a revolution of 532 years, and is produced by multiplying
the solar cycle 28, by the lunar cycle 19-

The Julian Period, is a revolution of 7980 years, and is
produced from the continual multiplication of the three
numbers 28, 19, 15, of the three former cycles, viz, the
solar, lunar, and indiction.

As there are certain fixed points in the heavens, from

which astronomers begin their computations, so there are

certain points of time, from which historians begin to

reckon; and these points or roots of time are called Eras

2S

C H R

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C H R

or Epochs. The most remarkable of these are, those of
the Creation, the Greek Olympiads, the building of Rome,
the era of Nabonnassar, the death of Alexander, thebirth
of Christ, the Arabian Hegira, or flight of Mahomet, and
the Persian Jesdegird. All of which, with some others of
less note, have their beginnings fixed by chronologers to
the years of the Julian period, to the age of the world, and
to the years before and after the birth of Christ.

The testimony of authors is the second principal part
of historical chronology. Though no man has a right to
be considered as infallible, it would however be making a
very unfair judgment of mankind, to treat them all as
dupes or impostors; and it would be an injury offered to
public integrity, to doubt the veracity of authors univer-
sally esteemed, and facts that are truly worthy of belief.
When the historian is allowed to be completely able to
judge of an event, and to have no intent of deceiving by
his relation, his testimony cannot be refused.

The Epochs form the 3d principal part of chronology ;
being those fixed points in history that have never been
contested, and of which there cannot reasonably be any
doubt. Notwithstanding that chronologers fix upon the
events which are to serve as epochs, in a manner quite ar-
bitrary ; yet this is of little consequence, provided the
dates of these epochs agree, and that there is no contra-
diction in the facts themselves.

Medals, Monuments, and Inscriptions, form the last of
the four principal parts of chronology; and this study is
but of very modern date, scarcely more than 200 years
having elapsed since close application has been made to
the study of these. To the celebrated Spanheim we owe
the greatest obligations, for the progress that is made in
this method ; and it is by the aid of medals that M. Vail-
lant has composed his judicious history of the kings of
Syria, from the time of Alexanderthe Great to thatofPom-
pey. Nor have they been of less service in elucidating all
ancient history, especially that of the Romans ; and even
sometimes that of the middle ages.

Besides the foregoing general account, there are some
few systems of chronology which may deserve some more
particular notice, as follows.

Sacred Chronology. There have been various systems
relating to sacred chronology ; which is not to be wondered
at, as the three chief copies of the Bible give a very dif-
ferent account of the first ages of the world. For while
the Hebrew text reckons about 4000 years from the crea-
tion to the birth of Christ, and to the flood 1656 years ;
the Samaritan makes the former much longer, though it
reckons from the creation to the flood only 1307 years;
and the Septuagint removes the creation of the world to
6000 years before Christ, and 2250 years before the flood.
Many attempts have been made to reconcile these diffe-
rences; though none of them are quite satisfactory. Wal-
ton and Vossius give the preference to the account of the
Septuagint; while others have defended the Hebrew text.
See an abstract of the different opinions of learned men on
this subject, in Strauchius's Brev. Chron. translated by
Sault, p. l66and 176.

The Chinese Chronology. No nation has boasted more
of its antiquity than the Chinese : but though they be al-
lowed to trace their origin as far back as the deluge, they
have few if any authentic records of their history for so
long a period as 500 years before the Christian era. This
indeed may be owing to the general destruction of ancient
remains by the tyrant Tsin-chi-hoang, in the year 213, or

some say 246, before Christ. From a chronology of the
Chinese history (for which we are obliged to an illustrious
Tartar who was viceroy of Canton in the year 1724, and
of which a Latin translation was published at Rome in
1730), we learn that the most remote epoch of theChinese
chronology does not surpass the first year of Guel-lie-
wang, or 424 years before our vulgar era. And this
opinion is confirmed by the practice of two of the most
approved historians of China, who admit nothing into their
histories previous to this period.

The Chinese, in their computation, make use of a cycle
of 60 years, called kia-tse, from the name given to the
first year of it, which serves as the basis of their whole
chronology. Every year of this cycle is marked with two
letters which distinguish it from the others; and all the
times of the emperors, for upwards of 2000 years, have
names in history common to them with the corresponding
years of the cycle. Philos. Trans. Abridg. vol. 9, pa. 343.

According to M. Freret, in his Essays, the Chinese date
the epocha of Yao, one of their first emperors, about the
year 2145, or 2057, before Christ; and they reckon that
their first astronomical observations, and the composition
of their calendar, preceded Yao 150 years : whence it is
inferred that the era of their astronomical observations
coincides with that of the Chaldeans. But later authors
date the rise and progress of the sciences in China from
the grand dynasty of Tcheou, about 1200 j'ears before
the Christian era, and show that all historical relations of
events prior to the reign of Yao are fabulous. Mem. de
l'Histoire des Sciences, &c. Chinois. vol. 1, Paris 1776.

The Hindu Chronology. The Hindus employ, in their
chronology, certain large periods, called Yoogs or Yergs.
These are 4 in number, the last of which, called cali yug,
or kalee yug, is only now attended to, and began 3101 be-
fore the Christian era. It comprehends within it the fol-
lowing, viz,
The era of Bikramajit, which began 3045 years before Christ,

Salaban, 78 years after Christ,

Bengal, not strictly Hindu, cycle of 60 years :

the present began ISO? of Christ.

Their solar-sidereal year contains 365d 6h 12m 30s, and
their tropical year 36'5d 5h 50m 353. They divide the
zodiac into 2S lunar, and into 12 solar constellations or
signs ; and their astronomical year commences when the
sun arrives at the first point of Aries. The length of their
months is determined by the time employed by the sun in
passing through each sign, and they are accordingly longer
in the apogee, and shorter in the perigee ; that which cor-
responds with the higher apsis being 31d 14h 3*)m, and that
with the lower 29a 8" 21m. For civil purposes, they count
the days by the age of the moon that happens to fall within
each month, and often from the simple phase of the moon.

Their fasts and festivals, like those of the Jews and Chris-
tians, being regulated by the lunar revolutions, they there-
fore employ a lunar year. This they make to consist of
12 months, of 30 lunar days, or each semi-lunation of 15
days, which are somewhat shorter than the natural day.
The commencement of the solar day is counted from sun-
rise, and the interval till the next sun-rise, is divided into
60 equal parts, each equal to 24 of our minutes. The
subdivisions also follow the sexagesimal scale.

The days of the week arc denominated from the planets,
as with the Europeans, and their arrangement is the same
also, proceeding from the sun and moon, to Mars, Mer-
cury, Jupiter, Venus, and Saturn, Friday, or Vcnus's

CHR

C 315 ]

CHR

day, appears as the first of the week in their calculations,
and probably because the cali yug began on that day; but
in common, the week is considered by the Hindus as be-
ginning with Sunday. From the view of their astronomy
and chronology, they appear to be quite astrological; and
they further appear to be the parents or originals of those
of the Europeans and other nations.

For more particular information, see the Philos. Trans.
Abridg. vol. 16, p. 742, and vol. 17, p. 250.

Babylonian, Egyptian, and Chaldean Annals. These, M.
Gibert has attempted to reduce to our chronology, in a
letter published at Amsterdam in 1743. He begins with
showing, by the authorities of Macrobius, Eudoxus, Varro,
Diodorus Siculus, Pliny, Plutarch, St. Augustin, &c,
that by a year, the ancients meant the revolution of any
planet in the heavens; so that it might consist sometimes
of only one day. Thus, according to him, the solar day
was the astronomical year of the Chaldeans ; and so the
boasted period of 473,000 years, assigned to their obser-
vations, is reduced to 1297 years 9 months; the number
of years which, according to Eusebius, elapsed from the
first discoveries of Atlas in astronomy, in the 384th year
of Abraham, to the march of Alexander into Asia in the
year 1682 of the same era. And the 17,000 years added
by Berosus to the observations of the Chaldeans, reduced
in the same manner, will give 46 years and 6 or 7 months;
being the exact interval between Alexander's march, and
the first year of the 123d Olympiad, or the time to which
Berosus carried his history.

Epigenius ascribes 720,000 years to the observations
preserved at Babylon ; but these, according to M. Gibert's
system, amount only to 1971 years 3 months ; which
differ from Callisthenes's period of 1903 years, allotted
to the same observations, only by 68 years, the period
elapsed from the taking of Babylon by Alexander, which
terminated the latter account, and to the time of Pto-
lemy Philadelphus, to which Epigenius extended his ac-
count.

The Newtonian Principles of Chronology. — Sir Isaac New-
ton has shown, that the chronology of ancient kingdoms is
involved in the greatest uncertainty : that the Europeans
in particular had no chronology before the Persian em-
pire, which commenced 536 years before the birth of
Christ, when Cyrus conquered Darius theMede : that the
antiquities of the Greeks are full of fables, because their
writings were in verse only, till the conquest of Asia by
Cyrus the Persian; about which time prose was intro-
duced by Pherecides Syrius and Cadmus Milesius. After
this time several of the Greek historians introduced the
computation by generations. The chronology of the Latins
was still more uncertain : their old records having been
burnt by the Gauls 120 years after the expulsion of their
kings, or 64 years before the death of Alexander the Great,
answering to 3S8 before the birth of Christ. The chrono-
logers of Gaul, Spain, Germany, Scythia, Sweden, Britain,
and Ireland, are of a still later date. For Scythia, beyond
the Danube, had no letters till Ulphilas, their bishop,
formed them, about the year 276. Germany had none till
it received them from the western empire of the Latins,
about the year 400. The Huns had none in the days of
Procopius, about the year 526. And Sweden and Norway
received them still later.

Newton, after a general account of the obscurity and
delicts of the ancient chronology, observes that, though
many of the ancients computed by successions and gene-

rations, yet the Egyptians, Greeks, and Latins, reckoned
the reigns of kings equal to generations of men, and three
of them to a hundred, and sometimes to 120 years; and
this was the foundation of their technical chronology. He
then proceeds from the ordinary course of nature, and a
detail of historical facts, to show the difference between
reigns and generations ; and that, though a generation from
father to son may at an average be reckoned about 33
years, or three of them equal to J 00 years, yet, when they
are taken by the eldest sons, three of them cannot be esti-
mated at more than about 75 or 80 years ; and the reigns
of kings are still shorter; so that 18 or 20 years may be
allowed a just medium. Sir Isaac then fixes on four re-
markable periods, viz, the return of the Heraclidse into
the Peloponnesus, the taking of Troy, the Argonautic ex-
pedition, and the return of Sesostris into Egypt, after his
wars in Thrace ; and he settles the epoch of each by the
true value of a generation. To instance only his estimate
of that of the Argonautic expedition : Having fixed the
return of the Heraclida3 to about the 159th year after the
death of Solomon, and the destruction of Troy to about
the 76th year after the same period, he observes, that Her-
cules the Argonaut was the father of Hyllus, the father of
Clerdius, the father of Aristomachus, and the father of
Aristodemus, who conducted the Heraclidae into Pelopon-
nesus; so that reckoning by the chief of the family, their
return was four generations later than the Argonautic ex-
pedition, which therefore happened about 43 years after
the death of Solomon. This is further confirmed by an-
other argument : y£sculapius and Hercules were Argo-
nauts: Hippocrates was the 18th inclusively from the for-
mer by the father's side, and the 19th from the latter by
the mother's side: now, allowing 28 or 30 years to each
of them, the 17 intervals by the father, and the 18 inter-
vals by the mother, will on a medium give 507 years; and
these, reckoning back from the commencement of the Pe-
loponnesian war, or the 431st year before Christ, when
Hippocrates began to flourish, will place the Argonautic
expedition in the 43d year after the death of Solomon, or
937 years before Christ.

The other kind of reasoning, by which Newton endea-
vours to establish this epoch, is purely astronomical. The
sphere was formed by Chiron and Musaeus, for the use of
the Argonautic expedition, as is plainly shown by several
of the asterisms referring to that event: and at the time of
the expedition the cardinal points of the equinoxes and
solstices were placed in the middle of the constellations
Aries, Cancer, Chela;, and Capricorn. This point is esta-
blished by Newton from the consideration of the ancient
Greek calendar, which consisted of 12 lunar months, and
each month of 30 days, which required an intercalary
month. Of course this luni-solar year with the intercalary
month, began sometimes a week or two before or after
the equinox or solstice; and hence the first astronomers
were led to the before-mentioned disposition of the equi-
noxes and solstices : and that this was really the case, is
confirmed by the testimonies of Eudoxus, Aratus, and
Hipparchus. Upon these principles Sir Isaac proceeds to
argue in the following manner. In the end of the year
l6'89 the star called the Prima Arietis, was in V 28° 51',
with north latitude 7° 8' 58"; and the star called the
Ultima cauda; Arietis was in 8 19° 3' 42", with north la-
titude 2° 34' 5" ; consequently the equinoctial eolure at
this time cut the ecliptic in 8 6° 44'; and by this he
reckons that the equinox had then gone back 36° 44' since
2S2

CHR [ 316 ]

the time of the Argonautic expedition. But it recedes 50'
in a year, or 1° in 72 years, and consequently 36° 44' in
2645 years ; and this, counted backwards from the begin-
ning of 1690, will place this expedition about 25 years
after the death of Solomon. But as there is no necessity
for allowing that the middle of the constellations, accord-
ing to the general account of the ancients, should be pre-
cisely the middle between the prima Arietis and ultima
Cauda?, our author proceeds to " examine what were those
stars through which Eudoxus made the colures to pass in
the primitive sphere, and in this way to fix the position of
the cardinal points." Now from the mean of five places
lie finds, that the great circle, which in the primitive
sphere, described by Eudoxus, or which at the time of
.the Argonautic expedition was the equinoctial colure, did
in the end of l6$9 cut, the ecliptic in q 6° 29' 15". In
the same manner our author determines that the mean
place of the solstitial colure is SI 6° 28' 46", and as it is at
right angles with the other, he concludes that it is rightly
drawn. And hence he infers that the cardinal points, in
the interval between that expedition and the year l689,
have receded from those colures-11 6° 29'; which, allow-
ing 72 years to a degree, amounts to 2627 years; and
these counted backwards, as above, will place the Argo-
nautic expedition 43 years after the death of Solomon.
Our author has, by other methods also of a similar nature,
established this epoch, and reduced the age of the world
500 years.

This elaborate system has not escaped censure; Mess.
Freret and Souciet having both attacked it, and on much
the same ground : but the former has confounded "reigns
and generations, which are carefully distinguished in this
system. The astronomical objections of both have been
answered by Sir Isaac Newton himself, and by Dr. Halley.
Philos. Trans, abr. vol. 7, pa. 172, 19 1 ; Newton's Chro-
nol. ch. 1.

CHRONOMETER, is any instrument or machine used
in measuring time ; such as dials, clocks, watches, &c.

The term is however more particularly used for a kind
of clock, so contrived as to measure a small portion of
time, even to the l6th, or the 40th part of a second ; one
of this latter kind I have seen, made by an ingenious ar-
tist; but it could not be stopped to the 10th part of the
proposed degree of accuracy. There is a description of
one also in Desagulier's Experimental Philosophy, invent-
ed by the late ingenious Mr. George Graham ; which
might be of great use for measuring small portions of time
in astronomical observations, the time of the fall of bodies,
the velocity of running waters, &c. But long intervals
of time cannot be measured by it with sufficient exact-
ness, unless its pendulum be made to vibrate in a cycloid;
for otherwise it is liable to err considerably, as is the case
of all clocks with short pendulums that swing in large
arches of the circle.

Various other contrivances, besides clocks, have been
used for measuring time for some particular purposes.
See a musical chronometer described in Malcolm's Trea-
tise of Music, pa. 407. The excellent watches that have
lately been made, for the purpose of finding the longitude,
by the ingenious artists, Harrison, Arnold, Mudge, Emery,
Hardy, &c, are sometimes called chronometers, also time-
keepers.

CHRONOSCOPE, a word sometimes used" for a pen-
dulum, or machine, to measure time.

CHRYSTALLINE. See Crystalline.

C I R

CIMA, or Sim a, in Architecture, a member, or mould-
ing, called also ogee, and cimatium.

CINCTURE, in Architecture, is a ring or list around
the shaft of a column, at its top and bottom.

CINTRE, in Building, the same as Centre; which see.
CIPHER, or Cypher, one of the numeral characters, or
figures, thus formed 0. The word comes from the Hebrew
saphar, to number.

' The cipher of itself signifies nothing, or implies a pri-
vation of value; but when combined with other numeral
characters, it alters their value in a tenfold proportion,
for every cipher so annexed; viz, when set after a figure
in common integral arithmetic, it increases its value in
that proportion, though it has no effect when placed be-
fore or to the left-hand side of such figures ; but on the
contrary, in decihial arithmetic, it decreases their value in
that proportion when set before the figures, yet makes no
alteration in their value when placed after them.
Thus, 5 is five,
but 50 is fifty,
and 500 is five hundred ;
whereas 05, or 005, &c, is still but 5 or five.
Also -5 is five tenths,
but '05 is five hundredths,
and "005 is five thousandths ;
whereas -50, or '500, &c, is still but '5 or five tenths.

The invention and use of the cipher, as in the common
arithmetic and notation of numbers, is one of the happiest
devices that can be imagined ; and is ascribed to the In-
dians, by the Arabians, by whom it was made known to
Europe at the revival of literature.

CIRCLE., a plane figure, bounded by a curve line which
returns into itself, called its circumference, and which is
every where equally distant from a certain point within,
called its centre. The circumference or periphery itself
is sometimes called the circle, but improperly, as that
name denotes the space contained within the circumfer-
ence.— A circle is described with a pair of compasses,
fixing one foot in the centre, and turning the other round
to trace out the circumference.

The circumference of every circle is supposed to be
divided into 360 equal parts, called degrees, and marked
°; each degree into 60 minutes or primes, marked ' ; each
minute into 6'0 seconds, marked"; and so on. So 24° 12'
15" 20'", is 24 degrees 12 minutes 15 seconds and 20
thirds.

Circles have many curious properties, some of the most
important of which are the following.

1. The circle is the most capacious of all plain figures,
or contains the greatest area within the same perimeter,
or has the least perimeter about the'same area; being the
limit and last of all regular polygons, having the number
of its sides infinite.

2. The area of a circle is always less than the area of
any regular polygon circumscribed about it, and its cir-
cumference always less than the perimeter of the polygon.
But on the other hand, its area is always greater than that
of its inscribed polygon, and its circumference greater
than the perimeter of the said inscribed polygon. How-
ever/ the area and perimeter of the circle approach al-
ways nearer and nearer to those of the two polygons, as
the number of the sides of these is greater ; the circle being
always limited between the two polygons.

3. The area of a circle is equal to that of a triangle
whose base is equal to the circumference, and perpendi-

C I R

[ 317 ]

C I R

* ular equal to the radius. And therefore the area of the
circle is found by multiplying half the circumference by
half the diameter, or the whole circumference by the whole
diameter, and taking the 4th part of the product; as de-
monstrated by Archimedes.

4. Circles, like other similar plane figures, are to one
another, as the squares of their diameters. And the area
of the circle is to the square of the diameter, as 11 to 14
nearly, as proved by Archimedes; or as '7S54 to 1 more
nearly ; or still more nearly as
■7853981633,9744830961,5660845819,8757210492,
9234984377,6455243736,1480769541,0157155224,
965700S706,335529266'9,9553702l62,S31S07666l,
7734611 + tol;
as it has been found by modern mathematicians.

In Wallis's Arithmetic of Infinites are contained the first
infinite series for expressing the ratio of a circle to the
square of its diameter : viz,

1st, The circle is to the square of its diameter,
a x 3 x 5x5x7x7 &c

as 2x4x4 x6x6x8&c'( discovered by Wallis

, , 9 95 49 „ f himself;

or 1 to - x — x — &c, \ '

-, by Ld. Brounker;

•2 + &c.

&c to 1, by

(4x5 2x4*6x7

Sir I. Newton; or as 1— ^ + j. — ^. + i — TrT&c to I, by
Gregory and Leibnitz ; and a great many other forms of
series have been invented also by other authors to express
the same ratio between the circle and circumscribed square.
5. The circumferences of circles are one to another, as
their diameters, or radii. Hence, as the areas of circles
are proportional to the rectangles of their radii and circum-
ferences; the quadrature of the circle will therefore be
effected by the rectification of its circumference; that is,
if the true length of the circumference could be found, the
true area could be determined also. But while several
mathematicians have endeavoured to investigate the true
area and circumference, others have even doubted of the
possibility of the same. Of this latter opinion is Dr. Isaac
Barrow: towards the end of his 15th mathematical lecture
he says, he is of opinion, that the radius and circumference
of a circle, are lines of such a nature, as to be not only
incommensurable in length and square, but even in length,
square, cube, biquadrate, and all other powers to infinity:
for, continues he, the side of the inscribed square is in-
commensurable to the radius, and the square of the side
of the inscribed octagon is incommensurable to the square
of the radius ; and consequently the square of the octago-
nal perimeter is also incommensurable to the. square of the
radius: and thus the ambits of all regular polygons, in-
scribed in a circle, may have their superior powers incom-
mensurate with the co-ordinate powers of the radius;
whence the last polygon, that is, the circle itself, appears
to have its periphery incommensurate with the radius ;
which, if true, will put a final stop to the quadrature of
the circle, since the ratio of the circumference to the ra-
dius is altogether inexplicable from the nature of the thing,
and consequently the problem requiring the explication of
iuch a ratio, is impossible to be solved, or rather it requires

that for its solution which is impossible to be apprehended.
He concludes by observing that this great mystery cannot
be explained in a few words: but if time and opportunity
had permitted, he would have endeavoured to produce
many things for the elucidation and confirmation of this
conjecture. Newton also, in bock 1 of his Principia, has
attempted to demonstrate the impossibility of the general
quadrature of oval figures, by the description of a spiral,
and likewise the impossibility of determining, by a finite
equation, the intersections of that oval and spiral, which
must be the case, if the oval be quadrable. And several
other authors have attempted to demonstrate the impossi-
bility of the general quadrature of the circle by any means
whatever. On the other hand, many mathematicians not
only believe in the possibility of the quadrature of the cir-
cle, but some have even professed to have discovered the
same, and have published to the world their pretended dis-
coveries: of which no one has rendered himself more con-
spicuous than our countryman Mr. Hobbes, though a great
scholar, and of excellent understanding in other matters.
See Quadrature.

The approximate quadrature of the circle however, or
the determination of the ratio between the diameter and
the circumference, is what the mathematicians of all ages
have successfully attempted, and with different degrees of
accuracy, according to the improved state of science.
Archimedes, in his book de Dimensione Circuli, first gave
a near value of that ratio in small numbers, being that of
7 to 22, which are still used as very convenient numbers
for this purpose in common measurements. Other and
nearer ratios have since been successively assigned, but in
larger numbers,

as 106 to 333,

or 1 13 to 355,,

or 1702 to 5347,

or 1815 to 5702, &c,
which are each more accurate than the foregoing: and se-
veral of these same ratios, as well as others, have lately
been found in the mathematical books of the Hindus, ac-
counts of which may be seen in my Tracts, vol- 2, pa. 158,
&c. Vieta, in his Universalium Inspectionum ad Canonum
Mathematicum, published' 1579, hy means of the inscribed.
and circumscribed polygons of 393216 sides, has carried
the ratio to ten places of figures, showing that if 'the dia-
meter of a circle be 1000 &c, the circumference will be
greater than 314,159,265,35,
but less than 314,159,265,37.

And Van Colen, in his book De CirCulb et Adscripts, has,
by the same means, carried that ratio to 36 places of fi-
gures; which were also recomputed and confirmed by
Willebrord Snell. After these, that indefatigable computer
Mr. Abraham Sharp extended the ratio to 72 places of.
figures, in a paper, published about the year 1706, by
means of the. series of Dr. Halley, from the tangent of an
arc of 30 degrees. And the ingenious Mr. Machin also
carried the same to a hundred places, by other series, de-
pending on the differences of arcs whose tangents have
certain relations to one another. Sec this method explain-
ed in my Mensuration, pa. 89, 4th edit. ; or my new Tracts,,
vol. 1, pa. 266. And, finally, M. De Lagny, in the Me-
moiresde l'Acad. 1719) by means of the tangent of the arc
of 30 degrees, has extended the same ratio to the astonish-
ing length of 12S places of figures; finding, that, if, the-
diameter be 1000 &c, the circumference will be

C I R

[ 318 ]

C I R

31415,92653,58979,32384,62643,38327,95028,
84197,l6939,93751,05820,97494,45923,07S 16,
40628,62089,98628,03482,53421, 17067.9S214,
80865,13272,30664,70938,446 ■+• or 447 -
From such methods as the foregoing, a variety of series
have been discovered for the length of the circumference
of a circle, such as the following, viz, If the diameter be
J, the circumference c will be variously expressed thus,
c = 4 x (1 _- + -__ + --_+--- &c),

4 &c),

*■ ~" 2.3 2.4.5 2.4.6.7 2.4.6.8.9 C^'
And many other series might here be added. See my
Mensuration in several places; also my paper on such se-
ries in the Philos. Trans. 1776, or my new Tracts, vol. 1,
pa. 268; Euler's Introductio in Analysin Infinitorum; and
many other authors.

6. Some of the more remarkable properties of the cir-
cle are as follow :

If two lines ab, cd cut the circle,
and intersect within it, the angle of in-
tersection e is measured by half the
sum of the intercepted arcs AC, PB.

But if the lines intersect without the
circle, the angle e is measured by half
the difference of the intercepted arcs

AC, DB.

7. The angle at the centre of a cir-
cle is double the angle at the circum-
ference, standing on the same arc ;
and all angles in the same segment are
equal. Also the angle at the centre is
measured by the arc it stands upon,
and the angle at the circumference by
half the same arc.

8. If the chords fg, in intersect
each other at right angles, the sums of
the opposite arcs are equal; viz, fh -+-

gi = FI -+- GH.

9. If one side no of a trapezium in-
scribed in a circle be produced out,
the external angle lop, will be equal
to the internal opposite angle M.

10. Any angle, as rqs, formed by
a tangent QR and chord qs, is mea-
sured by half the arc of the chord qs,
and is equal to any angle t in the al-
ternate segment qts.

'11. If vw be a diameter, and xzy a chord perpendi-
cular to it; then is xz or zy a mean proportional between

the segments vz, zw. So that if d de-
note the diameter vw, x the absciss
vz, and y the ordinate zx ; fhen is
y" = {d — x) x ior = <fc-r; which
is called the equation of the circle.

The chord vx is a mean proportional between the dia-
meter vw and the versed sine vz; and the chord wx is. a
mean proportional between the diameter and the versed
sine wz; also each versed sine is proportional to the square
of the corresponding chord ; viz, vz : wz : : vxa : wx".

12. If the diameter pq be divided
into two parts at any point r ; and if
rs be drawn perpendicular to pq, also
rt applied equal to the radius, and tr
produced to the circumference at v :
Then, between the two segments PR, rq,

kt is the arithmetical mean,
rs is the geometrical mean,
kv is the harmonial mean.

13. When two lines cut the circle, whether they inter-
sect within the circle, or without it, as in the two figures
to article 6, the segments between the common intersec-
tion and the two points where each line cuts the curve,
are reciprocally proportional, and their rectangles are
equal ; viz, ea : ec : : ed : eb,

or EA X EB = EC X ED.

14. In a trapezium inscribed in a
circle, the rectangle of the two diago-
nals is equal to the sum of the two
rectangles contained by each pair of
opposite sides ; viz, ac x bd = abx
dc ■+■ AD x BC

15. If any chords ef, eg, drawn
from the same point e in the circum-
ference, be cut by any other line hi,
the rectangles will be all equal which
are contained by each chord and the
part intercepted by this line, viz, ef x

EI = EG X EH = EK*.

16. In a circle whose centre is x
and radius no, if two points m, p, in
the radius produced, be so taken that ,
the three, km, no, np, be in continued
proportion ; then if from the points M
and p lines be drawn to any, or every
point in the circumference, as q ; these lines will be always
in the given ratio of mo to po; or mq : pq : : mo : po.

17. If vw be two points in the diameter, equidistant
from the centre t ; and if two lines be
drawn from these to any point x in the
circumference ; the sum of their squares
will be equal to the sum of the squares
of the segments of the diameter made
by either point; viz, vx1 -t- wx1 =
rv* -+- vs% or = aw -H- wsJ, or = 2RT1 ■+• 2VTl.

18. If a line fe perpendicular to the diameter ab,
meet any other chord cd in the point e; then is ax x rs

= CE X ED + El''.

C I R

[ 31Q ]

C I R

19. If upon the diameter gh of a
circle there be formed a rectangle
ghki, whose breadth gi or hk is equal
to gl or hl, the chord of a quadrant,
or side of the inscribed square; then if
from 1 and k lines be drawn to any
point m in the circle gmh, they will
cut the diameter gh in such a manner that go2 ■+• hn2
= GH2.

20. If the arcs PQ, qr, RS, &c,
be equal, and there be drawn the

chords pq, PR, ps, pt, &c, then it y///^/' _J^I
will be pq : pu : : PR : PQ + ps : :
ps : pr -+- pt : : pt : ps -+- pv, &c.

21. The centre of a circle being o, and p a point in the
radius, or in the radius produced ; if the circumference
be divided into as many equal parts ab, bc, cd, &c, as
there are units in 2n, and lines be drawn from p to all the
points of division ; then shall the continual product of all

C A? £ -J)

the alternate lines, viz, pa x pc x pe &c be = rn— in
when p is within the circle, or = xn — rn when p is with-
out the circle ; and the product of the rest of the lines,
viz pb x pd x pf &c = rn ■+■ xn: where r=AO the radius,
and x = op the distance of p from the centre.

22. To divide a circle into
any number of equal parts, by
means of concentric circles.—
adb being half the given circle,
divide its radius ac into the like
number of parts ; at the points
of division erect the perpen-
dicular ad, be, ef, meeting the semicircle \defc in the
points d, e, f, &c. Lastly, from the centre c, describe
circles through the points d, e,f, &c, and these will di-
vide the space in the manner required, cither all equal,
or in any ratios as the radius ac is first divided into. See
my Tracts, vol.3, p. 378, &c.

23. A circle may thus be divided into any number of
parts that shall be equal to one another both in area and
perimeter. Divide the diameter
qr into the same number of equal
parts at the points s, t, v, &c;
then, on one side of the diameter
describe semicircles on the diame-
ters qs, qt, qv, and on the other
side of it describe semicircles on
uv, rt, rs ; so shall the parts 1 7,
3 5, 5 3, 7 I, be all equal, both in area and perimeter. See
my Tracts, vol. 1, p. 254.

24. To describe a Circle either about or within a given

Regular Polygon. Bisect two of its angles, or two of its
sides, with perpendiculars, and the intersection of the
bisecting lines will, in either case, be the centre of the
circles.

Parallel, or Concentric Circles, are such as are equally
distant from each other in every point of their peripheries ;
or that have the same centre. As, on the other hand,
those are called the excentric circles, that have not the
same point for their centres.

The Quadrature of the Circle, is the manner of de-
scribing, or assigning, a square, whose surface shall be per-
fectly equal to that of a circle. This problem has exer-
cised the geometricians of all ages, but it is now generally
abandoned as a problem impossible to be effected, by most
persons that have any just claim to that rank, Descartes
insists on the impossibility of it, for this reason, that a
right line and a circle being of different natures, there
can be no strict proportion between them ; which is not
satisfactory. Dr. Barrow shows the strongest probability
of the same thing ; and not only that the diameter and
circumference themselves, but that all powers of them to
infinity, are incommensurate.

The Emperor Charles y offered a reward of 100,000
crowns to any person who should" resolve this celebrated
problem: and the Slates of Holland also proposed a re-
ward for the same thing. See Quadrature.

Circles of the Higher Orders,
are curves in which

IVY"

or wit"

: yz"
: yz"

: : yz
: : yz"

YX,

: yx"

xj

. (a — x) and

When m and n are each equal W, "Y" X

to 1, then wy : yz : : yz : yx, which is the property of
the common circle.

Put wy = x, yz =y, wx = a ; then is yx = a-
and the above proportions become

mm m+1 n

x : y : :y : (a-x) ory = x
m m 11 7t m + n m n

x :y : : y : (a : — .r) , or 3, =x .(a— x) , the

equations to curves of this kind.

Curves defined by this equation will be ovals when m
is an odd number. Thus suppose m = 1, then the equa-
tion becomes y'z = x. {a — x), or ax — x%, the equation
to the common circle. And if m = 3, the equation is
y1 = x3. (a — x) or «r3 — x*, which denotes a curve qf
the form ab, the first fig. below.

But when m denotes an even number, the curve will
have two infinite legs. So if m = 2, the equation becomes
y3 = xx. (a — x), or ax'2 — x3, for a circle of the 2d order
and which defines one of Newton's defective hyperbolas
being his 37th species of curves, whose asymptote is the
right line ef, making an angle of 40 degrees with the ab-
sciss ab, as in the 2d fig.

Circle of Curvature, or circle of cqui-curvature, is that
circle which has the same curvature with a given curve at
a certain point ; or that circle whose radius is equal to the
radius of curvature of the given curve at that point.

C I R

[ 320 ]

C I R

A circular arc is the only curve line that is equally
curved in every point. In all other curve lines, such as
the arc of an ellipse, or a parabola, or an hyperbola, or a
cycloid, the curvature is different in different points, and
the degree of curvature in any point is estimated by the
curvature of a circle which is said to have the same curva-
ture as the proposed curve line in that point; by which is
understood the circle which, .having the tangent of the
proposed curve in the said point for its tangent, approaches
so nearly to the proposed curve that no other circle what-
ever can be drawn between it and that curve.

This circle is also said to osculate the curve in the said
point, and is therefore often called the osculating circle, as
well as the circle of equal curvature with the curve in that
point. And the radius of this circle is called the radius of
curvature of the proposed curve in the said point ; also its
centre is called the centre of curvature.

Now there are some curve lines so very highly curved
in particular points, that every circle, of how small a ra-
dius soever, having the tangent to the curve in one of those
points for its tangent, will pass without the curve, or be-
tween the curve and its tangent. This, for example, is
the case with the curve of a cycloid in the two points
contiguous to its base, as also with the cissoid at its vertex.
And in such points the curvature of these curves is said

of the heavens, in their apparent diurnal rotation about
the earth.

Circle Equant, in the Ptolemaic Astronomy, is a circle
described on the centre of the equant. Its chief use is, to
find the variation of the first inequality.

Circles of Excursion, are lesser circles parallel to the
ecliptic, and at such a distance from it, that the excursions
of the planets towards the poles of the ecliptic, may be in-
cluded within them ; being usually fixed at about 10 d egrees.

It may here be observed, that all the circles of the
sphere, described above, are transferred from the heavens
to the earth : and thence come to have a place in geo-
graphy as well as in astronomy : all the points of each
circlebeing conceived as let fall perpendicularly on the
surface of the terrestrial globe, and thus tracing out circles
perfectly similar to them. So, the terrestrial equator is a
circle conceived precisely under the equinoctial line, which
is in the heavens ; and so of the rest.

Horary Circles, in Dialling, are the lines which show
the hours on dials. These are straight lines on the dials,
but called circles as being the projections of the meridians.

Horary Circle, or Hour Circle, on the globe, is asmall
brazen circle fixed to the north pole, divided into 24 hours,
and furnished with an index to point them out, thereby
showing the difference of meridians in time, and serving for

to be infinite, because it is greater than the curvature of the solution of many problems, on the artificial globes.

any circle, how small soever. Also the radius of the circle
of curvature in such points is nothing; the length of that
radius being always inversely or reciprocally as the de-
gree of curvature at any point.

The theory of these circles of equal curvature with
curves in particular points was first cultivated by Apol-
lonius in his Conic Sections : and it has since been carried
much farther by several great mathematicians of modern
times; particularly by Mr. Huygens in his doctrine of
Evolute Curves and Curves of Evolution, and by the great
Sir Isaac Newton. See Curvature.

Circles of the Sphere, are such as cut the mundane
sphere, and have their circumference in its surface.

These circles are either fixed or moveable.

The latter are those whose peripheries are in the move-
able or revolving surface; and which therefore move or
turn with it; as the meridians, &c. The former, having
their periphery in the immoveable surface, do not revolve ;
as the ecliptic, equator, and its parallels.

The circles of the sphere are either great or lesser circles.

A Great Circle of the Sphere, is that which divides it
into two equal parts or hemispheres, having the same centre
and diameter with it. As the horizon, meridian, equator,
ecliptic, the colures, and the azimuths.

A Lesser Circle of the Sphere, divides the, sphere into
two unequal parts, having neither the same centre nor di-
ameter with the sphere; its diameter being only some
chord of the sphere less than its axis. Such are the pa-
rallels of latitude, &c.

Circles of Altitude, or Alnmcantars, are lesser circles
parallel to the horizon, having their common pole in the
zenith, and still diminishing as they approach it. They are
so called from their use, which is to show the altitude of
a star above the horizon.

Ciuci.es of Declination, are great circles intersecting
each other in the poles of the world.

Ci acLB of Dissipation, in Optics. See Dissipation.

Diurnal Circles, are parallels to the equinoctial, sup-
posed to be described by the several stars, and other points

Circle of Illumination, is that imaginary circle on the
surface of the earth, which separates the illuminated side
or hemisphere of the earth from the dark side : and all
lines passing from the sun to the earth, being physically
parallel, are therefore perpendicular to the plane of this
circle.

Circles of Latitude, or Secondaries of the Ecliptic, are
great circles perpendicular to the plane of the ecliptic, in-
tersecting each other in its poles, and passing; through
every star and planet, &c. — These are so called, because
they serve to measure the latitude of the stars, which is
an arch of one of these circles, intersected between the
star and the ecliptic.

Circles of Longitude, are lesser circles parallel to the
ecliptic, diminishing more and more as they recede from
it, or as they approach the pole of that circle.

They are so called, because the longitudes of the stars
are reckoned upon them.

Circles of Perpetual Apparition, one of the lesser cir-
cles parallel to the equator, described by the most north-
ern point of the horizon, as the sphere revolves round by
its diurnal motion. — All the stars included within this cir-
cle, arc continually above the horizon, and consequently
never set.

Circle of Perpetual Occupation, is another lesser circle
at a like distance from the equator, but on the other side
of it, being described by the most southern point of the
horizon, and contains all those stars which never appear
in our hemisphere, or which never rise.

All other stars, being contained between these two cir-
cles, do alternately rise and set, at certain moments of the
diurnal rotation.

Polar Circles, are immoveable circles, parallel to the
equator, and at such a distance from the pole as is equal
to the greatest declination of the ecliptic, which is now
nearly 23° 28'. That next the northern pole is called the
arctic, and that next the southern one the antarctic.

Circles of Position, are circles passing through the
common intersections of the horizon and meridian, and

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[ 321

C 1 K

through any degree of the ecliptic, or the centre of any
star, or other point in the heavens ; and are used for find-
ing the situation or position of any star. These are usually
6" in number, cutting the equinoctial into 12 equal parts.

CIRCUIT, Electrical, denotes the course of the electric
fluid from the charged surface of an electric body, to the
opposite surface, into which the discharge is made. Some
electricians at first apprehended, that the same particles of
the electric fluid that were thrown on one side of the
charged glass, actually made the whole .circuit of the in-
tervening conductors, and arrived at the opposite side :
-whereas Dr. Franklin's theory only requires, that the re-
dundancy of electric matter on the charged surface should
pass into those bodies which form that part of the circuit
which is contiguous to it, driving forward that portion of
the fluid which they naturally possess ; and that the de-
ficiency of the exhausted surface should be supplied by
the neighbouring conductors, which form the last part of
the circuit, On this supposition, a vibrating motion is
successively communicated through the whole length of
the circuit.

Many attempts were made, both in France and England,
at an early period in the practice of electricity, to ascertain
the distance to which the electric shock might be carried,
and the velocity of its motion. The French philosophers,
at different times, caused it to pass through circuits of
900 and even 2000 toises, or about 2 English miles and a
half; and they discharged the Leyden phial through a
bason of water, whose surface was equal to one acre.
M. Monier found that the fluid, in passing through an iron
wire of 950 toises in length, did not spend a quarter of a
second ; and that its motion was instantaneous through a
wire of 1319 feet. In 1747, Dr. Watson, and other En-
glish philosophers, after many experiments of a similar
kind, conveyed the electric matter through a circuit of 4
miles ; and, from two several trials, they concluded that
its passage is instantaneous. By all which doubtless is
meant, that its motion is too rapid to be measured. Priest-
ley's Hist, of Elect, vol. 1, p. 128, Svoedit.

CIRCULAR, appertaining to a circle; as a circular
form, circular motion, &c.

Circular Instruments of Reflection, or Multiplying Cir-
cles, are such circles as, in a small size, by means of re-
flection, procure at pleasure all the advantages of the size
of large instruments, as well as diminishing the errors of
division. — Circular instruments are in fact considerable
improvements on Hadley's octant, and the marine sex-
tant, devised for the same purposes. We shall here de-
scribe the progress of these instruments under the contri-
vances of Tobias Mayer, Chevalier Borda, and Captain
Mendoza.

Mr. Mayer proposed to complete the limb of the sex-
tant to a whole circle, with the horizon glass moveable
round the centre, and with an additional index, which
may be called the horizon index, to distinguish it from
the centre index, to which the centre glass is attached.
This instrument is represented in plate 7, fig. 8 ; and the
manner of using it is as follows : After the index a is set at
0 (the beginning of the divisions) the two glasses are ren-
dered parallel, as in using Hadley's quadrant, by moving
the horizon index b, till the horizon of the sea (or the sun
or any other object), or its direct image, and the doubly
reflected image of the same, seen through the telescope,
coincide. After fixing the horizon index in that position,
the centre index a is to be mowd, in order to measure the

Vol. I.

distance of the two objects s and l (suppose the sun and
moon), by bringing into contact the double reflected image
of the sun with the direct image of the moon, seen through
the telescope. The centre image will then be at m, and
the arch 011 might give, as in the sextant, the angular di-
stance required ; but the construction of the circle ren-
ders it easy, in this position, to effect again the parallelism
of the glasses, and to make another observation of the con-
tact, in the like manner as from o; which operation will
bring the centre index to s. The index will then give osr,
or double the distance; which divided by 2, reduces the
errors of the instrument to one half. And it is obvious,
that by successive repetitions of the same process, triple,
quadruple, &c, the distance may be obtained, and the
said error further reduced, in the inverse ratio of the mul-
tiplication of the distance, to any degree of approximation
desired.

The method of rendering the glasses parallel, by means
of the horizon of the sea, is not accurate, on account of
the indistinctness of the images ; and when the sun is
used for that purpose, the observation becomes fatiguing
to the eye. The repetition of that observation remained
therefore a considerable inconvenience attached to this
circle. The author himself seems to have been aware of
it, as he proposed to provide the instrument with a diago-
nal ruler, fixed on one of the indexes, so that the other
index should touch it when the glasses were parallel : but
an adjustment of this nature must be subject to great er-
rors, and never was adopted in practice. The chevalier
Borda, wishing to remove that imperfection, contrived to
render the parallelism of the glasses unnecessary, by sub-
stituting the observation of the angular distance of the
two objects, for that of the coincidences of the images of
the same object. This constitutes the second great im-
provement of the reflecting circle.

In Borda's circle, the telescope is fixed at some distance
from the centre, and the horizon glass is carried near the
border of the instrument, as in plate 7. fig. 9. By this
arrangement, the rays of light can arrive at the centre
glass, both from the heavenly bodies situated to the right
of the horizon index, as s', and from those situated to the
left, as s. Thus, if the glasses are parallel to one an-
other, when the centre index is at 0, it is obvious that
there are two ways of making the observations. While
the direct image of the moon l is seen through the tele-
scope, the angular distance to the sun, if at s, may be
measured by moving the centre index to m, in order to
produce the contact ; or, if the sun be at 3', the same ope-
ration may be performed, by using the contrary motion to
m'. The first kind of observation, M. Borda calls obser-
vation to the left ; and the second observation to the right.
Suppose now, that (the horizon index being fixed in the
same position) the distance from l to s is observed to the
left, by bringing into contact the double reflected image
of s with the image of l, seen without reflection ; Jet us
then turn the instrument round, keeping it in the same
plane, so as to have the direct image of s through the te-
lescope, and thus make an observation of the same di-
stance to the right; the position of the centre index be-
ing in the first observation at m, and in the second at m\
it is clear, that if o be the point where the parallelism of
the glasses takes place, om will be equ;\l to om', and that
the arch mm', determined by the two positions, will give
double the distance. — It will be more convenient to have
the centre index at o, when the first observation is made,
2T

C I 11

[ 322 ]

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in order to take the double distance at one reading, after
the second observation. For tiiis purpose, the first part
of the process may be inverted, by previously fixing the
centre index at the beginning of the divisions, and moving
the horizon index h towards o, instead of moving the
centre index a to m, or towards h.

The last kind of observation, in which the incident ray,
which produces the first image on the centre glass, may
be conceived to run double the angular distance, passing
in its way over the line of collimation, has by M. Borda
been called the crossed observation. — The same process
may be repeated, by fixing alternately one of the indexes,
and moving the other, and continuing successive sets of
observations ; each set of two crossed observations, one
to the right and another to the left. The angle given by
the instrument, will be equal to double the angular di-
stance multiplied by the number of sets observed.

Whether a circle is used simply, as Mayer proposed it,
or in Borda's method, its peculiar advantages chiefly de-
pend on the multiplication of the distance sought. Capt.
Mendoza has therefore turned his attention to the im-
provement of this principle, and has according^ contrived
the following construction : In the crossed observations
made with Borda's circle, the indexes move alternately
through an arch which, in the divisions, is equal to double
the distance: for example, the centre index comes, in the
first crossed observation, from m to m ; in the third crossed
observation, from m' to m", &c ; and the horizon index,
in the 2d crossed observation, to h' ; in the 4th crossed
observation to h", &c ; and by each of the two indexes
may be found the same multiple of the distance sought.
Let us now place the nonius in a circle moving round the
centre, over, or adjacent to, the usual limb which con-
tains the divisions : then it will easily be conceived that,
by attaching that circle, which M. Mendoza calls the fly-
ing nonius, alternately to each of the indexes, it will serve
as nonius for both ; and that, after any number of obser-
vations, it will give the compound motion of the two in-
dexes. Thus, after the first observation, the flying nonius
will, at each crossed observation, advance double the di-
stance over the divisions, while each separate nonius, fixed
on the indexes, requires a set of two observations to pro-
duce the same effect in Borda's circle. The peculiar pro-
perty of the instrument being that of giving double the di-
stance, it was thought proper to divide the circle into 360
degrees, and not into' 720 according to the nature of the
sextant. Thus, after a crossed observation, the reading
of the nonius will, without reduction, exhibit the measure
of the simple distance. Mendoza has also extended the no-
nius round the circumference, so that, by the coincidence
of two divisions, the number of degrees will appear on the
limb, and that of the minutes and seconds on the flying
nonius. See Philos. Trans. 1801.

For more on the nature of circular instruments of re-
flection,see Montucla, Histoire des Mathematiques, torn. 3,
p. 527 : for the computation of the probable extent of
tiie errors to which they are liable, see Swanberg, Expo-
sition des operations faites en Lapponic &c, p. 29; and for
a catalogue of papers &c, connected with such instru-
ments, consult Dr. T. Young's Natural Philos. vol. 2,
p. 350. — For Sir Howard Douglas's Reflecting Circle or
Semicircle, sec the article Reflecting in this Dictio-
nary.

Circular Lines, a name given by some authors to
such straight lines as are divided by means of the divisions

made in the arch of a circle. Such as the Sines, Tan-
gents, Secants, ecc.

Circular Numbers, are such as have their powers
ending in the roots themselves. As the number 5, whose
square is 25, and its cube 125, &c.

Napier's Circular Parts, are five parts of a right-an-
gled or a quadrantal spherical triangle ; they are the two
legs, the complement of the hypothenuse, and, the com-
plements of the two oblique angles.

Concerning these circular parts, Napier gave a general
rule in his Logarithmorum Canonis Descriptio, which is
this : " The rectangle under the radius and the sine of the
middle part, is equal to the rectangle under the tangents
of the adjacent parts, and to the rectangle under the co-
sines of the opposite parts. The right angle or quadrantal
side being neglected, the two sides and the complements
of the other three natural parts are called the circular
parts; as they follow each other as it were in a circular
order. Of these, any one being fixed upon as the middle
part, those next it are' the adjacent, and those farthest
from it the opposite parts." Lord Buchan's Life of Na-
pier, pa. 9S.

This excellent, though very concise rule, Including all the
particular rules for thesolution of right-angled spherical tri-
angles, may be readily applied to those that are oblique, by
letting fall a perpendicular, excepting the two cases in which
there are given either the three sides, or the three angles.
— And for these a similar expedient has been devised by-
Lord Buchan and Dr. Minto. M. Pingre, in the Me-
moires de Mathematiqueet de Physique for the year 1756,
reduces the solution of all the cases of spherical triangles
to four analogies. These four analogies are in fact, under
another form, Napier's rule of the circular parts, and his
second or fundamental theorem, with its application to
the supplemental triangle. Though it would be no diffi-
cult matter to get by heart the four analogies of M. Pin-
gre, yet there are few persons blessed with a memory ca-
pable of retaining them for any considerable time. For
this reason, the rule for the circular parts ought to be
kept under its present form- If the reader attends to the
circumstance of the second letters of the words tangents
and cosines being the same with the first of the words ad-
jacent and opposite, he will find it almost impossible to
forget it. And the rule for the solution of the two cases
of spherical triangles, for which the former alone is in-
sufficient, may be thus expressed : Of the circular parts
of an oblique spherical triangle, the rectangle under the
tangents of half the sum and half the difference of the
segments at the middle part (formed by a perpendicular
drawn from an angle to the opposite side), is equal to the
rectangle under the tangents of half the sum and half the
difference of the opposite parts. By the circular parts of
an oblique spherical triangle, are meant ils three sides and
the supplements of its three angles. Any of these six be-
ing assumed as a middle part, the opposite parts are those
two of the same denomination with it; that is, if the mid-
dle part is one of the sides, the opposite parts are the other
two, and if the middle part is the supplement of one of
the angles, the opposite parts are the supplements of the
other two. — Since every plane triangle may be considered
as described on the surface of a sphere of an infinite i;i-
dius, these two rules may be applied to plane triangles,
provided the middle part be restricted to a side.

''Thus it appears that two simple rules suffice lor the
solution of all the possible cases of plane and spherical

C I R

triangles. These rules, from their neatness and the man-
ner in which they are expressed, cannot fail of engraving
themselves deeply on the memory of every one who is a
little versed in trigonometry. It is a circumstance wor-
thy of notice, that a person of a very weak memory may
carry the whole art of trigonometry in his head." Napier's
Life, pa. 102.

Circular Sailing, is that performed in the arch of a
great circle. — It is chic-fly on account of the shortest di-
stance that this method of sailing has been proposed ; and
- for the most part it is advantageous for a ship to reach her
port by the shortest course.

As the solutions of the cases in Mercator's sailing are
performed by plane triangles ; so the cases in great-circle
sailing are resolved by the solution of spherical triangles.
But, after all, the several cases in this kind of sailing serve
rather, for exercises in the solution of spherical triangles,
than for any real use towards the navigating of a ship.

Circular Spots, are made on pieces of metal by large
electrical explosions. See Philos. Trans, vol. 5S, pa. 68;
or my Abridgment, vol. 12, p. 510; also Priestley's Hist,
of Electricity, vol. 2, sect. 9> edit. 8vo.

These beautiful spots, produced by the moderate charge
of a large battery, discharged between two smooth sur-
faces of metals, or semi-metals, lying at a small distance
from each other, consist of one central spot, and several
concentric circles, which are more or less distinct, and
more or fewer in number, as the metal on which they are
marked is more easy or difficult of fusion, and as a greater
or less force is employed. They are composed of dots or
cavities, which indicate a real fusion. If the explosion
of a battery, issuing from a pointed body, be repeatedly
taken on the plain surface of a. piece of metal near the
point, or be received from the surface on a point, the me-
tal will be marked with a spot, consisting of all the pris-
matic colours disposed in circles, and formed by scales of
the metal separated by the force of the explosion.

Circular Velocity, a term in Astronomy, signifying
that velocity of a planet, or revolving body, which is
measured by the arch of a circle.

Circulating Decimals, called also recurring or re-
peating decimals, are those in which a figure or several fi-
gures are continually repeated. They are distinguished into
single and multiple, and these again into pure and mixed.

A pure single circulate, is that in which one figure only
is repeated ; as '222 &c, and is marked thus -2.

A pure multiple circulate, is that in which several figures
are continually repeated ; as -232323 &c, marked •23 ;
and '524524 &c, marked '524.

A mixed single circulate, is that which consists of a ter-l
minate part, and a single repeating figure; as 4'222 &c,
or 4s'2. And

A mixed multiple circulate, is that which contains a ter-
minate part with several repeating figures; as 45-524.

That part of the circulate which repeats, is called the
repetend : and the whole repctend, supposed infinitely
continued, is equal to a vulgar fraction, whose numerator
is the repeating number, or figures, and its denominator
the same number of nines : so '2 is = $; and -23 is =
U ; and -524 is = £§*.

It seems it was Dr. Wallis who first distinctly consi-
dered, or treated of infinite circulating decimals, as he

C 3S3 ] CIR

himself informs us in his Treatise of Infinites. Since his
time many other authors have treated on this part of arith-
metic ; the principal of these, however, to whom the art is
mostly indebted, are Messrs. Brown, Cunn, Martin, Emer-
son, Malcolm, Donn, and Henry Clarke, in whose writings
the nature and practice of this branch of arithmetic may be
fully seen, especially in the last-mentioned ingenious author.

CIRCUMFERENCE, in a general sense, denotes the
line or lines bounding any figure. But it is commonly
used in a more limited sense, to denote the curve line
which bounds a circle, and which is otherwise called the
periphery; the boundary of a right-lined figure, being ex-
pressed by the term perimeter.

The circumference of a circle is every where equidi-
stant from the centre. And the circumferences of different
circles are to one another as their radii or diameters, or
the ratio of the diameter to the circumference is the same
in every circle, which is nearly as 7 to 22, as it was found
by Archimedes, or, more nearly, as 1 to 3-1416. Under
the article Circle may be seen various other approxima-
tions to that ratio, one of which is carried to 128 places
of figures, viz, by M. De Lagny.

The circumference of every circle is supposed to be di-
vided into 360 equal parts, called degrees. — Any part of
a circumference is called an arc or arch; and a right line
drawn from one end of an arc to the other, is called its
chord. — The angle at the circumference is equal to half
the angle at the centre, standing on the same arc ; and
therefore it is measured by the half of that arc.

CIRCUMFERENTOR, a particular instrument used
by surveyors for taking angles. It consists of a brass cir-
cle and index all of one piece ; the diameter of the circle
is commonly about 7 inches ; the index about 14 inches
long, and an inch and a half broad. On the circle is a
card or compass, divided into 360 degrees; the meridian
line of which answers to the middle of the breadth of the
index. On the limb, or circumference of the circle, is
soldered a brass ring ; which, with another fitted with a
glass, forms a kind of box for the needle, which is sus*
pended on a pivot in the centre of the circle. There are
also two sights to screw on, and slide up and down the index,
as also a spangle and socket screwed on the underside of
the circle, to receive the head of the three-legged staff.

To take, or observe, the Quantity of an single by the Cir-
cumferentor. The angle proposed being ekg ; place the
instrument at k, with the flower-de-luce of the card to-
wards you ; then direct the sights to e, and observe what
degrees are cut by the south end of the needle, which let
be 295 ; then, turning the instrument about on its stand,
direct the sights to c, noting again what degrees are cut
by the south end of the needle, which suppose are 213.
This done, subtract the less number from the greater, viz,
2T2

CIS

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C L A

213 from-295, and the remainder, or 82 degrees, is the
quantity of the angle ekg sought.

CIRCUMGYRATION, is the revolving motion of any
body about a centre ; as of the planets about the sun, &c.

CIRCUM-Polar Stars, are those stars which, by rea-
son of their vicinity to the pole, move round it, without
setting.

CIRCUMSCRIBED Figure, is a figure which is drawn
about another, so that all its sides or planes touch the lat-
ter or inscribed figure.

The area and perimeter of every polygon that can be
circumscribed about a circle, are greater than those of the
circle ; and the area and perimeter of every inscribed po-
lygon, are less than those of the circle ; but they approach
always nearer to equality as the number of sides is greater.
And on these principles Archimedes, and some other au-
thors since his time, attempted the quadrature of the cir-
cle; which is nothing else, in effect, but the measuring
the area or capacity of a circle.

Circumscribed Hyperbola, is one of Newton's hyper-
bolas of the 2d order, that cuts its asymptotes, and con-
tains the parts cut off within its own space.

CIRCUMVALLATION, or Line of Oircumvallation, in
the Art of War, is a trench, bordered with a parapet,
thrown up around the besieger's camp, as a security
against any army that may attempt to relieve the place,
as well as to prevent desertion.

CIRCUMVOLUTION, in Architecture, the torus of
the spiral line of the Ionic order.

CISSOID, is a curve line of the second order, invented
by Diocles for the purpose of finding two continued mean
proportionals between two other given lines. The gene-
ration or description of this curve is as follows :

On the extremity b of the diame-
ter ab of the circle aob, erect the
indefinite perpendicular cbd, to
which from the other extremity a
draw several lines, cutting the circle
in i,o,n, &c; and on these lines set
off the corresponding equal distan-
ces, viz, hm = ai,fo = ao, and cl
= an, &c ; then, the curve line
drawn through all the points M, o, L,
&c, is the cissoid of Diocles, who
was an ancient Greek geometrician.

This curve is, by Newton, reck-
oned among the defective hyperbo-
las, being the 42d species in his Enu-
meratio Linearum Tertii Ordinis.
And in bisAppendix deEquationum
Constructione Lincari, at the end of
his Arithmctica Universalis, he gives
another elegant method of descri-
bing this curve by the continual mo-
tion of a square ruler. Other me-
thods have also been devised by different authors for the
same thing.

The Properties of the Cissoid are the following:

1. The curve has two infinite legs amol, Ainol meeting
in a cusp a, and tending towards the indefinite line cbd,
which is their common asymptote.

2. The curve passes through o and o, points in the cir-
cle equally distant from a and 15; or it bisects each semi-
circle.

3. Letting fall perpendiculars MP, IK from any corre-
sponding points 1, m; then is ap = bk, and am = hi,
because ai = jih.

4. ap : pb : : MP1 : ap!. So that, if the diameter ab
be == a, the absciss jp = i, and the ordinate pm =y;
then is a: : a — x : : y* : x", or x3 = (a — x) x y1, which is
the equation of the curve.

5. Sir Isaac Newton, in his last letter to M. Leibnitz, has
shown how to find a right line equal to one of the legs of
this curve, by means of the hypeibola; but he there sup-
pressed the investigation, which however may be seen in his
Fluxions.

6. The whole infinitely long cissoidal space, contained
between the infinite asymptote cbd, and the curves loao/
&c, of the cissoid, is equal to triple the generating circle.
aoboa. — See more of this curve in Dr. Wallis's Works,
vol. 1, pa. 545.

CIVIL Day. See Day.

Civil Month. See Month.

Civil Year, is the legal year, or annual account of time
which every government appoints to be used within its own
dominions. It is so called in contradistinction from the
natural year, which is measured exactly by the revolution
of the heavenly bodies.

CLAIRAUT (Alexis-Claude), a celebrated French
mathematician and academician, was born at Paris the 1 3th
of May 1713, and died the 17th of May 1765, at 52 years
of age. His father, a teacher of mathematics at Paris,
was his sole instructor; and such was the progress of our
author under him, that at the early age of 4 years, he
could read and write. The letters of the alphabet he
taught him in the figures of Euclid's Elements. At 9 years
of age he studied Guisniie's Application of Algebra to Geo-
metry; at 10 De l'Hopital's Conie Sections; and between
12 and 13 he read a memoir to the Academy of Sciences
concerning 4 new geometrical curves of his own invention.
About the same time he laid the first foundation of his
work upon curves that have a double curvature, which he
finished in 1729, at lfj years of age. He was named Ad-
joint-Mechanician to the Academy in 1731, at the age of
18, Associate in 1733, and Pensioner in 1738. During
his connexion with the Academy, he had a great number
of learned and ingenious communications inserted in their
Memoirs, besides several other works which he published
separately; the list of which is as follows:

1. On Curves of a Double Curvature; in 1730, 4to.

2. Elements of Geometry; 1741, 8vo.

3. Theory of the Figure of the Earth; 1743, 8vo.

4. Elements of Algebra; 1746, Svo.

5. Tables of the Moon; 1754, Svo.

His papers inserted in the Memoirs of the Academy are
too numerous to be particularized Jiere; but they may be
found from the year 1727, for almost every year till 1762;
being upon a variety of subjects, astronomical, mathema-
tical, optical, &c.

CLARKE (Dr. Samuel), a celebrated English divine,
philosopher, and metaphysician, was born at Norwich the
1 1th of Oct. 1675. He was the son of Edward Clarkr, esq,
alderman of that city, and for several years one of its repre-
sentatives in parliament. He received instruction in classi-
cal learning at the free-school of that town; and in 1691
removed thence to Caius-college, Cambridge, where his
shining abilities soon began to display themselves. —
Though the philosophy of Descartes was at that time the

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C L A

established system of the university, yet Clarke's sagacity
soon discovered its imperfections, and led him to the belief
of one more solid and rational, viz, the new system of New-
ton: this he soon mastered ; and in order to his first degree
of arts, performed a public exercise in the schools upon a
question taken from it. He greatly contributed to the
establishment of the Newtonian philosophy by an excel-
lent translation of Rohault's Phvsics, with notes, which he
finished before he was 22 years of age, a book which had
been for some time the system used in the university, and
founded on Cartesian principles. This was first published
in the year 1697, and it soon after went through several
other editions, all with improvements.

Mr. Whiston relates that, in that year, 1697, while he
was chaplain to Dr. Moore, bishop of Norwich, he met
with young Clarke, then wholly unknown to him, at a cof-
fee-house in that city, where they entered into a conver-
sation about the Cartesian philosophy, particularly Ro-
hault's Physics, which Clarke's tutor, as he tells us, had
put him upon translating. " The result of this conversa-
tion was," says Whiston, " that I was greatly surprised that
so young a man as Clarke then was, should know so much
of those sublime discoveries, which were then almost a se-
cret to all but a few particular mathematicians. Nor did
I remember (continues he) above one or two at the most,
whom I had then met with, that appeared to knolf so much
of that philosophy as Mr. Clarke."

He afterwards turned his thoughts to divinity; and hav-
ing taken holy orders, in 1698 he succeeded Mr. Whiston
as chaplain to Dr. Moore, bishop of Norwich, who was
ever after his constant friend and patron. Soon after he
published some theological treatises; and-Bishop Moore
gave him the rectory of Drayton, near Norwich, with a
lectureship in that city. In 1704, he was appointed to
preach Boyle's lecture; and the subject he chose was,
The Being and Attributes of God. He succeeded so well
in this, and gave so much satisfaction, that he was appoint-
ed to preach the same lecture the next year, when he chose
for his subject, The Evidences of Natural and Revealed
Religion. These sermons were first printed in two volumes,
in 1705 and 1706, and contained some remarks on such
objections that had been made by Hobbes and Spinoza, and
other opposers of natural and revealed religion. In the
6th edition was added, A Discourse concerning the Con-
nexion of the Prophecies of the Old Testament, and the
application of them to Christ.

About this time, Mr. Whiston informs us, he discovered
that Mr. Clarke (having read much of the primitive wri-
ters) began to suspect that the Athanasian doctrine of the
Trinity was not the doctrine of those early ages; and it was*
particularly remarked of him, that he never read the Atha-
nasian Creed at his parish church.

In the midst of his theological labours, Mr. Clarke found
time to show his regard to mathematical and philosophi-
cal researches, with his exact knowledge and skill in them.
And his natural affection and capacity for these studies
were not a little improved by the friendship of Newton ; at
whose request he translated his Optics into Latin in 1706.
With this version Sir Isaac was so highly pleased, that he
presented him with the sum of 500/ or 100/ to each of his
5 children.

The same year, also, Bishop Moore procured for him
the rectory of St. Bennett's, Paul's Wharf, in London; and
soon after carried him to court, and recommended him to

the favour of Queen Anne. She appointed him one of her
chaplains in ordinary; and also presented him to the rec-
tory of St. James's, Westminster, when it became vacant
in 1709. On this occasion he took the degree of d. d.
when the public exercise which he performed for it at
Cambridge was highly admired.

In 1715 and 1716", Dr. Clarke had a dispute with the
celebrated Leibnitz, concerning the principles of natural
philosophy and religion; and a collection of the papers
which passed between them "was published in 1717. This
work was addressed to Queen Caroline, then princess of
Wales, who was pleased to have the controversy pass
through her hands. It related chiefly to the subjects of
liberty and necessity.

About the year 1718 he was presented by the lord Lech-
mere to the mastership of Wigston's hospital in Leicester-
shire. In 1724 and 1725 he published 18 sermons,
preached on several occasions. In 1727, on the death of"
Newton, he had .the offer of succeeding him as Master of
the Mint, a place worth from 12 to 15 hundred a year;
but to this secular preferment he could not reconcile him-
self, and therefore absolutely refused it. In 172S was
published, A Letter from Dr. Clarke to Mr. Benjamin
Hoadley, occasioned by the Controversy relating to the
Proportion of Velocity and Force of Bodies in Motion,
which was printed in the Philosophical Transactions, No.
401. Dr. Clarke died the 17th of May, 1729, in the 54th
year of his age.

Queen Caroline took great pleasure in the doctor's con-
versation and friendship ; seldom a week passed in which
she did not receive some proof of the greatness of his ge-
nius and the force of his understanding.

As to the character of Dr. Clarke, he is represented as
possessing one of the most amiable and affectionate dispo-
sitions, remarkably free and easy in his conversation, cheer-
ful afld even playful in his manner. Bishop Hare says ofi
him, " He was a man who had all the good qualities that
could meet together to recommend him. He was pos-
sessed of all the parts of learning that are valuable in a
clergyman, in a degree that few possess any single one.
He had joined to a good skill in the three learned lan-
guages a great compass of the best philosophy and ma-
thematics, as appears by his Latin works; and his English
fines are such a proof of his own piety and of his know-
ledge in divinity, and have done so much service to religion,
as would make any other man, that was not under a sus-
picion of heresy, secure of the friendship of all good
churchmen, especially the clergy. — Much more may be
seen, related in his praise, by Bishop Hoadley, Dr. Sykes,
and Mr. Whiston, in their Memoirs of his life.

CLAVIUS (Chuistopheh), a German Jesuit, was
born at Bamberg in Germany, in 1537. He became a very
studious mathematician and elaborate writer; his works,
when collected, and closely printed, making 5 large folio
volumes, being a complete body or course of mathematics.
They are mostly elementary, with commentaries on Euclid
and others, possessing very little of invention of his own.
His talents and writings have been variously spoken of, and
it must be acknowledged that they are heavy and elaborate.
He was invited to Rome, to assist, with other learned men,
in the reformation of the calendar, by Pope Gregory, of
which he afterwards undertook a defence against Sca-
liger, Vieta, and others, who attacked it. He died at
Rome the 6th of February, 1612, at 75 years of age, after

CLE

[ 32fi ]

CLE

more than 50 years close application to the mathematical
sciences.

CLEF, or Cliff, in Music, a mark at the beginning of
the lines of a song, which shows the tone or key in which
the piece is to begin. Or, it is a letter marked on any line,
which explains and points out the name of all the rest.

Anciently, every line had a letter marked for a clef; but
now a letter on one line suffices; since by this all the rest
are. known, reckoning up and down, in the order of the
letters.

It is called the clef, or key, because that thereby are
known the names of all the other lines and spaces; and
consequently the quantity of every degree or interval. But
because every note in. the octave is called a key, though,
in another sense, this letter marked is called peculiarly the
signed clef; since, being written on any line, it not only
signs and marks that one, but it also explains all the rest.
By clef, therefore, for distinction sake, is meant that let-
ter, signed on a line, which explains the rest; and by key,
the principal note of a song, in which the melody closes.

There are three of these signed clefs, c,f, g. The clef
of the highest part in a song, called treble, or alt, is g, set
on the second line, reckoning upwards. The clef of the
bass, or the lowest part, is/, on the 4th line upwards. For
all the other mean parts, the clef is c, sometimes on one,
and at others on another line. Indeed, some that are
really mean parts, are sometimes set with the g clef. It
must however be observed, that the ordinary signatures of
clefs bear little resemblance to those letters. Mr. Mal-
colm is of opinion that it would be well if the letters them-
selves were used. Kepler takes great pains to show, that
the common signatures are only corruptions of the letters
they represent.

The clefs are always taken fifths to one another. So
the clef/ being lowest, c is a fifth above it, and g a fifth
above c.

When the place of the clef is changed, which is not fre-
quent in the mean clef, it is with a design to make the sys-
tem comprehend as many notes of the song as possible,
and so to have the fewer notes above or below it. So that,
if there be many lines above the clef, and few below it,
this purpose is answered by placing the clef in the first or
second line; but if there be many notes below the clef, it
is placed lower in the system. In effect, according to the
relation of the other notes to the clef note, the particular
system is taken differently in the scale, the clef line ma-
king one in all the variety.

But still, in whatever line of the particular system any
clef is found, it must be understood to belong to the same
of the general system, and likewise to be the same indivi-
dual note or sound in the scale. By this constant relation
of clefs, we learn how to compare the several particular
systems of the several parts, and to know how they com-
municate in the scale, that is, which lines are unison and
which are not ; for it is not to be supposed that each part
has certain bounds, within which another may never
enter — some notes of the treble, for example, must be low-
er than some of the mean parts, or even of the bass. There-
fore, to join together into one system all the parts of a com-
position written separately, the note of each particular part
must be placed at the same distances above and below the
proper clef, as they stand in the separate system: and be-
cause all the notes that are consonant, or heard together,
must stand directly over each other, that the notes belong-

ing to each part may be distinctly ascertained, they may
be made with such differences as shall not confound or
alter their significations with respect to time, but only show
their relation to this or that part. Thus we shall discover
in what manner the parts change and pass through one
another; and which, in every note, is highest, lowest, or
unison.

Jt must here be observed, that for the performance of
an}1 single piece, the clef only serves for explaining the in-
tervals in the lines and spaces : so that it need not be re-
garded what part of any greater system it is; but the first
note may be taken as high or low as we please. For as
the proper use of the scale is not to limit the absolute de-
gree of tone, so the proper use of the signed clef is not to
bound the pitch at which the first note of any part is to
be taken, but to determine the tune of the rest, with respect
to the first; and considering all the parts together, to de-
termine the relation of their several notes by the relations
of their clefs in the scale : thus, their pitch of tune being
determined in a certain note of one part, the other notes of
that part are determined by the constant relations of the
letters of the scale, and the notes of other parts by the re-
lations of their clefs.

In effect, for performing any single part, the clef note
may be taken in an octave, that is, at any note of the same
name; provided we do not go too high or too low for find-
ing the rest of the notes of a song. But in a concert of
several parts, all the clefs must be taken, not only in the
relations, but also in the places of the system abovemen-
tioned, that every part may be comprehended in it.

The natural and artificial note expressed by the same
letter, as c and eft, are both set on the same line or space.
When there is no character of flat or sharp, at the begin-
ning with the clef, all the notes are natural; and if in any
particular place the artificial note be required, it is denoted
by the sign of a flat or sharp, set on the line a space before
that note.

If a sharp or flat be placed at the beginning in any line
or space with the clef, all the notes on that line or space
are artificial ones; that is, are to be taken a semitone higher
or lower than they would be without such sign. And the
same affects all their octaves above and below, though they
be not marked so. In the course of the song, if the natu-
ral note be sometimes required, it is signified by the cha-
racter ^.

CLEPSYDRA, a kind of water-clock, or an hour-glass
serving to measure time by the fall of a certain quantity,
commonly out of one vessel into another. — There have
been also clepsydrae made with quicksilver; and the term
is also used for hour-glasses of sand.

By this instrument the Egyptians measured their time '
and the course of the sun. Also Tycho Brahe, in mo-
dern times, made use of it to measure the motion of the
stars, &c; and Dudley used the same contrivance in ma-
king all his maritime observations.

The use of clepsydrae is very ancient. They were pro-
bably invented in Egypt under the Ptolemies; though some
authors ascribe the invention of them to the Greeks, and
others to the Romans. Pliny informs us, that Scipio Na-
sica, about 150 years. before Christ, gave the first hint for
the construction of them : and Pancirollus lias particu-
larly described them. According to his account, the clep-
sydra was a vessel made of glass, with a small hole in the
bottom, edged with gold: in the upper part of this vessel

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C L I

a line was drawn, and marked with the 12 hours : the ves-
sel was filled with water, and a cork with a pin fixed in it
Moated on the surface, pointing to the first hour; and as
the water sunk in the vessel by issuing out of the small
hole, the pin indicated the other hours as it descended.
The same kind of instruments were also used by the Hin-
dus, for measuring time ; and from the invention might
come into all the countries westward from them. See my
Tracts, vol. 2, p. 178.

Clepsydrae were chiefly used in the winter; as sun-dials
served for the summer. They had however two great de-
fects; the one, that the water ran out more or less easily,
as the air was more or less dense; the other, that the wa-
ter flowed more rapidly at the beginning, than towards the
conclusion when its quantity and pressure were much de-
creased. Amontons has invented a clepsydra which, it is
said, is free from both these inconveniences ; and the same
effect is produced by one described by Mr Hamilton, in
the Philos. Trans, vol. 44, pa. 171, or my Abridg. vol. 9,
pa. 236. Varignon also, in the Memoires del'Acad. 1699,
delivers a general geometrical method of making clepsy-
dras, or water-clocks, with any kind of vessels, and with
any given orifices for the water to run through.

Vitruvius, in lib. 9 of his Architecture, treats of these
instruments ; and Pliny in chap. 6"0, lib. 7, says that Sci-
pio Nasica was the first who measured time at Rome by
clepsydra, or water-clocks. Gesner, in his Pandects,
pa. 91) gives several contrivances for these instruments.
Solomon de Caus also treats on this subject in his Reasons
of Moving Forces, &c. So also does Ozanam, in his Ma-
thematical Recreations, in which is contained a Treatise
on Elementary Clocks, translated from the Italian of Do-
minique Martinelli. There is likewise a treatise on Hour-
Glasses by Arcangelo Maria Radi, called Nova Scienza de
Horologi Polvere. See also the Technica Curiosa of Gas-
per Schottus; and Amsntons Remarques et Experiences
Physiques sur la Construction d'une nouvelle Clepsydre,
exempte des defauts des autres.

CLERC (John le), a celebrated writer and universal
scholar, was born at Geneva in 1657. After passing
through the usual course of study at Geneva, he went to
France in 1678; but returning the year after, he took
holy orders. In 1682 Lc Clerc visited England, to learn
the language : but the smoky air of London not agreeing
with his constitution, he soon returned to Holland, where
he settled ; and was appointed professor of philosophy, po-
lite literature, and the Hebrew tongue, in the school at
Amsterdam. Here he long continued to read lectures ;
for which purpose he drew up and published his Logic,
Ontology, Pneumatology, and Natural Philosophy. He
published also Ars Critica ; a Commentary on the Old
Testament; a Compendium of Universal History ; an Ec-
clesiastical History of the first two centuries ; a French
translation of the New Testament, and other works. In
1686, he began, jointly with M.dc la Crose, his Bibliothe-
que Universellc et Historique, in imitation of other lite-
rary journals ; which was continued to the year 1693,
making 26 volumes. In 1/03 he commenced his Biblio-
theque Choisie, and continued it to 1714, when he began
another work on the same plan, called Bibliotheque Anci-
< nne et Moderne, which he continued to the year 1728 ;
all of 1 hem justly esteemed excellent stores of useful know-
ledge, lie published also, in 1713, a small treatise on
Practical Geometry, in 2 vols, small Svo, called Pratique

de la Geometrie, sur le papier et sur le terrain. In 1728
he was seized with a palsy and fever; and after spending
the last six years, of his life with little or no understand-
ing, he died in 1736, at 79 years of age.

CLIFF, in Music, the same as Clef, which see.

CLIMACTERIC, a critical year in a person's life.

According to some, this is every 7th year : but others
allow it only to those years produced by multiplying 7 by
the odd numbers 3, 5, 7, .9- These years, observe they,
bring with them some remarkable change with respect to
health, life, or fortune: the grand climacteric is the 63d
year; but some add also the 81st to it : the other remark-
able climacterics are the 7th, 21st, 35th, 49th, and 56th.

CLIMATE, or Clitne, in Geography, a part of the sur-
face of the earth bounded by two lesser circles parallel
to the equator; and of such a breadth, as that the longest
day in the parallel nearer the pole exceeds the longest
day in that next the equator, by some certain space, as
half an hour, or an hour, or a month.

The beginning of a climate, is a parallel circle in which
the day is the shortest; and the end of the climate, is
that in which the day is the longest. The climates there-
fore are reckoned from the equator to the pole ; and are
so many zones or bands, terminated by lines parallel to the
equator : though, in strictness, there are several climates,
or different degrees of light or temperature, in the breadth
of one zone. Each climate only differs from its contigu-
ous ones, in that the longest day in summer is longer or
shorter, by half an hour, for instance, in the one place
than in the other.

As the climates commence at the equator, at the beoin-
ning of the first climate, that is at the equator, the day is
just 12 hours long; but at the end of it, or at the begin-
ning of the 2d climate, the longest day is 12 hours and a
half long; and at the end of the 2d, or beginning of the
3d climate, the longest day is 13 hours long; and so of
the rest, as far as the polar circles, where the hour cli-
mates terminate, and month climates commence. And as
an hour climate is a space comprised between two paral-
lels of the equator, in the first of which the longest day
exceeds that in the latter by an hour; so the month
climate is a space contained between two circles parallel
to the polar circles, and having its longest day longer or
shorter than that of its contiguous one, by a month, or 30
days. But some authors, as Riccioli, make the longest -
day of the contiguous climates to differ by half hours, to
about the latitude of 45 degrees; then to differ by an
hour, or sometimes 2 hours, to the polar circle ; and after
that by a month each. See tables of climates in Vare-
nius, chap. 25, prop. 13.

The ancients, who confined the climates to what they
thought the habitable parts of the earth, reckoned only-
seven, the middles of which they made to pass through
some remarkable places ; as the 1st through Meroe, the
2d through Sienna, the 3d through Alexandria, the 4th
through Rhodes, the 5th throughRome, the 6th through
Pontus, and the 7th through the mouth of the Borysthe-
nes. But the moderns, who have sailed farther towards
the poles, make 30 climates on each side.

Vulgarly the term climate is bestowed on any country
or region differing from another cither in respect of the
seasons, the quality of the soil, or even the manners of
the inhabitants; without any regard to the length of the.'
longest day. Abulfeda, an Arabic author, distinguishes.

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C L O

the first kind of climates by the term Real Climates, and
the latter by that of Apparent Climates.

CLOCK, a machine now constructed in such a manner,
and so regulated by the uniform motion of a pendulum,
•as to measure time, and all its subdivisions, with great
exactness. Before the invention of the pendulum, a ba-
lance, not unlike the fly of a kitchen-jack, was used in-
stead of it. — They were at first called nocturnal dials, to
distinguish them from sun dials, which showed the hour
by the shadow of the sun.

The invention of clocks with wheels is ascribed to Pa-
cificus, archdeacon of Verona, in the 9th century, on the
credit of an epitaph quoted by Ughelli, and borrowed by
him from Panvinius. Others attribute the invention to
Boethius, about the year 510.

Mr. Derham however reduces clock-work to a much
older date; ranking Archimedes's sphere, made 200 years
before Christ, as mentioned by Claudian, and that of Po-
sidonius, mentioned by Cicero, among machines of this
kind: not that either their form or use were the same with
those of modern mechanics ; but that they had their mo-
tion from some hidden weights, or springs, with wheels or
pulleys, or some such clock-work principle.

In the Disquisitiones Monasticae of Benedictus Haef-
ten, published in the year 1644, we are informed, that
clocks were invented by Silvester iv, a monk of his or-
der, about the year 998,' as Dithmarus and Bozius have
also shown ; for before that time, they had nothing butsun-
dials and clepsydras for pointing out the hour. — Conrade
Gesner, in his Epitome, pa. 604, says, that Richard Wal-
lingford, an English abbot of St. Albans, who flourished
in the year 1326, made a wonderful clock by a most ex-
cellent art, the like of which could not be produced by
all Europe. — Moreri, under the word Horologe du
Palais, observes, that Charles v, called the wise king
of France, ordered at Paris the first large clock to be
made by Henry de Vie, whom he sent for from Germany,
and set it upon the tower of his palace, in the year 1372.
—John Froissart, in his Histoire- et Chronique, vol. 2,
ch. 28, says, the duke of Bourgogne had a clock, which
sounded the hour, taken away from the city of Courtray,
in the year 1382 : and the same thing is said by Wm. Pa-
radin, in his Annates de Bourgogne.

Clock-makers were first introduced into England in
1368, when Edward m granted a licence for three artists
to come over from Delft in Holland, and practise their
occupation in this country.

The water-clocks, or clepsydra?, and sun-dials, have
both a much better claim to antiquity. The French an-
nals mention one of the former kind, sent by Aaron king
of Persia, to Charlemagne, about the year 807, which it
would seem bore some resemblance to the modern clocks :
it was of brass, and showed the hours by 12 little balls of
the same metal, which at the end of each hour fell upon a
bell, and made a sound. There were also figures of 12
cavaliers, which at the end of each hour came out through
certain apertures, or windows, in the side of the clock,
and shut them again, &c.

The invention of pendulum clocks is owing to the
happy industry of tlie 1 7 th century; and the honour of
that discovery is disputed between Galileo and Huygcns.
The latter, who wrote an excellent volume on the subject,
declares it was first put in practice in the year 1657, and
tkc description of it printed in 1658. Becher, De Nova

Temporis dimetiendi Theoria, anno 1680, contends for
Galileo ; and relates, though at second-hand, the whole
history of the invention; adding that one Trefler, clock-
maker to the father of the then grand duke of Tuscany,
made the first pendulum clock at Florence, under the di-
rection of Galileo Galilei, a pattern of which was brought
to Holland. And the Academy del Cimento expressly
observes, that the application of the pendulum to the
movement of a clock was first proposed by Galileo, and
put in practice by his son Vicenzo Galilei, in 1649. But
whoever may have been the inventor, it is certain that the
invention never flourished till it came into the hands of
Huygens, who insists, that if ever Galileo thought of such
a thing, he never brought it to any degree of perfection.
The first pendulum clock made in England was in the year
l662, by one Fromantil, a Dutchman.

Among the modern clocks, those ofStrasburg and Lyons
are very eminent for the richness of their furniture, and
the variety of their motions and figures. In the former,
a cock claps his wings, and proclaims the hour : the angel
opens a door, and salutes the Virgin, &c. In the latter,
two horsemen encounter, and beat the hour upon each
other : a door opens, and there appears on the theatre the
Virgin, with Jesus in her arms ; the Magi, with their retinue,
marching in order, and presenting their gifts ; two trum-
peters sounding all the while to proclaim the procession.

These however are far excelled by two that have lately-
been made by English artists, as a present from the East-
India company to the emperor of China. These two
clocks are in the form of chariots, in each of which a
lady is placed, in a fine attitude, leaning her right hand
on a part of the chariot, under which appears a clock of
curious workmanship, little larger than a shilling, that
strikes and repeats, and goes for eight days. On the lady's
finger sits a bird, finely modelled, and set with diamonds
and rubies, with its wings expanded in a flying posture,
and actually flutters for a considerable time, on touching
a diamond button below it; the body of the bird, in which
are contained part of the wheels that animate it as it were,
is less than the l6th part of an inch. The lady holds in
her left-hand a golden tube little thicker than a large pin,
on the top of which is a small round box, to which is fixed
a circular ornament not larger than a sixpence, set with
diamonds, which goes round in three hours in a constant
regular motion. Over the lady's head is a double um-
brella, supported by a small fluted pillar not thicker than
a quill, and under the larger of which a bell is fixed at a
considerable distance from the clock, with which it seems
not to have any connexion ; but from which a commu-
nication is secretly conveyed to a hammer, that regularly
strikes the hour, and repeats the same at pleasure, by
touching a diamond button fixed to the clock below. At
the feet of the lady is a golden dog ; before which, from
the point of the chariot, are two birds fixed on spiral
springs, the wings and feathers of which are set with
stones of various colours; and they appear as if flying
away with the chariot, which, from another secret mo-
tion, is contrived to run in any direction, either straight
or circular, &e ; while a boy, that lays hold of the cha-
riot behind, appears also to push it forwards. Above the
umbrella are flowers and ornaments of precious Stones;
and it terminates with a flying dragon set in the same
manner. The whole is of gold, most curiously executed,
and embellished with rubies and pearls.

C L O

[

]

CLO

The ingenious Dr. Franklin contriver! a clock to show
the hours, minutes, and seconds, with only three wheels
and two piniops in the whole movement. The dial-plate
had the hours engraven on it in spiral spaces along two
diameters of a circle, containing four times fjO minutes.
The index goes round in lour hours, and points out the
minutes from any hour by which it has passed to the next
following hour. The small hand, in an arch at top, goes
round once in a minute, and shows the seconds. The
clock is wound up by a line going over a pulley, on the
axis of the great wheel, like a common 30-hour clock.
Many of these very simple machines have since been con-
structed, that measure time exceedingly well. This clock
is subject, however, to the inconvenience of requiring fre-
quent winding, by drawing up the weight; as also to some
uncertainty as to the particular hour shown by the index.
Mr. Ferguson proposed to remedy these inconveni-
ences by another construction, which is described in his
Select Exercises, pa. 4. This clock will go a week with-
out winding, and always shows the precise hour; but, as
Mr. Ferguson acknowledges, it has two disadvantages
which do not belong to Dr. Franklin's clock : when the
minute hand is adjusted, the hour plate must also be set
right, by means.of a pin : and the smallness of the teeth
in the pendulum wheel will cause the pendulum ball to
describe but small arcs in its vibrations; and therefore
the momentum of the ball will be less, and the times of
the vibrations will be more affected by any unequal im-
pulse of the pendulum wheel on the pallets. Besides, the
weight of the flat ring, on which the seconds are engraven,
will load the pivots of the axis of the pendulum wheel
with a great deal of friction, which ought by all possible
means to be avoided. To remedy this inconvenience, the
seconds plate might be omitted;

Mr. Ferguson also contrived a clock, showing the ap-
parent diurnal motions of the sun and moon, the age and
phases of the moon, with the. time of her coming to the
meridian, and the times of high and low water ; and all
this by having only two wheels and a pinion added to the
common movement. See, his Select Exercises before men-
tioned. In this clock the figure of the sun serves as an
hour index, by going round the dial-plate in 24 hours ;
and a figure of the moon goes round in 24 h. 50| min.
the time of her going round in the heavens from any me-
ridian to the same meridian again. A clock of this kind
was adapted by Mr. Fcrguscm to the movement of an old
watch. See also a description and drawing of an astro-
nomical clock, showing the apparent daily motions of the
sun, moon, and stars, with the times of their rising, south-
ing, and setting; the places of the sun and moon in the
ecliptic, and the age and phases of the moon for every
duy in the year, in the same book, p. IQ.

There have been several treatises upon clocks ; the prin-
cipal of which are the following: Hieronymus Cardan,
de Varictate Rerum, libri 17. — Conrade Dasypodius, De-
scriptio Ilorologii Astronomici Argentinensis in summa
Templi erecti. — Guido Pancirollus, Antiqua deperdita et
nova reperta. — L'Usage du Cadran, ou dc l'Horloge Phy-
sique Universellc, par Galilee, Mathematicien du Due de
Florence. — Oughtred's Opuscula Mathematica. — Huy-
gens's Horologium Oscillatoriuin. — Pendule perpctuelle,
par 1'Abbe de Hautefeuille. — J..]. Bechcri Theoriaet Ex-
perientia de nova Temporis dimetiendi Ratione et Horo-
logiorum Constructione. — Clark's Oughtrcdus explica-
tes, ubi de Constructione Horologiorum. — Horological

Vol. I.

Disquisitions. — Iiuygens's Posthumous Works.-— Sully's
Regie Artificielle du Temps, &c. — Serviere's Recueil d'Ou-
vrages Curieux. — Dcrham's Artificial Clock-maker. — Ca-
mus's Trailes des Forces Mouvantes. — Alexandre's Traite
General des Horologies. — Also Treatises and Principles of
Clock-making, by Hatton, Cuming, &c, &c.

CLOUD, a visible collection of vapours, suspended in
the atmosphere.

Though it be generally allowed that the, clouds are
formed from the aqueous vapours, which before were so
closely united' with the atmosp'here as to T)e invisible; it
is however not easy to account for the long continuance
of some very opaque clouds without dissolving ; or to as-
sign the reason why the vapours, when they have once be-
gun to condense, do not continue to do so till they at last
fall to the ground in the form of rain or snow, &c. It is
now known, that a separation of the latent heat from the
water, of which vapour is composed, is attended with a
condensation of that vapour in some degree ; in such case,
it will first appear as a smoke, mist, or fog; which, if in-
terposed between the sun and earth, will form a cloud ;
and the same causes continuing to operate, the cloud will
produce rain or snow. It is however abundantly evident
that some other cause, besides mere heat or cold, is con-
cerned in the formation of clouds, and the condensation
of atmospherical vapours. This cause is esteemed in a
great measure the electrical fluid : indeed electricity is
now so generally admitted as an agent in all the great opeT
rations of nature, that it is no wonder to find the forma-
tion of clouds attributed to it; and this has accordingly
been given by Beccaria as the cause of the formation of
all clouds whatsoever, whether of thunder, rain, hail, or
snow.

Brit whether the clouds arc produced, that is, the at-
mospheric vapours rendered visible, by means of electri-
city or not, it is certain that they do often contain the
electric fluid in prodigious quantities, and many terrible
and destructive accidents have been occasioned by clouds
very highly electrified. The most extraordinary instance
of this kind perhaps on record happened in the island of
Java, in the East-Indies, in August, 1772. On the 11th
of that month, at midnight, a bright cloud was observed
covering a mountain in the district called Cheribou, and
several reports like those of a gun were heard at the same
time. The people who dwelt on the upper part of the
mountain not being able to fly fast enough, a great part of
the cloud, eight or nine miles in circumference, detached
itself under them, and was seen at a distance, rising and
falling like the waves of the sea, and emitting globes of
fire so luminous, that the night became as bright as day.
The effects of it were astonishing ; every thing was de-
stroyed for 20 miles round ; the houses were demolished;
plantations were buried in the earth; and 2140 people
lost their lives, besides 1500 head of cattle, and a vast
number of horses, goats, &c. Another remarkable in-
stance of the dreadful effects of electric clouds, which
happened at Malta the 2J)th of October 1757, is also re-
lated in Brydone's Tour through Malta.

The height of the clouds is not usually great; the sum-
mits of high mountains being commonly quite free from
them, as many travellers have experienced inpassing these
mountains. It is found that the most highly electrified
clouds descend lowest, their height being often not more
than 7 or 800 yards above the earth's surface ; and some-
times thunder-clouds appear actually to touch the ground
2 U

C L U

I 330 ]

C O F

with one of their edges; but the generality of clouds are
suspended at the height of a mile, or little more, above the
earth.

The motions of the clouds, though often directed by the
wind, are not always so, especially when thunder is about
to ensue. In this case they are seen to move very slowly,
or even to appear quite stationary for some time. The
reason of this probably is, that they are impelled by two
-opposite streams of air nearly of equal strength; and in
such cases it appears that both the aerial currents ascend
to a considerable height ; for Messrs. Charles and Robert,
while endeavouring to avoid a thunder-cloud, in one of
their aerial voyages with a balloon, could find no altera-
tion in the course of the current, though they ascended to
the height of 4000 feet above the earth. In some cases
the motions of the clouds evidently depend on their elec-
tricity, independent of any current of air whatever. Thus,
in a calm and warm day, small clouds are often seen meet-
ing each other in opposite directions, and setting out from
such short distances, that it cannot be supposed that any
opposite winds are the cause. Such clouds, when they
meet, instead of forming a larger one, become much
smaller, and sometimes entirely vanish ; a circumstance
most probably owing to the discharge of opposite electri-
cities into each other. And this serves also to throw some
light on the true cause of the formation of clouds; for if
two clouds, the one electrified positively, and the other
negatively, destroy each other on contact, it follows that
any quantity of vapour suspended in the atmosphere, will,
while it retains its natural quantity of electricity, remain
invisible, but becomes a cloud when electrified either plus
or minus.

The shapes of the clouds arealso probably owing to their
electricity; for in those seasons in which a great commo-
tion has been excited in the atmospherical electricity, the
clouds are seen assuming strange and whimsical forms,
that are continually varying. This, as well as the meet-
ing of small clouds in the air, and vanishing upon contaet,
is a sure indication of thunder.

The uses of the clouds are evident, as from them pro-
ceeds the rain that fructifies the earth, and without which,
according to the present state of nature, the whole surface
of the earth would soon become a mere desert. They are
also useful as a screen interposed between the earth and
the scorching rays of the sun, which are often so power-
ful as to destroy the grass and other tender vegetables. In
the more secret operations of nature also, where the elec-
tric fluid is concerned, the clouds bear a principal share;
and chiefly serve as a medium for conveying that fluid from
the atmosphere into the earth, and from the earth into the
atmosphere: in doing which, when electrified to a great
degree, they sometimes produce very terrible effects; an
instance of which is related above. See the articles At-
mospherical Electricity, Evaporation, Rain, &c.
CLOUTS, in Artillery; are thin plates of iron nailed on
that part of the axle-tree of a gun-carriage which comes
through (he nave, and through which the linspingoes.

CLUVIER, or Cluverius, (Philip), a celebrated
geographer, was born at Dantzic in 1580. After an edu-
i cation at home, he. travelled into Poland, Germany, and
the Netherlands, to improve himself in the knowledge of
the law. Rut, when at Leyden, Joseph Scaliger persua-
ded him to give way to his genius for geography. In pur-
suance of this advice, Cluvier visited the greatest part of
the European states. He was well skilled in many lan-

guages, speaking ten with facility, viz, Greek, Latin, Ger-
man, French, English, Dutch, Italian, Hungarian, Polish,
and Bohemian. On his return to Leyden, he taught there
with great applause ; and died in l6"23, being only 43 years
of age, justly esteemed the first geographer who had put
his researches in order, and reduced them to certain prin-
ciples. He was author of several ingenious works in geo-
graphy, viz :

1. De Tribus Rhcni Alveis.

2. Germania Antiqua.

3. Italia Antiqua, Sicilia, Sardinia, et Corsica.

4. Introductio in Universam Geographiam.
COASTING, is that part of Navigation in which the

places are not far asunder, so that a ship may sail in sight
of land, or within soundings, between them.

COCHLEA, one of the five Mechanical powers, other-
wise called the Screw ; being so named from the resem-
blance a screw bears to the spiral shell of a snail, in Latin
Cochlea. See Screw, and Mechanical Powers.

COCK of a Dial, the pin, stile, or gnomon.

COEFFICIENTS, in Algebra, are numbers, or given
quantities, usually prefixed to letters, or unknown quan-
tities, by which it is supposed they are multiplied ; and so,
with such letters, or quantities, making a» product, or co-
efficient production ; whence the name.

Thus, in 3a the coefficient is 3, in bx it is b, and in cr1
it is<\ If a quantity have no number prefixed, unity or
1 is understood ; as a is the same as la, and be the same as
lbc. The name coefficient was first given by Vieta.

In any equation so reduced as that its highest power or
term has 1 for its coefficient; then the coefficient of the
2d term is equal to the sum of all the roots, both positive
and negative ; so that if the 2d term is wanting in an equa-
tion, the sum of the positive roots of that equation is equal
to the sum of the negative roots, as they mutually balance
and cancel each other. Also the coefficient of the 3d
term of an equation is equal to the sum of all the reef-
angles arising by the multiplication of every two of the
roots, how many ways soever they can be combined by
twos; as once in the quadratic, 3 times in the cubic, 6
times in the biquadratic equation, &c. And the coeffi-
cient of the 4th term of an equation, is the sum of all the
solids made by the continual multiplication of every three
of the roots, how often soever such a ternary can be had ;
as once in a cubic, 4 times in a biquadratic, 10 times in an
equation of 5 dimensions, &c. And thus it goes on infi-
nitely.

Coefficients of the same Order, is a term sometimes
used for the coefficients prefixed to the same unknown
quantities, in different equations.

Cax ■+■ by -+- cz = m,

Thus in the equations -J dx -+- ey -+- ft = n,
igx +■ hy ■+- kz = p,
the coefficients a, d, g, are of the same order, being the
coefficients of the same letter x ; also b, e, h, are of the
same order, being the coefficients ofy; and so on.

O/DpowVeCoEFFiciENTS, such as are taken each from a
different equation, and from a different order of coefficients.
Thus, in the foregoing equations, a, e, k, or u, h,f, or d, b, k,
&C, are opposite coefficients.

OEI.KSTIAL. See Celestial.

COFFER, in Architecture, a square depressurc or sink-
ing, in each interval between the modillions of the Corin-
thian cornice; usually filled up with a rose) sometimes
with a pomegranate, or other enrichment.

C O H

[ 331 ]

C O H

Coffer, in Fortification, denotes a. hollow lodgment,
athwart a dry moat, 6 or 7 feet deep, and 16 or 18 broad.
The upper part of it is made of pieces (xf timber, raised 2
feet above the level of the moat ; the elevation having hur-
dles laden with earth for its covering, and serving as a pa-
rapet with embrazures.

The coffer is nearly the same with the caponiere, except-
ing that this last is sometimes made beyond the counterscarp
on the glacis, and the coffer always in the moat, taking up
its whole breadth, which the caponiere does not. It dif-
fers from the traverse and gallery, in chat these are made
by the besiegers, and the coffer by the besieged.

The besieged commonly make use of coffers to repulse
the besiegers, when they endeavour to pass the ditch. And,
on fhe other hand, the besiegers, to save themselves from
the fire of these coffers, throw up the earth on that side
towards the coffer. '

Coff ER-Dams, or Batardeaux, in Bridge-building, are
inclosures formed for laying the foundation of piers, and
for other works in water, to exclude the surrounding water,
and so prevent it from interrupting the workmen.

These inclosures are sometimes single, and sometimes
double, with clay rammed between them; sometimes they
are made with piles driven close together, and sometimes
the piles are notched or dove-tailed into one another; but
the moat usual method is, to drive piles with grooves in
them, at the distance of five or six feet from each other,
and then boards are let down between them, after which
the water is pumped out.

COGGESHALL'S Sliding-Rule, an instrument used in
Gauging, and so called from its inventor. See the descrip-
tion and use under Sliding-/?u/<;.

COHESION, in Natural Philosophy, as distinguished
from adhesion, is that species of attraction which, uni-
ting particle to particle, retains together the component
parts of the same mass or kind. Or it is that action or
power by which the homogeneous particles of bodies are
kept attached to each other.

This power was first considered by Ncw'ton as one of
the properties essential to all matter, and the cause of all
that variety observed in the texture of different terrestrial
bodies. He docs not attempt to investigate the nature of
this power, but wisely contents himself with calling it a
mutual attraction peculiar to the particles of matter. His
words arc as follow : " The particles of all hard homoge-
neous bodies, which touch one another, cohorewith a great
force; to account for which, some philosophers have re-
course to a kind of hooked atoms, which in effect is no-
thing else but to beg the thing in question. Others ima-
gine that the particles of bodies are connected by rest, i.e.
in effect by nothing at all; and others by conspiring mo-
tions, i.e. by a relative rest among themselves. For-my-
sclf, it rather appears to me, that, the particles of bodies
cohere by an attractive force, whereby they tend mutually
toward each other: which force, in the very point of con-
tact, is very great ; at little distances is less, and at a little
tarther distance is quite insensible."

It is uncertain in what proportion this force decreases
as the distance increases : Desaguliers conjectures, from
some phenomena, that it decreases as tin biquadratic or
4th power of the distance, so that at twice the distance it
acts 16 times more weakly, &c.

" Now if compound bodies be so hard, as by experience
we find some of them to be, and yet have a great many
hidden pores within them, and consist of parts only laid

together ; no doubt those simple particles which have no
pores within them, and which were never divided into
parts, must be vastly harder. For such hard particles,
gathered into a mass, cannot possibly touch in more than
a few points; and therefore much less force is required to
sever them, than to break a solid particle, whose parts
touch throughout all their surfaces, without any interme-
diate pores or interstices. But how such hard particles,
only laid together, and touching only in a few points,
should come to cohere so firmly, as in fact we find they do,
is inconceivable; unless there be some cause, whereby they
are attracted and pressed together. Now the smallest
particles of matter may cohere by the strongest attrac-
tions, ana! constitute larger, whoseattracting force is feebler:
and again, many of these larger particles cohering, may
constitute others still larger, whose attractive force is still
weaker; and so on for several successions, till the progres-
sion end in the biggest particle, on which the operations in
chemistry, and the colours of natural bodies depend; and
which by cohering compose bodies of a sensible magni-
tude."

Again, the opinion maintained by many, is that which is
so strongly defende/1 by J. Bernoulli, De Gravitate jEthe-
ris; who attributes the cohesion of the parts of matter to
the uniform pressure of the atmosphere : confirming this
opinton by the known experiment of two polished marble
planes, which cohere very stongly in the open air, but
easily drop asunder in an exhausted receiver. However,
if two plates of this kind be smeared with oil, to fill up the
pores in their surfaces, and prevent the lodgment of air,
and one of them be gently rubbed upon the other, they
will adhere so strongly, even when suspended in an ex-
hausted receiver, that the weight of the lower plate will not
be able to separate it from the upper one. And though
this theory might serve tolerably well to explain the co-
hesion of compositions, or greater collections of matter;
yet it falls far short of accounting for that first cohesion
of the atoms, or primitive corpuscules, of which the par-
ticles of hard bodies are composed ; for a body does not
fall in pieces in the exhausted receiver.

Again, some philosophers have positively asserted, that
the powers, or means, arc immaterial, by which matter
coheres; and, in consequence of this supposition, they
have so refined upon attractions and repulsions, that their
systems appear but little short of scepticism, or denying
the existence of matter altogether. A system of this kind
is adopted by Dr. Priestley, from Messrs. Boscovich and
Michell, to solve some difficulties concerning the New-
tonian doctrine of light. See his History of Vision, vol. ] ,
pa. 392. "The easiest method," says Priestley, " of solving
all difficulties, is to adopt the hypothesis of Mr. Boscovich,
who supposes that matter is not impenetrable, as has been
perhaps universally taken for granted; but that it con-
sists of physical points only, endued with powers of at-
traction and repulsion in the same manner as solid matler
is generally supposed to be : provided therefore that any
body moves with a sufficient degree of velocity, or has a
sufficient momentum to overcome any powers of repul-
sion that it may meet with, it will find no difficulty in ma-
king its way through any body whatever; for nothing
else will penetrate each other but powers, such as we
know do in fact exist in the same place, and counter-
balance or over-rule each other. The most obvious dif-
ficulty, and indeed almost the only one, that attends this
hypothesis, as it supposes the mutual penetrability of
2U2

C O H

[ 332 ]

C O H

matter, arises from the idea of the nature of matter, and
the perplexity we meet with in attempting to force two-
bodies into the same place. But it is demonstrable, that
the first obstruction arises from no actual contact of mat-
ter, but from mere powers of repulsion. This difficulty
•we can overcome ; and having got within one sphere of
repulsion, we fancy that we are now impeded by the solid
matter itself. But the very same is the opinion of the
generality of mankind with respect to the first obstruction.
Why, therefore, may not the next be another sphere of
repulsion, which may only require a greater force than
we can apply to make it yield without disordering the
arrangement of the constituent particles ; but which may-
be overcome, by a body moving with the astonishing 've-
locity of light."

Other philosophers have supposed that the powers, both
of gravitation and cohesion, are material ; and that they
are only different actions of the ethereal fluid, or elemen-
tary fire. In proof of this doctrine, they allege the ex-
periment with the Magdeburg hemispheres, as they are
called. The pressure of the atmosphere, we know, is, in
this case, capable of producing a very strong cohesion ;
and if there be in nature any fluid more penetrating, as
well as more powerful in its effects, than the air we breathe,
it is possible that what is called the attraction of cohesion
may in some measure be an effect of the action of that
fluid. Such a fluid as this is the element of fire. Its ac-
tivity is such as to penetrate all bodies whatever : and in
the state in which it is commonly called fire, it acts ac-
cording to the quantity of solid matter contained in the
body. In this state it is capable of dissolving the strong-
est cohesions observed in nature. Fire, therefore, being
able to dissolve cohesions, must also be capable of causing
them, provided its power be exerted for that purpose,
which possibly it may be, when we consider its various
modes or appearances, viz, as fire or heat, in which state
it consumes, destroys, and dissolves ; or as light, when it
seems deprived of that destructive power; and as the
electric fluid, when it attracts, repels, and moves bodies,
in a great variety of ways. In the Philos. Trans, for 1777
this hypothesis is noticed, and in some measure adopted
by Mr. Henley. " Some gentlemen (he observes) have
supposed that the electric matter is the cause of the co-
hesion of the particles of bodies. If the electric matter
be, as I suspect, a real elementary fire inherent in all bo-
dies, that opinion may probably be well founded ; and
perhaps the soldering of metals, and the cementation of
iron, by fire, may be considered as strong proofs of the
truth of their hypothesis."

But whatever the cause of cohesion may be, its effects
are evident and certain. The different degrees thereof,
constitute bodies of different forms and properties. Thus,
Newton observes, the particles of fluids, which do not co-
here too strongly, and are small enough to render them
susceptible of those agitations which keep liquors in a
fluor, arc most easily separated and rarefied into vapour,
and make what the chemists call volatile bodies ; being
rarefied with an easy heat, and again condensed with
a moderate cold. Those that have grosser particles,
and so are less susceptible of agitation, or cohere by a
stronger attraction, are not separable without a greater
degree of heat; and some of them not without fermenta-
tion ; and these constitute what the chemists call fixed
bodies.

Air, in its fixed state, possesses the interstices of solid

substances, and probably serves as a bond of union to their
constituent parts ; for when these parts are separated, the
air is discharged, and recovers its elasticity. And this
kind of attraction is evinced by a variety of familiar ex-
periments ; as, by the union of two contiguous drops of
mercury; by the mutual approach of two pieces of cork,
floating near each other in a bason of water; by the ad-
hesion of two leaden balls, whose surfaces are scraped and
joined together with a gentle twist, which is so considera-
ble, that, if the surfaces are about a quarter of an inch in
diameter, they will not be separated by a weight of 100 lb .
by the ascent of oil or water between two glass planes, so
as to form the hyperbolic curve, when they are made to
touch on one side, and kept separate at a small distance
on the other; by the depression of mercury, and by the
rise of water in capillary tubes, and on the sides of glas*
vessels ; also in sugar, sponge, and all porous substances
And where this cohesive attraction ends, a power of re-
pulsion begins.

To determine the force of Cohesion, in a variety of different
substances, many experiments have been made, and par-
ticularly by Professor Muschenbroek. The adhesion of
polished planes, about two inches in diameter, heated in
boiling water, and smeared with grease, required the fol4 -
lowing weights to separate them :

Cold Grease Hot Grease

Plane of Glass - 1301b - 3001b

Brass - 150 - 800

Copper - 200 - 850

Marble - 225 - 600

Silver - 150 - 250

Iron ' -. 300 • -. 950

But when the brass planes were made to adhere by
other sorts of matter, the results were as in the following
table :

With Water - - 12 ox

Oil - - IS

Venice Turpentine - 24

Tallow Candle - - 800

Rosin - - 850

Pitch - - 1400

In estimating the Absolute Cohesion of solid pieces of
bodies, he applied weights to separate them according to
their length : his pieces of wood were long square paral-
lelopipedons, each side of which was '26 of an inch, or
about J, and they were drawn asunder by the following
weights :

Fir - - 600 lb

Elm ' - - 950

Alder - - 1000

Linden-tree - - 1000

Oak - - 1150

Beech - - 1250

Ash - - 1250

He tried also wires of metal, 1-1 Oth of a Hhinland inch

in diameter : the metals and weights were as follows :

Of Lead - - 29$ lb

Tin - - 40$

Copper - - 299*

Yellow Brass - - 36'0

Silver - - 370

Iron - - 450

Gold - - 500

lie then tried the Relative Cohesion, or the force with

which bodies resist an action applied to them in a direction

COL

[ 333 ]

COL

Pine

9*

Fir

9

Beech

7

Elm

9

Oak

8|

Alder

9i

perpendicular to their length. For this purpose he fixed
pieces of wood by one end into a square hole in a metal
plate, and hung weights towards the other end, till they
broke at the hole: the weights and distances from the hole
are exhibited in the following table :

Distance Weight

:. - 36f oz.

40
561
44
48
48
See his Elem. Nat. Philos. cap. 19.

The inquisitive reader may further consult Ritter on
Cohesion ; Gilbert's Journal, iv, 1 ; Benzenberg on Co-
hesion, xvi, 76; Fontana on Solidity and Fluidity, Soc.
Ital. i, 89 ; and Dr. T. Young on the Cohesion of Fluids,
in the Philos. Trans. 1S05, or in the 2d vol. of his Nat.
Philos.

COIGNET (Michael), a writer on navigation, and a
native of Antwerp, who flourished in the latter part of the
l6th century, and died in 1623. He published a work
on navigation hi the Flemish language; which was after-
wards enlarged, and published in French, under the title
of "Instruction nouvelle des Points plus excellents et ne-
cessaires touchant l'Art de Naviger." This served as a
supplement to Medina's book on the same subject, whose
mistakes are well exposed by Coignet. In treating of the
sun's declination, he took notice of the gradual decrease in
the obliquity of the ecliptic, a point long disputed, but
now settled from the theory of attraction, as well as long
observation. He also described the cross-staff with three
transverse pieces, which he mentioned as being then in
common use among mariners ; but he preferred that of
Gemma Frisius. He mentioned likewise some instruments
of his own invention, which however are now quite disused.
As the old sea-table, above noticed, erred more and more
in advancing towards the poles, he set down another, to
be used by such as sailed beyond the 60th degree of lati-
tude. At the end of the book is delivered a method of
sailing on a parallel of latitude, by means of a ring dial,
and a 24-hour glass; on which the author very much
values himself.

COLD, the privation of heat, or the opposite to it.
As it is supposed that heat consists in a particular mo-
tion of the parts of a hot body, hence the nature of cold,
which is its opposite, is deduced ; for it is found that cold
extinguishes, or rather abates heat ; hence it would seem
to follow, that those bodies are cold, which check and
restrain the motion of the particles in which heat consists.
In general, cold contracts most bodies, and heat ex-
pands them : though there are some instances to the con-
trary, especially in the extreme cases or states of these
qualities of bodies. Thus, though iron, in common with
other bodies, expand with heat, yet, when melted, it is
always found to expand in cooling again. So also, though
water always is found to expand gradually as it is heated,
and to contract as it cools, yet in ,the act of freezing, it
suddenly expands again, and that with a most enormous
force, capable of rending rocks, or bursting the very thick
shells of metal, ike. A computation of the force of
freezing water has been made by the Florentine Acade-
micians, from the bursting of a very strong brass globe or
shell, by freezing water in it ; when, from the known
thickness and tenacity of the metal, it was found that the

expansive power of a spherule of water only one inch in
diameter, was sufficient to overcome a resistance of more
than 27,000 pounds, or 13 tons and a half. See also ex-
periments on bursting thick iron mortar-shells by freezing
water in them, by Major Edward Williams of the Royal
Artillery, in the Edinb. Philos. Trans, vol.2.

Such a prodigious power of expansion, almost double
that of the most powerful steam-engines, and exerted in so
small a mass, seemingly by the force of cold, was thought
a very powerful argument in favour of those who sup-
posed that cold, like heat, is a positive substance. Dr.
Black's discovery of latent heat, however, has now af-
forded a very easy and natural explication of this phe-
nomenon. He has shown, that, in the act of congelation,
water is not cooled more than it was before, but rather
becomes wanner: that as much heat is discharged, and
passes from a latent to a sensible state, as, had it been
applied to water in its fluid state, would have heated it to
135°. In this process, the expansion is occasioned by a
great number of minute bubbles suddenly produced. For-
merly these were conjectured to be cold in the abstract;
and to be so subtile, that, insinuating themselves into the
substances of the fluid, they augmented its bulk, at the
same time that, by impeding the motion of its particles on
each other, they changed it from a fluid to a solid. But
Dr. Black shows that these are only air extricated during
the congelation ; and to the extrication of which he
ascribes the great expansive force exerted by freezing wa-
ter. The only question therefore now remaining, is, by
what means this air comes to be extricated, and to take
up more room than it naturally does in the fluid. To this
it may be answered, that perhaps part of the heat which
is discharged from the freezing water, combines with the
air in its unelastic state, and, by restoring its elasticity,
gives it that extraordinary force, as is seen also in the
case of air suddenly extricated in the explosion of gun-
powder.

Cold also usually tends to make bodies electric, which
are not so naturally, and to increase the electric proper-
ties of such as are so. And it is farther found, -that all
substances do- not transmit cold with equal facility; but
that the best conductors of electricity, viz, metals, are also
the best conductors of cold. It may farther be added,
that when cold has been carried to such an extremity as
to render any body an electric, it then ceases to conduct
the cold so well as before. This is exemplified in the
practice of the Laplanders and Siberians; where, to ex-
clude the extreme cold of the winters from their habita-
tions the more effectually, and to admit at the same time
a small portion of light, they cut pieces of ice, which in
the winter season must always be electric in those coun-
tries, and place them in their windows ; which they find
to be much more effectual in keeping out the cold, than
any other substance.

Cold is the destroyer of all vegetable life, when increased
to an excessive degree. It is found that many garden
plants and flowers, which seem to be very stout and hardy,
go off at a little increase of cold beyond the ordinary stan-
dard. And in severe winters, nature has provided the best
natural defence for the corn-fields and gardens,. by a co-
vering of snow, which preserves such parts green and
healthy as are under it, while such as are not covered by
it are either killed or greatly injured.

Dr. Clarke is of opinion, that cold is owing to certain
nitrous, and other saline particles, endued with particular

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figures proper to produce sucH effects. Hence, sal-am-
rnoniac, saltpetre, or salt of urine, and many other volatile
and alkalizate salts, mixed with water, very much increase
its degree of cold. In the Philos. Trans, number 274,
M. Geoffroy relates some remarkable experiments with re-
gard to the production of cold. Four ounces of sal-am-
moniac dissolved in a pint of water, made his thermometer
descend 2 inches and £ in less than 15 minutes. An ounce
of the same salt put into 4 or 5 ounces of distilled water,
made the thermometer descend 2 inches and i. Half an
ounce of sal-ammoniac mixed with 3 ounces of spirit of
nitre, made the thermometer descend 2 inches and TST ;
but, on using spirit of vitriol instead of nitre, it sunk 2
inches and f . In this last experiment it was remarked,
that the vapours raised from the mixture had a consider-
able degree of heat, though the liquid itself was so extreme-
ly cold. Four ounces of saltpetre mixed with a pint of
water, sank the thermometer an inch and \; but a like'
quantity.of sea salt sank it only ■£ of an inch. Acids al-
ways produced heat, even common salt with its own spirit.
Volatile alkaline salts produced cold in proportion to their
purity, but fixed alkalies heat.

But a far greater degree of cold produced by the mix-
ture of salts and aqueous fluids, was that shown by Hom-
berg; who gives the following receipt for making the ex-
periment-: Take a pound of corrosive sublimate, and as
much sal-ammoniac ; powder them separately, and mix
the powders well; put the mixture into a phial, pouring
upon it a pint and a half of distilled vinegar, shaking all
well together. This composition becomes so cold, that it
can scarce be held in the hand in summer; and it hap-
pened, as M. Homberg was making the experiment, that
the matter froze. The same thing once happened to M.
Geoffroy, in making an experiment with sal-ammoniac and
water, but it never was in his power to make it succeed a
second time.

If, instead of making these experiments with fluid water,
it be taken in its congealed state of ice, or rather snow,
degrees of cold will be produced greatlj' superior to any
that have yet been mentioned. A mixture of snow and
common salt sinks a Fahrenheit's thermometer to 0; pot-
ashes and pounded ice sank it 8 degrees farther; two af-
fusions of spirit of salt on pounded ice, sank it 14| below
0; and by repeated affusions of spirit of nitre M. Fahren-
heit sank it to 40° below 0. This is the ultimate degree
of cold which the mercurial thermometer will measure;
for the mercury itself begins then to congeal; and there-
fore recourse must afterwards be had to spirit of wine,
naphtha, or some other fluid that will not congeal. The
greatest degree of cold hitherto produced by artificial
means, has been SO0 below 0; which was done at Hud-
son's Bay by means of snow and vitriolic acid, the ther-
mometer standing naturally at 20° below 0. Indeed
greater degrees of cold than this have been supposed: Mr.
Martin, in his Treatise on Heat, relates, that at Kirenga in
Siberia, the mercurial thermometer sunk to 1)8° below
0; and Professor Braun at Petersburg, when he made the
first experiment of congealing quicksilver, fixed the point
of congelation al 350° below 0; but from later experi-
ments it has been more accurately determined, that 40n
below 0 is the freezing point of quicksilver.

The most remarkable experiment however was made by
Mr. Walker of Oxford, with spirit of nitre poured ( n
Glauber's salt, the effect of which was found to be similar
to that of the same spirit poured on ice or snow; ai»d the

addition of sal-ammoniac rendered the cold still more in-
tense. The proportions of these ingredients recommended
by Mr. Walker, are concentrated nitrous acid two parts by
weight, water one part; of this mixture, cooled to the
temperature of the atmosphere, 18 ounces; of Glauber's
salt, a pound and a half avoirdupoise ; and of sal-ammo-
niac, 12 ounces. On adding the Glauber's salt to the ni-
trous acid, the thermometer fell 52°, viz, from 50 to —2;
and on the addition of the sal-ammoniac it fell to — 9.
Thus Mr. Walker was able to freeze quicksilver without
either ice or snow, when the thermometer stood at 45°;
viz, by putting the ingredients in 4 different pans, and in-
closing these within each other.

Excessive degrees of cold also occur naturally in many-
parts of the globe in the winter season.

Though the thermometer in this country seldom de-
scends so low as 0, yet in the winter of 17S0, Mr. Wilson
of Glasgow observed, that a thermometer laid on the snow
sunk to 25° below 0; and Mr. Derham, in the year 1708,
observed in England, that the mercury stood within one-
tenth of an inch of its station when plunged into a mix-
ture of snow and salt. At Petersburg, in 1732, the ther-
mometer stood at 28° below 0; and when the French aca-
demicians wintered near the polar circle, they observed
the thermometer at 33° below 0 ; and in the Asiatic and
American continents still greater degrees of cold are often
observed. Lately, Professor Leslie, of Edinburgh, has fro-
zen water, merely by evaporation in the exhausted receiver
of an air-pump; and, it is said, he can freeze mercury in
the same way. Mercury has also been frozen by Dr.
Marcct, by the evaporation of ether. See Nicholson's
Journal, v.34, pa. 119; also the article Freezing in thi*
Dictionary. Further, a paper of Dr. Marcer/s was read at
the Royal Society, July 8, 1813, on the intense degree of
cold produced by the sulphuret of carbon. From the au-
thor's experiments it appears that this liquor is the most
evaporable of all known fluids, or at least to produce, by its
evaporation, the most intense degree of cold. If the bulb
of a spirit-thermometer, closely enveloped in fine flannel,
or cotton wool, be moistened with the fluid, its temperature
falls to about 0; but if the thermometer be exposed to the
effect of a vacuum, by being enclosed in the receiver of a
good air-pump, it sinks to — 80°, in a minute or two. The
congelation of mercury in glass tubes may therefore be
most quickly and easily performed by this process at all
seasons, and under any atmospheric temperature.

In the last bulletin of the retreat of the French from
Moscow, in November and December 1812, the cold, tl\eie
stated at 16 and IS below freezing, is perhaps to be under-
stood as being on the scale of Reaumur. The thermometer
most in use through England is Fahrenheit's; and upon
that scale the cold above mentioned by the French would be
from 8 to 1 1 below zero; a degree in England very rarely
known, but common enough amidst the horrors ot Russia;
where, in 1773, the Empress Catharine made the famous
experiment on the ordnance shell, of the largest dianii'ter,
IS inches. The shell was filled with water, in her pre-
sence, as she states, and in less than an hour the whole
was so frozen as to burst, and with no small explosion! —
" Avec beaucoup de fracas ! " Some leu- years before,
there had been another curious experiment by Prince Oi-
loffj who, in a foundation for a massy brick coach gateway
to his court-yard, in a soil so loose and wet, that piles
must have been driven, filled so many corresponding ca-
vities with water, on which, when turned by the frost into

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columns, he raised the projected superstructure ! And tit
that time (1773) it had lasted four years, and so might be
likely to last for ever ! The words of the report are, " Une
porte cochere voutee en briques, et tres solide ! Elle ex-
iste depuis quatre ans; et elle existera, je crois, jusqu'a
cc qu'on l'abatte!"

The effects of these extreme degrees of cold are very sur-
prising. Trees are burst, rocks rent, and rivers and lakes
frozen several feet deep : metallic substances blister the
skin, like red-hot iron: the air, when drawn in by breath-
ing, hurts the lungs, and excites a cough: even the effects
of fire in a great measure seems to cease; and it is observed,
that though metals are kept for a considerable time before
a. strong fire, they will still freeze water when thrown upon
them. When the French mathematicians wintered at Tor-
nea in Lapland, the external air, when suddenly admitted
into their room's, converted the moisture of the air into
whirls of snow ; their breasts seemed to be rent when they
breathed it, and the contact thereof was intolerable to their
bodies; and the spiric of wine, which had not been highly
rectified, burst some of their thermometers by the conge-
lation of the aqueous part.

Extreme cold often proves fatal to animals in those
countries where the winters are very severe; thus 7000
Swedes perished at' once in attempting to pass the
mountains which divide Norway from Sweden. But it is
not necessary that the cold, in order to destroy human life,
should be so very intense as has just been mentioned; it is
only requisite to be a little below 32° of Fahrenheit, or the
freezing point, accompanied with snow or hail, from which
shelter cannot be obtained. The snow which falls upon
the clothes, or the uncovered parts of the body, then melts,
and by a continual evaporation carries off the animal heat
to such a degree, that a sufficient quantity is not left for
the support of life. In such cases, the peraon first feels
himself extremely chill and uneasy; he turns listless, un-
willing to walk or use exercise to' keep himself warm, and
at last turns drowsy, sits down to refresh himself with sleep,
but awakes no more. See Freezing.

A striking illustration of these effects of cold is related
by captain Cook, in an occurrence which took place du-
ring a botanical excursion of Sir Joseph Banks and Dr.
Solander among the hills of Terra del Fuego. The party,
consisting of 11 persons, were overtaken by darkness, and
obliged to spend the night on the hills, during extreme
cold. Dr. Solander, who had more than once crossed the
mountains which divide Sweden from Norway, well knew
that extreme cold, especially when joined with fatigue, pro-
duces a torpor and sleepiness that are almost irresistible;
he therefore conjured the company to keep moving, what-
ever pains it might cost them, and whatever relief they
might be promised by an inclination to rest: " Whoever
sits down," said he, " will sleep; and whoever sleeps, will
wake no more.''' Thus at once admonished and alarmed,
they set forward ; but while they were still upon the naked
rock, and before they had got among the bushes, the cold
became suddenly so intense, as to produce (he effect that
had been most dreaded ; Dr. Solander himself was the first
who found the inclination, against which he had warned
others, irresistible, and insisted upon being suffered to lie
down. Mr, Banks entreated and remonstrated in vain ;
down he lay upon the ground, though it was covered with
snow; and it was with great difficulty that his friend kept
liim from sleeping. Richmond also, one of the black ser-
vants, began to linger, having suffered from the cold in the

same manner as the Doctor. Mr. Banks therefore sent 5
of the company forward to get a fire ready at the first con-
venient place they could find; and himself, with 4 others,
remained with the Doctor and Richmond, whom partlyby
persuasion and entreaty, and partly by force, they brought
in; but when they had got through the greater part of the
birch and swamp, they both declared they could go no
farther. Mr. Banks had recourse again to entreaty and
expostulation, but they produced no effect. When Rich-
mond was told that if he did not go on he would in a short
time be frozen to death, he answered that he desired no-
thing but to lie down and die. The Doctor did not so ex-
plicitly renounce his life; he said he was willing to go on,
but that he must first take some sleep, though he had be-
fore told the company that to sleep was to perish. Mr.
Banks and the rest found it impossible to carry them, and
there being no remedy, they were both suffered to sit down,
being partly supported by the bushes, and in a few minutes
they fell into a profound sleep. Soon after some of the
people who had been sent forward returned with the wel-
come news, that a fire was kindled about a quarter of a
mile farther on the way. Mr. Banks then endeavoured to
wake Dr. Solander, and happily succeeded ; but, though
he had not slept five minutes, he had almost lost the use of
his limbs, and the muscles were so shrunk, that his shoes
fell from his feet ; — he. consented to go forward with such
assistance as could be given him ; but no attempts to relieve
poor Richmond were successful. He, together with ano-
ther black left with him, died. Several others began to
lose their sensibility, having been exposed to the cold and
snow near an hour and a half, but the fire recovered
them. See Captain Cook's first Voyage; also Rees's Cy-
clopaedia, art. Cold.

COLLECTOR, in Electricity, is a small appendage to
the prime conductor of the electrical machine, and gene-
rally consisting of pointed wires, affixed to that end of the
prime conductor which stands contiguous to the glass
globe, or cylinder, or other electric of the machine. Its
office is to receive the electricity, whether positive or ne-
gative, from the excited electric, much more readily than
the blunt end of the prime conductor would be able to
receive it, without that appendage.

COLLEGE (Royal Military), was instituted in
1799» under the direction of 12 commissioners, a governor,
and professor of mathematics. It consists of two depart-
ments; the senior, established first at High Wycombe, audi
the junior at Great Marlow, in Buckinghamshire, each
under the care of a commandant, supcrintendant, and ad-
jutant, and under the instruction of a number of able ma-
sters in different branches of science. But in the year
1813 the senior department has been removed to Farnham,
and the junior to Sandhurst, near Bagshot, Surrey. The
pupils in the senior department are young officers who
have received their commissions; those in the junior de-
partment are gentlemen cadets, amounting to about 3 or
400 in number, are educated and fitted for commissions
in the army, each cadet payin'g a certain annual sum, to-
wards the expense of his education and maintenance, &c.

College, Navrrf, the new name of the Nftval Academy
at Portsmouth. See Academy.

COLLIMAT10N, Line of, in a telescope, is a line pass-
ing through the intersection of those wires that are fixed
in the focus of the object-glass, and the centre of the same
glass.

COLLINS (John), an eminent accountant and mathe-

COL

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matician, was born at Wood Eaton near Oxford, March 5,
lt>24. He received his education from his father, a dis-
senting minister, and at 16 years of age was put appren-
tice to a bookseller at Oxford ; but soon after the com-
mencement of the civil wars he became a clerk of the
Prince of Wales's kitchen. In this situatiou;be was under
the superintendence of Mr. Marr, a good mathematician,
and famous for the dials with which he adorned the king's
gardens. From hence he travelled abroad, to prosecute
his favourite stud)'; and on his return he took upon him
the profession of an accountant, and published, in the year
l6'52, a large work entitled, An Introduction to Merchants
Accompts; which was followed by several other publica-
tions on different branches of accounts. In l6'58, he pub-
lished a treatise called The Sector on a Quadrant; con-
taining the description and use of four several quadrants,
each accommodated to the making of sun-dials, &c; to
which he afterwards added an appendix concerning re-
flected dialling, from a glass placed reclining. — In l65£),
he published bis Geometrical Dialling ; and the same year
also his Mariner's Plain Scale new plained.— (Collins now
became a Fellow of the Royal Society inLondon, to which
he made various communications; particularly some in-
genious chronological rules for the calendar, printed in
the Philos. Trans, number 46, for April 1669: also a cu-
rious dissertation concerning the resolution of equations
in numbers, printed also in the Trans, for March 1671 .'
an elegant construction of the curious problem, Having
given the mutual distances of three objects in a plane, with
the angles made by them at a fourth place in that plane,
to find the distance of this place from each of the three
former, vol. 6, pa. 2093: and thoughts about some defects
in algebra, vol. 14, pa. 375.

Collins wrote also several commercial tracts, highly ac-
ceptable to the public; viz, A Plea for bringing over Irish
cattle, and keeping out the fish caught by foreigners: —
For the promotion of the English fishery : — For the work-
ing of the Tin-mines : — A discourse on Salt and Fishery.
He was frequently consulted in nice and critical cases of
accounts, of commerce, and engineering. On one of these
occasions, being appointed to inspect the ground for cut-
ting a canal or river between the Isis and the Avon, he con-
tracted a disorder by drinking cider when he was too
warm, which ended in his death the 10th of November
16'83, at 59 years of age.

Mr. Collins was a very useful man to the sciences, keep-
ing up a constant correspondence with the most learned
men, both at home and abroad, and promoting the publi-
cation of many valuable works, which, but for him, would
never have been introduced to the public; particularly
Dr. Barrow's optical and geometrical Lectures; his abridg-
ment of the Works of Archimedes, Apollonius, and Theo-
dosius; Branker's translation of Rhronius's Algebra, with
Dr. Pell's additions, &c; which were procured by his fre-
quent solicitations.

After a considerable time had elapsed, his papers were
all delivered into the hands of the learned and ingenious
Mr. William Jones, f. r. s, among which were found ma-
nuscripts, on mathematical subjects, of Briggs, Oughtrcd,
Barrow, Newton, Pell, and many others. From a variety
of letters from these, and many other celebrated mathe-
maticians, it appears that Collins spared neither pains nor
expense to procure what tended to promote real science :
and even many of the late discoveries in physical know-
ledge are indebted to him for their improvement; for

while he excited some to make known every new and
useful invention, he employed others to improve them.
Sometimes he was peculiarly useful, by showing where
the defect was in any useful branch of science, pointing
out the difficulties attending the inquiry, and at other
times setting forth the advantages, and keeping up a spirit
and warm desire for improvement. Mr. Collins was also
as it were tlie register of all the new improvements made
in the mathematical sciences; the magazine to which the
curious had frequent recourse : in so much that he ac-
quired the appellation of the English Mersennus. If some
of his correspondents had not obliged him to conceal their
communications, there could have been no dispute about
the priority of the invention of a method of analysis, the
honour of which evidently belongs to Newton ; as appears
undeniably from the papers printed in the Commercium
Epistolicum D. Joannis Collins et aliorum de Analyst
promota; jussu SocietatisRegia? in lucem editum, 1712;
a work that was composed from the letters in the pos-
session of our author.

Collins's Quadrant. See Quadrant.

COLLISION, is the percussion or striking of bodies
against each other. Striking bodies are considered either
as elastic, or non-elastic. They may also be cither both
in motion, or one of them in motion and the other at rest.
When non-elastic bodies strike, they adhere together as one
mass ; which, after collision, either remains at rest, or
moves forward as one body. But when elastic bodies
strike, they always separate after the stroke. The prin-
cipal theorems relating to the collision of bodies, are the
following :

-1. If any body impinge or act obliquely on a plane sur-
face; the force or energy of the stroke, or action, is as the
sine of the angle of incidence. Or the force upon the sur-
face, is to the same when acting perpendicularly, as the
sine of incidence is to radius.

2. If one body act on another, in any direction, and by
any kind of force ; the action of that force on the second
body is made only in a direction perpendicular to the sur-
face on which it acts.

3. If the plane, acted on, be not absolutely fixed, it
will move, after the stroke, in the direction perpendicular
to its surface.

4. If a body a strike another body b, which is either at
rest, or else in motion, either towards a or from it; then
the momenta, or quantities of motion, of the two bodies,
estimated in any one direction, will be the very same after
the stroke that they were before it.

ABC

O ; o- ■ 1

Thus, first, if a with a momentum of 10, strike B at rest,
and communicate to it a momentum of 4, in the direction
ab. Then there will remain in a only a momentum off)
in that direction: which together with the momentum of
n, viz 4, makes up still the same momentum between them
as before. — But if B were in motion before the stroke, with
a momentum of 5, in the same direction, and receive hum
a an additional momentum of 2: then "the motion (if a
after the stroke will be 8, and that of n, 7 ; which toge-
ther make 15, the same as 10 and 5, the motions before
the stroke. — Lastly, if the bodies move in opposite direc-
tions, and meet each other, namely a with a motion of 10,
and n, of 5; and a communicate to b a motion of 6' in
the direction ab of its motion: then, before the stroke, the
whole motion from both, in the direction a b, is 10— 5, or

COL

[

5: but after the stroke the motion of a is 4 in the direc-
tion ab, and the motion of b is 6 — 5, or 1, in the same
direction ab; therefore the sum 4-1-1, or 5, is still the
same motion from both as it was before.

5. If a hard and fixed plane be struck either by a soft or
a hard unelastic body ; the body will adhere to it. But
if the plane be struck by a perfectly elastic body, it will
rebound from it withthesame velocity with which itstruck
the plane.

6. The effect of the blow of the elastic body, upon the
plane, is double to that of the non-elastic one ; the velo-
city and mass being the same in both.

7. Hence, non-elastic bodies lose, by their collision, but
half the motion that is lost by elastic bodies ; the masses
and velocities being equal.

)C

8. If an elastic body a impinge upon a firm plane de at
the point b, it will rebound from it in an angle equal to
that in which itstruck it; or the angle of incidence will be
equal to the angle of reflection : namely, the angle abd

= CBE.

e

9. If the non-elastic body b, moving with the velocity
v in the direction b6, and the body 6 with the velocity v,
strike each other, the direction of the motion being in the
line BC ; then they will move after the stroke with a com-
mon velocity, which will be more or less according as, be-
fore the stroke, 6 moved towards b, or from b, or was at
rest; and that common velocity, in each of these cases, will
be as follows : viz,

when b moved from b,

B + b
BV — bv

when b moved towards b,

s + b

when b was at rest.

For example, if the bodies or weights, b and b, be 5lb
and 31b ; and their velocities v and v, 60 feet and 40 feet
per second; then 300 and 120 will be their momenta bv
and bv, and 8 = b •+- b the sum of the weights. Conse-
quently the common velocity after the stroke, in the three
cases above-mentioned, will be thus, viz,
000 + 120 420 „„T . , _
-> = — or 52J in the first case,

8 8 '

300 — 120 160 „„T . , ,

-; — = — or 22| in the second case,

300

— - - - or 371. in the third case.

10. If two perfectly clastic bodies impinge on each
other; their relative velocity is the same both before and
after the impulse; that is, they will recede from each other
with the same velocity with which they approached and
met.

It is not meant however by this theorem, that each body
will have the very same velocity after the impulse as it had
before; for that will be varied according to the relation of
the masses of the two bodies ; but that the velocity of the
one, after the stroke, will be so much increased, and the
other decreased, as to have the same difference as before,

Vol. I.

337 ] COL

in one and the same direction. So, if the elastic body b
move with the velocity v, and overtake the elastic body b,
moving the same way, with the velocity v; then their re-
lative velocity, or that with which they strike, is only
v— 0; and it is with this same velocity that they separate
from each other after the stroke : but if they meet each
other, or the body b move, contrary to the body b; then
they meet and strike with the velocity v -+• v, and it is with
the same velocity that they separate again, and recede from
each other after the stroke : in like manner, they would
separate with the velocity v of b, if b were at rest before
the stroke. Also the sum of the velocities of the one body,
is equal to the sum of the others. But whether they move
forwards or backwards after the impulse, and with what
particular velocities, are circumstances that depend on the
various masses and velocities of the bodies before the stroke,
and are as specified in the next theorem.

11. If the two elastic bodies b and b move directly to-
wards each other, or directly from each other, the former
with the velocity v, and the latter with the velocity v;
then, after their meeting and impulse, the respective velo-
cities of b and b in the direction bc, in the three cases of
motion, will be as follow : viz,

2bu + (b— b)v ., , . r

■ — the velocity of b,

B + b J '

— ^ — the velocity of b,

B + b J

when the bodies both moved towards c before the stroke ;

and

— 2fo + (B— b)v , , . e

- — the velocity of b,

B+b . .

2BV + (B — b)v , , . c,

; the velocity 01 0,

B + b . .

when b moved towards c, and b towards b before the stroke;
also,

B — b . , .

x v the velocity of b,

a + b

x v the velocity of b,

when b was at rest before the stroke. .

12. The motions of bodies after impact, that strike each
other obliquely, are thus determined.

•pqv jE

;: ^s. g!--- -— ^qi

'"£ 'Si' Va?\^ %
\ / g| ^^oj

C^ ;F

Let the two bodies b, b, move in the oblique direction^
ba, 6a, and strike each other at a, with velocities which
are in proportion to the lines ba, 6a. LetCAH represent
the plane in which the bodies touch in the point of con-
course; to which draw the perpendiculars bc, 6d, and
complete the rectangles ce, df. Now the motion in ba is
resolved into the two bc, ca; and the motion in 6a is re-
solved into the two bv, da; of which the antecedents bc,
6d are the velocities with which they directly meet, and
the consequents ca, da arc parallel, and therefore by
these the bodies do not impinge on each other, and conse-
quently the motions according to these directions will not
be changed by the impulse; so that the volocitics with
which the bodies meet, arc as bc or ea, and 6d or ia.

U

COL

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COM

The motions therefore of the bodies B, b, directly striking
each .other with the celerities ea, fa, will be determined
by art. 11 or 9, according as the bodies are elastic or non-
elastic; which being done, let AG be the velocity, so de-
termined, of one of them, as a ; and since there remains
also in the same body a force of moving in the direction
parallel to be, with a velocity as be, make ah equal to
be, and complete the rectangle gh : then the two mo-
tions in ah and ag, or hi, are compounded into the dia-
gonal a i, which therefore will be the path and celerity of
the body b after the stroke. And after the same manner
is the motion of the other body b determined after the im-
pact.

13. The state of the common centre of gravity of bodies
is not affected by the collision or other actions of those
bodies on each other. That is, if it were at rest before
their collision, so will it be also at rest after collision : and
if it were moving in any direction, and with any velocity,
before collision; it will do the very same after collision.

See more on this subject under the article Percussion.

COLONNADE, a peristyle of a circular figure; or a"
series of columns disposed in a circle, and insulated within-
side.

COLOUR, a property inherent in light, by which, ac-
cording to the various sizes of its parts, or from some other
cause, it excites different vibrations in the optic nerve;
which, propagated to the' sensorium, affect the mind with
different sensations. See the doctrine of colours fully ex-
plained under Chromatics. See also Optics, Achromatic,
and Telescope.

COLSON (John), a respectable mathematician, who
flourished about the beginning of the 18th century, and
who appears to have been a Cambridge scholar. We first
notice him as author of some papers in the Philos. Trans,
viz, one in vol. 25 (an. 1707) on The Universal Resolution
of Cubic and Biquadratic Equations; .a second in vol. 34,
on Negative-Affirmative Arithmetic; and 3dly, in vol.39,
on the Construction and Use of Spherical Maps. He was
also author of several separate works; as, The British
Hemisphere, being a map of a new contrivance, in the
form of a half globe, of about 15 inches in diameter, but
comprehending the whole known surface of the habitable
earth, the city of London being the centre or vertex of the
map. In 1736 he published a translation from the Latin
of Sir Isaac Newton's Method of Fluxions and Infinite
Series, with a large comment on the whole work, consist-
ing of annotations, illustrations, and supplements. At the
date of this publication Mr. Colson was in the situation of
master of Sir Joseph Williamson's free mathematical
school at Rochester, and a f. r. s. Three years after this
we find he succeeded Mr. Sanderson as Lucasian Professor
of Mathematics at Cambridge, a situation which he held
for 20 years, after which he was succeeded, in 1759, by
Dr. Waring; though Mr. Colson lived till Dec. 20, 1760.
The character of our author seems to have been chiefly
that of minute precision and patient laborious industry.
His paper on the solution of cubic and biquadratic equa-
tions above-mentioned, is founded nearly on the idea of
the solution by Descartes, by assuming the biquadratic as
equal to the product of two quadratics with indeterminate
coefficients. The construction of the equations also, by
means of the circle and parabola, is in imitation of the
like constructions of Descartes in his Geometry, and of Mr.
Baker in his Geometrical Key. Mr. Colson, it seems, was
so well pleased with the Analytical Institutions of theSig-

nora Agnesi, that he made an entire translation of that
lady's work, from the Italian, the copy of which was found
among his papers, and published at the expense of Mr.
Baron Maseres in 2 volumes 4to.

COLUMBA Noachi, Noah's Dove, a small constellation
in the southern hemisphere, consisting of 10 stars.

COLUMN, in Architecture, a round pillar, made to
support or adorn a building.

The column is the principal part of an architectonical
order, and is composed of three parts, the base, the shaft,
and the capital ; each of which is subdivided into a num-
ber of lesser parts, called members, or mouldings.

Columns are different according to the different orders
in which they are used ; and also according to their mat-
ter, construction, form, disposition, and purpose. The
proportion of the length of each to its diameter, and the
diminution of the diameter upwards, are diversly stated by
different authors.. The medium of them is nearly as fol-
lows :

The Tuscan is the simplest and shortest of all : its height
3i diameters, or 7 modules; and it diminishes 5 part of
its diameter.

The Doric is more delicate, and adorned with flutings;
its height 1\ or S diameters.

The Ionic is more delicate still, being 9 diameters long.
It is distinguished from the rest by the volutes, or curled
scrolls in its capital, and by its base, which is peculiar to it.

The Corinthian is the richest and most delicate of all
the columns, being 10 diameters in length, and adorned
with two rows of leaves, and stalks or stems, from whence
spring out small volutes.

The Composite column is also 10 diameters long, its
capital adorned with rows of leaves like the Corinthian,
and with angular volutes like the Ionic.

COLURES, are two great circles imagined to intersect
at right angles in the poles of the world, and to pass, the
one through the equinoctial points Aries and Libra, and
the other through the solstitial points Cancer and Capri-
corn; whence they are called the Equinoctial and Solsti-
tial Colures. By thus dividing the ecliptic into four equal
parts, they mark the four seasons, or quarters of the year.

It is a matter of dispute over what part of the back of
Aries the equinoctial colure passed in the time of Hip-
parchus. Newton, in his Chronology, takes it to have
been over the middle of this constellation. Souciet insists
that it passed over the dodecatemorion of Aries, or mid-
way between the rump and first of the tail. There are
some observations in the Philos. Trans, number 466, con-
cerning the position of this colure in the ancient sphere,
from a draught of the constellation Aries, in the Aratasa
published at Leyden and Amsterdam in l6"52, which ap-
pears to confirm Newton's opinion ; but the antiquity and
authority of the original draught may still be questioned.

COMA Berenices, Berenice's Hair, a modern con-
stellation of the northern hemisphere; composed of un-
formed stars between the Lion's tail and Bootes.

It is said that this constellation was formed by Conon,
an astronomer, to console the queen of Ptolemy Eueigetes,
for the loss of a lock of her hair, which was stolen out of
the temple of Venus, where she had dedicated it on ac-
count of a victory obtained by her husband.

The stars in this constellation are, in Tycho's cata-
logue 1*, in Hevelius's 21, and in the Britannic cata-
logue -13.
. COMBINATIONS, denote the alternations or vuiia-

COM

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COM

tions of any number of quantities, letters, sounds, or the
^like, in all possible ways. Mersenne give* the combina-
tions of all the notes and sounds in music, as far as 64;
the sum of which amounts to a number expressed by 90
places of figures. And the number of possible combina-
tions of the 24 letters of the alphabet, taken first two by
two, then three by three, and so on, according to Prestet's
calculation, amounts to

1391724288887252999425128493402200.

Father Truchet, in Mem. de l'Acad. shows, that two
square pieces, each divided diagonally into two colours,
may be arranged and combined 64 different ways, so as to
form as many different kinds of chequer-work : a circum-
stance worthy the attention of masons, paviours, &c.

Doctrine o/Combinations.

I. Having given any number of things, with the number
in each combination ; to find the number of combinations.
1. When only two are combined together.

One thing admits of no combination.

Two, a and b, admit of one only, viz, ab.

Three, a, b, c, admit of three, viz, ab, ac, be.

Four admit of six, viz, ab, ac, ad, be, bd, cd.

Five admit of ten, viz, ab, ac, ad, ae, be, bd, be, cd,
ce, de.

Whence it appears that the numbers of combinations,
of two and two only, proceed according to the triangular
numbers 1, 3, 6, 10, 15, 21, &c, which are produced by
the continual addition of the arithmetical series 0, 1, 2,
3, 4, 5, &c. So that if n be the number of things, then
the general formula for expressing the sum of all their

» ■ - i -ii , n ■ (n — *)

combinations by twos, will be .

J ' 1 .'2

Thus, if n = 2 ; this becomes —— = 1.
If re = 3; it is3 '2 - - =3.

2

If n = 4; it is — — - - = 6. &c.

2

2. When three are combined together ; then

Three things admit of one order, abc.

Four admit of 4 ; viz, abc, abd, acd, bed.

Five admit of 10 ; viz, abc, abd, abe, acd, ace, aide, bed,
bee, bde, cde. And so on according to the first pyramidal
numbers 1, 4, 10, 20, &c, which are formed by the con-
tinual addition of thle former, or triangular numbers 1, 3,
6, 10, &c. And the general formula for any number n

of combinations, taken by threes, is,- x x '■.

So, if n = 3: it is -i^— = 1.

1.2.3

If n = 4; it is — — — = 4.

If» = 5; it is5— ^= 10. &c'.
Proceeding thus, it is found that a general formula for
any number n of things, combined by in at each time, is
s = - - x ~ . x '?—■ — x &c, continued to m factors,

12 3 4 '

or terms, or till the last factor in the denominator be m.
So, in 6 things, combined by 4's, the number of com-

, ■ .6.5.4.3

bmations is = 1 5.

1.2.3.4

3. By adding all these series together, their sum will
be the whole number of possible combinations of n things
combined both by twos, by threes, by fours, &c. And as
the said scries are evidently the coefficients of the power
n of a binomial, wanting only the first two 1 and n; there-

fore the said sum, or whole number of all such combina-
tions, will be

(1 ■+- l)n - n - 1 , or 2" — n — 1 . Thus if the number
of things be 5 ; then 2s — 5 — 1 = 32 — 6 = 26.

II. To find the number of Changes and Alterations which
any number of quantities can undergo, when combined in all
possible varieties of ways, with themselves and each other, both
as to the things themselves, and the Order or Position of them.

One thing admits but of one order or position.

Two things may be varied four ways; thus, aa, ab,
ba, bb.

Three quantities, taken by twos, may be varied nine
ways; thus, aa, ab, ac, ba, ca, bb, be, cb, cc.

In like manner four things, taken by twos, maybe varied,
42 or l6 ways; and 5 things, by tw.os, 5'2or 25 ways ; and,
in general, n things, taken by twos, may be changed or va-
ried n2 different ways.

For the same reason, when taken by threes, the changes
will be n'°; and when taken by fours, they will be n4; and
so generally, when taken by n's, the changes will be nn.

Hence, then, adding all these together, the whole num-
ber of changes, or combinations in n things, taken both by
2's, by 3,'s, by 4's, &c, ton's, will be the sum of the geo-
metrical series, n -+- n2 ■+- n? -+- n* nn, which sum is =

nn — 1

x n.

71 — 1

For example, if the number of things n be 4; this gives

4* — 1 255

x 4 = — x 4 =;840.

4—1 3

And if n be 24, the number of letters in the alphabet ;
the theorem gives

^x24 = (24"-l) ^ =

1391724288S87252999425128493402200. In so many
different ways, therefore, may the 24 letters of the alpha-
bet be varied or combined among themselves, or so many
different words may be made out of them.

COMBUST, or Combustion, is said of a planet when
it is in conjunction with the sun, or not distant from it
above half their disc. But according to Argol, a planet
is Combust, or in Combustion, when it is within 8|° of
the sun.

COMBUSTION, in Chemical and Natural Philosophy,
a term denoting the decomposition of certain substances
accompanied by light and heat.

When a stone or a brick is heated, it undergoes no
change, except an augmentation of temperature; and when
left to itself it soon cools again, and becomes as at first.
But with combustible bodies the case is very different.
When heated to a certain degree in the open air, they sud-
denly become much hotter of themselves ; continue for a
considerable time intensely hot, sending out a copious
stream of caloric and light to the surrounding bodies. This
emission after a certain period begins to diminish, and at
last ceases altogether. The combustible has now under-
gone a complete change; it is converted into a substance
possessing very different properties, and no longer capable
of combustion. Thus, when charcoal is kept for some time
at the temperature of about 800°, it kindles, becomes in-
tensely hot, and continues to emit light and caloric for a
long time. When the emission ceases, the charcoal has all
disappeared, except an inconsiderable residuum of ashes;
being almost entirely converted into carbonic acid gas,
which makes its escape, unless the experiment be conduct-
ed in proper vessels. If it be collected, it is found to ex^
cced greatly in weight the whole of the charcoal consumed.
6 J 2XJ

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C 0 M

Previous to the 17th century, the theory of combustion
was but little understood or attended to. Since that pe-
riod the labours of Bacon and Boyle, and Hooke and
Mayow, together with those of Stahl and Lavoisier, have
thrown much light on the subject, particularly the theory
of Lavoisier, which is by far the most rational that have
yet been offered to the public. He corrects the errors of
his predecessors, and has advanced before them one very
important stop; but many new steps are still wanting to
render his theory of combustion complete. It explains
indeed, in a satisfactory manner, why during the process of
combustion the burning body gradually wastes away; but
it gives no explanation of the constant emission of heat and
light, though a circumstance as worthy of attention as the
wasting of the body.

Dr. Thomson, admitting the truth and accuracy of the
Lavoiserian theory as far as it proceeds, divides the bodies
which occupy the attention of chemists into: — 1. Com-
bustibles; 2. Supporters of combustion; and 3. Incom-
bustibles.'

The combustibles, or those bodies which in common lan-
guage are said to burn, may be divided into, 1. Simple
combustibles; 2. Compound combustibles; and 3. Com-
bustible oxides. Simple combustibles are, sulphur, phos-
phorus, carbon, hydrogen, and all the metals, except per-
haps gold, silver, and mercury. Compound combustibles
consist of compounds formed by the simple combustibles
uniting together two and two; and combustible oxides are
composed of one or more simple combustibles combined
with a dose of oxygen. These oxides may be arranged
under two heads: viz-, those containing only a single base
combined with oxygen may therefore be termed simple
combustible oxides; and those containing more than one
base may therefore be termed compound combustible ox-
ides. The simple combustible oxides are only four in
number : namely, oxide of sulphur, oxide of phosphorus,
charcoal, and carbonic oxide gas. All the simple com-
bustible oxides are by combustion converted into acids.
The compound combustible oxides include by far the
greater number of combustible bodies; for almost all the
animal and vegetable substances belong to them, and the
double base is usually carbon and hydrogen.

The supporters of combustion are a set of bodies which
are not of themselves, strictly speaking, capable of under-
going combustion, but which are absolutely necessary for
the process. All the supporters known at present are six;
viz, — 1. Oxygen gas; 2. Air; 3. Gaseous oxide of azote;
4. Nitrous gas; 5. Nitric acid ; and 6. Oxy-muriatic acid.
There are other substances to which the author gives the
name of partial supporters ; but all supporters contain one
common principle, namely, oxygen.

The incombustible bodies are neither capable of under-
going combustion themselves, nor of supporting the com-
bustion of bodies that are. Of course they are not imme-
diately connected with combustion; but they arc noticed
here because some of the alkalies and earths which belong
to this class possess certain properties in common with
combustibles, and are capable of exhibiting phenomena
somewhat analogous to combustion ; phenomena which
Dr. Thomson describes under the title of semi-combustion.
From these observations it is obvious, that in every case
of combustion there must be present a combustible and a
supporter; and Lavoisier ascertained beyond a doubt, that,
during the process, the combustible always unites with the.
oxygen oi the supporter. This new compound Dr. Thom-

son calls a product of combustion ; and maintains that every
such product is either water, or an acid, or a metallic ox-
ide. He admits indeed that other bodies sometimes make
their appearance during combustion; but affirms that these,
on examination, will be found neither to be products, nor
to have undergone combustion. See Nicholson's Journal,
vol.2, 2d series; also Dr. Thomson's Chemistry.

CO MITT, a heavenly body in the planetary region, ap-
pearing suddenly, and again disappearing; and during the
time of its appearance moving in a proper, though very
excentric orbit, like a planet.

Comets are vulgarly called Blazing Stars, and have this
to distinguish them from other stars, that they are usually
attended with a long train of light, tending always oppo-
site to the sun, and being of a fainter lustre, the farther it
is from the body of the comet. And hence arises a po-
pular division of comets, into three kinds, viz, bearded,
tailed, and hairy comets ; though in reality, this division
rather relates to the several circumstances of the same
comet, than to the phenomena of several. Thus, when
the comet is eastward of the sun, and moves from him, it
is said to be bearded, because the light precedes it in the
manner of a beard : when the comet is westward of the
sun, and sets after him, it is said to be tailed, because the
train of light follows it in the manner of a tail : and lastly,
when the sun and comet are diametrically opposite, the
earth being between them, the train is hid behind the
body of the comet, excepting the extremities, which, be-
ing broader than the body of the comet, appear as it were
around it, like a border of hair, or coma, from which it is
called hairy, and a comet. But there have been comets
whose disc was as clear, round, and well defined, as that
of Jupiter, without either tail, beard, or coma.

Of the Nature of Comets. — Philosophers and astrono-
mers, of all ages, have been much divided in their opinions
as to the nature of comets. Their strange appearance
has always been matter of terror to the vulgar, who have
uniformly considered them as evil omens, and forerunners
of war, pestilence, &c. Diodorus Siculus and Apollinus
Myndius, in Seneca, inform us, that many of the Chal-
deans held them to be lasting bodies, having stated revo-
lutions as well as the planets, but in orbits vastly more
extensive; on which account they arc only visible while
near the earth, but disappear again when they go into the
higher regions. Others were of opinion, that the comets
were only meteors raised very high in the air, which blaze
for a while, and disappear when the matter of which they
consist is consumed or dispersed.

Some of the Greeks, before Aristotle, supposed that a
comet was a vast heap or assemblage of very small stars
meeting together, by reason of the inequality of their mo-
tions, and so uniting into a visible mass, by the union of
all their small lights ; which must again disappear, as
those stars separated, and each proceeded in its course.
Pythagoras, however, accounted them a kind of planets
or wandering stars, disappearing in the superior parts of
their orbits, and becoming visible only in the lower parts
of them.

But Aristotle held, that comets were only a kind of
transient fires, or meteors, consisting of exhalations raised
to the. upper region of the air, and there set on fire ; far
below the course of the moon.

Seneca, who lived in the first century, and who had seen
two or three comets himself, plainly intimates that he
thought them above the moon; and argues strongly against

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those who supposed them to be meteor;., or maintained
other absurd opinions concerning them ; declaring his be-
lief that they were not tires suddenly kindled, but the
eternal productions of nature. He points out also the
only way to arrive at a certainty on this subject, viz, by
collecting a number of observations concerning their ap-
pearance, in order to discover whether they return peri-
odically or not. " For this purpose," he observes, "one
age is not sufficient ; but the time will come when the na-
ture of comets and tjieir magnitudes will be demonstrated,
and the routes they take, so different from the planets, ex-
plained. Posterity will then wonder, that the preceding
ages should be ignorant of matters so plain and easy to be
known."

For a long time this prediction of Seneca seemed not
like'ly to be fulfilled ; and Tycho Brahe was the first
among the moderns, who restored the comets to their true
rank in the creation ; for after diligently observing the
comet of 1577, and finding that it had no sensible diurnal
parallax, he assigned it its true place in the planetary re-
gions. See his book De Cometa anni 1577-

Before this, however, there were various opinions con-
cerning them. In the dark and superstitious ages, comets
were held to be forerunners of every kind of calamity,
and it was supposed they had different degrees of malig-
nity, according to the shape they assumed ; whence also
they were differently denominated. Thus, it was said that
some were bearded, others hairy ; that some represented
a beam, sword, or spear ; others a target, &c ; whereas
modern astronomers acknowledge only one species of co-
mets, and account for their different appearances from
their different situations with respect to the sun and the
earth.

Kepler, in other respects a very great genius, indulged
the most extravagant conjectures, not only concerning
comets, but the whole system of nature in general. , The
planeis he imagined were huge animals swimmiiig round
the sun ; and the comets monstrous and uncommon ani-
mals generated in the celestial spaces.

A still more ridiculous opinion, if possible, was that of
John Bodin, a learned Frenchman in the l6th century;
who maintained that comets "are spirits, which having
lived on the earth innumerable ages, and being at last ar-
rived on the confines of death, celebrate their last triumph,
or are recalled to the firmament like shining stars ! This
is followed by famine, plague, &c, because the cities and
people destroy the governors and chiefs who appease the
wrath of God." — Others again have denied even the exist-
ence of comets, and maintained that they were only false
appearances, occasioned by the refraction or reflection of
light.

Hevelius, from a great number of observations, pro-
posed it as his opinion, that the comets, like the solar ma-
cula? or spots, are formed or condensed out of the grosser
exhalations of his body ; in which he diffcre but little from
the opinion of Kepler.

James Bernoulli, in his Systema Cometarum, imagined
that comets were no other than the satellites of some very
distant planet, which was itself invisible to us on account
of its distance, as were also the satellites unless when in a
certain part of their orbits.

Descartes advances another opinion : He conjectures
that, comets are only stars, formerly fixed, like the rest,
in the heavens; but which becoming gradually covered
with macula: or spots, and at length wholly deprived of

their light, cannot keep their places, but are carried off by
the vortices of the circumjacent stars; and in proportion
to their magnitude and solidity, moved in such a manner,
as to be brought nearer the orbof Saturn; and thus com-
ing within reach of the sun's light, rendered visible.

But the absurdity of all these hypotheses now abun-
dantly appears from the observed phenomena of comets,
and from the doctrine of Newton, which is as follows :

The comets, he says, are compact, solid, fixed, and du-
rable bodies; in fact a kind of planets, which move in very
oblique and excentric orbits, every way with the greatest
freedom; persevering in their motions, even against the
course and direction of the planets: and their tail is a
very thin and slender vapour, emitted by the head or nu-
cleus of the comet, ignited or heated by the sun. This
theory of the comets at once solves their principal pheno-
mena, which are as below.

The Principal Phenomena of the Comets.

1. First then, those comets which move according to
the order of the signs, do all, a little before they disap-
pear, either advance slower than usual, or else go retro-
grade, if the earth be between them and the sun; but more
swiftly, if the earth be placed in a contrary part. On the
other hand, those which proceed contrary to the order of
the signs, move more swiftly than usual, if the earth be
between them and the sun ; and more slowly, or else re-
trograde, when the earth is in a contrary part. — For since
this course is not among the fixed stars, but among the
planets ; as the motion of the earth either conspires with
them, or goes against them ; their appearance, with re-
spect to the earth, must be changed ; and, like the pla-
nets, they must sometimes appear to move swifter, some-
times slower, and sometimes retrograde.

2. So long as their velocity is increased, they nearly
move in great circles; but towards theend of their course,
they deviate from those circles; and when the earth pro-
ceeds one way, they go the contrary way. Because, in
the end of their course, when they recede almost directly
from the sun, that part of the apparent motion which a-
rises from the parallax, must bear a greater proportion to
the whole apparent motion.

3. The comets move in ellipses, having one of their fo-
ci in the centre of the sun ; and by radii drawn to the
sun, describe areas proportional to the times. Because
they do not wander precariously from one fictitious vor-
tex to another ; but, making a part of the solar system,
return perpetually, and run a constant round. Hence,
their elliptic orbits being very long and excentric, they
become invisible when in that part which is most remote
from the sun. And from the curvity of the paths of co-
mets, Newton concludes, that when they disappear, they
are. much beyond the orbit of Jupiter; and that in their
perihelion they frequently descend within the orbits of
Mars and the inferior planets.

4. The light of their nuclei, or bodies, increases as they
recede from the earth towards the sun; and on the con-
trary, it decreases as they recede from the sun. Because,
as they are in the regions of the planets, their access to-
wards the sun bears a considerable proportion to their
whole distance.

5. Their tails appear the largest and brightest, imme-
diately after their transit through the region of the sun,
or after their perihelion. Because then, their heads being
the most heated, will emit the most vapours. — From the
light of the nucleus we infer their vicinity to the earth,

COM

[

and that they are by no means in the region of the fixed
stars, as some have imagined ; since, in that case, their
heads would be no more illuminated by the sun, than the
planets are by the fixed stars.

6. The tails always decline from a just opposition to
the sun towards those parts which the nuclei or bodies
pass over, in their progress through their orbits. Because
all smoke, or vapour, emitted from a body, in motion,
tends obliquely upwards, still receding from that part to-
wards which the smoking body proceeds.

7. This declination, ceteris paribus, is the smallest when
the nuclei approach nearest the sun : and it is also less
near the nucleus, or head, than towards t«he extremity of
the tail. Because the vapour ascends more swiftly near
the hrad of the comet, than in the higher extremity of its
tail; and also when the comet is nearer the sun, than
when it-is farther off.

8. The tails are somewhat brighter, and more distinctly
defined in their convex, than in their concave part. Be-
cause the vapour in the convex part, which goes first, being
somewhat nearer and denser reflects thelight more copiously.

g. The tails always appear broader at their upper ex-
tremity, than near the centre of the comet. Because the
vapour in a free space continually rarefies and dilates.

10. The tails are always transparent, and the smallest
stars appear through them; because they consist of infi-
nitely thin vapour.

The Phases of Coynets. — The nuclei, which are also call-
ed the heads, and bodies, of comets, viewed through a
telescope, exhibit faces very different from those of the
fixed stars or planets. They are liable to apparent alte-
rations, which Newton ascribes to changes in the atmo-
sphere of comets: and this opinion was confirmed by ob-

342 ] COM

And from these observations it was that Hevelius formed
his opinion, that comets are like macula; or spots formed
out of the solar exhalations.

Of the Magnitude of Comets. — The estimates that have
been given of the magnitude of comets by Tycho Brahe,
Hevelius, and other astronomers, are not very accurate;
as it does not appear that they distinguished between the
nucleus and the surrounding atmosphere. Thus Tycho
computes that the true diameter of the cometin 1577, was
in proportion to the diameter of the earth, as 3 to 14;
and Hevelius made the diameter of the comet of 1652, to
that of the earth, as 52 to 100. But the diameter of the
atmosphere is often 10 or la times as great as that of the
nucleus : the former, in the comet of l6'82, was measured
by Flamsteed, and found to be 2', when the diameter of
the nucleus alone was only 11 or 12". Though some co-
mets, estimated by a comparison of their distance and ap-
parent magnitude, have been reckoned much larger than
the moon, and even equal to some of the primary planets.
The diameter of that of 17-14, when at the distance of the
sun from us, measured about 1', which makes its diameter
about three times that of the earth : at another time the
diameter of its nucleus was nearly equal to that of the
planet Jupiter.

Of the Tails of Comets. — There have been various con-
jectures about the nature of the tails of comets, the prin-
cipal of which are those of Newton, and his followers.
Newton shows that the atmospheres of comets will furnish
vapour sufficient to form their tails. This he argues from
their wonderful rarefaction in our air at a distance from
the earth ; which is such that a cubic inch of common
air, expanded to the rarity of that at the distance of half
the earth's diameter, or 4000 miles, would fill a space

servations. of the comet in 1744; Hist. Acad. Scienc. . larger than the whole region of the stars. Since then the

1744. Sturmius says that, observing the comet of l6"S0
with a telescope; it appeared like a coal dimly glowing, or
a rude mass of matter illuminated with a dusky fumid
light, less sensible towards the extremes than in the mid-
dle; whereas a star appears with a round disc, and a vivid

Hsht-

Of the comet of \66l, Hevelius observes, that its body
was of a yellowish colour, very bright and conspicuous,
but without any glittering light: in the middle was a dense
ruddy nucleus, almost equal to Jupiter, encompassed by
a much fainter, thinner matter. February 5th, its head
was somewhat larger and brighter, and of a gold colour ;
butjts light more dusky than that of the stars: and here
the nucleus appeared divided into several parts. Feb. 6th,
the disc was lessened ; the parts of the nucleus still ex-
isted, though less than before : one of them, on the lower
part of the disc, on the left, much denser and brighter
than the rest; its body round, and representing a very lu-
cid small star: the nucleus still encompassed with ano-
ther kind of matter. Feb. 10th, the head somewhat more
obscure, and the nucleus more confused, but brighter at
top than bottom. Feb. 13th, the head diminished much
both in size and splendor. March 2d, its roundness a lit-
tle impaired, and its edges lacerated, &c. March 2Sth,
very pale, and exceeding thin; its matter much dispersed;
and no distinct nucleus at ail appearing.

Weigelius, who s:iw the comet of l6'64, as also the moon,
and a small cloud in the horizon illuminated by the sun
at the same time, observi d, that through the telescope the
moon appeared of a continued luminous surface: but the
comet very different ; being exactly like the small cloud.

coma, or atmosphere of a comet, is ten times higher than the
surface of the nucleus, from the centre; the tail, ascending
still much higher, must necessarily be immensely rare: so
that it is no wonder the stars are visible through it.

Now the ascent of vapours into the tail of the comet,
he supposes to be occasioned by the rarefaction of the
matter of the atmosphere at the time of the perihelion.
Smoke, it is observed, ascends the chimney by the impulse
of the air in which it floats; and air, rarefied by heat,
ascends by the diminution of its specific gravity, carrying
up the smoke along with it: in the same manner then it
may be supposed that the tail of a comet is raised by
the sun.

The tails therefore thus produced in the perihelions of
comets, will go oft' along with their head, into remote re-
gions; and either return from thence, together with the
comets, after a long series of years ; or rather be there
lost, and vanish "by little and little, and the comet be left
bare; till at its return, descending towards the sun, some
short tails arc again gradually produced from the head :
which afterwards, in the perihelion, descending into the
sun's atmosphere, will be immensely increased.

Newton farther observes, that the vapours, when thus
dilated, rarefied, and diffused through all the celestial le-
gions, may probably, by means of their own gravity, be
gradually attracted down to the planets, and become in-
termingled with their atmospheres. He adds that this in-
termixture, may be useful and necessary for the conserva-
tion of the water and moisture of the planets, dried up or
consumed in various ways. And I suspect, adds our au-
thor, that the spirit, which makes the finest, subtilest, and

COM

[ 343 ]

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best part of our air, and which is absolutely requisite for
the life and being of all things, is derived principally from
the comets. — On this principle there may appear to be
some foundation for the popular opinion of presages from
comets; since the tail of a comet thus intermingled with
our atmosphere, may produce changes very sensible in ani-
mal and vegetable bodies.

It may here be added, that another use which Newton
conjectures comets may be designed to serve, is that of
recruiting the sun with fresh fuel, and repairing the con-
sumption of his light by the streams continually sent forth
in every direction from that luminary. In support of this
conjecture he observes, that comets in their perihelion
may suffer a diminution of their projectile force, by the
resistance of the solar atmosphere ; so that by degrees
their gravitation towards the sun may be so far increased,
as to precipitate their fall into his body.

Various opinions have been entertained respecting the
generation of the tails of comets. Apian, Tycho Brahe,
and some others, think they are produced by the sun's
rays transmitted through the nucleus of the comet, which
they supposed was transparent, and there refracted as in
a glass lens, so as to form a beam of light behind the co-
met. Descartes accounted for the phenomenon of the
tail by the refraction of light from the head of the comet
to the spectator's eye. Mai ran supposes that the tails are
formed from the luminous matter composing the sun's at-
mosphere : and M. De Lalande combines this hypothesis
with that of Newton recited above. But Mr. Rowning,
not satisfied with Newton's opinion, accounts for the tails
of comets in the following manner: It is well known, says
he, that when the sun's light pas-.es through the atmo-
sphere of any body, as the earth, that which passes on one
side, is by the refraction made to converge towards that
which passes on the opposite side; and this convergeucy
is not wholly effected either at the entrance of the light
into the atmosphere, or at its egression from it ; but be-
ginning at its entrance, it increases in every point of its
progress: It is also agreed, that the atmospheres of the
comets are very large and dense : he therefore supposes
that by such time as the light of the sun has passed through
a considerable part of the atmosphere of the comet, the
rays are so far refracted towards each other, that they
then begin sensibly to illuminate it, or rather the vapours
floating in it; and so render that part they have yet to
pass through, visible to us: and that this portion of the
atmosphere of a comet thus illuminated, appears to us in
form of a beam of the sun's light, and passes under the
denomination of a comet's tail. Rowning's Nat. Philos.
part 4. chap. 1 1 .

M. Euler, Mem. Berlin, torn. 2, pa. 117, thinks there
is a great affinity between the tails of comets, the zodiacal
light, and the aurora borealis, and that the common cause
of all of them, is the action of the sun's light on the atmo-
spheres of the comets, of the sun, and of the earth. He
supposes that the impulse of the rays of light on the at-
mosphere of a comet, may drive some of the finer parti-
cles of that atmosphere far beyond its limits; and that
this force of impulse combined with that of gravity to-
wards the comet, would produce a tail, which would al-
ways be in opposition to the sun, if the comet did not
move. But the motion of the comet in its orbit, and
about an axis, must vary the position and figure of the
tail, giving it a curvature, and deviation from a line join-

ing the centres of the sun and comet; and that this devia-
tion will be greater, as the orbit of the comet has the
greater curvature, and as the motion of the comet is more
rapid. It may even happen, that the velocity of the co-
met, in its perihelion, may be so great, that the force of
the sun's rays may produce a new tail, before the old one
can follow; in which case the comet might have two or
more tails. The possibility of this is confirmed by the
comet of 1744, which was observed to have several tails
while it was in its perihelion.

Dr. Hamilton urges several objections against the New-
tonian hypothesis; and concludes that the tail of a comet
is formed of matter which has not the power of refract-
ing or reflecting the rays of light ; but that it is a lucid or
self-shining substance: and from its similarity to the au-
rora borealis, that it is produced by the same cause, and
is properly an electrical phenomenon. Dr. Halley like-
wise was doubtless inclined to this hypothesis, when he
said, that the streams of light in an aurora borealis so
much resembled the long tails of comets, that at first sight
they might well be taken for such : and that this light
seems to have a greater affinity to that which the effluvia
of electric bodies emit in the dark. Philos. Trans. No.
347. Hamilton's Philos. Essays, pa. 91-

The Motion of Comets. — If it be supposed that the paths
of comets are perfectly parabolical, as some have ima-
gined, it will follow that, being impelled towards the sun
by a centripetal force, they descend as from spaces infi-
nitely distant; and that by their falls they acquire such a
velocity as will carry them off again into the remotest re-
gions, never more to return. But the frequency of their
appearance, and their degree of velocity, which does not
exceed what they might acquire by their gravity towards
the sun, seem to put it past doubt that they move like the
planets, in elliptic orbits, though exceedingly excentric ;
and so return again after very long periods.

The apparent velocity of the comet of 1472, as ob-
served by Regiomontanus, was such as to carry it through
40° of a great circle in 24 hours : and it was observed
that the comet of 1770 moved through more than 45° in
the last 25 hours.

Respecting the return of comets there have been dif-
ferent opinions. Newton, Flamsleed, Halley, and other
English astronomers, appear quite satisfied of the return of
these bodies. Cassini and some of the French think it
highly probable; but De la Hire and others oppose it.
Those on the affirmative side suppose that the comets de-
scribe orbits prodigiously excentric, insomuch that '.ve can
observe them only in a very small part of their revolu-
tion : out of this, they are lost in the immensity of space;
hid not only from our eyes, but our telescopes: that small
part of their orbit next us passing sometimes within those
of all the inferior planets.

Cassini gives the following reasons in favour of the re-
turn of comets. 1. It is found that they move a consi-
derable time in the arch of a great circle, when referred
to the fixed stars, that is, a circle whose plane passes
through the centre of the earth; deviating but a little
from it chiefly towards the end of their appearance; a de-
viation however common to them with the planets. — -2.
Comets, as well as planets, appear to move so much the
faster as they arc nearer the earth ; and when they are at
equal distances from their perigee, their velocities are
nearly the same. By subtracting from their motion the

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C O M

apparent inequality of velocity occasioned by their different
distance from the earth, their equal motion might be
found : but we should not still be certain that this is their
true motion; because they might have considerable in-
equalities, not distinguishable in that small part of their
orbit visible to us. It is rather probable that their real
motion, as well as that of the planets, is unequal in itself ;
and hence we have a reason why the observation made
during the appearance of a comet, cannot give the just
period of its revolution. — 3. There are no two different
planets whose orbits cut the ecliptic in the same angle, and
whose nodes are in the same points of the ecliptic, and
having the same apparent velocity in their perigee : con-
sequently, two comets seen at different times, yet agreeing
in all those three circumstances, can only be one and the
same comet. Not that this exact agreement, in these cir-
cumstances, is absolutely necessary to determine their
identity : for the moon herself is irregular in all of them,
so that it appears there may be cases in which the same
comet, at different periods of revolution, may disagree in
these points.

As to the objections against the return of comets, the
principal is that of the unfrequency of their appearance,
with regard to the number of revolutions assigned to them.
In 1702 there was a comet, or rather the tail of one, seen
at Rome, which M. Cassini takes to be the same with that
observed by Aristotle, and again lately in the year 1 668 ;
which would imply a period of 34 years : Now, it may
appear strange that a star which has so short a revolution,
and of consequence such frequent returns, should be so
seldom seen. Again, in April of the same year 1702, a
comet was observed by Messrs. Bianchini and Maraldi,
which the latter supposed was the same with that of 1664,
both on account of its motion, velocity, and direction.
M. de la Hire thought it had some relation to another he
had observed in 169S, which Cassini refers to that of 1652;
which would make it a period of 43 months, and the
number of revolutions between 1652 and 1692, 14: now,
it is hard to suppose, that in this age, when the heavens
are so narrowly watched, a star should make 14 revolu-
tions unperceived ; especially such a star as this, which
might appear above a month together ; and consequently
be often disengaged from the crepuscula. For this reason
M. Cassini is very reserved in maintaining the hypothesis
of the return of comets, and only proposes those for pla-
nets where the motions are easy and simple, and are solved
without straining, or allowing any irregularities.

M. de la Hire proposes one general difficulty against the
whole system of the return of comets, which would seem
to prevent any comet from returning as a planet ; which
is this; that by the disposition necessarily given to their
courses, they ought to appear as small at first as at last ;
and always increase till they arrive at their nearest proxi-
mity to the earth; or if they should chance not to be ob-
served, as soon as they are capable of being seen, it is yet
hardly possible but they must often show themselves be-
fore they have arrived at their full magnitude and bright-
ness: but, adds he, none were ever yet observed till they
had arrived at it. However, the appearance of a comet
in the month of October 1723, while at a great distance,

so as to be too small and dim to be viewed without a tele-
scope, as well as the observations of several others since,
may serve to remove this obstacle, and set the com ets still
on the same footing with the planets.

It is a conjecture of Newton, that as those planets
which are nearest to the sun, and revolve in the least or-
bits, are the smallest ; so among the comets, such as in
their perihelion come nearest the sun, are the smallest, and
revolve in the least orbits.

Of the Writings and Lists of Comets.

There have been many writings on the subject of co-
mets, beside the notices of historians as to the appearance
of certain particular ones.

Regiomontanus first showed how to find the magnitudes
of comets, their distance from the earth, and their true
place in the heavens. His 16 problems De Cometa; Mag-
nituiline, Longitudine, ac Loco, are to be found in a book
published in the year 1 544, with the title of Scripta Joan-
nis Regiomontank

Peter Apian observed and wrote on the comets of 1631,
1632, &c. Other writers are Tycho Brahe, in his Pro-
gymnasmata Astronomia? Instauratas. — Kepler, of the co-
met in the year 1607, and deCometis Libelli tres. — Ric-
ciolus, in his Almagestum Novum, published 1651, enu-
merates 154 comets cited by historians down to the year
l6lS. — Hevelius's Prodromus Cometicus, containing the
history of the comet of the year 1664. Also his Cometo-
graphia. — Lubienietz, in a large folio work expressly on
this subject, published 1667, extracts, with immense la-
bour, from the passages of all historians, an account of
415 comets, ending with that of 1665. — Dr. Hooke, in his
Posthumous works. — M. Cassini's little Tract of Comets.
— Sturmius's Dissertatio de Cometarum Natura. — New-
ton, in his Principia, lib. 3 ; who first assigned their pro-
per orbits, and by calculations compared the observations
of the great comet of 168O with his theory. — Dr. Halley,
in his Synopsis Cometica, in the Philos. Trans. No. 218,
&c ; who computed the elements and orbits of 24 comets,
and who first ventured to predict the return of one in
17-59, which happened accordingly. — De Lalande, Theo-
rie des Cometes, 1759; also, in his Astronomie, vol.3 —
Clairaut, Th6orie du mouvement des Cometes, 1760. —
D'Alembert, Opuscules Mathfematiques, vol. 2, pa. 97. —
M. Albert Euler, 1762. — Sejour, Essai sur les Cometes,
1775.— Besides Boscovich, De la Grange, De la Place,
Frisi, Lexel, Barker, Hancocks, Cole, with many others.
— And M. Pringre's Com6tographie, in 2 vols. 4to, 1784;
in which is contained the most ample list of such conies
as have been well observed, and their elements computed,
to the number of 67. And accounts of a very few more
that have been observed since that time, may be. seen in
the Mem. de l'Acad. and in the Philos. Trans. A very in-
genious and ample work on comets is also published by
Sir Henry Englefield, entitled, On the determinations of
the Orbits of Comets.

The whole list of comets that have been noticed, on re-
cord, amount to upwards of 500, but the following is a
complete list of all that have been properly observed, and
their elements computed, the mean distance of the earth
from the sun being supposed 100,000.

COM

[ 345 ]

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TABLE OF THE ELEMENTS OF COMETS.

83?
1231
1264

.1299
1301
1337

1456
1472
1531
1.532
1533
1556
1577
1580

I<582

1585
1590
1593
1596

1607
1618
1618
1652
1661
1664
1665
1672
1677
1678
1680

1682

16S3
1684
1686
1689
1698
1699
1702
1706

1707

1723
1729

Ascending
Node

6 26 33
0 13 30
5 19 0
5 2S 45
3 17 8
0 1 5 abou
2 24 21

6 22

18 30
II 46

19 25

20 27
5 44

25 42
0 25 52
0 18 57a

0 19 8

7 21 7
7 4 43

1 7 43
5 15 31
5 14 15

10 12 13

10 15 37

1 20 21

9 23 25

Inclin. of
Orbit

Vol

16 l
28 10

22 31
21 14
18 2
27 31
26 49
11 40

2 2
2 2
2 59
1 53
1 57
21 17
20 48

23 23
8 28 15

"ll 20 35
10 23 45
8 27 44
10 21 46
6 9 25
0 13 12

0 13 II

1 22 8
1 22 47
1 22 50
4 8 43
4 7 55
0 14 16

10 10 35
10 10 33
10 10 17
I.

1 ] about

6 5
36 30
30 25
68 57
70 about

32 11
32 H
17 56

5 20
17 56
32 36
35 49
32 7
74 33
64 40
64 52
61 28

59 29

6 4
29 41
87 58
55 12
52 10
17 2
21 28
37 34
79 28
32 36
21 19

76 5
83 22
79 3

3 4

60 56

61 7
58 40
61 20
61 23
17 56
17 42
83 11
65 49
31 22
69 17
11 46

69 20

4 30
55 14
55 14
88 50
88 36
88 38

30 20

31 13
49 59

77 2
76 58

70 53

19 3

14 48

21 0

5 45

3 20

about

7 59

0 20 0
10 1 0

1 15 34
10 1 39

3 21 7
27 16

8 50

9 22
19 6
19 12

5 23
11 27

8 51
6 55
26 19
18 16
28 31
2 16
10 18 20
0 2 14

0 28 19

3 25 59

4 10 41
2 11 55

1 17 0
4 17 37

10 27 46
8 22 40
8 22 44
8 23 27
8 23 43
8 22 40
10 2 53
10 1 36

2 25 30

7 28 62
2 17 1

8 23 45

9 0 51
2 31

18 41
12 29
12 36
17 4

19 55
19 58

1 30

1 27

1 12 52

10 22 17

10 22 40

10 27 22

Perihel.
Dist.

58000
94776
44500
41081
31793
45700
40666
64450
58550
54273
56700
50910
20280
66390
1S342
59628
59553
22570
4006

109358
57661
8911
51293
54942
58680
51298
37975
84750
44851
102575
10649
69739
2S059
123801
6l2f
617
656
592
603
58328
58250
56020
96015
32500
1689
69129
74400
64590
42581
42686
86350
85974
85904
102655
102565
99865
406980
426140
416927

Time of Perihelion

March

Jan.

July

March

Oct.

June

June

Feb.

Aug.

Oct.

June

April

Oct.

Nov.

May

New
Oct.
Feb.
July
Aug.

Oct.

Aug.

Nov.

Nov.

Jan.

Dec.

April

March

May

Aug.

Dec.

Sept.

July

June

Sept.

Dec.

Oct.

Jan.

March

Jan.

Dec.

Jan.

Sept.
June

17 6
31 8

22 about

2 7

1 1

8 22
28 23
24 21

19 22
16 20
21 20

20 19
28 15
28 14

6 16

7 9
Style

7 20

8 4
18 14
10 20

8 16

16 4

17 3
8 13

12 16
26 24

4 12

24 5

1 9

6 1

26 14

15 0
17 23

17 21

18 0
18 0
14 8

14 22

13 3
8 10

16 15
1 15

18 17

13 9

13 14

30 5

30 5

12 0
12 0

15 0
15 1
27 16
23 7
25 11
23 0

Calculated by

Ret.
Dir.
Dir.
Dir.
Ret.
Ret.
Ret.
Ret.
Ret.
Ret.
Ret.
Dir.
Ret.
Dir.
Ret.
Dir.
Dir.
Ret.
Ret.

Dir.

Ret.
Dir.
Ret.
Ret.
Ret.
Dir.
Dir.
Dir.
Dir.
Ret.
Ret.
Dir.
Ret.
Dir.
Dir.

Ret.

Ret.
Dir.
Dir.
Ret.
Ret.
Ret.
Dir.
Dir.

Dir.

Ret.

Ret.
Dir.

Pingre

Pingre

Dunthorn

Pingre

Pingre

Pingre

Halley

Pingre

Pingre"

Halley

Halley

Halley

Douwes

Halley

Halley

Halley

Pingre

Pingre

Pingre

Halley

Halley

La Caille

Halley

Pingre

Halley

Pingre

Halley

Halley

Halley

Halley

Halley

Halley

Halley

Douwes

Halley

Halley

Euler

Newton

Pingre

Halley

Halley

Halley

Halley

Halley
Pingre

Halley
La Caille
La Caille
La Caille
Struyck
Houtteryn
La Caille
Struyck
La Caille
Douwes
Bradley
Douwes
La Caille
Maraldi

2Y

COM

t 346 ]

C O M

1737
1739

1742

1743

1743
1744

1747
1748

1748
1757

1758
1750

1759

1759

17C2

1763
1764
1766
1766
1769

Ascending
Node

Inclin. of |
Orbit

10-10 52
10 10 33
7 16 22
27 18
27 25
5 35
5 33
5 38
5 43
16 9
9 32
5 47
5 29
5 42
8 11
8 21
5 16
15 45
15 47
]5 46

15 51

16 5
1 15 46
1 15 48
1 15 49
1 15 49
4 26 58
4 27 19
4 27 19

22 52
22 46

22 52
4 40
4 13
4 6
4 4
4 7

20 50

23 48
1 23 46
1 23 49
1 23 49
1 23 49
1 23 46
1 24 7

1 23 45
4 19 40
4 19 39
4 19 40

2 19 51
2 19 20

18 56

19 20
18 58

18 35

19 2
26 23

0 5
4 11
14 23
25 1
25 2
25 3

77 19
77 1
18 21
55 53

55 43

67 4

66 59

66 52

56 35
61 44

68 14

67 11
66 51

2 16
2 20
45 48
47 9
47 4
47 5
47 18
47 50
47 11
47 8
47 18
47 14

77 57
79 6
79 6
85 27
85 35
85 28
56 59
12 50
12 39
12 48
12 41

68 19
17 38
17 40
17 35
17 38
17 39
17 40
17 29
17 41
79 7

78 59

79 3
4 52
4 42

85 22

84 45

85 12
85 40
85 3
72 41
52 54
40 50
11 8
40 38
40 43
40 47

10 16 27

10 22 37

10 25 55

3 12 34

12 39

7 33

7 32

7 35

7 39

16 42

10 49

7 33

7 26

7 38

2 58

3 2 42

8 6 34
6 17 13
6 17 6
6 17 10
6 17 18
6 17 19
6 17 12
6 17 13
6 17 15
6 17 16

9 10 6
9 7
9 7

5
4

Perihel.
Dist.

Time of Perihelion

2 58
2 39
2 49
2 36
8 27 38
10 3 14
3
3
3
3
3
1
3
23 34

23 24.
1 23 38
4 18 25

19 4
15 22
15 15
14 24

13 43

14 30

24 52
0 15 15
4 23 15
8 2 18
4 24 6
4 24 14
4 24 11

394927
408165
22282
67160
67358
76555
76550
76568
76530
75766
73668
76890
76620
76545
838 II
83501
52057
22206
22322
22250
22156
22192
22222
22223 .
22200
22176
229388
219859
219851
84066
84150
84040
65525
33754
33907
33797
33932
21535
58255
58490
58360
58380
58350
5829S
59708
58234
80139
•79851
8020S
96599
96 180
101415
101249
101065
100686
100936
49876
55522
50532
33275
12376
12287
12258

May
June
Jan.
June

Feb.

Feb.

Jan.

Sept.
March

March

Feb.
March

April

June
Oct.

June
March

22 11

25 9

30 9

17 11

17 10

8 5

8 4

8 5

8 8

1 22

7 4

7 22

8 5
8- 7

10 21
10 21
20 21

Calculated by

Nov.

Dec.
Dec.

May

Nov.
Feb.
Feb.
April
Oct.

9

28 12
3 10
3 7

28 20

29 .1
2S 19
18 2
21 8
21 10
21 10

12 14
12 14
12 13
12 13.
12 13%

12 13

13 10
12 13
27 0
27 2

27 1
16 21

16 13
29 0

28 15

29 2
2S 2
2S 7

1 20
12 14

17 9
22 21

7 13
7 12
7 13

Dir.
Ret.

Ret.

Ret.

Dir.

Ret.
Dir.

Dir.

Ret.

Ret.

Dir.
Dir.

Dir.
Ret.

Dir.

Ret.
Ret.
Dir.

Dir.
Ret.
Ret.
Dir.
Dir.

Kies

De l'Isle

Bradley •

Zanotti

La Caille

Struyck

Le Monnier

La Caille

Zanotti

Euler

Euler

Wrigt

Klinkenberg

Houtteryn

Struyck

La Caille

Klinkenberg

Bets and Bliss-

Maraldi

La Caille

Zanotti

Cheseaux

Euler

Pingre

Klinkenberg

Hiorter

Cheseaux

Maraldi

La Caille

Maraldi

Le Monnier

La Caille

Struyck

Bradley

La Caille

Pingre

De Ratte

Pingre

Messier

Lai and e

Maraldi

La Caille

La Caille

Klinkenberg

Klinkenberg

Bailly

Pingre

La Caille

Chappe

La Caille

Chappe

Maraldi

Lalande

Bailly

Klinkenberg

Struvck

Pingre

Pingre

Pingrr

Pingre

Lalande

Wallot

Cassin, jun-

COM

[ 347 ]

COM

1769

1770
1771

1772
1773

1774

1779

1781

1781
1781
1783
1784
1784
1785
1785
1786
1787
1788
1788
1790
1790
1790
1792
1795

1796
1797

Ascendin
Node

25 7
25 3
25 11
19 41
25 5

24 42

25 14
25 3
25 5
25 6
16 39
13 39
15 29

4 15 4
4 14 30
4 12 .56
4 12. "o
4 12 17
4 16 14
4 12 0
4 14 22

3 18 42
0 27 51
0 27 50
8 12 43

4 1 16
4 3 15
4 3 35
4 1 12
4 1 5
6 0 57
6 o 50
6 1 22
6 0 50
0 25 3
0 25 6
0 25 4
4 4 0
4 4 30
4 4 9

2 25 13

23 1
17 23

24 14
26 50
26 52
24 12

4 34

14 23

16 52

7 11

11 21 42

5 26 12
8 27 9
1 3 11

6 10 46
II 29 11
1113 23

0 17

0 17 2

10 29 17

4- 2 9

Inclin. of
Orbit

40 49

40 41

41 1
29 41

40 41

41 28
40 43
40 50
40 50
40 47

1 44
1 41
1 47

1 55
1 49

31 26
11 15

11 17

19 0

61 25

62 36

62 36
61 21
61 13
82 48
82 49

82 21

83 0

32 26
32 24
32 26
53 56
53 15
53 48

5 16

81 43
27 13
53 9
51 9

47 55
70 14
87 32
50 54

48 16

12 28
64 53
31 54
56 58

63 52
39 47
24 17

20 3
24 42

64 55
50 36
43 52

4 24 11
4 24 9

■ 4 24 '33
4 13 15
4 24 7

■ 4 25 46
4 24 22
4 24 16
4 24 11

4 24 16
U 26 7
11 25 5
11 26 7
11 22 51

0 7 14
1 1 29 45
11 26 16
11 26 26
II 26 13
11 25 57
11 26 19

6 28 23
3 13 28
3 13 48
3 IS 6
2 15 36
2 21 40
2 20 43
2 15 16
2 14 58

10 16 28
10 16 48
10 17 26
10 17 22
2 27 14
2 27 13
2 27 14
8 6 30
8 6 19

8 6 21

5 22 17
5 28 13

7 29 11

0 16 3

1 1.5 25

2 20 44
10 28 55

3 19 52

9 27 30
5 9 26
0 7 44
3 9 8

0 23 12

2 0 15

3 21 45
9. 3 43

1 6 30
5 13 37
5 7 37

5 15 34

6 12 44
I 19 35
3 14 59

Perihel.
Dist.

12272

12289

12100

15880

12303

'11640

12280

12264

12269

12274

62959

65800

62955

64456

71717

64946

67438

67689

62872

63100

62758

52824

90576

90188

101814

113390

123800

121550

113010'

112530

142525

142525,

1 42600*

142860

71322

71313

71319

9781

10047

9926

1027558

944040

77586

96101

156530

70786

65053

114340

42730

41010

34891

106301

76691

75810

106328

79796

129300

22700

25800

21500

157800

52545

4847G

Time of Perihelion

Oct.

Aug.

Aug.

Nov.
April

Feb.

Sept.

Aug.

Jan.

Sept.

March

Jan.

July

Nov.

Nov.

Jan.

April

Jan.

April

July

May

Nov.

Jan.

May
Jan.
Dec.

April
July
April

7 14
7 14
7 12
16 10
7 14
7 11
7 18
7 15
7 16

7 16
9 0

10 22
9 0

8 9
25 2

12 21

13 13

14 0

9 1
9 4

8 19
22 6

18 22

19 1
18 21

5 11
2 12
2- 19
5 p
5 9
14 4

14 18

15 5
15 11

4 3
4 2
4 2
30 20
30 16
30 18
13 0
27 6

7 5
29 13
15 6
21 5

9 21

27 8

8 9
7 22

10 20
10 7

20 9
15 5

28 8

21 6

13 14

14 23

15 15
14 19

2 20

9 3
4 12

Dir.
Dir.

Dir.

Ret.
Dir.

Dir.
Dir.

Dii

Dir.

Ret.

Dir."

Dir."
Ret.
Dir.

Ret.

Dir.
Ret.
Dir.
Ret.

Dir.
Ret.
Dir.
Ret.
Dir.

Calculated by

Ret.

Dir.

2Y 2

Prosperin

Audiffredi

Slop

Zanotti

Asclepi

Lambert

Widder

Euler

Lexell

Pingre

Pingre

Pingre

Prosperin

Prosperin

Prosperin

Widder

Lexell

Pingre

Slop

Lambert

Rittenhouse

Pingre

Pingre

Prosperin

Lalande

Pingre

Lambert

Scultz

Lexell

Pingre

De Saron

De Saron

Boscovich

Mechain

De Saron

Mechain

D'Angos

Lexell

Lexell

Mechain

Boscovich

La Place

Mechain

Mechain

Mechain

Mechain

Chev. d'Angos

Mechain

Mechain

Mechain

P. de Saron

Mechain

Mechain

De Saron

Mechain

Lalande

Zach

Bouvard

Prosperin

Olbers

Bouvard

Burckhardt

COM

[ 348 ]

C O M

1798
1799

1799
1801
1803
1804

1805
1805
1807

Ascending
Node "

8 9 31
3 9 32
3 9 27
3 9 34
10 26 49
1 14 2S

10 10 17
5 26 48
5 26 50

11 15 7
8 10 35
S 26 46

Inclin. of
Orbit

42 23
50 56
50 57
50 51
77 2
21 20
57 0
56 29
56 44

15 58

16 25
63 11

Perihelion

1 4 30
0 3 39
0 3 39
0 3 36
6 10 20
6 3 49
2 8
28 45

28 54

29 0
19 24

0 56

Perihel.
Dist.

7796S
83987
84018

62581

26170

109420

107117

107227

37567

89159

64648

Time of Perihelion

Dec.

Sept.

Dec.

Aug.
Sept.
Feb.
Feb.
Nov.
Dec.
Sept.

31 13
7 6
7 6

7 4
25 22

8 13

9 21
13 14
13 16
IS 0
31 6
18 8

Ret.

Ret.
Dir.

Dir.

Calculated by

Burckhardt

Mechain

Von Zach

Burckhardt

Mechain

Burckhardt

Mechain

Gauss

Legend re
Legendre
Olbers

M. Facio has suggested, that some of the comets have
their nodes so very near the annual orbit of the earth, that
if the earth should happen to be found in that part next
the node at the time of a comet's passing by ; as the ap-
parent motion of the comet will be immensely swift, so its
parallax will become very sensible; and its proportion to
that of the sun will be given : whence, such transits of
comets will afford the best means of determining the dis-
tance between the earth and sun.

The comet of 1472, for instance, had a parallax above
20 times greater than the sun's; and if that of l6l8 had
come down in the beginning of March to its descending
node, it would have been much nearer the earth, and its
parallax much more notable. But hitherto none has
threatened the earth with a nearer appulse than that of
1680: for Dr. Halley finds, by calculation, that Nov. 11th,
at 1 h. 6 min. afternoon, that comet was not more than
one semidiameter of the earth to the northward of the
earth's path ; at which time had the earth been in that
part of its orbit, the comet would have had a parallax
equal to that of the moon. — What might have been the
consequence of so near an appulse, a contact, or lastly, a
shock of these bodies ? Mr. Whiston says, a deluge !

To determine the Place and Course of a Comet. — Observe
the distance of the comet from two fixed stars, whose lon-
gitudes and latitudes are known : then with the distances
thus found, calculate the place of the comet by spherical
trigonometry.

Longomontanus shows an easy method of finding and
tracing out the places of a comet mechanically, which is,
to find two stars in the same line with the comet, by stretch-
ing a thread before the eye over all the three; then do the
same by two other stars and the comet: this done, take a
celestial globe, or a planisphere, and draw a line upon it
first through the former two stars, and then through the
latter two; so shall the intersection of the two lines be the
place of the comet at that time. If this be repeated from
time to time, and all the points of intersection connected,
it will show the path of the comet in the heavens.

The last comet which has been observed, was that of
1807. It was distinctly seen by the naked eye for many
weeks in September and October. Its elements are as re-
corded above; to whic.li it may be added that its diameter,
according to Dr. Herschel, is 538 miles.

The following observations on this comet have also been
published by M. Hurckhardt, of the Institute, on Sept. 22,
1 80? ■ " The comet approached nearest the sun on the 1 2th
ol September,at 10 o'clock in the evening; itwasthen dis-
tant 3y millions of leagues from the sun, and 54 from the

earth. Since that time the distance of the comet from
the sun has increased, but its distance from the earth will
diminish till the middle of October, when its least distance
will exceed 41 millions of leagues. ' It is not probable
that the comet will become more visible, as it loses some-
what more of its light by receding from the sun, than it
gains by approaching nearer the earth. In general, the
orbit of this comet is such, that it can never approach very
near the earth. The body of the comet appears to be se-
parated from the luminous matter, which surrounds it in
the shape of a parabola. This phenomenon, which has
not been perceived in other comets, has been ascertained
by the observations of all the astronomers in Paris, and
without doubt will not escape the notice of other astro-
nomers. It does not follow, however, that the body of the
comet is wholly separated from the luminous appearance,
as the space that seems empty may be filled up by parti-
cles less luminous than the rest."

COMETARIUM, a machine adapted to give a repre-
sentation of the revolution of a comet about the sun. It
is so contrived as, by elliptical wheels, to show the unequal
motion of a comet in every part of its orbit. The comet
is represented by a small brass ball, carried by a radius
vector, or wire, in an elliptical groove about the sun in
one of its foci ; and the years of its period are shown by
an index moving with an equable motion over a graduated
silvered circle. See a particular description, with a figure
of it, in Fergusson's Astron. Svo. pa. 400.

COMMANDINE (Frederick), a celebrated mathe-
matician and linguist, was born at Urbino in Italy, in
1509; and died in 1575, at 66" years of age. He was fa-
mous for 'his learning and knowledge in the sciences. To
a great depth, and just taste in the mathematics, he joined
a critical skill in the Greek language, a happy conjunction
that well qualified him for translating and expounding the
writings of the Greek mathematicians. And accordingly,
with a most laudable zeal and industry, he translated and
published several of their works, to which no former writer
had done that good office. On which account, Francis
Moria, duke of Urbino, who was very conversant in those
sciences, proved a very affectionate patron to him. He is
greatly applauded by Bianchnnus, and other writers; and
he justly deserved their encomiums.

Of his own works he published the following:

1. Commentarius in Planisphasrium Ptolemaei: 1 558,
in 4to.

2. DeCentroGravitatis Solid. Bonon. 1565, in 4to. •

3. Horologiorum Descriptio: Rom. 1562, in 4to.

He translated and illustrated with notes, the following

C O M

[ 349 ]

C O M

works, most of them beautifully printed in 4to, by the cele-
brated printer Aldus :

1. Archimedis Circuli Dimensio; de Lineis Spiralibus;
Quadratura Parabola?; deConoidibus et Sphairoidibus; de
Arena? Numero : 1558.

2. Ptolemaei Planisphajrium;etPlanisph. Jordani: 1558.

3. Ptolemoai Analemma: 1562.

4. Archimedis de iis quse vehuntur in aqua: 1565.

5. Apollonii Pergasi Conicorum libri quatuor, una cum
Pappi Alexandrini Lemmatibus, et Commentariis Eutocii
Ascalonitee, &c: 1566.

6. Machometes Bagdadinus de Superficierum Divisioni-
bus: 1570.

7. Elementa Euclidis: 1572.

8. Aristarchus de Magnitudinibus et Distantiis Solis et
Lunae: 1572.

9. Heronis Alexandrini Spiritualium liber: 1583.

10. Pappi Alexandrini Collectiones Mathem. 1588.
COMMANDING Ground, in Fortification, an eminence

or rising ground, overlooking any post or strong place. —
This is of three sorts; 1st, A Front Commanding Ground,
or a height opposite to. the face of the post, which batters
upon its front: 2dly, A Reverse Commanding Ground, or
an eminence that can play upon the rear or back of the
post : 3dly, An Enfilade Commanding Ground, or an
eminence in flank which can, with its shot, scour the whole
length of a straight line.

COMMENSURABLE Quantities, or Magnitudes, are
such as have some common aliquot part, or which may' be
measured or divided, without a remainder, by one and the
same measure or divisor, called their common measure.
Thus, a foot and a yard are commensurable, because there
is a third quantity that will measure each, viz, an inch;
which is 12 times contained in the foot, and 36 times in
the yard. — Commensurables are to each other, as one ra-
tional whole number is to another; but incommensurables
are not so : and therefore the ratio of commensurables is
rational, but that of incommensurables, irrational; hence
also the exponent of the ratio of commensurables is a
rational number.

COMMENSURABLE Numbers, whether integers, or
fractions, or surds, are such as have some other number,
which will measure or divide them exactly, or without a
remainder. Thus, 6 and 8. are commensurable, because
2 measures or divides them both. And § and |, or -^ and
J>z are commensurable fractions, because the fraction £ or
-jij, &c, will measure them both ; and in this sense, all
fractions may be said to be commensurable. Also the
surds 2^/2 and 3i/2 are commensurable, being measured
by a/2, or being to each other as 2 to 3.

Commensurable in Power. Euclid says, right lines
are commensurable in power, when their squares are mea-
sured by one and the same- space or superficies.

COMMON, is applied to an angle, line, measure, or
the like, that belongs to two or more figures or other things.
As a common angle, a common side, a common base, a
common measure, &c.

Common Measure, or divisor, is that which measures
two or more numbers without a remainder. So of 8 and 12,
a common measure is 2, and so is 1.

The greatest Common Measure is the greatest number
that can measure two other numbers.. So of 8 and 12, the
greatest common measure is 4.

To find the greatest common measure of two numbers. Di-

vide the greater term by the less; then divide the divisor
by the remainder, if there be any; and so on continually,
always dividing the last divisor by the last remainder, till
nothing remains; then is the last divisor the greatest com-
mon measure sought.

Thus, to find the greatest common measure of Sl6 and
1488.

Sl6) 1488 ( 1

816 ,

672) 816 ( 1
6'72

144" ) 6"2 ( 4
576

96 ) 144 ( 1
96
The common measure, 48 ) 96 ( 2
• 96
Therefore 48 is the greatest common measure of 81.6 and
1488, thus:

4S ) 816 (17 4S ) 14SS (31

48 ' 144

336 4S

336 4S

The common measure is useful in fractions, to reduce a
fraction to its least terms, by dividing those that are given,
by their greatest common measure. So -j^Vf reduces to
II-, by dividing 8l6 and 1488 both by their greatest com-
mon measure 48.

COMMUNICATION of Motion, that act of a moving
body, by which it gives motion, or transfers its motion to
another body. Father Mailebranche considers the com-
munication of motion as something metaphysical; that is,
as not necessarily arising from any physical principles, or
any properties of bodies, but flowing from the immediate
agency of God.

The communication of motion results from, and is an
evidence of the impenetrability and inertia of matter, as
such; unless we admit the hypothesis of the penetrability
of matter, advanced by Boscovich and Michell, and ascribe
to the power of repulsion those effects which have been
usually ascribed to its solidity and actual resistance.

Newton shows that action and reaction are equal and
opposite; so that one body striking or acting against ano-
ther, and thence causing a change in its motion, does itself
undergo the very same change in its own motion, the con-
trary way. And hence, a moving body striking directly
another at rest, it loses just as much of its motion as it
communicates to the other. For the laws and quantity of
motion so communicated, either in elastic or non-elastic
bodies, see Collision.

COMMUTATION, Angle of, is the distance between
the sun's true place seen from the earth, and the place of
a planet reduced to the ecliptic: which therefore is found
by taking the difference between the sun's longitude and
the heliocentric longitude of the planet.

COMPANY, Rule of, or Rule of Fellowship, in Arithr
metic, is a rule by which arc determined the true shares
of profit or loss, due to the several partners, or associates,
in any enterprise, or trade, in due proportion to the stock
contributed by each, and the tfme it was employed. To
do which properly, see the Rule of Fellowship.

COMPARTMENT, a design composed of several dif-

C O M

[ 350 ]

C O M

ferent figures, disposed with symmetry; to adorn a parterre,
a cieling, pannel of joinery, or -the like.

COM PARTITION, the useful and graceful distribution
of the whole ground-plot of an edifice, into rooms of office,
and of reception, or entertainment.

COMPASS, or Mariner's Compass, is an instrument used
at sea by manners to direct and ascertain the course of
their ships. It consists of a circular brass box, which con-
tains a paper card with the 32 points of the compass, or
winds, fixed on a magnetic needle that always turns to the
north, excepting a small deviation, which is variable at
different places, and at the same place at different times.
See Variation of the Compass.

The needle with the card turns on an upright pin fixed
in the centre of the box. To the middle of the needle
is fixed a brass conical socket or cap, by which the card
hanging on the pin turns freely round the centre.

The top of the box is covered with a glass, to prevent
the wind from disturbing the motion of the card. The
whole is inclosed in another box of wood, where it is sus-
pended by gimbals, or brass hoops, to keep the card in a
horizontal position during the various motions of the ship.
The whole ought to be so placed in the ship, that the
middle section of the box, parallel to its sides, may be pa-
rallel to the middle section of the ship along its keel.

The invention .of the compass is ascribed by some wri-
ters to Flavio Gioia, or'Flavio of Malphi, about the year
1302 ; and hence it is that the territory of Principato, the
part of the kingdom of Naples where he was born, has a
compass for its arms. He divided his compass only into 8
points. Others are of opinion that the invention is due to
the Chinese; and Gilbert, in his book De Magnete, affirms
that Marcus Paulus, a Venetian, making a journey to
China, brought back the invention with him in 1260. YVhat
strengthens this conjecture is, that at first they used the
compass, in the same manner as the Chinese still do, viz,
letting it float on a small piece of cork, instead of suspend-
ing it on a pivot. It is also added, that their emperor Chi-
ning'us, a celebrated astrologer, had a knowledge of it 1120
years before Christ. The Chinese divide their compass
only into 24 points. But Ludi Vertomanus affirms, that
when he was in the East Indies, about the year 1500, he
saw a pilot of a ship direct his course by a compass, fas-
tened and framed as those now commonly used. And Bar-
lov?, in his book called The Navigator's Supply, anno 1597,
informs us, that in a personal conference with two East
Indians, they affirmed, that instead of our compass, they
use a magnetical needle of 6* inches, and longer, upon a
pin in a dish of white China earth, filled with water; in
the bottom of which they have two cross lines for the 4
principal winds, the rest of the divisions being left to the
skill of their pilots. Also in the same book he observes,
that the Portuguese, in their first discovery of the East
Indies, got a pilot of Mahinde, who brought them from
thence in 33 days, within sight of Calicut.

But Fauchette relates some verses of Guoyot de Pro-
vence, who lived in France about the year 1200, which
seem to make mention of the compass under the name of
Marinette, or Mariner's Stone; which show it was used in
France near 100 years before either the Malfitc or Vene-
tian one. The French even lay claim to the invention, from
tin- fleur de lis with which most people distinguish the
north point of the card. With as much reason Dr. Wallis
ascribe-, it u> the English, from its name Compass, by which

name most nations call it, and which he observes is used in
many parts of England to signify a circle.

The mariner's compass long continued very rude and im-
perfect, but at length received great improvement from the
invention and experiments of Dr. Knight, who discovered
the useful practice of making artificial magnets ; and the
further emendations of Mr. Smeaton, and Mr. M'Culloch,
by which the needles are larger and stronger than former-
ly, and instead of swinging in gimbals, the compass is sup-
ported in its very centre upon a prop, and the centres of
motion, gravity, and magnetism, are brought almost all to
the same point.

After the discovery of that most useful property of the.
magnet, or loadstone, viz, its giving a polarity to hardened
iron or steel, the compass was many years in use before it
was known in anywise to deviate from the poles of the
world. About the middle of the l6th century, so confident
were some persons that the needle invariably pointed due
north, that they treated with contempt the notion of the
variation, which about that time began to be suspected.
However, careful observations soon discovered that in
England, and its neighbourhood, the needle pointed to the
eastward of the true north line; and the quantity of this
deviation being known, mariners became as well satisfied
as if the compass had none; because the true course could
readily be obtained by making allowance for the true va-
riation.

From succeeding observations it was afterwards disco-
vered, that the deviation of the needle from the north was
not a constant quantity, but that it gradually diminished,
and at last, namely, about the year 1657, it was found that
the needle pointed due north at London, and has ever
since been going to the westward, till now the variation is
upwards of two points of the compass : indeed it was 22°
41' about the middle of the year 1781, as appears by the
Philos. Trans, pa. 225, for that year, and is probably now
somewhat more, which it would be of consequence to know.
So that in any one place it may be suspected the varia-
tion has a kind of libratory motion, traversing through
the north, to unknown limits eastward and westward. But
the settling of this point must be left to time and future ex-
periments. Since the above was printed in the former edi-
tion of this work, the magnetic observations have been par-
tially resumed by the Royal Society; inconsequence, the
variation of the needle was there found to be (in September
1811) 24° 14' west, and the dip or inclination 70° 32f.
See Variation, also Inclination, and Dip. Also for
a further description of different compasses and their uses,
see that useful book, Robertson's Navigation, vol. 2, pa-
231.

The Azimuth Compass differs from the common sea com-
pass in this, that' the circumference of the card, or box is
divided into degrees; and there is fitted to the box an in-
dex with two sights, which are upright pieces of brass,
placed diametrically opposite to each other, having a slit
down the middle of them, through which the sun or star
is to be viewed at the time of observation.

The Use of the Azimuth Compass, is to take the bearing
of any celestial object, when it is in, or above the horizon,
that from the magnetical azimuth or amplitude, the varia-
tion of the needle' may be known. Sec Azimuth, and
Am i'i.itude.

The figure of the compass card, with the names of the
32 points or winds, are as in fig. 5, plate 8; where other

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compasses are also exhibited. See also a new one in the or pencil, put in its place. There are in use compasses of

Supplement to Gavallo's Treatise on Magnetism. various kinds and contrivances, adapted to the various

As there are 32 whole points cpjite around the circle, purposes they are intended for; as,
which contains 360 degrees, therefore each point of the Compasses, of Three Legs, or Triangular Compasses;

compass contains the 32d part of 36*0, that is 1 1-f- degrees, the construction of which is like that of the common eom-

or 11° 15'; consequently the half point is 5° 37' 30", and passes, with the addition of a third leg or point, which has

the quarter point 2° 48' 45". . a motion every way. Their use is to take three points at

The points of the compass are otherwise called rhumbs; once, and so to form triangles, and lay down three po-

and the numbers of degrees, minutes, and seconds, made by
every quarter point with the meridian, are exhibited in the
following table :

A Table of Rhumbs, showing the Degrees, Minutes, and
Seconds, that every Point and Quarter-point of the Compass
makes with the Meridian.

NbyE

NEbN

P.s.c,,

48

45

37

30

26

15

15

0

3

45

52

30

41

15

30

0

18

45

7

30

56

15

22 30
11 15

48

45

4

37

30

4

26

15

4

15

-o

5

3

45

5

52

30

5

41

15

5

30

0

6

18

45'

6

7

30

6

56

15

6

45

0

7

33

45

7

22

30

7

11

15

7

sitions of a map to be copied at once.

Beam Compasses' consist of a long straight beam or
bar, carrying two brass cursors ; one of these being fixed
at one end, the other sliding along the beam, with a
screw to fasten it on occasionally. To the cursors may
be screwed points of any kind, whether steel, jiencils, or
the like.. To the fixed cursor is sometimes applied an
adjusting or micrometer screw, by which an extent is
obtained to very great nicety. The beam compasses
are used to draw large circles, to take great extents, or the
like.

Bow Compasses, or Bows, are a small sort of com-
passes, that shut up in a hoop, which serves for a handle.
Their use is to describe arcs or circumferences with a very
small radius.

Caliber Compasses. See Caliber.

Clockmakers Compasses, are jointed like the common
compasses,- with a quadrant or bow, like the spring com-
passes, only of different use, serving here to keep the in-
strument firm at any opening. They are made very-
strong, with the points of— their legs of well-tempered
steel, as being used to draw or cut lines in pasteboard, or
copper, &c.

Cylindrical and Spherical Compasses, consist of four
branches, joined in a centre, two of them being circular
and two flat, a little bent at the ends. The use of them is
to take the diameter, thickness, or caliber of round or cy-
lindrical bodies ; as cannons, balls, pipes, &c.

There are also spherical compasses, differing in nothing
from the common ones, but that their legs are arched ;
serving to take the diameters of round bodies.

Another sort of compasses also has lately been invented
for measuring the diameter of round bodies, as balls, &c,
which consist of two flat pieces of metal set at right angles
on a straight bar or beam of the same ; the one piece be-
liding thereon, so far as just to

Compass Dials, are small dials, fitted in boxes, for the

pocket, to show the hour of the day by the direction of ing fixed,~and the other

the needle that indicates how to place, them right, by receive the round body between them; and then its di-

turning the dial about till the cock or stile stand directly ameter, or distance between the two pieces, is shown by

over the needle. But these can never be very exact, be- the divisions marked on the beam.

cause of the variation of the needle itself; unless that Elliptical Compasses, are used to draw ellipses or

variation be allowed for, in making and placing the in- ovals of any kind. The instrument consists of a beam ab

strument. (Plate 8, fig. 6,) about a foot long, bearing three cursors ;

COMPASSES, or Pair of Compasses, a mathematical to one of which may be screwed points of any kind; and

instrument for describing circles, measuring amd dividing to the bottom of the other two are riveted two sliding

lines or figures, &c. dove-tails, adjusted in grooves made in the cross branches

The common compasses consist of two sharp-pointed of the beam. The dove-tails having a motion every way,

branches or legs of iron, steel, brass, or other metal, by turning about the long branch, they go backward and

joined together at the top by a rivet, about which they forward along the cross; so that when the beam has gone

move as on a centre. Those compasses are of the best half-way round, one of these will have moved the whole

kind in which the pin or axle, on which the joint turns, is length of one of the branches; and when the beam has

made of steel, and also half the joint itself, as the opposite gone quite round, the same dove-tail has gone back the

metals wear more equally : the points should also be made whole length of the branch. The same is understood of

of hard steel, well polished ; and the joint should open the other dove-tail.

and shut with a smooth, easy, and uniform motion. In Note, the, distance between the two sliding dove-tails, is

some compasses, the points arc both fixed ; but in others, the distance between the two foci of the ellipse ; so that

one is made to take out occasionally, and a drawing-pen, by changing that distance, the ellipse will be more circular

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or slender. Under the ends of the branches of the cross,
are placed four steel points to keep it fast.

The use of this compass is easy : by turning round the
lourr branch, the pen, pencil, or other points will draw the
ellipse required.

Its figure shows both its use and construction.

German Compasses have their legs a little bent out-
wards, near thetop; so that when shut, the points only meet.

Hair Compasses are so contrived withinside by a
small adjusting screw to one of the legs, as to take an ex-
tent to a hair's breadth.

Proportional Compasses are those whose joint lies,
not at the end of the legs, but between the points termi-
nating each leg. These are either simple, or compound.
In tbe former kind the centre, or place of the joint is fixed;
so that one pair of these serves only for one proportion.

Compound Proportional Compasses have the joint or
centre moveable. They consist of two parts or sides of
brass, which lie upon each other so exactly as to appear
but one when they are shut. These sides easily open,
and move about the centre, which is itself moveable in a
hollow canal cut through the greatest part of their length.
To this centre on each side is fixed a sliding piece, of a
small length, with a fine line drawn thereon serving as an
index, to be set against other lines or divisions placed up-
on the compasses on both sides. These lines are, 1, A line
of lines; 2, a line of superficies, areas, or planes, the
numbers on which answer to the squares of those on the
line of lines ; 3, a line of solids, the numbers on which
answer to the cubes of those on the line of lines ; 4, a line
of circles, or rather of polygons to be inscribed in circles.
These lines are all unequally divided, the first three from
1 to 20, and the last from 6 to 20. The use of the first
is to divide a line into any number of equal parts ; by the
2d and 3d are found the sides of like planes or solids in
any given proportion ; and by the 4th, circles are divided
into any number of equal parts, or any polygons inscribed
in them. See Plate 8, fig. 7-

Spring Compasses, or Dividers, are made of hardened
steel, with an arched head, which by its spring opens the
legs ; the opening being directed by a circular screw fast-
ened to one of the legs, let through the other, and worked
with a nut.

Trisecting Compasses, for the trisecting of angles me-
chanically, for which purpose they were invented by M.
Tarragon. The instrument consists of two central rules,
and an arch of a circle of 120 degrees, immoveable, with
its radius: the radius is fastened with one of the central
rules, like the two legs of a sector, that the central rule
may be carried through all the points of the circumference
of the arch. The radius and rule should be as thin as
possible; and the rule fastened to the radius should be
hammered cold, to be more elastic; and the breadth of
the other central rule must be triple the breadth of
the radius : in this rule also is a groove, with a dove-tail
fastened on it, for its motion ; there must also be a hole
in the centre of each rule.

Turn-up Compasses, a late contrivance to save the
trouble of changing the points ; the body is like the com-
mon compasses ; and towards the bottom of the legs with-
outsidc, are added two other points, besides the usual
ones ; the one carrying a drawing-pen point, and the other
a port-crayon ; both being adjusted to turn up, to be used
or not, a8 occasion may require.

COMPLEMENT in general, is what is wanting, or
necessary, to complete sonie certain quantity or thing.
As, the

Complement of an arch or angle, as of 90° or a qua-
drant, is what any given arch or angle wants of it; so the
complement of 50° is 40°, and the complement of 100° is
— 10°, a negative quantity. — The complement to ISO0 is
usually called the supplement, to distinguish it from the
complement to 90°, properly so called.- — -The sine of the
complement of an arc, is contracted into the word cbsine ;
the tangent of the complement, into cotangent ; &c.

Arithmetical Complement, is what a number or loga-
rithm wants of unity or 1 with some number of ciphers.
It is best found, by beginning at the left-hand side, and
subtracting every figure from 9) except the last, or right-
hand figure, which must be subtracted from 10.
So, the arithmetical comp. of the log. 9'5329714,
by subtracting from 9's, &c, is 0'46702S6\

The arithmetical complements are much used in opera-
tions by logarithms, to change subtractions into additions,
which are more conveniently performed, especially when
there are more than one of them in the operation.

Complement, in Astronomy, is used for the distance
of a star from the zenith ; or the arc contained between
the zenith and the place of a star which is above the ho-
rizon. It is the Same as the complement of the altitude,
or co-altitude, or the zenith distance.

Complement of the Course, in Navigation, is the quan-
tity which the course wants of 90°, or 8 points, viz, a quar-
ter of the compass.

Complement of the Curtain, in Fortification, is that
part of the anterior side of the curtain, which makes the
demigorge.

Complements of the Line of Defence, is the remainder of
that line, after the angle of the flank is taken away.

Complements of a Parallelogram, or in a Parallelogram.
are the two lesser parallelograms, made by drawing two
right lines parallel to each side of
the given parallelogram, through the -
same point in the diagonal. So p
and Q are the complements in the
parallelogram abcd.

In every case, these complements
are always equal, viz, the parallelo-
gram p = Q.

Complement of Life, a term much used, in the doe-
trine of Life Annuities, by De Moivre, and, according to
him, it denotes the number of years which a given life
wants of 86, this being the age which he considered as the
utmost probable extent of life. So 56 is the complement
of 30, and 30 is the complement of 56.

That author supposed an equal annual decrement of
life through all its stages, till the age of 86. Thus, if
there be 56" persons living at 30 years of age, it is supposed
that one will die every year, till they be all dead in 56
years. This hypothesis in some cases is near the truth,
for the value of single lives. Demoivre had another hy-
pothesis, that of life decreasing in geometrical progression,
for calculating the value of joint lives; but more inac-
curate than the former. See Life Annuities; De Moi-
vre's Treatise on Annuities, pa. 83 ; Simpson's Life An-
nuities; Price on Reversionary Payments, pa. 1; and
Baily's Life Annuities.

COMPOSITE Number, is one that is compounded of,

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or made up by the multiplication of two other numbers,
greater than 1, or which can be measured by some other
number greater than 1. As 12, which is composed, or
compounded of 2 and 6, or 3 and 4, viz, by multiplying
together 2 and 6, or 3 and 4, both products making the
same number 12; which therefore is a composite number.
Composites are opposed to prime numbers, or primes,
which cannot be exactly measured by any other number
except unity ; or be produced by multiplying together two
other factors.

Composite Numbers between themselves, are the same with
commensurable numbers, or such as have a common mea-
sure or factor ; as 15 and"12, which have the common
term 3. The doctrine of Prime and Composite numbers
is pretty fully treated in the 7th and 8th books of Euclid's
Elements.

Composite Numbers, in musical calculations, are such
as are composed by the multiplication of the prime in-
tegers 2, 3, 5. All primes larger than these, or com-
posite numbers into which such enter, are not musical
numbers, but are irrational, or surd, in a musicar sense.

Composite Order, in Architecture, is the last of the
five orders of columns; and is so called, because its capital
is composed out of those of the other orders. Thus, it
borrows a quarter-round from the Tuscan and Doric ; a
double row of leaves from the Corinthian ; and volutes
from the Ionic. Its cornice has single modillions, or
dentils; and its column is 10 diameters in height. This
order is also called the Roman order, and Italic order, as
having been invented by the Romans, like as the other
orders are denominated from the people among whom
they had their rise.

COMPOSITION, is a species of reasoning by which
we proceed from things that are known and given, step by
step, till we arrive at such as were before unknown or re-
quired ; viz, proceeding upon principles self-evident, on
definitions, postulates, and axioms, with a previously de-
monstrated series of propositions, step by step, till it gives
a clear knowledge of the thing to be known or demon-
strated. Composition, otherwise called the synthetical
method, is opposed to Resolution, or the analytical me-
thod, and is chiefly used by the ancients, Euclid, Apol-
lonius, &c. See Pappus ; also the term Analysis.

Composition of forces, or of motion, is the union or as-
semblage of several forces or motions that are oblique to
one another, into an equivalent one in another direction.

J. When several forces or motions are united, that act
in the same line of direction, the combined force or motion
will be in the same line of direction also. But when ob-
lique forces arc united, the compounded force takesanew di-
rection, different from both, and is either a right line or a
curve, according to the nature of the forces compounded.

2. If two compounding motions be both equable ones,
whether equal to each other or not, the line of the com-
pound motion will be a straight one. Thus, if the one
equable in the direction ab be suffi-
cient to carry a body over the space
ab in any time, and the other equable
motion sufficient to pass over ac in the
same time ; then by the compound mo-
tional- both acting on the body together,
it would in the same time pass over the
diagonal ad of the parallelogram abdc.
For because the motions are uniform,

Vol. I.

any spaces -Ab, ac passed over in the same time, are pro-
portional to the velocities, or to ab and ac ; and conse-
quently all the points a, d, d, d, of the path are in the
same right line.

3. And though the compounding motions be not equa-
ble, but variable, either accelerated or retarded, provided
they do but vary in a similar manner, the compounded mo-
tion will still be in a straight line. Thus, suppose for in-
stance, that the motions both vary in such a manner, as
that the spaces passed over in the same time, whether they
be equal to each other or not, are both as the same power
n of the time; then ab" : ac" : : Ab" : ac", and hence
ab : ac : : a* : ac, and therefore, as before, aA/d is still a
right line.

4. But if the compounding motions be not similar to
each other, as when the one is equable and the other va-
riable, or when they are varied in a dissimilar manner,
then the compounded motion is in some curve line: so if
the motion in the one direction ef be in a less proportion,
with respect to the time, than that in the direction eg is
then the path will be a curve line e/jh
convex towards ef; butif the motion in
ef be in a greater proportion than that
in eg, then the path of the compound
motion will be a curve ezh concave
towards ef : that is, in general, the
curvilineal path is convex towards that
direction in which the motion is in the
less proportion to the time. Hence,
for a particular instance, if the mo-
tion in the direction ef be a motion of projection, which
is an equable motion, and the motion in the direction eg
that arising from gravity, which is a uniformly accelerated
motion, or in proportion to the squares of the times; then
is eg as oh2, and Eg as g!f, that is eg : Eg : : or : gA%
which is the property of the parabola ; and therefore the
path e/(H of any body projected, is the common parabola.

5. If there be three forces united, or acting against the
same point a at the same time, viz, the force or weight b
in the direction ab, and the forces or tensions in the di-
rections ac, a d ; and if these three forces mutually balance
each other so as to keep the common point a in equilibrio,
then are these forces directly proportional to the respective
sides of a triangle, formed by drawing lines parallel to the
directions of these forces; or indeed perpendicular to those
directions, or making any one and the same angle with
them. So that, if be be drawn parallel, for instance, to ad,
and meet ca produced in e, forming the triangle abe, then
are the three forces in the directions ab, ac, ad, respec-
tively proportional to the sides ab, ae, be.

And this theorem, with its corollaries, Dr. Keil observes,
is the foundation of all the new mechanics of M. Varignon;
2Z

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by help of which may the force of the muscles be com-
puted, and most of the mechanic theorems in Borelli, De
Motu Animalium, may immediately be deduced. And
hence also may any number of forces be compounded to-
gether into one only. See Collision, Composition of
Forces, &c.

Composition of Numbers and Quantities. See Com-
bination.

Composition of Proportion, according to the 15th de-
finition of the 5th book of Euclid's Elements, is when, of
4 proportionals, the sum of the 1st and 2d is to the 2d, as
the sum of the 3d and 4th is to the 4th:

as if it be a : b : : c : d,
then by composition a -t- b : b : : c + d : d.
Or, in numbers, if 2 : 4 : : 9 : IS,
then by composition 6 : 4 : : 27 : IS.

Composition of Ratios, is the adding of ratios toge-
ther; which is performed by multiplying together their
corresponding terms, viz, the antecedents together, and the
Consequents together, for the antecedent and consequent
of the compounded ratio; like as the addition of logarithms
is the same thing as the multiplication of their correspond-
ing numbers. Or, if the terms of the ratios be placed
fraction-wise, then the addition or composition of the ra-
tios is performed by multiplying the fractions together.
Thus, the ratio of a : b, or of 2 : 4,
added to the ratio of c : d, or of 6 : S,
makes the ratio of ac : bd, or of 12 : 32; and so the ra-
tio of ac to bd is said to be compounded of the ratios of
a to b, and c to d. So likewise, if it were required to
compound together the three ratios, viz, of a to b, c to d,

and e to /; then - x - x - = — - are the terms of the
b d f bdf

compound ratio; or the ratio of ace to bdf Is compounded

or made up of the ratios of a to b, c to d, and e to/.

Hence, if the consequent of each ratio be the same as

the antecedent of the preceding ratio, then is the ratio of

the first term to the last, compounded, or made up of all

the other ratios, viz, the ratio of a to e, equal to the sum

of all the ratios of a to b, of b to c, of c to d, and of d to

e: for — x — x — x — = - the terms or exponents of the

b c d e e
compounded ratio.

Hence also, in a series of continual proportionals, the
ratio of the first term to the third is double of the ratio of
the first to the second,

and the ratio of the 1st to the 4th is triple of it,
and the ratio of the 1st to the 5th is quadruple of it,
and so on ; that is, the exponents are double, triple, qua-
druple, &c, of the first exponent: as in the series 1, a, a",
a1', a*, &c ; where the ratio I to a1 is double, of 1 to a3
triple, &c, of the ratio of 1 to a; or the exponent of a?,
a?, a4, &c, double, triple, quadruple, &C, of a.

COMPOUND Interest, called also Interest upon In-
terest, is that which, is reckoned not only upon the princi-
pal, but on the interest itself forborn, which thus becomes
a sort of secondary principal.

If ;• be the amount of 1 pound for 1 year, that is the
sum of the principal and interest together for one year;
then is r2 the amount for 2 years,
and r3 the amount for 3 years,

and in general j' the amount for t years; that is rl is the
sum or total amount of all the principals and interests to-
gether of 1/ for the whole time or number of years c; con-

sequently, if p be any other principal sum, forborn fori
years, then its amount in that time at compound interest,
is a =pr*.

The Rule' therefore in words is this: to one pound add
its interest for one year, or half \ ear, or for the first time
at which the interest is reckoned; raise the sum r to the
power denoted by the time or number of terms; then this
power multiplied by the principal, or first sum lent, will
produce the whole amount.

For example, To find how much 50/ will amount to in
5 years at 5 per cent, per annum, compound interest. —
Here the interest of 1/ for 1 year is '05, and therefore,
r =. ] -05 ; hence the 5th power of
it for 5 years, is r5 = T2762S &c;
multiply this by p or - 50
gives the amount/))' or 63'S14/
or 631 \6s o~d for the amount sought.

If the compound interest, or amount of any sum, be
required for the parts of a year, it may be determined as
follows :

1. When the time is any aliquot part of a year.

Rule. Find the amount of 1/ for 1 year as before, and
that root of it which is denoted by the aliquot part will
be the amount of \l for that time; which multiplied
by the principal, will be the amount of the given sum re-
quired.

2. When the time is not an aliquot part of a year.
Rule. 1. Reduce the time into days, and the 365th

root of the amount of 1/ for 1 year, is the amount of the
same sum for one. day. 2. Raise this amount to that
power whose index is equal to the number of days, and it
will be the amount of U for the given time. 3. Multiply
this amount by the principal, and it will be the amount of
the given sum required.

Note. The readiest way of extracting the roots, espe-
cially high ones, is by logarithms, thus : Divide the loga-
rithm of the rate, or amount of 1/, for a year, by the de-
nominator of the given aliquot part, and the quotient will
be the logarithm of the root sought.

But Compound Interest for any given number of whole
years, is best computed by means of such a table as the
following, being the amounts of \l for any number of
years, and at several rates of compound interest.

As an example of the use of this table, suppose it were
required to find the amount of 250/ for 35 years at 4 per
cent, compound interest.

In the column of 4 per cent, and line of 35 years,

is 3'y46oo,

which multiplied by the principal

ojves ----- 986'52250

or - ----- - 986/ 10s 5|tf,

which is the amount sought.

Note. By a bare inspection of this table, it appears
how many years are required for any sum of money to
double itself, at any rate of compound interest; viz, by
looking in the columns when the amount becomes the
number 2. So it is found that at theseveral rates the re-
spective times requisite- for doubling any sum, ate lie trl
thus; viz,

Rate 3 3* 4 14 5 (>'

Years 23* 20* 171 15| I'M 12

Table of the Amount
Years and

COM [ 855 ]

of 1 / at Compound Interest for many
several Rates of Interest.

COM

k! at a per

3^ pir | 4 per j 4j per

5 per

b per

?

cent.

cent. cent. J cent.

cent.

cent.

1

1-03000

1-03500 1 1-04000 j 1-04500

1-05000

1 -06000

2

1 -06090

1-07123 1 1-08160

1-09203

1-10250

1-12360

3

1-09273

1-10872

1-12486

1-14117

1-15763

1-19102

4

1-12551

1-14752

1-16986

1-19252

1-21551

1-26248

5

1-15927

1-18769

1-21665

1-24618

1-27628

1-33823

6

1-19405

1-22926

1-26532

1-30226

1-34010^

1-41852

7

1-22987

1-27228

1 -3 1 593

1-36086

1-40710

1-50363

8

1-26677

1-31681

1-36857

1-42210

1-47746

1-59385

9

1-30477

1-36290

1-42331

1-48610

1-55133

1-68948

ID

1-34392

1-41060

1-48024

1-55297

1-62890

1-79085

11

1-38423

1-45997

1-53945

1-62285

1-71034

1-89830

12

1-42576

1-51107

1-60103

1-69588

1-79586

2-01220

13

1-46853

1 -56395

1-66507

1-77220

1-88565

2-13293

14

1-51259

1-61869

1-73168

1-85194

1-97993

2-26090

15

1-55797

1-67535

1-80094

1-93528

2-07893

2-39656

16

1-60471

1-73399

1-87298

2-02237

2-18287

2-54035

17

1-65285

1-79468

1-94790

2-11338

2-29202

2-69277

18

1-70243

1-85749

2-02582

2-20848

2-40662

2-85434

19

1-75351

1-92250

2-10685

2-30786

2-52695

3-02560

20

1-80611

1-98979

2-19112

2-41171

2-65330

3-20714

•21

1-86029

2-05943

2-27877

2-52024

2-78596

3-39956

22

1-91610

2-13151

2-36992

2-63365

2-92526

3-60354

23

1-97359

2-20611

2-46472

2-75217

3-07152

3-81975

24

2.03279

2-28333

2-56330

2-87601

3-22510

4-04893

25

2-09378

2-36324

2-66584

3-00543

3-38635

4-29187

26

2-15659

2-44596

2-77247

3-14068

3-55567

4-54938

27

2-22129

2-53157

2-88337

3-28201

3-73346.

4-82235

28

2-28793

2-62017

2-99870

3-42970

3-92013

5-11169

23

2-35657

2-71 188

3-11865

3-58404

4-11614

5-41839

30

2-42726

2-80679

3-24340

3-74532

4-32194

5-74349

31

2-50008

2-90503

3-37.313

3-91386

4-53804

6-08810

32

2-57508

3-00671

3-50806

4-08993

4-76494

6-45339

33

2-65234

3-11194

3-64838

4-27403

5-00319

6-84059

34

2-73191

3-22086

3-79432

4-46636

5-25335

7-25103

35

2-81386

3-33359

3-94609

4-66735

5-51602

7-68609

36

2-89828

3-45027

4-10393

4-87738

5-79182

8-14725

37

2-98523

3-57103

4-26809

5-09686

6-08141

8-63609

38

3-07478

3-69601

4-43881

5-32622

6-38548

9-15425

39

3-16703

3-82537

4-61637

5-56590

6-70475

9-70351

40

3-26204

3-95926

4-80102

5-81636

7-03999

10-28572

41

3-35990

4-09783

4-99306

6-07810

7-39199

10-90286

42

3-46070

4-24126

5-19278

6-35162

7-76159

11-55703

43

3-56452

4-38970

5-40050

6-63744

8-14967

12-25045

44

3-67145

4-54334

5-61652

6-93612

8-55715

12-98548

45

3-78160

4-70236

5-84118

7-24825

8-98501

13-76461

46

3-89504

4-86694

6-07482

7-57442

9-43426

14-59049

47

4-01190

5-03728

6-31782

7-91527

9-90597

15-46592

48

4-13225

5-21359

6-57053

8-27146

10-40127

16-39387

49

4-25622

5-39606

6-83335

8-64367

10-92133

17-37750

50

4-38391

5-58493

7-10668

903264

11-46740

18-42015

COMPOUND Motion, that motion which is the effect
of several conspiring powers or forces, viz, such forces as
are not directly opposite to each other: as when the ra-
dius of a circle is considered as revolving about a centre,
and at the same time a point moving along it ; which pro-
duces a kind of a spiral for the path of the point. And hence
it is easily perceived, that all curvilinear motion is com-
pound, or the effect of two or more forces ; though every
compound motion is not curvilinear.

It is a popular theorem in Mechanics, that in uniform
compound motions, the velocity produced by the con-
spiring powers or forces, is to that of either of the two
compounding powers separately, as the diagonal of a pa-
rallelogram, according to the direction of whose sides they
act separately, is to either of the sides. See Composition
of Motion, and Collision.

Compound Numbers, those composed of the multipli-

cation of two or more numbers: as J 2, composed of 3
times 4. See Composite.

Compound Pendulum, that which consists of several
weights constantly keeping the same distance, both from
each other, and from the centre about which they oscil-
late.

Compound Quantities, in Algebra, -arc such as are con-
nected together by the signs -+- or — . Thus, a -+- b, or
a — c -+- d, or aa — la, are compound quantities. Com-
pound quantities are distinguished into binomials, tri-
nomials, quadrinomials, &c, according to the number of
terms in them ; viz, the binomial having two terms ; the
trinomial, three ; the quadrinomial, 4 ; &c. Also, those
that have more than two terms, are called by the general
name of multinomials, as also polynomials.

Compound Ratio, is that which is made, by adding
two or more ratios together ; viz, by multiplying all
their antecedents together for the antecedent, and all
the consequents together for the consequent of the com-
pound ratio. So 6" to 72 is a ratio compounded of the

ratios 2 to 6, and 3 to 12; because — = - x — ■

73 6 12

also ab to cd is a ratio compounded of the ratio of a to c,

and b to d ; for — - = - x -. See Composition of Ratios,
cd c d

COMPRESSION, the act of pressing, or squeezing
something, so as to bring its parts nearer together, and
make it occupy less space. Compression differs from
condensation as the cause from the effect, compression be-
ing the action of any force on a body, without regarding
its effects ; whereas condensation denotes the state of a
body that is actually reduced into a less bulk, and is an
effect of compression, though it may be effected also by
other means. Nevertheless, compression and condensation
are often confounded.

Pumps, which the ancients imagined to act by suction,
do in reality act by compression ; the piston, in working
in the narrow pipe, compresses the inclosed air, so as to
enable it, by the force of its increased elasticity, to raise
the valve, and make its escape ; on which, the balance
being destroyed, the pressure of the atmosphere on the
stagnant surface, forces up the water in the pipe, thus
evacuated of its air.

It was long thought that water was not compressible in-
to less bulk : and it was believed, till lately, that after
the air had been purged out of it, no art or violence was
able to press it into less space. In an experiment made by
the Academy del Cimento, water, when violently squeezed,
made its way through the fine pores of a globe of gold, ra-
ther than yield to the compression.

But the ingenious Mr. Canton, attentively considering
this experiment, found that it was not sufficiently accu-
rate to justify the conclusion which had always been
drawn from it; since the Florentine philosophers had no
method of determining that the alteration of figure in their
globe of gold, occasioned such a diminution of its internal
capacity, as was exactly equal to the quantity of water
forced into its pores. To bring this matter therefore to a
more accurate and decisive trial, he procured a small
glass tube of about two feet long, with a ball at one end,
of an inch and a quarter in diameter. Having filled the
ball and part of the tube with mercury, and brought it
exactly to the heat of 50° of Fahrenheit's thermometer,
he marked the place where the mercury stood in the Mtbe,
2Z2

C O M

[ 356 ]

CON

which was about six inches and a half above the ball ; he
then raised the mercury by heat to the top of the tube,
and there seated the tube hermetically; then on reducing
the mercury to the same degree of heat as before, it stood
in the tube T3^ of an inch higher than the mark. The
same experiment was repeated with water exhausted of
air, instead of mercury, and the water stood in the tube
-f%3_ of an inch above the mark. Since the weight of the
atmosphere on the outside of the ball, without any coun-
terbalance from within, will compress the ball, and equally
raise both the mercury and water, it appears that the
water expands -^^ of an inch more than the mercury by
removing the weight of the atmosphere. Having thus de-
termined that water is really compressible, he proceeded
to estimate the degree of condensation corresponding to
any given weight. For this purpose he prepared another
ball, with a tube joined thereto ; and finding that the mer-
cury in .j%3o of an inch of the tube was the hundred thou-
sandth part of that contained in the ball, he divided the
tube accordingly. He then filled the ball and part of the
tube with water exhausted of air; and leaving the tube
open, placed this apparatus under the receiver of an air-
pump, and observed the degree of expansion of the water
answering to any degree of rarefaction of the air ; and
again by putting it into the glass receiver of a condensing
engine, he noted the degree of condensation of the water
corresponding to any degree of condensation of the air.
He thus found, by repeated trials, that, in a temperature
of 50*, and.when the mercury has been at its mean height
in the barometer, the water expands one part in 21740;
and is as much compressed by the weight of an additional
atmosphere ; or the compression of water by twice the
weight of the atmosphere, is one part in 10870 of its whole
bulk. Should it be objected, that the compressibility of
the water was owing to any air which it might be supposed
to contain, he answers, that more air would make it more
compressible; he therefore let into the ball a bubble of
air, and found that the water was not more compressed by
the same weight than before.

In some further experiments of the same kind, Mr. Can-
ton determined that water is more compressible in winter
than in summer; but he observed the contrary in spirit
of wine, and oil of olives. The following table was form-
ed, when the barometer was at 29 inches and a half, and
the thermometer at 50 degrees.
Compression of

Spirit of wine

Oil of olives

Rain water

Sea water

Mercury
He infers that these fluids are not only compressible, but
elastic ; and that the compressions of them, by the same
weight, are not in the inverse ratio of their densities, or
specific gravities, as might be supposed. Phil. Trans, vol.lii.
176'2, art. 103; and vol. liv. 176'4, art. 47-

The compression of tlie air, by its own weight, is sur-
prisingly great : but the air may be still farther compress-
ed by art. See Elasticity of Air. This immense com-
pression and dilatation, Newton observes, cannot be ac-
counted for in any other way, but by a repelling force,
with which the particles of air are endued; by virtue of
which, when at liberty, they mutually recede from each
other.

onth pts.

Spec, grav

66

846

48

918

46'

1000

40

1028

3

13595

He also further informs us, that this repelling power is
stronger and more sensible in air, than in other bodies ;
because air is generated out of very fixed bodies, but not
without great difficulty, and by the help of fermentation :
now those particles always recede from each other with
the greatest violence, and are compressed with the "reatest
difficulty, which, when contiguous, cohere the most strongly.
See Air-, Attraction, Cohesion, Dilatation, and
Repulsion.

COMPUTATION, the manner of accounting and esti-
mating time, weights, measures, money, &c. See Cal-
culation, which it is also used for.

CONCAVE, an appellation used in speaking of the inner
surface of hollow bodies, more especially of spherical or
circular ones.

Concave glasses, lenses, and mirrors, have either one
side or both sides concave.

The- property of all concave lenses is, that the rays of
light, in passing through them, are deflected, or made to
recede from one another; as in convex lenses they are
inflected towards each other; and that the more as the con-
cavity or convexity has a smaller radius. Hence parallel
rays, as those of the sun, b}' passing through a concave
lens, become diverging; diverging rays are made to di-
verge more; and converging rays are made either to con-
verge less, or to become parallel, or; go out diverging. And
hence it is, that objects viewed through concave lenses,
appear diminished; and the more so, as they are portions
of less spheres. See Lens.

Concave mirrors have the contrary effect to lenses :
they reflect the rays which fall on them, so as to make
them approach more to, or recede from each other, than
before, according to the situation of the object; and that
the more as the concavity is greater, or as the radius of
concavity is less. Hence it is that concave mirrors mag-
nify objects that are presented to them ; and that in a
greater proportion, as they are portions of smaller spheres.
And hence also concave mirrors have the effect of burnin<*
glasses. See Mirror, and Burning-Glass.

CONCAVITY, that side of a figure or body which is
hollow. — An arch of a curve has its concavity turned all
the same way, when the right lines that join airy two of its
points are all on the same side of the arch.

Archimedes, intending to include in his definition such
lines as have rectilinear parts, says, a line has its concavity
turned one way, when the right lines that join any two of
its points, are either all on one side of it, or while some
fall upon the line itself, none fall on the opposite side.
Archim. de Sphajr. et Cyl. Def. 2; and Maclaurin's Flux-
ions, art. 180. .

When two lines that have their concavity turned the
same way, have the same extremes, and the one includes
the other, or has its concavity towards it, the perimeter of
that which includes, is greater than that which is included.
Archiin. ib. ax. 2.

CONCENTRIC, having the same centre. It is opposed
to excentric, or having different centres. The word is
chiefly used in speaking of round bodies and figures, such
as circular, and elliptic ones; but it may also be used for
polygons that are drawn parallel to each oilier, from the
same centre. — Nonius's method of graduating instruments
consists in describing with the same quadrant 45 concen-
tric arches, dividing the outermost into 90 equal parts, the
next into 89, and so on.

CON

C 3S7 ]

CON

CONCHOID, or Cofchiles, the name of a curve in-
vented by Nicomedes. It was much used by the ancients
in the construction of solid problems, as appears by what
Pappus says. — It is thus constructed : ap and bd beingtwo
lines intersecting at right angles; from p draw a number
of other lines pfde, &c, on which take always de = df
= ABorBC; so shall the curve line drawn through all
the points e, e, e, be the first conchoid, or that of Nico-
medes; and the curve drawn through all the other points
f, f, f, is called the second conchoid ; though in reality,
they are both but parts of the same curve, having the same
pole p, and four infinite legs, to which the line dbd is a
common asymptote.

£ , E.A gj ***'

The fixed point p is called the pole or centre of the
conchoid. And the line ed is calted the directrix.
To determine the equation of the curve : putAB = BC

= DE = DF = a, PB ■= b, BG = EH = X, and GE = BH

=zy\ then the equation to the first conchoid will be x"1 x
(b -+- x)x -h xlyL — a'1 x (J+ x)2, or X* -+- 2bx3 -+- b2x2 •+•
x2y2 = arb2 ■+- 2irbx -+- a?x2 ; and, changingonly the sign
of x, as being negative in the other curve, the equation
to the 2d conchoid will be x"' x (b — x)2 -+■ x2y2 =az x
(6 — a:)2 or x* — 2bx3 + bzx2 ■+■ x2y2 = azb2 — 2azbx ■+■
a2x2.

Of the whole conchoid, expressed by these two equa-
tions, or rather one equation only, with different signs,
there are three cases or species ; as first,
when BC is less than bp, the conchoid will be as in fig. 1 ;
when bc is equal to bp, the conchoid will be as in fig. 2;
and when bc is greater than bp, the conchoid will be as
in fig. 3.

Newton approves of the use of the conchoid for trisect-
ing angles, or finding two mean proportionals, or for con-
structing other solid problems. Thus, in the Linear Con-
struction of equations, towards the end of his Universal
Arithmetic, he says, " The ancients at first endeavoured
in vain at the trisection of an angle, and the finding of
two mean proportionals by a right line and a circle. Af-
terwards they began to consider several other lines, as the
conchoid, the cissoid, and the conic sections, and by some
of these to solve these problems." Again, " Either there-
fore the trochoid is not to be admitted at all into geo-
metry, or else, in the construction of problems, it is to be
preferred to all lines of a more difficult description: and
there is the same reason for other curves; for which rea-
son we approve of the trisections of an angle by a con-
choid, which Archimedes in his Lemmas, and Pappus in
his Collections, have preferred to the inventions of all
others in this case; because we ought either to exclude

all lines, besides the circle and right line, out of geometry,
or admit them according to the simplicity of their descrip-
tions, in which case the conchoid yields to none, except
the circle/' Lastly, " That is arithmetically more sim-
ple which is determined by the more simple equations,
but that is geometrically more simple which is determined
by the more simple drawing of lines;1 and in geometry,
that ought to be reckoned best which is geometrically
most simple: wherefore I ought not to be blamed, if, with
that prince of mathematicians, Archimedes, and other an-
cients, I make use of the conchoid for the construction of
solid problems."

CONCRETE Numbers are those that are applied to ex-
press or denote any particular subject; as 3 men, 2 pounds,
&g. Whereas, if nothing be connected with a number,
it is taken abstractedly or universally: thus, 4 signifies
only an aggregate of 4 units, without any regard to a par-
ticular subject, whether men or pounds, or any thing else.

CONCURRING, or Congruent Figures, in Geo-
metry, are such as, being laid upon one another, do ex-
actly correspond to, and cover one another, and conse-
quently must be equal among themselves. Thus, trian-
gles having two sides and the contained angle equal, each
to each, are congruent or equal to each other in all respects.

CONDAMINE (Charles Marie de la), a French
mathematician, and chevalier of the order of St. Lazare.
He was born at Paris in 1701 ; and began his career in
the military service; but, soon yielding to his inclination
for natural curiosities, he travelled over the East. After-
wards, in 1735, 1736", &c, he was employed, along with
M. Godin, Bouguer, and other mathematicians, to mea-.
sure the meridian in Peru, near Quito; from which he re-
turned by the river Amazon, through vast dangers and
difficulties, as may be seen in the account he published
of the voyage, which was not finished under the course of
ten years. After in some degree recovering his health he
travelled through Italy and England, &c, and died in 1774.
Condamine was admitted into the Academy of Sciences
in 1730, the French Academy in 176*0, and was also a
member of several foreign academies. The published works
of Condamine are,

1. Voyage dans 1'Amerique Meridionale, 1745, Svo.

2. Figure de la Terre, 1749> in 4to.

3. Voyage a l'Equateur, avec le Supplement, ct l'Hi-
stoire des Piramides de Quito, 1751 and 1752, 2 vols. 4to.

4. Three "Memoirs on Inoculation, and Letters on the
same subject.

Besides numerous scientific articles in most of the vo- .
lumes of Memoirs of" the Academy of Sciences, from the
year 1731 to 1768 inclusively.

CONDENSATION, is the compressing or reducing a
body into a loss bulk or space; by which means it is ren-
dered more dense and compact. Compression and Con-
densation (though sometimes used the one for the other)
differ in this; viz, that the former denotes a diminution
of bulk occasioned by the application of external force;
whereas, the latter expresses the same effect when pro-
duced without that external application, as is the case
with most bodies in cooling. But the word Condensation
has been more commonly used for denoting the conver-
sion of vapour into water, or of vapours in general into
liquids.

Notwithstanding this definition, it is to be remarked,
that in every case of condensation, or of the contraction of
a body into a narrower space, the effect is produced by

CON

[ 358 ]

CON

the escape of something; and in bodies which appear to
be the simplest in nature, the contraction which they un-
dergo in cooling, is occasioned by the escape of the ca-
loric, which the present.state of philosophical knowledge
reckons among the elementary substances. But the re-
verse of this is not always true; viz, bodies are not always
contracted in their dimensions when the caloric escapes
from them ; and such has been found to be the case with
water, with iron, and with some other substances in cer-
tain temperatures.

It is however evident that condensation in general con-
sists in bringing the parts closer to each other, and increa-
sing their contact, whatever may be their form or state of
previous existence, or the means by which it is effected :
in opposition to rarefaction, which renders the body lighter
and looser, -by setting the parts farther asunder, and di-
minishing their contact, and of consequence their cohesion.

Air condenses, either by cold, or by pressure, but much
more by the latter; and most of all by chemical process.
Water condenses also under the same circumstances of ac-
tion ; but it suddenly expands by congelation : indeed al-
most all matter, both solids and fluids, has the same pro-
perty of condensation by those means. See Compres-
sion. So also vapouris condensed, orconverted into water,
by distillation, or naturally in the clouds. The way in
which vapour commonly condenses, is by the escape of the
caloric. On touching it, the vapour parts with this heat:
and on doing so, it immediately loses the proper charac-
teristics of vapour, and becomes water. But though this
be the most common and usual way in which vapour in
general is condensed, yet nature certainly proceeds after
another manner; since we often observe the vapours most
plentifully condensed when the weather is really warmer
than at other times. — -Most bodies are susceptible of three
successive states of existence ; namely, the solid, the li-
quid, and the elastic or vaporous ; and all these are ef-
fected by the introduction of caloric. During every one
of those states, a different degree of condensation is pro-
duced by the intermediate gradations of temperature, viz,
such as are not quite sufficient to induce a different state
of existence in the body concerned. Thus, a quantity of
the vapour of water, which at the temperature of 242°
(Fahrenheit's thermometer), and under the mean gravity
of the atmosphere, occupies the space of 3600 cubic
inches; if it be gradually cooled until the temperature
becomes equal to about 212°, its bulk will be contracted
so as to occupy the half of the space it did before, viz,
about 1800 cubic inches. If the temperature be lowered
below212°, the vapour will be condensed into liquid water,
which will occupy the space of not more than a single cu-
bic inch. If the cooling be continued, the water will be
contracted in its bulk, but not very regularly (that is, the
decrements of bulk will not be exactly proportional to the
decrements of heat) until the temperature descends to a-
bout 42°. Below that degree the water, instead of con-
tracting its bulk, is expanded by further cooling; viz, by
a further abstraction of caloric. This is a very remark-
able property of water, on which some interesting pheno-
mena of nature are depending. The water, though cx-
paudinghelow the temperature of 42°, still continues fluid,
as far as about the temperature of 32°; but below this
last-mentioned point, by further cooling it becomes a solid,
namely, ice ; and in this form water occupies a greater
space than it did in a liquid state.

Since the enlargement of the bulk of water commences

at about the temperature of 42°. which is ten degrees above
the point of melting ice, commonly called the freezing
point, which is 32°, it is evident that the particles of water
begin to arrange themselves in a particular order long be-
fore the freezing takes place. And, by following the ana-
logy, it may be supposed that in every state of existence,
the particles of bodies have, more or less, a tendency to
arrange themselves in a certain order; to which tendency
the irregularities in the condensations of bodies are pro-
bably to be attributed.

CONDENSER, a pneumatic engine, or syringe, by
which an extraordinary quantity of air may be forced into
a given space; so that frequently ten atmospheres, or ten
times as much air as the space naturally contains, without
the engine, may be thrown in by means of it, and its egress
prevented by valves properly disposed.

The condenser is made either of metal or glass, and
either in a cylindrical or globular form, into which the air
is thrown with an injecting syringe. The receiver, or ves-
sel containing the condensed air, should be made very
strong, to bear the force of the air's elasticity thus in-
creased ; for which reason it is commonly made of brass.
When glass is used, it will not bear so great a condensa-
tion of air; but then the experiment will be more enter-
taining, as the effect may be viewed of the condensed air
upon any subject put therein.

CONDITION, Equations of, in the Integral Cal-
culus, are certain equations of this form, - = -, useful in
* y x

ascertaining whether a proposed fluxion will admit of tinite
integration, or a finite'fluent. This method was given by
M. Clairaut, in the Memoirs of the Academy of Sciences
for the year 1740, and may be explained in this manner.
When a given fluxional equation is very complex, it is
often difficult to determine whether it be intcgrable or not ;
the object of this method is then to ascertain that point.
In order to which, Clairaut demonstrates, that if there be
a fluxional equation of the first order, expressed generally
by these two terms ax + bj), where a denotes all the
terms or quantities that multiply ■*, and b all those that
multiply y, being fluxions of x and y ,• and if in such
equation, there be taken the fluxion of a, making y only
variable; next in like manner the fluxion of b, making ar
only variable ; then if that fluxion of a when divided by
y, give the same result as the fluxion of b when divided
by x, the proposed equation will be integrated. Suppose,
for instance, the general proposed quantity be axmyn: its
fluxion is amx ml yn x -t- any "'l xmy. Here then a is
amxm~'y", and b is any'"' xm. Now, by taking the
fluxions of these two quantities again, making only y vari-
able in a, and x only in B, striking out j in the former,
and •* from the latter, there results the same quantity
amnxm''y"~l. This rule is expressed more briefly by
these kinds of symbols: if the fluxion of ax -+- ny be inte-

arable, then must^ =-, that is, the fluxion of a, making
° y x a

only y variable, and dividing byy, will be equal to the
fluxion of B, making only X variable, and dividing by x.

Clairaut shows, in like manner, that in a fluxional ex-
pression from three variables, x, y, z, thus expressed,
£x -+- ny ■+■ cz, in order that it may be intcgrable, these.

three following equations must be true, i = -1 and - = ?,

y x z. x

and -r = -.; whence it is sufficiently suggested that if there

CON [ 359

be a fluxional expression from four variables, it must be
satisfied by six such equations, viz, as many as are the
combinations of 4 things by 2 at a time; and so on for
more complex cases. And this is what is meant by Equa-
tions of Condition.

For example, suppose the fluxional expression x^/y ■+■
.tv . t ayx .

-£y ayf/x — -— be proposed, which is sufficiently

complex not to be easily judged whether it be integrable
or not. Here, taking all the terms that multiply x, they

give */y —2 = A, and all the terms that multiply j give

— a^/x = B. Then taking the fluxion of a, making

]

CON

'Wv

V.V

only y variable, and suppressing y,

taking ' the fluxion of b, making only x variable, and

dividing by ■*, gives also — ; -, the same as the for-

mer; that is, 4 = -, and consequently the proposed ex-
pression is quadrable.

Hence the integration is easy and evident, for it is only
to integrate either a* making?/ constant, or bj> making x;

that is either x*/v : — or —. ajl/x, both of which

v J 3x1 i^/y J v

give, by their integration, the same quantity x^/y — ay

%/x, which is therefore the fluent of the proposed fluxion

■ / xy -s, avx

vv 3a;3

CONDORCET (John-Anthony Nicholas de Ca-
ritat, Marquis of), member of the Institute of Bologna,
of the Academies of Turin, Berlin, Stockholm, Upsal, Phi-
ladelphia, Petersburg, Padua, &c, and secretary to tiie
Academy of Sciences at Paris, was bom at Ribemont in
Picardic, the 17th of September 1743. His early attach-
ment to the sciences, and progress in them, soon rendered
him a conspicuous character in the commonwealth of let-
ters. He was received as a member of the Academy of
Sciences at 25 years of age, namely, in March 1769, as
Adjunct-Mechanician; afterwards, he became Associate
in 1770, Adjunct-Secretary in 1773, and sole Secretary
soon after, which he enjoyed till his death, or till the dis-
solution of the Academy by the Convention.

Condorcet soon became an author, and that in the most
sublime branches of mathematics. He published his Es-
sais d Analyse in several parts ; the first part in 17(55, at
22 years of age ; the second, in 1767 ; and the third, in
1768. These works are chiefly on the Integral Calculus,
or the finding of Fluents, and make one volume 4to.

He published the Eloges of the Academicians or mem-
bers of the Academy of Sciences, from the year 1666 till
1700, in several volumes. He wrote also similar Eloges
of the Academicians who died during the time that he dis-
charged the important office of Secretary to the Academy;
as well as the very useful histories of the different branches
of science commonly prefixed to the volumes of Memoirs,
till the volume for the year 17S3, when it is to be la-
mented that so useful a part of the plan of the Academy
was discontinued.

I lis other memoirs contained in the volumes of the Aca-
demy, are the following.

1. Tract on the Integral Calculus; 1765.

2. On the problem of Three Bodies; 1767.

3. Observations on the Integral Calculus; 1767-

4. On the Nature of Infinite Series; on the Extent of

the Solutions which they give; and on a new method of Ap-
proximation for Differential Equations of all Orders; 1769.

5. On Equations for Finite Differences; 1770.

6. On Equations for Partial Differences; 1770.
7- On Differential Equations; 1770.
8. Additions to the foregoing Tracts; 1770.
9- On the Determination of Arbitrary Functions which

enterthelntegralsof Equations to Partial Differences ; 1 771.

10. Reflexions on the Methods of Approximation hi-
therto known for Differential Equations; 1771.

11. Theorem concerning Quadratures ; 1771.

12. Inquiry concerning the Integral Calculus; 1772.

13. Calculation of Probabilities, part 1 and 2; 1781.

14. Continuation of the same, part 3 ; 1782.

15. Ditto, part 4; 1783.

16. Ditto, part 5; 1784.

'Condorcet had the character of being a very worthy
honest man, and a respectable author, though perhaps not
a first-rate one, and produced an excellent set of Eloges
of the deceased Academicians, during the time of his se-
cretaryship. A late French political writer has observed
of him, that he laboured to succeed to the literary throne
of d'Alembert, but that he cannot be ranked among illus-
trious authors ; that his works have neither animation nor
depth, and that his style is dull and dry ; that some bold
attacks on religion and declamations against despotism
have chiefly given a degree of fame to his writings.

On the breaking out of the troubles in France, our au-
thor took a decided part on the side of the people, and
steadily maintained the cause he had espoused amid all
the shocks and intrigues of contending parties; till, under
the tyranny of Robespierre, he was driven from the con-
vention, being one of those members proscribed on the
31st of May 1793, and he died March 28, 1794. The
manner of his death is thus described by the public prints
of that time. He was obliged to conceal himself with the
greatest care for the purpose of avoiding the fate of Bris-
sot and the other deputies who were executed. He did
not however attempt to quit Paris, but concealed himself
in the house of a female, who, though she knew him only
by name, did not hesitate to risk her own life for the pur-
pose of preserving that of Condorcet. In her house he
remained till the month of March 1794, when it was ru-
moured that a domiciliary visit was to be made, which
obliged him to leave Paris. Though he had neither pass-
port nor civic card, he escaped through the barrier, and
arrived at the plain of Mont Rouge, where he expected to
find an asylum in the country-house of an intimate friend.
Unfortunately this friend had set out for Paris, where he
was to remain for three days. — During all this period,
Condorcet wandered about the;. fields and in the woods,
not daring to enter an inn on account of not having a
civic card. Half dead with hunger, fatigue, and feary and
scarcely able to walk on account of a wound in his foot,
he passed the night under a tree.

At length his friend returned, and received him with
great cordiality; but as it \vas deemed imprudent that he
should enter the house in the day-time, he returned to the
woods till night. In this short interval between morning
and night his caution forsook him, and he went to an inn
at Clamars, where he ordered an omlette. His torn
clothes, his dirty cap, his meagre and pale countenance,
and the greediness with which he devoured the omlette,
fixed the attention of the persons in the inn, aniong whom
was a member of the revolutionary committee of Cla-

CON

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CON

mars. This man conceiving him to be Condorcet, who
had effected his escape from the Bicetre, asked him whi-
ther he was going, and whether he had a passport. The
confused manner in which he replied to these questions,
induced the member to order him to be conveyed before
the committee, who, after an examination, sent him to
the district of Boury la Reine. He was there interrogated
again; and the unsatisfactory answers which he gave, de-
termined the directors of the district to send him to pri-
son on the succeeding day. — During the night he was con-
fined in a kind of dungeon. On the next morning, when
his keeper entered with some bread and water for him, he
found him stretched on the ground without any signs
of life.

On inspecting the body, the immediate cause of his
death could not be discovered, but it was conjectured that
he had poisoned himself. Condorcet indeed always car-
ried a dose of poison in his pocket; and he said to the
friend who was to have received him into his house, that
he had been often tempted to make use of it, but that the
idea of a wife and daughter, whom he loved tenderly, re-
strained him. During the time that he was concealed at
Paris, he wrote a history of the Progress of the Human
Mind, in two volumes.

CONDUCTOR, in Electricity, a term first introduced
in this science by Dr. Desaguliers, and used to denote
those substances which are capable of receiving and trans-
mitting electricity ; in opposition to electrics, in which
the matter or virtue of electricity may be excited and ac-
cumulated, or retained. The former are also called non-
electrics, and the latter non-conductors. And all bodies
are ranked under one or other of these two classes, though
none of them are perfect electrics, nor perfect conductors,
so as wholly to retain, or freely and without resistance to
transmit the electric fluid.

To the class of conductors, belong all metals and semi-
metals, ores, and all fluids (except air and oils), together
with the substances containing them, the effluvia of fla-
ming bodies, ice (unless very hard frozen), and snow, most
saline and stony substances, charcoals, of which the best
are those that have been exposed to the greatest heat,
smoke, and the vapour of hot water.

It seems probable that the electric fluid passes through
the substance, and not merely over the surfaces of metal-
lic conductors ; because, if a wire of any kind of metal
be covered with some electric substance, as resin, sealing-
wax, &c, and ajar be discharged through it, the charge
will be conducted quite as well as without the electric
coating.

Some philosophers have even alleged, that electricity
will pervade a vacuum, and be transmitted through it al-
most as freely as through the substance of the best con-
ductor : but Mr. Walsh found, that the electric spark or
shock would no more pass through a perfect vacuum, than
through a piece of solid glass. In other instances how-
ever, when the vacuum has been made with all possible
care, the experiment has not succeeded.

It may also be observed, that many of the forementioned
substances are capable of being electrified, and that their
conducting power may be destroyed and recovered by dif-
ferent processes : for example, green wood is a conduc-
tor; but baked, it becomes a non-conductor; again its
conducting power is restored by charring it; and lastly it
is destroyed by reducing this to ashes.

Again, many electric substances, as glass, resin, air, &c,

become conductors by being made very hot : however, air
heated by glass must be excepted.

See, on this subject, Priestley's Hist, of Electricity,
vol. 1 ; Franklin's Letters &c, pa. 96 and 262, edit. 1769;&
Cavallo's Complete Treat, of Electr. chap. 2 ; Henley's
Exper. and Obser. in Electr. also Philos. Trans, vol.67,
p. 122; and elsewhere in the different volumes of the
Transactions.

Prime Conductor, is an insulated conductor, so con-
nected with the electrical machine, as to receive the elec-
tricity immediately from the excited electric. Mr. Grer
first employed metallic conductors in this way, in 1734- ;
and these were several pieces of metal suspended on silken
strings, which he charged with electricity. M. Du Fay
fastened to the end of an iron bar, which he used ashisprime
conductor, a bundle of linen threads, to which, he ap-
plied the excited tube; but these were afterwards changed
for small wires suspended from a common gun-barrel, or
other metallic rod.

In the present advanced state of the science, this part
of the electrical apparatus has been considerably im-
proved. The prime conductor is made of hollow brass,
and usually of a cylindrical form. Care should be taken,
that it be perfectly smooth and round, without points and
sharp edges. The ends of the conductor are spherical;
and it is necessary, that the part most remote from the
electric should be made round and much larger than the
rest, the better to prevent the electric matter from escaping,
which it always endeavours most to do at the greatest di-
stance from the electric : and the other end also should
be furnished with several pointed wires or needles, either
suspended from, or fixed to an open metallic ring, and
pointing to the globe, or cylinder, or plate, to collect
the electricity. It is best supported by pillars of solid
glass, covered with sealing-wax or good varnish. Prime
conductors of a large size are usually made of paste-
board, covered with tin-foil or gilt paper; these being use-
ful for throwing off a longer and denser spark than those
of a smaller size: they should terminate in a smaller knob
or obtuse edge, at which the sparks should be solicited.
Mr. Naime prepared a conductor 6 feet in length, and 1
foot in diameter, from which he drew electrical sparks at
the distance of 16, 17, or 18 inches; and Dr. Van Ma-
rum still far exceeded this, with a conductor of 8 inches
diameter, and upwards of 20 feet long, formed of different
pieces, and applied to the large electrical machine in Tey-
ler's Museum at Haerlem, the most powerful machine of
the kind ever yet constructed. But the size of the con-
ductor is always limited by that of the electric, there be-
ing a maximum which the size of the former should not
exceed ; for it may be so large, that the dissipation of the
electricity from its surface may be greater than that which
the electric is capable of supplying.

Dr. Priestley recommends a prime conductor of polish-
ed copper, in the form of a pear, supported by a pillar and
a firm basis of baked wood : this receives its fire by a long
arched wire of soft brass, which may be easily bent, and
raised or lowered to the globe : it is terminated by an open
ring, in which some sharp-pointed wires are suspended. In
the body of this conductor are holes for the insertion of
metalline rods. This, he observes, collects the fire per-
fectly well, and retains it equally every where. See Phi-
losoph. Transact, vol. 6±, art. 7 ; and Hist. Elect, vol. 2,
§2.

Mr. Henley has contrived a new kind of prime conduce

C O N

[ 361 ]

CON

or, which, from its use, is called the luminous conductor.
It consists of a glass tube ]8 inches long, and 2 inches
diameter. The tube is furnished at both ends with brass
caps and ferrules about 2 inches long, cemented and made
air-tight, and terminated by brass balls. In one of these
caps is drilled a small, hole, which is covered by a strong
\alve, and serves for exhausting the tube of its air. Within
the tube at each end there is a knobbed wire, projecting to
the distance of 2 inches and a half from the brass caps.
• To one of the balls is' annexed a fine-pointed wire for re-
ceiving and collecting the electricity, and to the other a
■wire with a knob or ball for discharging it. The conduc-
tor, thus prepared, is supported on pillars of sealing-wax
or glass. Resides the common purposes of a 'prime con-
ductor to an electrical machine, this apparatus serves to
exhibit and ascertain the direction of the electric matter
in its passage through it. See a figure of this conductor
in the Philos. Trans, with a description of experiments,
&c, with it, vol. 64, pa. 403.

Conductors of Lightning, are pointed metallic rods,
fixed to the upper parts of buildings, to secure them from
strokes of lightning. ' These were invented and proposed
by Dr. Franklin for this purpose, soon after the identity
of electricity and lightning was ascertained; and they ex-
hibit a very important and useful application of modern
discoveries in this science. This ingenious philosopher,
having found that -pointed bodies are better fitted for re-
ceiving and throwing off the electric fire, than such as are
terminated by blunt ends or flat surfaces, and that metals
are the readiest and best conductors, soon discovered that
lightning and electricity resembled each other in this and
other distinguishing properties : he therefore recommended
a pointed metalline rod, to be raised some feet above the
highest part of a building, and to be continued down into
the ground, or the nearest water. The lightning, should
it ever come within a certain distance of this rod or wire,
would be attracted by it, and pass through it, preferably
to any other part of the building, and be conveyed into
the earth or water, and there dissipated, without doing
any damage to the building. Many facts have occurred
to evince the utility of this simple and apparently trifling
apparatus. And yet some electricians, of whom Mr. Wil-
son was the chief, have objected to the pointed termina-
tion1 of this conductor; preferring rather a blunt end:
because, they pretend, a point invites the electricity from
the clouds, and attracts it at a greater distance than a
blunt conductor. Philos. Trans, vol. 54, pa. 234; vol. 63,
pa. 49; and vol. 68, pa. 232.

This subject has indeed been very accurately examined
and discussed ; and pointed conductors are almost uni-
versally, and for the best reasons, recommended as the
most proper and eligible. A sharp-pointed conductor, as
it attracts the electric matter of a cloud at a greater di-
stance than the other, draws it off gradually : and by con-
veying it away gently, and in a continued stream, prevents
an accumulation and a stroke; whereas a conductor with
a blunt termination receives the whole discharge of a cloud
at once, and is much more likely to be exploded, when-
. ever a cloud comes within a striking distance. To this
may be added experience; for buildings guarded by either
natural or artificial conductors terminating in a point,
have very seldom been struck by lightning; but others,
having flat or blunt terminations, have often been struck
and damaged by it.

The best conductor for this purpose, is a rod of iron,
Vol. I.

or rather of copper, as being a better conductor of elec-
tricity, and less liable to rust, about 3 quaiters of an inch
thick, which is either to be. fastened to the walls of a
building by wooden cramps, or supported by wooden
posts, at the distance of a foot or two from the wall ;
though less may do : its upper end should terminate in a
pyramidal form, with a sharp point and edges ; and, when
made of iron, gilt or painted near the top, or else pointed
with copper ; and be elevated 5 or 6 feet above the highest
part of the building, or chimneys, to which it may be
fastened. The lower end should be driven 5 or 6 feet into
the ground, and directed from the foundations of the
building, or continued till it communicates with the nearest
water: and if this part be made of lead, it will be less apt
to decay. When the conductor is formed of different
pieces of metal, care should be taken that they are well
joined : and it is farther recommended, that a communi-
cation be made from the conductor by plates of lead, 8 or
10 inches broad, with the lead on the ridges and gutters,
and with the pipes that carry down the rain, which should
be continued to the bottom of the building, and be made
to communicate either with water or moist earth, or with
the main pipe which supplies the house with water. If
the building be large, two, three, or more conductors
should be applied to different parts of it, in proportion to
its extent. Philos. Trans, vol. 64, pa. 403.

Chains have been used as conductors for preservinu-
ships; but as the electric matter does not pass readily
through the links composing them, copper wires, a little
thicker than a goose-quill, have been preferred, and are
now generally used. They should reach 2 or 3 feet above
the highest mast, and be continued down in any conve-
nient direction, so as always to touch the sea water. Phi-
los. Trans, vol. 52, pa. 633. See also Franklin's Letters &c
1769, pa. 65, 124, 479, &c ; and Cavallo's Electr. ch. 9. '
For the Construction and management of Electrical
Kites, and Conductors or Machines for drawing electricity
from the clouds, see Priestley's Hist, of Electr. vol. 2
pa. 103, edit. 1775.

CONE, a kind of round pyramid, or a solid body hav-
ing a circle for its base, and its sides formed by right lines
drawn from the circumference of the base to a point at
top, being the vertex or apex of the cone.

Euclid defines a cone to be a solid figure, whose base is
a circle, and is produced by the entire revolution of a
right-angled triangle about its perpendicular leo-, called
the axis of the cone. If this leg, or axis, be greater than
the base of the triangle, or radius of the circular base of
the cone, then the cone is acute-angled, that is, the angle
at its vertex is an acute angle ; but if the axis be less than
the radius of the base, it is an obtuse-angled cone; and
if they be equal, it is a right-angled cone.

But Euclid's definition only ex-
tends to aright cone, that is, to a cone
whose axis is perpendicular or at
right-angles to its base; and not to
oblique ones, in which the axis is in-
clined to the same, the general defi-
nition, or description of which may
be this : If a line va continually
pass through the point v, turning on
that point as a joint, and the lower
part of it be carried round the cir-
cumference aec of a circle; then
the space inclosed between that circle

CON

[ 362 ]

CON

and the path of the line, is a cone. The circle abc is
the base of the cone; v is its vertex; and the line vd,
from the vertex to the centre of the base, is the axis of
the cone. Also the other part of the revolving line, pro-
duced above v, will describe another cone wacb, called
the opposite cone, and having the same common axis
produced Dvsf, and vertex v.

Properties of the Cone. — l.The area or surface of every
right cone, exclusive of its base, is equal to a triangle
whose base is the periphery, and its height the slant side
of the cone. Or, the curve superficies of a right cone, is
to the area of its circular base, as the slant side is to the
radius of the base. And therefore, the same curve sur-
face of the cone is equal to the sector of a circle whose
radius is the slant side, and its arch equal to the circum-
ference of the base of the cone.

2. Every cone, whether right or oblique, is equal to
one-third part of a cylinder of equal base and altitude;
and therefore the solid content is found by multiplying
the base by the altitude, and taking | of the product;
and hence also all cones of the same or equal base and
altitude, arc equal.

3. Though the solidity of an oblique cone be obtained
in the same manner as that of a right one, it is otherwise
with regard to the surface, since this cannot be reduced to
the measure of a sector of a circle, because all the lines
drawn from the vertex to the base arc not equal. See a
Memoir on this subject, by M. Euler, in the Nouv. Mem.
de Petersburg, vol. 1. Dr. Barrow has demonstrated, in
his Lectiones Geometrical, that the solidity of a cone
with an elliptic base, forming part of a right cone, is equal
to the product of its surface by a third part of one of the
perpendiculars drawn from the point in which the axis of
the right cone intersects the ellipse; and that it is also
equal to ■§■ of the height of the cone multiplied by the el-
liptic base : consequently that the perpendicular is to the
height of the cone, as the elliptic base is to the curve sur-
face. For the curve surface of all the oblique parts of a
cone, see my Mensur. pa. 174, &c.

4. To find the Curve Surface of the Frustum of a Cone.
Multiply half the sum of the circumferences of the two
ends, by the slant side, or distance between these circum-
ferences.

5. For the Solidity of the Frustum of a Cone, add into
one sum the areas of the two ends and the mean propor-
tional between them, multiply that sum by the perpendi-
cular height, and •§■ of the product will be the solidity.
See also my Mensuration, pa. 136, &c.

6. The Centre of Gravity of a cone is | of the axis di-
stant from the vertex.

Cones of the Higher Kinds, are those whose bases are
circles of the higher kinds; and are generated, like the
common cone, by conceiving a line turning on a point or
vertex on high, and revolving round the circle of the
higher kind.

Cone of Rays, in Optics, includes all the several rays
which fall from any point of a radiant object, on the sur-
face of a glass.

Double Cone, ox Spindle, in Mechanics, is a solid form-
ed of two equal cones joined at their bases. If this be
laid on the lower part of two rulers, making an angle with
each other, and elevated in a certain degree above the ho-
rizontal plane, the cones will roll up towards the superior
ends, and appear to ascend, though in reality its centre
of gravity descends perpendicularly lower.

CONFIGURATION, the exterior surface or shape that
bounds bodies, and gives them their particular figure.

CONFUSED Vision. See Vision.

CONGELATION, or Freezing, the act of fixing
the fluidity of any liquid, by cold, or the application of
cold bodies: in which it differs from coagulation, which
is produced by other causes. See Freezing, Frost, '
and !ce.

CONGRUITY, in Geometry, is applied to lines and
figures, which exactly correspond- when laid over each
other; as having the same terms, or bounds. It is as-
sumed, as an axiom, that those things are equal and simi-
lar, betw'een which there is acongruity. Euclid, and most
geometricians alter him, demonstrate great part of their
elements from the principle of congruity: though Leib-
nitz and Wolfius substitute the notion of Similitude in-
stead of that of congruity.

CONIC Sections, are the figures made by cutting a
cone by a plane.

2. According to the different positions of the cutting
plane, there arise five different figures or sections, viz, a
Triangle, a Circle, an Ellipse, a Parabola, and an Hyper-
bola: the last three of which only are peculiarly called
conic sections.

3. If the cutting plane pass
through the vertex of the cone, and
any part of the base, the section will
evidently be a Triangle; as vab.

4. If the plane cut the cone pa-
rallel to the base, or make no angle
with it, the section will be a Circle,
as AED.

5. The section dab is an Ellipse,
when the cone is cut obliquely
through both sides, or when the
plane is inclined to the base in a less
anule than the side of the cone is.

6. The section is a Parabola, when
the cone is cut by a plane parallel
to the side, or when the cutting plane
and the side of the cone make equal
angles with the base.

7. The section is an Hyperbola, when the cutting plane
makes a greater angle with the base than the side of the
cone does. And if the plane be continued to cut the op-
posite cone, this latter section is called the opposite hyper-
bola to the former; as due.

CON

8. The Vertices of any section, arc
tlie points where the cutting plane
meets the opposite sides of the cone,
or the sides of the vertical triangular
section ; as a and b.— Hence, the el-
lipse and the opposite hyperbolas
have each two vertices ; but the pa-
rabola only one ; unless we consider
the other as at an infinite distance.

9- The Axis, or Transverse Dia-
meter, of a conic section, is the line'
or distance ab between the vertices.
— Hence the axis of a parabola is in-
finite in length.

363 ]

CON

10. The Centre c, is the middle point of the axis. —
Hence the centre of a parabola is infinitely distant from
the vertex. And of an ellipse, the axis and centre lie
within the curve ; but of an hyperbola, without.

11. A Diameter is any right line, as ab or de, drawn
through the centre, and terminated on each side by the
curve : and the extremities of the diameter, or its inter-
sections with the curve, are its Vertices. — Hence all the
diameters of a parabola are parallel to the axis, and infi-
nite in length ; because drawn through the centre, a point
at an infinite distance. And hence also every diameter of
the ellipse and hyperbola has two vertices ; but of the pa-
rabola, only one; unless we consider the other as at an in-
finite distance.

12. The Conjugate to any diameter, is the line drawn
through the centre, and parallel (o the tangent of the curve
at the vertex of the diameter. So fg, parallel to the fan-
gent at i>, is the conjugate to de; and hi, parallel to the
tangent at a, is the conjugate to ab. — Hence the conju-
gate hi, of the axis a b, is perpendicular to the same; but
the conjugates of other diameters arc oblique to them.

13. An Ordinate to any diameter, is a line parallel to
its conjugate, or to the tangent at its vertex, and terminated
by the diameter and curve. So dk and el are ordinales
to the axis ab ; and mn and no ordinates to the diameter
de. — Hence the ordinates of the axis are also perpendi-
cular to it ; but of other diameters, the ordinates are ob-
lique to them.

14. An Absciss is any part of a diameter, contained
between its vertex and an ordinate to it; as ak or bk, and
dnoi'en. — Hence, in the ellipse and hyperbola, every
ordinate has two abscisses ; but in the parabola, only one ;
the other vertex of the diameter being infinitely distant.

15. The Parameter of any diameter, is a third propor-
tional to that diameter and its conjugate, in the ellipse and
hyperbola, or to any absciss and its ordinate in the para-
bola.

16. The Focus is the point in the axis where the ordi-
nate is equal to half the parameter : as K and l, where dk
or el is equal to the semiparameter. — Hence, the ellipse
and hyperbola have each two foci ; but the parabola only

one. — The foci, or burning points, were so called, because
ail rays are united or reflected into one of them, which pro-
ceed from the other focus, and are reflected from the curve.

d H a I

17. If dae, fbg, be two opposite hyperbolas, having
ab for their first or transverse axis, and ab for their second
or conjugate axis; and if due,fbg-be two other opposite
hyperbolas, having the same axes, but in the contrary or-
der, viz, ab their first axis, and ab their second; then these
two latter curves dae, fbg, are called the Conjugate Hy-
perbolas to the former two, dae, fbg; and each pair of
opposite curves mutually conjugate to the other.

18. And if tangents be drawn to the four vertices of the
curves, or extremities of the axis, forming the inscribed
rectangle hikl; the diagonals hck and icl, of this rect-
angle, are called the Asymptotes of the curves.

19. Scholium. The rectangle inscribed between the
four conjugate hyperbolas, is similar to a rectangle circum-
scribed about anellipse, by drawing tangents, in like man-
ner, to the four extremities of the two axes ; also the
asymptotes or diagonals in the hyperbola, are analogous
to those in the ellipse, cutting this curve in similar points,
and making the pair of equal conjugate diameters. More-
over, the whole figure, formed by the four hyperbolas, is
as it were, an ellipse turned inside out, cut open at the
extremities D, E, r, G, of the said equal conjugate dia-
meters, and those four points drawn out to an infinite di-
stance, the curvature being turned the contrary way, but
the' axes, and the rectangle passing through their extre-
mities, remaining fixed, or unaltered.

Further if there be placed or conjoined four cones, in
such a manner* as to have all their vertices meet in a com-
mon point, also all their axes in the same plane, and their
sides touching and coinciding through their whole length,
in two lines crossing in the said common vertex point ;
then if these four cones be all cut by one plane, parallel
to the common plane of their four axes, the section will
exhibit a very beautiful appearance of four hyperbolic
sections, of which each pair of opposite ones are equals,
and each pair has also the resemblance of what are usu-
ally called the conjugates to the other pair; but are not
really so, except only when the four cones are all equal ;
in which case the four hyperbolic sections are all equal also.

From the foregoing definitions arc easily derived the fol-
lowing general corollaries to the sections.

Ellipse. Hyperbola. Parabola,

/

3 A2

CON

[ 364 ]

CON

20. Corol. 1. In the ellipse, the semiconjugate axis,
CD, or ce, is a mean proportional between co and cp, the
parts of the diameter op of a circular section of the cone,
drawn through the centre c of the ellipse, and parallel to
the base of the cone. For de is a double ordinate in this
circle, being perpendicular to op as well as to ab.

21.- In like manner, in the hyperbola, the length of the
semiconjugate axis, CD or ce, is a mean proportional be-
tween co and cp, drawn parallel to the base, and meeting
the sides of the cone in o and p. Or, if ao' be drawn pa-
rallel to the side vb, and meet PC produced in o', making
co' = co; and on this diameter o'p a circle be drawn pa-
rallel to the base: then the semiconjugate cd or ce will
be an ordinate of this circle, being perpendicular to o'p as
well as to ab.

Or, in both figures, the whole conjugate axis de is a
mean proportional between qa and br, parallel to the
base of the cone. See my Conic Sections, pa. 6.

In the parabola, both the transverse and conjugate are
infinite; for ab and br are both infinite.

22. Corol. 2- In all the sections, as will be equal to
the parameter of the axis, if qg be drawn making the angle
aqg equal to the angle bar. In like manner Bg will be
equal to the same parameter, if rj be drawn to make the
angle BRg = the angle abq.

23. Corol. 3. Hence the upper hyperbolic section, or
section of the opposite cone, is equal and similar 'to the
lower one. For the two sections have the same transverse
or first axis ab, and the same conjugate or second axis
de, which is the mean proportional between aq and br;
and they have also equal parameters ag, Bg. So that the
two opposite sections make, as it were, but the two oppo-
site ends of one entire section or hyperbola, the two being
everywhere mutually equal and similar. Like the two
halves of an ellipse, with their ends turned the contrary
way.

24. Corol. 4. And hence, though both the transverse
and conjugate axis in the parabola be infinite, yet the for-
mer is infinitely greater than the latter, or has an infinite
ratio to it. For the transverse has the same ratio to the
conjugate, at the conjugate has to the parameter, that is,
as an infinite to a finite quantity, which is an infinite ratio.

The peculiar properties of each particular curve, will
be best referred to the particularwords Ellipse, Hyper-
bola, Parabola ; and therefore it will only be proper
here to notice a few of the properties that are common to
all the conic sections.

Some other General Properties.

25. From the foregoing definitions, &c, it appears, that
the conic sections are in themselves a system of regular
curves, naturally allied to each other; and that one is
changed into another perpetually, when it is either in-
creased, or diminished, in infinitum. Thus, the curvature
of a circle being ever so little increased or diminished,
passes into an ellipse ; and again, the centre of the ellipse
going off infinitely, and the curvature being thereby dimi-
nished, is changed into a parabola ; and lastly, the curva-
ture ofa parabola beingevcr so little changed, there ariseth
the first of the hyperbolas; the innumerable species of
which will all of them arise orderly by a gradual diminu-
tion of the curvature; till this quite vanishing, the last
hyperbola ends in a right. line. Hence it is manifest, that
-•very regular curvature, like that of a circle, from the
circle itself to a right line, is a conical curvature, and is

distinguished with its peculiarname, according to the divers
degrees of that curvature.

26". That all diameters in a circle and ellipse intersect
one another in the centre of the figure within the section:
that in the parabola they are all parallel among them-
selves, and to the axis : but in the hyperbola, they inter-
sect one another, without the figure, in the common centre
of the opposite and conjugate sections.

27- In the circle, the latus rectum, or parameter, is
double the distance from the vertex to the focus, which is
also the centre. But in ellipses, the parameters are in all
proportions to that distance, between the double and qua-
druple, according to their different species. While, in the
parabola, the parameter is just quadruple that distance.
And, lastly, in hyperbolas the parameters are in all pro-
portions beyond the quadruple, according to their various
kinds.

28. The first general property of the conic sections,
with regard to the abscisses and ordinates of any diameter,
is, that the rectangles of the abscisses are to each other,
as the squares of their corresponding ordinates. Or, which
is the same thing, that the square of any diameter is to the
square of its conjugate, as the rectangle of two abscisses
of that diameter, to the square of the ordinate which divides
them. That is, in all the figures,

the rect

But as in the parabola the infinites cb and eb are in a
ratio of equality, for this curve the same property becomes
ac : ae : : CD" : efs, that is, in the parabola, the abscisses
are as the squares of their ordinates.

Or, when one of the ordinates is the semiconjugate Gil,
dividing the diameter equally in the centre, the same gene-
ral property becomes,
ag . gb or ag2 : ac . cb : : Gir . cd'2,
or ab'j : in'2 : : ac . cb : cd'2.

29. Hence is derived the equation of the curves of the
conic sections; thus, putting the diameter a b = d , its con-
jugate 111 = c, absciss AC =zx, and its ordinate cd =y;
then is the other absciss cb = d — x in the ellipse, ord-t- x
in the hyperbola, or d in the parabola; and hence the last
analogy above, becomes d'1 : c'2 : : x{d IP x) or tlx ^f x'-:y,
or d'-y- = c\dx ^F x~) 's tno general equation for all the
conic sections ; and, in particular it is
dty = c'(dx — x'2) in the ellipse,
d<y* = c'(dx -+-u"'2) in the hyperbola, and
d'f- = c"dx, or dyr = c:x in the parabola: Or the three
equations may be otherwise expressed thus :

y' = —(dx ~ xJ) 'n tne ellipse, '

y"- = j,(dx -*- .r") in the hyperbola, and

■»■ = —£ or =px in the parabola, where the parameter

p — — tho third proportional to the diameter and its

conjugate, by the definition of it.
And from this one general proposition alone, which is

CON

[

easily derived from the section in the solid cone itself, to-
gether with the definitions only, as laid down above, all
the other properties of all the sections may easily be de-
rived, without any further reference to the cone, and with-
out mechanical descriptions of the curves in piano; as is
done in my Treatise on Conic Sections, for the use of the
Royal Mil. Acad.; in which also all the similar proposi-
tions in the ellipse and hyperbola are carried on word for
word in them both.

The more ancient mathematicians, before the time of
Apollonius Pergsus, admitted only the right cone into
their geometry, and they supposed the section of it to be
made by a plane perpendicular to one of its sides ; and as
the vertical angle of a right cone may be either right,
acute, or obtuse, the same method of cutting these several
, cones, viz, by a plane perpendicular to one side, produced
all the three conic sections. The parabola was called the
seetion of a right-angled cone; the ellipse, the section of
the acute-angled cone ; and the hyperbola, the section of
the obtuse-angled coiie. But Apollonius, who, on ac-
count of his writings on this subject, obtained the appella-
tion of Magnus Geometra, the Great Geometrician, ob-
served, that these three sections might be obtained in every
cone, both oblique and right, and that they depended on
the different inclinations of the plane of the section to the
cone itself Apollon. Con. Halley's edit. lib. 1, p. Q.

Instead of considering these curves as sections cut from
the solid cone, which is the true genuine way of all the
ancients, and of the most elegant writers among the mo-
derns, Descartes and some others of the moderns have
given arbitrary constructions of curves on a plane, from
which constructions they have demonstrated the properties
of these, and have afterwards proved that some principal
property of them belongs to such curves or sections as are
cut from a cone; and hence it is inferred by them that
those curves, so described on a plane, are the same with
the conic sections.

The doctrine of the conic sections is of great use in
physical and geometrical astronomy, as well as in the
physico-mathematical sciences. It has been much culti-
vated by both ancient and modern geometricians, who
have left many good treatises on the subject. The most
ancient of these is that of Apollonius Pcrgaeus himself,
containing S books, the first 4 of which have often been
published ; but Dr. Halley's edition has all the eight.
Pappus, in his Collect. Mathem. lib. 7, informs us, that
the first four of these were written by Euclid, though
perfected by Apollonius, who added the other 4 to them.
Among the moderns, the chief writers are Mydorgius de
Sectionibus Conicis ; Gregory St. Vincent's Quadratura
Circuli et Sectionum Coni ; Lahirc de Sectionibus Co-
nicis; Trevigar Elem. Section. Con.; De Witt's Elementa
Curvarum ; Dr. Wallis's Conic Sections; De 1' Hospi-
tal's Anal. Treat, of Conic Sections ; Dr. Simson's Sec-
tion. Con. ; Milne's Elementa Section. Gonicarum; Bos-
covich's excellent treatise ; Muller's Conic Sections ;
Steel's Conic Sections; Emerson's Conic Sections; Dr.
Hamilton's elegant treatise; my own treatise, above cited ;
and my friend Dr. Abram Robertson of Oxford has also
published a curious work on this subject, containing not
only a treatise on the science, but a history of the writings
relating to it.

COMICS, that part of the higher geometry, or geometry
of curves, which considers the cone, and the several curve
Jines arising from the sections of it.

365 ] CON

CONJUGATED.™, or Diameter, in the Conic Sections,
is the axis, or a diameter parallel to a tangent to the
curve at the vertex of another axis, or diameter, to which
that is a conjugate. Indeed the two are mutually conju-
gates to each other, and each is parallel to the tangent at
the vertex of the other.

Conjugate Hyperbolas, also called Adjacent Hyper-
bolas, are such as have the same axes, but in the contrary
order; the first or principal axis of the one being the 2d
axis of the other, and the 2d axis of the former, the 1st
axis of the latter. See art. 17 of Conic Sections.

CONJUNCTION, in Astronomy, is the meeting of the
stars and planets in the same point or place in the heavens;
and is either true or apparent.

True Conjunction is when the line drawn through the
centres of the two stars passes also through the centre of
the earth. _ And Apparent Conjunction is when that
line does not pass through the earth's centre, but having
only the same apparent longitude.

CONOID, is a figure resembling a cone, except that
the slant sides from the base to the vertex are not straight
lines as in the cone, but curved. It is generated by the
revolution of a conic section about its axis; and it is
therefore threefold, answering to the three sections of the
cone, viz, the Elliptical Conoid, or spheroid, the Hyperbolic
Conoid, and the Parabolic Conoid.

If a conoid be cut by a plane in any position, the sec-
tion will be of the figure of some one of the conic sections;
and all parallel sections, of the same conoid, are like and
similar figures. When the section of the solid returns
into itself, it is an ellipse; which is always the case in the
sections of the spheroid, except when it is perpendicular
to the axis ; which position is also to be excepted in the
other solids, the section being always a circle in that po-
sition. In the parabolic conoid, the section is always an
ellipse, except when it is parallel to the axis, when it is
a parabola. And in the hyperbolic conoid, the section is an
ellipse, when its axis makes with the axis of the solid, an
angle greater than that made by the said axe of the solid
and the asymptote of the generating hyperbola; the section
being an hyperbola in all other cases, but when those
angles are equal, in which case it is a parabola.

But when the section is parallel to the fixed axis, it is
of the same kind with, and similar to the generating plane
itself; that is, the section parallel to the axis, in the sphe-
roid, is an ellipse similar to the generating ellipse ; in the
parabolic conoid it is a parabola, similar to the generating
one; and in the hyperbolic conoid, it is an hyperbola
similar to the generating one.

The section through the axis, which is the generating
plane, is, in the spheroid the greatest of the parallel sec-
tions, but in the hyperboloid it is the least, and in the pa-
raboloid those parallel sections are all equal.

-The analogy of the sections of the hyperboloid to those
of the cone, are very remarkable, all the three conic sec-
tions being formed by cutting an hyperboloid in the same
positions as the cone is cut. Thus, let an hyperbola and
its asymptote revolve together about the transverse axis,
the former describing an hyperboloid, and the latter a
cone circumscribing it : then let it be supposed that they
arc both cut by one plane in any position ; so shall the
two sections belike, similar, and concentric figures ; that
is, if the plane cut both the sides of each, the sections will
be concentric and similar ellipses ; but if the cutting plane
be parallel to the asymptote, or to the side of the cone, the

CON

[ 366 ]

C O N

sections will be parabolas; and in all other positions, the
sections will be similar and concentric hyperbolas.

And this analogy of the sections will not appear strange,
when it is considered that -a cone is a species of the hy-
perboloid ; or a triangle a species of- the hyperbola, the
axes being infinitely small. See my Mensuration, prop. 1,
part 3, sect 4, pa. 201 ; or my Tracts, vol. 2, pa. 137.

CONON (of Samos), a respectable mathematician
and philosopher, who flourished about the 130th olympiad,
being a contemporary and friend of Archimedes, to whom
Conon communicated his writings, and sent him some
problems, which Archimedes received with approbation,
observing they ought to be published while Conon was
living, for he comprehends them with ease, and can give a
proper demonstration of them.

At another time he laments the loss of Conon, thus ad-
miring his genius: "How many theorems in geometry,"
says he, "which at first seemed impossible, would in time
have been brought to perfection ! Alas ! Conon, though
he invented many, with which he enriched geometry, had
not time to perfect them, but left many in the dark, being
prevented by death." He had an uncommon skill in ma-
thematics, joined to an extraordinary patience and appli-
cation. This is farther confirmed by a letter sent to Ar-
chimedes by a friend of Conon's. " Having heard of Co-
non's death, with whose friendship 1 was honoured, and
with whom you kept an intimate correspondence ; as he
was thoroughly versed in geometry, I greatly lament the
loss of a sincere friend, and a person of surprising know-
ledge in mathematics. 1 then determined to send to you,
as I had before done to him, a theorem in geometry, hi-
therto observed by no one."

Conon had some disputes with Nicoteles, who wrote
against him, and treated him with too much contempt.
Apollonius confesses it ; though he acknowledges that
Conon was not fortunate in his demonstrations.

Our author invented a kind of volute, or spiral, dif-
ferent from that of Dinostratus; but because Archimedes
explained the properties of it more clearly, the name of
the inventor was forgotten, and it was hence called Ar-
ch imedes's volute or spiral.

CONSECTARY, or Corollary, a consequence deduced
from some foregoing principles.

CONEQUENT, is the latter of the two terms of a
ratio; or that to which the antecedent is referred and com-
pared. Thus, in the ratio a : b, or a to i,-the latter term
b is the consequent, and a is the antecedent.

CONSISTENT Bodies,. is a term much used by Mr.
Boyle, for such as are usually called firm, or fixed bodies ;
in opposition to fluid ones.

CONSOLE, in Architecture, is an ornament cut upon
the key of an arch, having a projecture or jetting, and oc-
casionally serving to support small cornices, busts, and
bases.

CONSONANCE, in Music, is commonly used in the
same sense with concord, viz, for the union or agreement
of two sounds produced at the same time, the one grave,
the other -acute, which is compounded together by such a
proportion of each, as proves agreeable to the ear.

An unison is the first consonance, an eighth is the 2d,
a fifth is the 3d ; and then follow the fourth, with the third
and sixths, major and minor.

CONSTANT Quantities are such as remain invariably
the same, while others increase or decrease. Thus, the
diameter of a circle is a constant quantity; for it remains

the same while the abscisses and ordinates, or the sines,
tangents, &c, are variable. These are sometimes called
given, or invariable, or permanent quantities ; and in al-
gebra it is now usual to .represent them by the leading
letters of the alphabet, a, b, c, &c ; while the variable
ones are denoted by the last letters, 2, y, x, &c.

CONSTELLATIONS, certain imaginary figures of
birds, beasts, fishes, and other animals in the heavens,
within which are arranged certain stars. These assemblages
are also sometimes called asterisms.

The ancients portioned out the firmament into several
parts, or constellations ; reducing a certain number of stars
under the representation of certain images, to assist the
imagination and memory, to conceive or retain their num-
ber, order, and disposition, or even to distinguish the vir-
tues they attributed to them.

The division of the heavens into constellations is very
ancient; being known to the most early authors, whether
sacred or profane. In the Book of Job the names of some
of them are mentioned ; witness that sublime expostulation,
" Canst thou restrain the sweet influence of the Pleiades,
or loosen the bands of Orion ?" And the same may be ob-
served of the oldest among the heathen writers, Hesiod and
Homer.

The ancients in their division took in only the visible
firmament, or so much as came under their notice, as
visible to the naked eye. The first or earliest of these, is
contained in the catalogue of Ptolemy, given in the 7th
book of his Almagest, prepared, as he assures us, from his
own observations, compared with those of Hipparchus,
and the other ancient astronomers. In this catalogue
Ptolemy has formed 48 constellations. Of these, 12 are
about the ecliptic, commonly called the 12-signs; 21 to
the north of it; and 15 to the south. The northern con-
stellations are, the Little Bear, the Great Bear, the Dra-
gon, Cepheus, Bootes, the Northern Crown, Hercules,
the Harp, the Swan, Cassiopeia, Perseus, Auriga, Ophiu-
cus or Scrpentary, the Serpent, the Arrow, the Eagle,
the Dolphin, the Horse, Pegasus, Andromeda, and the
Triangle.

The constellations about the ecliptic are Aries, Taurus,
Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius,
Capricorn, Aquarius, and Pisces: or according to the
English names, the Ram, the Bull, the Twins, the Crab,
the Lion, the Virgin, the Balance, tbe Scorpion, theArcher,
the Goat, the Water-hearer, and the Fishes.

The Southern constellations are, the Whale, Orion, the
Eridanus, the Hare, the Great Dog, the Little Dog, the
Ship, the Hydra, the Cup. the Raven, the Centaur, the
Wolf, the Altar, the Southern Crown, and the Southern
Fish.

The other stars not comprehended under these con-
stellations, yet visible to the naked eye, the ancients called
Informes, or Sporades, some of which the modern astro-
nomers have since reduced into new figures, or constella-
tions. But on the modern celestial globes, the constella-
tions are made to include all the unformed stars. Pto-
lemy has set down the longitude and latitude of all these
stars to about the year of Christ 137, amounting to the
number of 1022, viz,

in the northern constellations - ' 360

in the zodiacal constellations - 346"

in the southern constellations - SlfJ

in all of Ptolemy's catalogue - 1022

Among the modern astronomers, Tycho Biahe is the

CON

[ 367 ]

CON

first who determined, with exactness, and in consequence
of his own observations, the long, and lat. of the fixed
stars, out of which he formed 45 constellations; of these,
43 were of the old ones described by Ptolemy, to which
Tycho added the Coma Berenices, and Antinous; but he
omits 5 of the old southern constellations, viz, the Cen-
taur, the Wolf, the Altar, the Southern Crown, and South-
ern Fish ; which he could not observe, because of the high
northern latitude of Uranibourg.

After Tycho, Bayer gave the figures of (50 constellations,
very exactly represented, and with tables annexed, having
added to the 48 old ones of Ptolemy the following 12
about the south pole; viz, the Peacock, the Toucan, the
Crane, the Phoenix, the Dorado, the Flying Fish, the Hy-
dra, the Chameleon, the Bee, the Bird of Paradise, the
Triangle, and .the Indian. Besides accurately distinguish-
ing the relative size and situation of every star, Bayer
marks the stars in each constellation with the letters of
the Greek and Roman alphabets, setting the first letter a.
to the first or principal star in each constellation, (3 to the
2d in order, y to the 3d, and so on; a Very useful method
of noting and describing the stars, which has been used by
all astronomers since, and who have farther enlarged this
method, by adding the ordinal numbers 1, 2, 3, &c, to the
other stars discovered since his time, when any constella-
tion contains more than can be marked by the two alpha-
bets. The number and order of the stars, as mentioned
, by Bayer, are,

of the 1st magnitude

of the 2d magnitude

of the 3d magnitude

of the 4th magnitude

of the 5th magnitude

of the 6th magnitude

of the unformed stars

all

17

63

196.

415
348
341
326

1706"

After Bayer, a catalogue, with new constellations, was
published by Schiller, in l6'27, in a work entitled Coelum
Stellatum Christianum, the Christian Starry Heaven, in
which he substitutes, very improperly, other figures of the
constellations, and names, taken from the sacred scrip-
tures, instead' of the old ones.

In the year 1065, Riccioli published his Astronomy Re-
formed, containing a catalogue of the stars in 62 constel-
lations, viz, the 60 of Bayer, with the Coma Berenices and
Antinous of Tycho. He distributes the stars in all the
constellations into four classes. In the first of these classes
are contained those stars determined by his own observa-
tions, and those of Grimaldi. In the second are those stars
which had been ascertained by Tycho Brahe and Kepler.
In the third are the stars determined by Hipparchus and
Ptolemy. And the 4th class consists of those of the
southern hemisphere discovered by navigators, who have
ascertained their places in a more or less accurate man-
ner; in which he has marked the longitudes and latitudes
for the year 1700, the period to which he has reduced all
his observations. This catalogue was followed by a num-
ber of celestial schemes and maps of the heavens, publish-
ed in l673 by Pardies, who has represented very carefully
all the constellations, with the stars they contain. After
this, Vitalis published a catalogue of the fixed stars in his
Tables of the Primum Mobile, in which their longitudes
and latitudes, with the right ascensions and declinations
are set down for the year 16'75.

Some time after this, Royer published maps of the hea-

vens, reduced into 4 tables, with a catalogue of the fixed
stars for the year 1700. To the stars marked by Bayer,
he adds a number of stars not before seen, with others ta-
ken from the tables of Riccioli, and not mentioned by
Bayer: he also formed, out of the unformed stars, eleven
other constellations ; five of which are to the north, and
are called the Giraffe, the River Jordan, the River Tigris,
the Sceptre, and the Flower-de-luce; with 6 on the south
part, which are the Dove, the Unicom, the Cross, the
Great Cloud, the Little Cloud, and the Rhomboide. To
this work Royer has joined the catalogue of the southern
stars observed by Dr. Halley at the island of St. Helena.

Hevelius has also improved on the labours of those who
went before him, and collected together several stars of
the before unformed class into some new constellations.
These are, the Unicorn, the Camelopardalis, described by
Bartschius, the Sextant of Urania, the Dogs, the Little
Lion, the Lynx, the Fox and Goose, the Sobieski's Crown,
the Lizard, the Little Triangle, and the Cerberus; to which
Gregory has added the Ring and the Armilla. Some of
these new constellations however answer to those of Royer,
as the Camelopardal to the Giraffe, the Dogs to the PJver
Jordan, and the Fox to the River Tigris. The latitudes
and longitudes arc added for the year 1700.

Finally, Flamstecd has given a catalogue of the fixed
stars, not only much more correct, but much larger than
any of those that had appeared before his time. He has
set down the longitude, latitude, right ascension, and polar
distance of 2934 stars, as they were at the beginning of
169O, all determined from his own observations. He dis-
tinguishes all the stars into seven classes, or orders of mag-
nitude, distinguishing those of Bayer by his letters, and
marking their variation in right ascension, for showing
their situation in the succeeding years. See the term Ca-
talogue.

This catalogue was followed by an Atlas Ccelestis, pub-
lished at London in the year 1729, describing, in several
schemes, the figures of the constellations seen in our hemi-
sphere, with the exact position of the fixed stars, with re-
spect to the circles of the sphere, as resulting from the last
catalogue corrected by Flamsteed. And still later obser-
vations, made with farther improved telescopes, have
greatly enlarged the number and accuracy of tine stars;
but the number of the constellations remains the same as
above described, except that an attempt was afterwards
made by Dr. Hill to add to the list 14 new ones, formed
out of more of the clusters of unformed stars.

Beside the literal marks of the stars introduced by
Bayer, it is usual also to distinguish them by that part of
the constellation in which they are placed; and many of
them again have their peculiar names; as Arcturus, be-
tween the knees of Bootes; Geinina, or Lucida, in the Co-
rona Septentrionalis, or Northern Crown ; Palilitium, or
Aldebaran, in the Bull's eye, Pleiades in his neck, and
Hyades in his forehead ; Castor and Pollux in the heads of
Gemini; Capella, with the Hcedi in the shoulder of Au-
riga; Regulus, or Cor Leonis, the Lion's Heart; Spica
Virginis in the hand, and Vindemiatrix in the shoulder of
Virgo; Antares or Cor Scorpionis, the Scorpion's Heart;
Fomalhaut, in the mouth of Piscis Australis, or Southern
Fish ; Regel, in the foot of Orion; Sirius, in the mouth of
Canis Major, the Great Dog; Procyon, in the back of Ca-
nis Minor, the Little Dog; and the Pole Star, the last in
the tail of Ursa Minor, the Little Bear.

The Greek and Roman poets, from the ancient theology,

CON [ 368

give wild and romantic Cables about the origin of Lhe con-
stellations, probably derived from the hieroglyphics of the
Egyptians, and transmitted, with some alterations, from
them to the Greeks, who probably obscured them greatly
with their own fables. See Hyginus's Poeticon Astron. ;
Riccioli Almagest, lib. 6', cap. 3, 4, 5; Shelburne's Notes
upon Manilius ; Bailly's Ancient Astronomy; and Gebe-
lin's Monde Primitif, vol. 4: from the whole of which it
appears probable,, that the invention of the signs of the zo-
diac, and probably of most of the other constellations of
the sphere, is to be ascribed to some very ancient nation,
inhabiting the northern temperate zone, probably what is
now called Tartary, or the parts to the northward of
Persia and China; and from thence transmitted through
China, India, Bab) Ion, Arabia, Egypt, Greece, &c.

It is a very probable conjecture, that the figures of the
signs in the zodiac are descriptive of the seasons of the
year, or months, in the sun's path : thus, the first sign
Aries, denotes, that about the time when the sun enters
that part of the ecliptic, the lambs begin to follow the
sheep; that on the sun's approach to the 2d constellation,
Taurus, the Bull, is about the time of the cows bringing
forth their young. The third sign, now Gemini, was ori-
ginally two kids, and signified the time of the goats bring-
ing forth their young, which are usually two at a birth,
while the former, the sheep and cow, commonly produce
only one. The 4-th sign, Cancer, the Crab, an animal
that goes sideways and backwards, was placed at the
northern solstice, the point where the sun begins to return
back again from the north to the southward. The 5th sign,
Leo, the Lion, as being a very furious animal, was thought
to denote the heat and fury of the burning sun, when he
has left Cancer, and entered the next sign Leo. The suc-
ceeding constellation, the 6th in order, received the sun at
the time of the ripening of corn and the approach of har-
vest; which was aptly expressed by one of the female
reapers, with an ear of corn in her hand ; viz, Virgo, the
maid. Libra, or the Balance, which is the following sign,
evidently denotes the equality of days and nights, that
takes place at that season: and Scorpio, the next in order,
denotes the time of gathering in the fruits of the earth,
which being generally an unhealthy season, is represented
by this venomous animal, extending his long claws, threa-
tening the mischief that is likely to follow. The fall of the
leaf was the season of the ancient hunting; for which rea-
son the constellation Sagittary represents a huntsman with
his arrows and his club, the weapons of destruction for the
creatures he pursues. The reason of the Goat's being
chosen to mark the southern solstice, when the sun has
attained his extreme limit in that quarter, and begins to
return and mount again to the northward, is obvious
enough; the character of that animal being, that it is
mostly found climbing, and ascending some mountain, as
it browzes. There yet remain two of the signs of the zo-
diac to be considered with regard to their origin, viz, Aqua-
rius and Pisces. As to the former, it is to be considered
that the winter is a wet and uncomfortable season; this
therefore was expressed by Aquarius, the figure of a man
pouring out water from an urn. The last of the zodincal
constellations was Pisces, a couple of fishes, tied together,
that had been caught: The lesson was, the severe season is
over, your (locks do not yet yield their store; but the seas
and rivers are open, and there you may take fish in
abundance.

Through a vain and blind zeal, rather than from any

]

CON

love for the science, some persons have been induced to
alter either the figures of the constellations, or their names.
Thus, venerable Bede, instead of the profane names and
figures of the twelve zodiacal constellations, substituted
those of the 12 apostles; which example was followed by
Schiller, who completed the reformation, and gave Scrip-
ture names to all the constellations in the heavens. Thus,
Aries, or the Ram, was changed into Peter; Taurus, or the
Bull, into St. Andrew; Andromeda, into the Sepulchre of
Christ; Lyra, into the Manger of Christ; Hercules, into
the Magi coming from the East; the Great Dog, into Da-
vid; and so on. And Weigel, professor of mathematics
in the university of Jena, made a new order of constella-
tions; changing the firmament into a Ccelum Heraldicum :
and introducing the arms of all the princes in Europe, by-
way of constellations. Thus Ursa Major, the Great Bear,
he transformed into the elephant of the kingdom of Den-
mark; the Swan, into the Ruta with swords of the House
of Saxony; Ophiuchus,intotheCross of Cologne; the Tri-
angle, into Compasses, which he calls the Symbol of Arti-
ficers; and the Pleiades into the Abacus Pythagoricus,
which he calls that of merchants; &c.

But the more judicious among astronomers never ap-
proved of these innovations; as they only tend to intio-
duce confusion into the science. The old constellations
are therefore still retained; both because better could not
be substituted, and also to keep up the greater correspon-
dence and uniformity between the old astronomy and the
new. See Catalogue.

CONSTRUCTION, in Geometry, the art or manner of
drawing or describing figures, schemes, the lines of a pro-
blem, or such like.

Construction of Equations, in Algebra, is the finding
the roots or unknown quantities of an equation, by geo-
metrical construction of right lines or curves; or the re-
ducing given equations into geometrical figures; which is
effected by lines or curves according to the order or rank of
the equation.

The roots of any equation may be determined, that is,
the equation may be constructed, by the intersections of a
straight line with another line or curve of the same di-
mensions as the equation to be constructed : for the loots
of the equation are the ordinates of the curve at the points
of intersection with the right line; and it is well known
that a curve may be cut by a right line in as many points
as its dimensions amount to. Thus, then, a simple equa-
tion will be constructed by the intersection of one right
line with another: a> quadratic equation, or an affected
equation of the 2d rank, by the intersections of a light line
with a circle, or any of the conic sections, which are all
lines of the 2d order; and which may be cut, by the right
line, in two points, thereby giving the two roots of the qua-
dratic equation. A cubic equation may he constructed by
the intersection of the right line with a line of the Sid order:
and so on.

But if, instead of the right line, some other line of a
higher order be used ; then the 2d line, whose intersec-
tions with the former are to determine the loots of the
equation, may be taken as many dimensions lower, as the
former is taken higher. And, in general, an equation of
any degree will be constructed by the inu (-sections of two
lines whose dimensions, multiplied together, produce the
dimension of the given equation. Thus, the intersections
of a circle with the conic sections, or of these with each
other, will construct biquadratic equations, or those of

CON

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the 4th power, because 2 x 2 = 4.; and the intersections
of the circle or conic sections with a line of the 3d order,
will construct equations of the 5th and 6th power; and
so on. — For example,

To construct a Simple Equation. This is done by re-
solving the given simple equation into a proportion, or
finding a third or fourth proportional, &c. Thus, 1. If the

equation be ax = be; then a : b : : c : x = — , the fourth

proportional to a, b, c.

2. If ax = bz : then a : b : : b : x = — , a third propor-
tional to a and b.

3. If ax = b" —

then, since b'2

x (6 — c), it will be a :b -+■ c : : b — c :

a fourth proportional to a, b -+- c and b -

4. If ax = b1 ■+■ c2; then construct
the right-angled triangle abc, whose
base is 6, and perpendicular is c, so
shall the square of the hypothenuse
be b2 -+- c'2, which call h2; then the

(ft + 0

equation i3 ax = li2, and x :
proportional to a and /(.

h?

, a third

To construct a Quadratic Equation.

1. If it be a simple quadratic, it may be reduced to
this form x2 = ab; and hence a : x : : x ; b, or x =t/ab

a mean proportional between a and b. P.

Therefore upon a straight line take ..•*"" I ""••.

ab = a, and bc = 6; then upon the

diameter ac describe a semicircle, ; a \ 3 \
and raise the perpendicular bd to A B C

meet it in d; so shall bd be = x the mean proportional
sought between ab and bc, or between a and b. •

2. If the quadratic be affected, let it first be x2 -+-
Zax = b2; then form the right-angled triangle abc, whose
base ab is a, and perpendicular bc is b; and with the
centre a and radius ac describe the semicircle dce; so
shall db and be be the two roots of the given quadratic
equation x2 -+- 2ax = b2.

3. If the quadratic be x2 — lax = b1, then the con-
struction will be the very same as of the preceding one
x2 ■+- lax = b2.

4. But if the form be lax — x2 = b2: form a right-
;mgled triangle whose hypothenuse ro is a, and perpen-
dicular Gil is b; then with the radius FG and centre p
describe a semicircle igk ; so shall ih and 11 k be the two
roots of the given equation lax — x2 = b2, or xz — 2ax
= — b2. See Maclaurin's Algebra, part 3, cap. 2, and
Simpson's Algebra, pa. 267.

To construct Cubic and Biquadratic Equations. — These
are constructed by the intersections of two conic sections ;
for the equation will rise to 4 dimensions, by which are.
dtrfcrmined the ordinatts from the 4 points in which these
conic sections may cut one another ; and the conic sec-
tions may be assumed in such a manner, as to make this

Vol. I.

equation coincide with any proposed biquadratic : so that
the ordinates from these 4 intersections will be equal to
the roots of the proposed biquadratic. When one of the
intersections of the conic section falls upon the axis, then
one of the ordinates vanishes, and the equation, by which
these ordinates are determined, will then be of 3 dimen-
sions only, or a cubic; to which any proposed cubic
equation may be accommodated. So that the three re-
maining ordinates will be the roots of that proposed cubic.
The conic sections for this purpose should be such as are
most easily described ; the circle may be one, and the pa-
rabola is usually assumed for the other.

Vieta, in his Canonica Recensione Effectionum Geome-
tricarum, and Ghetaldus, in his Opus Posthumum de Re-
solutione et Compositione Malhematica, as also Descartes,
in his Geornetria, have shown how to construct simple
and quadratic equations. Descartes has also shown how
to construct cubic and biquadratic equations, by the in-
tersection of a circle and a parabola : And the same has
been done more generally by Baker, in his Clavis Geonie-
trica, or Geometrical Key. But the genuine foundation
of all these constructions was first laid and explained bv
Slusius in his Mesolabium, part 2. This doctrine is also
pretty well handled by Lahire, in a small treatise, call-
ed La Construction des Equations Analytiques, annexed to
his Conic Sections. Newton, at the end of his Algebra,
has given the construction of cubic and biquadratic equa-
tions mechanically ; as also by the conchoid and cissoid,
as well as the conic sections. See also Dr. Halley's Con-
struction of Cubic and Biquadratic Equations ; Colson's,
in the Philos. Trans. ; the Marquis de l'Hospital's Traite
Analytique des Sections Coniques ; Maclaurin's Algebra,
part 3, c. 3, &c.

CONTACT, the relative state of two things that touch
each other, but without cutting or entering; or whose
surfaces join to each other without any interstice.

The contact of curve lines or surfaces, with either
straight or curved ones, is only in points; and yet these
points have different proportions to one another, as is
shown by Mr. Robartes, in the Philos. Trans, vol. 27, pa.
470 ; or my Abr. vol. 5, pa. 678. Because few or no
surfaces are capable of touching in all points, and the co-
hesion of bodies is in proportion to their contact, those
bodies will adhere fastest together, that are capable of the
greatest contact.

Angle of Contact, is the opening
between a curve line and a tangent
to it, particularly the circle and its
tangent ; as the angle formed at a be-
tween b a and ac, at the point of con-
tact a. It is demonstrated by Eu-
clid, that the line ca standing perpen-
dicular on the radius da, touches the
circle only in one point: and that no right line can be
drawn between the tangent and the circle. Hence, the
angle of contact is less than any rectilinear angle ; and
the angle of the semicircle between the radius da and the
arch ab, is greater than any rectilinear acute angle. This
seeming paradox of Euclid has exercised the wits of ma-
thematicians: it was the subject of a long controversy be-
tween Peletarius and Clavius; the former of whom main-
tained that the angle of contact is heterogeneous to a rec-
tilinear one, as a line is to a surface; the latter maintained
the contrary. Dr. Wallis has a formal treatise on the
3B

CON

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COP

angie of contact, and of the semicircle ; where, with other
great mathematicians, he approves of the opinion of Pele-
tarius.

CONTENT, a term often used for the measurement of
bodies and surfaces, whether solid or superficial ; or the
capacity of a vessel and the area of a space; being the
quantity either of matter or space included within certain
bounds or limits.

CONTIGUITY, the relation of bodies touching one
another.

CONTIGUOUS, a relative term, understood of things
so disposed, that their" surfaces touch each other.

Contiguous Angles, are such as have one leg or side
common to each angle ; and are otherwise called adjacent
angles; in contradistinction to those made by continuing
their legs through the point of contact, which are called
opposite or vertical angles.

CONTINENT, a terra firma, main-land, or a large ex-
tent of country, not interrupted by seas : so called, in op-
position to island, peninsula, &c.

Ourearth is usually divided into two grand continents,
the old and the new : the old continent comprehends Eu-
rope, Asia, and Africa ; the new continent comprehends
North and South America. Since the discovery of New
Holland and New South Wales, it is a doubt with many
whether to call that vast country an island or a continent.
CONTINGENT Line, the same with tangent line, in
Dialling, being the intersection of the planes of the dial
and equinoctial, and at right angles to the substilar line.

CONTINUAL, or Continued Proportionals, are
a series of three or more quantities compared together,
so that the ratio is the same between every two adjacent
terms, viz. between the 1st and 2d, the 2d and 3d, the 3d
and 4th, &c. As 1, 2,4, S, 10, &c, where the terms con-
tinually increase in a double ratio; or 12, 4, 4, |-, where
the terms decrease in a triple ratio. A scries of continual or
continued proportionals, is otherwise called a Progression.
Continued Fractions. See Fractions.
Continued Quantity, or Body, is that whose parts are
joined and united together.

Continued Proportion, is that in which the conse-
quent of the first ratio is the same with the antecedent of
the second ; as in these, 3 : 6* : : 6 : 12. See Continual
Proportion.

On the contrary, if the consequent of the first ratio be
different from the antecedent of the second, the propor-
tion is called Discrete : as 3 : 6 : : 4 : 8.

CONTRACTION, in Arithmetic, is the shortening or
contracting certain operations, as in Multiplication, Divi-
sion, Extraction of the Square Root, &c : the method for
which see under the respective heads.

CONTRA-Harmonical Proportion, that relation of
three terms, in which the difference of the first and second
is to the difference of the 2d and 3d, as the 3d is to the
first; Thus, for instance, 3, 5, and 6, are numbers con-
tra-harmonically proportional ; for 2 : 1 : : 6 : 3.

CONTRA-Muuii, in Fortification, is a little wall built
before another partition wall, to strengthen it, so that it
may receive no damage from the adjacent buildings.

CONTRATE-Wiieel, is that wheel in watches which
is next to the crown, whose teeth and hoop lie contrary to
those of the other wheels ; from whence comes its name.

CONTRAVALLATION, Line of, in Fortification, is a
trench, guarded with a parapet; being made by the be-

siegers, between them and the place besieged, to .secure
themselves on that side, and stop the sallies of the garri-
son. It is made beyond the reach of musket-shot from
the town ; sometimes going quite around it, and some-
times not, as occasion may require. The besiegers lie be-
tween the lines of circumvallation and contravallation :
but it is now seldom used.

CONVERGING Curves. . See Curve.
Converging, ot'Convergent Lines, in Geometry,
are those that continually approximate, or whose distance
becomes continually less and less the farther they are con-
tinued, till they meet: in opposition to divergent lines,
whose distance becomes continually greater.

Lines that converge the one way, diverge the other.
Converging Rays, in Optics, are such as incline to-
wards one another in their passage, and in Dioptrics, are
those rays which, in their passage out of one medium into
another of a different density, aie refracted towards one
another; so that, if far enough continued, they will meet
in a point or focus.

Converging Series, a series of terms or quantities, that
always decrease the further they proceed, or which tend to
a certain magnitude or limit: in opposition to diverging
series, or such as become larger and larger continually.
-See Series.

CONVERSE. A proposition is said to be the converse
of another, when, after drawing a conclusion from some-
thing first supposed, we return again, by making a suppo-
sition of what had before been concluded, and draw from
thence as a conclusion what before was made the supposi-
tion. Thus, when it is supposed that the two sides of a
triangle are equal, and thence demonstrate or conclude
that the two angles opposite to those sides are equal also ;
then Jhe converse is to suppose that the two angles of a
triangle are equal, and thence to prove or conclude that
the sides opposite to those angles are also equal.

CONVERSION, or Convertendo, is when there are
four proportionals, and it is inferred, that the first is to its
excess above the 2d, as the 3d to its excess above the 4th :
according to Euclid, lib. 5, def. 17.
Thus, if it be - - S:

then convertendo, or by conversion, - S :
Or if there be - r a :

then convertendo, or by conversion, a : a — b

CONVEX, round or curved and protuberant outwards,
as the outside of a globular body.

Convej!: Lens, Mirror, &c. See Lens, Mirror, &c.
CONVEXITY, the exterior or outward surface of a
convex or round body.

COORDINATES, in the Higher Geometry, is the gene-
ral term used, when the absciss and ordinates of a curve arc
considered connectedly, whether they are at right angles
with eacli other or not ; and the nature of a curve is de-
termined by the equation between its coordinates.

COPERNTCAN, something relating to Copernicus.
As, the

Copeunican Sphere. See SPHE-KE.
Copernican System, is that system of the world, in
which it is supposed that the sun is at rest in the centre,
and the earth and planets all moving around him in their
own orbits. Here it is supposed, that the heavens and
stars are at rest; and the diurnal motion which they ap-
pear to have, from east to west, is imputed to the earth's
diurnal motion from west to east.

6 : : 4 : 3
2 : : 4 : 1
b : : c : d
: c : c — d

COP

[ 371 ]

COR

This system was maintained by many of the ancients;
particularly Ecphantus, Seleucus, Aristarchus, Philolaus,
Cleanthes Samius, Nicetas, Heraclides Ponticus, Plato,
and Pythagoras; from the last of whom it was anciently
called the Pythagoric, or Pythagorean System. The same
system was also held by Archimedes, in his book of the
number of the Grains of Sand ; but after him it became
neglected, and even forgotten, for many ages; till about
.300 years since, when Copernicus revived it; from whom
it took the new name of the Copernican System. See the
next article.

COPERNICUS (Nicholas), an eminent astronomer,
was born at Thorn in Prussia, January 19, 1 473. He
was instructed in the Latin and Greek languages at home;
and afterward sent to Cracow, where he studied philo-
sophy, mathematics, and medicine: though his genius was
naturally turned to mathematics, which he chiefly stu-
died, and pursued through all its various branches.

He set out for Italy at 23 years of age ; stopping at
Bologna, that he might converse with the celebrated as-
tronomer of that place, Dominic Maria, whom he assisted
for some time in making his observations. From hence
he passed to Rome, where he was presently considered as
not inferior to the famous Regiomontanus. Here he soon
acquired so great a reputation, that he was chosen pro-
fessor of mathematics, which he taught there for a long
time with the greatest applause ; and here also he made
some astronomical observations about the year 1500.

Afterward, returning to his own country, he began to
apply his fund of observations and mathematical know-
ledge, to correcting the system of astronomy which then
prevailed. He set about collecting all the books that had
been written by philosophers and astronomers, and to ex-
amine all the various hypotheses they had invented* for
the solution of the celestial phenomena ; to try if a more
symmetrical order and constitution of the parts of the
world could not be discovered, and a more just and ex-
quisite harmony in its motions established, than what the
astronomers of those times so generally admitted. But
of all their hypotheses, none pleased him so well as the
Pythagorean, which made the sun to be the centre of the
system, and supposed the earth to move both round the
sun, and also round its own axis. He thought he dis-
cerned much beautiful order and proportion in this; and
that all the embarrassment and perplexity, from epicycles
and excentrics, which attended the Ptolemaic hypotheses,
would here be entirely removed.

He began to consider this system, and to write upon it,
when he was about 35 years of age. He carefully con-
templated the phenomena, made mathematical calcula-
tions, examined the observations of the ancients, and made
new ones of his own ; till, after more than 20 years chiefly
spent in this manner, he brought his scheme to perfection,
establishing that system of the world which goes by his
name, and is now universally received by all philosophers.

This system however was at first considered as a most
dangerous heresy, and his work had long been finished
and perfected, before he could be prevailed on to give it
to the world, notwithstanding being strongly urged to it
by his friends. At length, yielding to their entreaties, it
was printed, and he had but just received a perfect copy,
when he died the 24th of May 1543, at 70 years of age ;
by which it is probable be was happily relieved from the
violent fanatical persecutions of the church, which were

but too likely to follow the publication of his astrono-
mical opinions; and which indeed was afterwards the fate
of Galileo, for adopting and defending them. This work
of Copernicus, first printed at Norimbcrg in folio, 1543,
and of which there have been other editions since, is in-
titled De Revolutionists Orbium Ccelestium, being a large
body of astronomy, in 6 books.

When Rheticus, the disciple of our author., returned
out of Prussia, he brought with him a tract of Coperni-
cus, on plane and spherical trigonometry, which he had
printed at Norimberg, and which contained a table of
sines. It was afterward printed at the end of the first
book of the Revolutions. An edition of our author's great
work was also published in 4to at Amsterdam in 1 6" 1 7 ,
under the title of Astronomia lnstaurata, illustrated with
notes by Nicolas Muler of Groningeff:

COPERNICUS, the name of an astronomical instru-
ment, invented by Whiston, to show the motion and phe-
nomena of the planets, both primary and secondary. It
is founded on the Copernican system, and therefore called
by his name.

COPPER, is a malleable and ductile metal, of a pale-
red colour, with a tinge of yellow. It is soluble with
effervescence in nitrous acid, to which it gives a sky-blue,
colour: on the addition of liquid ammonia, a green pre-
cipitate is thrown down, which is resoluble in an excess
of ammonia, forming a very rich purplish-blue fluid, and
from this, if slightly supersaturated with muriatic acid,
the copper is again precipitable in its metallic form, by a
bar of clean iron. See Aikin's Chemical Dictionary, Ar-
ticle Copper.

COR Caroli, Charles's Heart, an extra-constellated
star of the 2d magnitude in the northern hemisphere, be-
tween the Coma Berenices and Ursa Major; so called by
Sir Charles Scarbbrough, in honour of King Charles I.

Cor Hydr*, the Hydra's Heart, a star of the 2d
magnitude, in the Heart of the constellation Hydra.

Cor Leonis, Lion's Heart, or Regulus, a star of the
first magnitude in the constellation Leo.

Cor Scorpii. See Antares.

CORBEILS, in Fortification, are little baskets about
a foot and a half high, 8 inches broad at the bottom, and
12 at the top; which being filled with earth, are set against
one another on the parapet, or elsewhere, leaving cer-
tain port-holes, from whence to fire under cover upon
the enemy.

CORBEL, in Architecture, the representation of a bas-
ket, sometimes seen on the heads of caryatides.

Corbel, or Corbil, is also used, in Building, for a
short piece of timber placed in a wall, with its end pro-
jecting out 6 or 8 inches, as occasion serves, in the man-
ner of a shouldering-piece.

CORBET, the same as Corbel.

CORDON, in Fortification, a row of stones jutting out
between the rampart and the basis of the paraptt, like
the tore of a column. The cordon ranges round the
whole fortress, and serves to join the rampart, which is a-
slope, and the parapet, which is perpendicular, more a-
greeably together. In fortifications raised of earth, this
space is filled up with pointed stakes instead of a cordon.

CORDS, in Music, arc the sounds produced by an in-
strument or the voice.

CORIDOR, or Corridor, in Fortification, is the co-
vert-way lying entirely round the whole compass of the
3B2

COR

t 372 ]

COR

works of a place, between the outside of the moat and the
pullisadoes, being about 20 yards broad.

Cokidor, is also used, in Architecture, for a gallery,
or long aisle, around a building, leading to several cham-
bers at a distance from each other, sometimes wholly in-
closed, and sometimes open on one side.

CORINTHIAN Order, of Architecture, is the 4th in
order, or the 5th and last according to Scamozzi and Le
Clerc. This order was invented by an Athenian archi-
tect, and is the richest and most delicate of them all ; its
capital is adorned with rows of leaves, and of S volutas,
which support the abacus. The height of its column is
10 diameters, and its cornice is supported by modillions.

CORNEA Tunica, the second coat of the eye; so call-
ed from its substance resembling the horn of a lantern.
This is situated in the fore-part ; and is surrounded by
the sclerotica. It has a greater convexity than the rest
of the eye, and is a portion of a small sphere, or rather
spheroid, and consolidates the whole eye.

CORNICE, Corniche, or Cornish, the third and
uppermost part of the entablature of a column, or the
uppermost ornament of any wainscoting, &c.

COROLLARY, oi-Consectary, a consequence drawn
from some proposition or principles already advanced or
demonstrated, without the aid of any other proposition :
thus from this theorem, That a triangle which has two
equal sides, has also two equal angles, this consequence
may be drawn, That a triangle which hath the three sides
equal, has also its three angles equal. — What is now call-
ed a corollary, was by the earlier geometers named a
Porism.

CORONA, Crown or Crowning, in Architecture, the
flat and most advanced part of the cornice ; so called be-
cause it crowns the cornice and entablature: by the work-
men it is called the drip, as serving by its projecture to
screen the rest of the building from the rain.

CORONA, in Optics, a luminous circle, usually co-
loured, which is seen round the sun, moon, or largest pla-
nets. See Halo.

Corona Borealis, or Septentrionalis, the Northern Crown
or Garland, a constellation of the northern hemisphere,
being one of the 48 old ones. It contains S stars accord-
ing to the catalogue of Ptolemy, Tycho, and Hevelius ;
but according to the Britannic Catalogue, 21.

Corona Australis, or Meridionalis, the Southern Crown,
a constellation of the southern hemisphere, whose stars in
Ptolemy's catalogue are 13; in the British catalogue, 12.

CORPUSCLE, the diminutive of corpus, used to ex-
press the minute parts, or particles, that constitute natu-
ral bodies; meaning much the same as atoms. Newton
shows a method of determining the sizes of the corpuscles
of bodies, from their colours.

CORPUSCULAR Attraction, denotes that power by
which the minute component particles of bodies are uni-
ted, and adhere to each other. It may be distinguished
into attraction of aggregation, viz, that power by which
the homogeneous particles of bodies are united ; and at-
traction of affinity or of composition, viz, that power by
which the heterogeneous particles of bodies are united.

Corpuscular Philosophy, that scheme or system of
physics, in which the phenomena of bodies are accounted
for, from the motion, rest, position, &c, of the corpuscles
or atoms of which bodies consist.

The Corpuscular philosophy, which now flourishes un-

der the name of the mechanical philosophy, is very an-
cient. Leucippus and Democritus taught it in Greece;
from them Epicurus received it, and improved it; and
from him it was called the Epicurean Philosophy.

Leucippus, it is said, received it from one Mochus, a
Phenician physiologist, before the time of the Trojan war,
and the~first who philosophized about atoms: which Mo-
chus is, according to the opinion of some, the Moses of
the Scriptures.

After Epicurus, the corpuscular philosophy gave way
to the peripatetic, which became the popular system.
Thus, instead of atoms, were introduced specific and sub-
stantial forms, qualities, sympathies, &c, which amused
the world, till Gassendus, Charleton, Descartes, Boyle,
Newton, and others, retrieved the corpuscularian hypo-
thesis ; which is now become the basis of the mechanical
and experimental philosophy.

Boyle reduces the principles of the corpuscular philo-
sophy to the 4 following heads.

1. That there is but one universal kind of matter, which
is an extended, impenetrable, and divisible substance, com-
mon to all bodies, and capable of all forms. — On this
head, Newton finely remarks thus : " All things consi-
dered, it appears probable to me, that God in the begin-
ning created matter in solid, hard, impenetrable, move-
able particles; of such sizes and figures, and with such
other properties, as most conduced to the end for which
he formed them : and that these primitive particles, being
solids, are incomparably harder than any of the sensible
porous bodies compounded of them ; even so hard as never
to wear, or break in pieces : no other power being able to
divide what God made one in the first creation. While
these corpuscles remain entire, they may compose bodies
of one and the same nature and texture in all ages: but
should they wear away, or break in pieces, the nature of
things depending on them would be changed : water and
earth, composed of old worn particles, of fragments of par-
ticles, would not be of the same nature and texture now,
with water and earth composed of entire particles at the
beginning. And therefore, that nature may be lasting,
the changes of corporeal things are to be placed only in
the various separations, and new associations, of these per-
manent corpuscles."

2. That this matter, in order to form the vast variety
of natural bodies, must have motion in some, or all its
assignable parts; and that this motion was given to matter
by God, the creator of all things ; and has all manner of
directions and tendencies. — "These corpuscles (says New-
ton) have not only a vis inertia?, accompanied with such
passive laws of motion as naturally result from that force;
but also are moved by certain active principles ; such as
that of gravity, and that which causes fermentation, and
the cohesion of bodies."

3. That matter must also be actually divided into parts;
and each of these primitive particles, fragments, or atoms
of matter, must have its proper magnitude, figure, and
shape.

4. That these differently sized and shaped particles,
have different orders, positions, situations) and postures,
from whence all the variety of compound bodies arises.

CORRECTION of a Fluent. See Fluent.
CORRIDOR. SeeCoiuuou.

CORTES (Martin), a Spanish author on Navigation,
who flourished in the middle of the lo'th century. In 1551

COS

[ 373 3

COT

was, published at Seville, his book entitled, Breve de la Twickenham, Jan. 10, 1782, in the 72d year of his age.

Sphera y de la Arte de Nauegar con nueuos Instrumentos Mr. Costard was a man of great learning ; deeply read in

y Reglas; which he says he composed at Cadiz in 1545. divinity and astronomy; well versed in the Greek, and a

In the dedication to the emperor, Cortes boasts that he perfect master of the Hebrew and other oriental languages,

was the first who had reduced navigation into a Compen- — His private character was amiable, and his correspon-

dium, enlarging much on what he had performed. He
briefly and clearly exposed the errors of plane-charts, and
seemed to reflect on what had been said against the varia-
tion of the compass by Medina, when he advised the rhari-
ner, rather to be guided by experience, than to mind sub-

dence with the literati was extensive both at home and
abroad. He had a great veneration for the Greeks and
their literature; and was generally skilled in the learning
of most of the ancient nations. Besides various publica-
tions on divinity, &c, he was author of the History of

tie reasonings. Besides, he endeavoured to account for Astronomy, 4to, 1767, with its application to geography,
this variation, in imagining the needle to be influenced by history, and chronology. His communications to the

a magnetic pole (which he called the Point Attractive)
different from that of the world ; a notion which has been
farther prosecuted by others. — A translation of Cortes's
book by Richard Eden, was, on the recommendation of
that great navigator Steven Burrough, and the encourage-
ment of the Society for making discoveries at sea, pub-
lished at London in 156l.

CORVUS, the Raven, a southern constellation, fabled
by the Greeks, as taken up to heaven by Apollo, to whom
it tattled that the beautiful maid Coronis, the daughter of
Phlegeos, and mother of Esculapius by Apollo, played
the deity false with Ischys, under a tree upon which the
animal happened to be perched. The stars in this con-
stellation, in Ptolemy's and Tycho's catalogues, are 7; but
in the Britanniccatalogue, 9-

COSECaNT*COSINE,COTANGENT,COVERSED

Royal Society, which are chiefly on astronomical subjects,
are -contained invbls. 43, 44,48 and 6'8of the Philosophical
Transactions.

COTES (Roger), a very eminent mathematician, phi-
losopher, and astronomer, was born July 10, l6S2, at
Burbach in Leicestershire, where his father Robert was
rector. He was first placed at Leicester school ; where,
at 12 years of -age, he discovered a strong inclination to
the mathematics. This being observed by his uncle, the
Rev. Mr. John Smith, he gave him all the encouragement
he could; and prevailed on his father to send him for
some time to his house in Lincolnshire, that he might
assist him in those studies: and here he laid the founda-
tion of that deep and extensive knowledge in that science,
for which he was afterwards so deservedly famous. He
was hence removed to St. Paul's school, London, where

SINE, are the secant, sine, tangent, and versed sine of the he made a great progress in classical learning ; and yet he
complement of an arch or angle ; Co being, in this case, a found so much leisure as to support a constant eorrespon-
contraction of the word complement, and was first intro- dence with his uncle, not only in mathematics, but also in

duced by Gunter.

COSM1CAL Aspect, among astrologers, is the aspect
of a planet with respect to the earth.

Cosmical Rising, or Setting, is said of a star when it
rises or sets at the same time when the sun rises.

But, according to Kepler, to rise or set cosmically, is

metaphysics, philosophy, and divinity. His next remove
was to Trinity-college, Cambridge, where he took his de-
grees, and became fellow.

Jan. 1 706, he was appointed professor of astronomy
and experimental philosophy, on the foundation of Dr.
Thomas Plume, archdeacon of Rochester; being the first

only simply to rise or set, that is, to ascend above, or de- that enjoyed that office, to which he, was unanimously
scend below, the horizon ; as much as to say, to rise or chosen, on account of his high reputation and merits. He
set to the world. . entered into orders in 1713; and the same year, at the
COSMOGONY, the science of the formation of the uni- desire of Dr. Bentley, he published at Cambridge the se-
verse ; as distinguished from cosmography, which is the cond edition of Newton's Mathematica Principia ; insert-
science of the parts of the universe, supposing it formed, ing all the improvements which the author had made to
and in the state as we behold it ; and from cosmology, that time. To this edition he prefixed an excellent pre-
which reasons on the actual arid permanent state of the face, in which he pointed out the true method of philoso-
world as it now is; whereas cosmogony reasons on the phizing, showing the foundation on which the Newtonian
variable state of the world at the lime of its formation. philosophy was raised, and refuting the objections of the

COSMOGRAPHY, the description of the world; or Cartesians and all other philosophers against it.

the art that teaches the construction, figure, disposition, The publication of this edition of Newton's Principia

and relation of all the parts of the world, with the manner added greatly to his reputation ; nor was the high opinion

of representing them on a plane. It consists chiefly of two the public now conceived of him in the least diminished,

parts; viz, Astronomy, which shows the structure of the but rather much increased, by several productions of his

heavens, with the disposition of the stars; and Geography, own, which afterward appeared. He gave in the Philos.

which shows those of the earth. Transactions, two papers, viz, 1, Logometria, in vol. 29;

COSMOLOGY, the science of the world in general. and a Description of the great fiery meteor that was seen

COSS, Rule of, meant the same as Algebra, by which March 6, 1716, in vol. 31.

name it was for some time called, when first introduced This extraordinary genius in the mathematics died, to

into Europe through the Italians, who named it Regola de the great regret of the university, and all the lovers of the

cosa, the Rule of the thing; the unknown quantity, or sciences, June 5, 1716", in the very prime of his life, being

that which was required in any question, being called cosa, not quite 34 years of age.

the thing; the terms whence we have Coss, and Cossic Mr. Cotes left behind him some very ingenious, and in-

numbers, &c. deed admirable tracts, part of which, with the Logometria

COSTARD (George), was born at Shrewsbury 1710, above mentioned, were published, in 1722, by Dr. Robert

and educated at Wadham-college, Oxford, of which he Smith, his cousin and successor in his professorship, aftcr-

became a fellow and a tutor; he died at his vicarage at wards master of Trinity-college, under the title of Har-

GOV

C 374 ]

c o u

monia Mensurarum, which contains a number of very in- it is usual for the besieged to palisade it along the middle*

genious and learned works : See the Introduction to my and undermine it on all sides.

Logarithms. He wrote also a Compendium of Arith- COVING, in Building, is when houses are built pro-

metic ; on the Resolution of Equations; on Dioptrics; jecting over the (Hfcund plot, and the turned prefecture

and on the Nature of Curves. Besides these pieces, he formed into an arch.

drew up, in the time of his lectures, a course of Hydro- Coving Cornice, is one that has a large casemate or

statical and Pueumatical Lectures, in English, which we
published also by Dr. Smith in 8vo, 1737, and are held
in great estimation.

So high an opinion had Sir Isaac Newton of our au-
thor's genius, that he used to say, " If Cotes had lived, we
might have known something."

COTESIAN theorem, in Geometry, an appellation used

hollow in it.

COUNT-Wheel, is a wheel in the striking part of a
clock, moving round once in 12 or 2-A hours. It it some-
times called the locking-wheel, because it has usually ] 1
notches in it at unequal distances from one another, to
make the clock strike.

COUNTER-Approaches, in Fortification, lines or

for an elegant property of the circle discovered by Mr. trenches made by the besieged, where they come out to at

Cotes. The theorem is this: If the factors of the binomial
ac ttsX^ be required, the index c being an integer number.
With the centre o, and radius AO = a, describe a circle,

C J) — C

and divide its circumference into as many equal parts as
there are units in 2c, at the points a, b, c, d, &c ; then in
the radius, produced if necessa^, take op = x, and from
the point p, to all the points of division in the circum-
ference, draw the lines pa, pb, pc,&c; so shall these lines
taken alternately, be the factors sought; viz,
pb x pd x pf &c = oc -+■ xc, and

tack the lines of the besiegers in form.

Counter-Battery, a battery raised to play on an-
other, to dismount the guns, &c.

Counter-Breast-work, the same as Fausse-Braye.

Counter-Forts, Buttresses, or Spurs, are pillars
of masonry serving to prop or sustain walls, or terraces,
subject to bulge, or be thrown down.

Counter-Fugue, in Music, is when fugues proceed
contrary to one another.

Counter-Guard, in Fortification, a work commonly
serving to cover a bastion. It is composed of two faces,
forming a salient angle before the flanked angle of a bas-
tion. ^

CoUNTER-HARMONICAL. SeeCoNTRA-HARMONICAL.

Counter-Mine, a subterraneous passage, made by the
besieged, in search of the enemy's mine, to give air to it, to
take away the powder; or by any other means to frustrate
the effect of it.

Counter-Part, a term in Music, only denoting that
one part is opposite to another: so, the bass and treble are
counter- parts to each other.

Counter- Point, in Music, the art of composing har-

pa x pc x pe &c = ac ~ -cc, viz,ac — xc oric
according as the point p is within or without the circle, mony ; or disposing and concerting several parts so toge-
For instance, if c = 5, divide the circumf. into 10 equal ther, as that they may make an agreeable whole,
parts, and the point p being within the circle, then will Counter-Poise, any thing serving to weigh against an-

other; particularly a piece of metal, usually of brass or
iron, making an appendage to the Roman statera, or steel-
yard. It is contrived to slide along the beam; and from
the division at which it keeps the balance in equilibrio,
the weight of the body is determined. It is sometimes
called the Pear, on account of its figure ; and Mass, by
reason of its weight. Rope-dancers make use of a pole by
way of counterpoise, to keep their bodies in equilibrio.
COUNTERSCARP, is that side of the ditch that is

0A5 -+- OP = BP X DP X FP X HP'X KP,

and oas — op5 = ap x cp x ep x gp x ip.

In like manner, if c = 6, having divided the circum-
ference into 12 equal parts, then will

* 0Ab •+- 0P°= BP X DP X FP X HP X KP X MP,
OA° — OP°=AP X CP X EP X GP X IP X LP.

The demonstration of this theorem may be seen in Dr.
Pemberton's Epist. de Cotesii inventis. See also Dr.
Smith's Theoremata Logometrica and Trigonometrica,

added to Cotes's Harm. Mens, pa, 114; De Moivre Mis- next the country; or properly the talus that supports the
eel. Analyt. pa. 1 7 ; Waring's Letter to Dr. Powell, pa. 39, earth of the covert-way : though by this word is often inl-
and Simpson's Essays, p. 113. derstood the whole covert-way, with its parapet and glacis.

By means of this theorem, the acute and elegant au- And so it must be understood when it is said, The enemy

thor was enabled to make a farther progress in the inverse lodged themselves on the counterscarp.
method of Fluxions, than had been done before. But in Counter-Sw allows-tail, is an outwork in Fortifi-

tbe application of his discovery there still remained a li- cation, in form of a single tenaille, wider towards the place,

mitation, which was removed by Dcmoivre. Vide ut or at the gorge, than at the head, or next the country,
supra. Counter-tenor, one of the mean or middle parts of

COVERT-Way, in Fortification, a space of ground music; so called, as being opposite to the tenor.
IcTel with the adjoining country, on the outer edge of the COURSI''., in Navigation, the point of the compass, or

ditch, ranging quite round all the works. This is other- horizon, which a ship steers on; or the angle which the

wise called the Corridor, and has a parapet with its ban- rhumb-line on which it sails makes with the meridian;

quette and glacis, which form the height of the parapet. It being sometimes reckoned in degrees, and sometimes in

is sometimes also called the Counterscarp, because it is on points of the compass.

the edge of the scarp. — One of the greatest difficulties in When a ship sails either due north or south, she sails on

a siege, is to make a lodgment on the covert-way ; because a meridian, makes no departure, and her distance and dif-

CRA

[ 375 ]

CRA

ference of latitude are the same. When she sails due east
or west, her course makes right-angles with the meridian,
and she sails either upon the equator, or a parallel to it;
in which case she makes no difference of latitude, but her
distance and departure are the same. But when the ship
sails between the cardinal points, on a course making al-
ways the same oblique angle with the meridians, her path
is then the loxodromic curve, being a spiral cutting all
the meridians in the sameangle, and terminating in the pole.

Hence arises a kind of geographical, or nautical para-
dox; at least to those who have but a limited knowledge
of those subjects ; which is this. If we suppose two
places having different latitudes and longitudes, and that a
person leaves one of those places on a certain point of the
compass, bearing directly towards the other; and having
arrived there, leaves this last place upon the direct oppo-
site point of the compass to that on which he first set out,
it will be found on his pursuing this course, that he will
never return again to the place whence he first departed ;
but will leave it more or less on the right or left, according
to the relative situations of the two places.

The cause of this deviation from bis original track is
however easily explained : for in. the first case, as the tra-
veller is supposed to go directly from one place to the
other, he does not continue, in any two successive in-
stances, to pursue the same point of the compass; where-
as on his return, he is considered as constantly observing
the same point, and hence arises the deviation in question.

For the same reason, if a vessel depart from a port situ-
ated in north latitude, and constantly pursue the same
north-easterly, or north-westerly course, it will cominually
approach towards the north pole in a spiral traced on the
"terrestrial surface, but it will never arrive there, however
far the voyage may be pursued. And exactly the same
thing will take place with respect to the south pole, if the
course be directed towards the south-east, or south-west.

COURTAIN. See Curtin.

CRAB, in Mechanics, an engine used for mounting guns
on their carriages. See Gin.

Crab, oh ship-board, is a wooden pillar, having its
lower end let down through the ship's decks, and resting
on a socket, like the capstan : in its upper end are three or
four holes, at different heights, through the middle of it,
one above another, to receive long bars, against which men
act by pushing or thrusting. — It is employed to wind-in
the cable, and for other purposes requiring a great mecha-
nical power.

The Crab with three claws is used to launch ships, and
to heave them into the dock, or off the. key.

CRABTRLE (Wm.), an ingenious English astronomer,
who resided at Broughton near Manchester in Lancashire,
in the early part of the 17th century. He was the friend
of the congenial young astronomer Horrox, near Liver-
pool, with whom he carried on a learned astronomical cor-
respondence for the several years l6'36', 16'37, 16'38, 1639,
1640; by which means he was very instrumental in en-
couraging Mr. Horrox to prosecute his useful astronomical
observations and writings. A collection of those of Mr.
Crabtree, mostly for the same years as tljose above, was
published by Dr. Wallis, along with the works of his friend
Horrox, in one vol. 4to, 1672. A long and learned letter
of his to Mr.Gascoigne (the inventor of the Micrometer),
on the nature and appearances of the solar spots, in the
Pbilos. Trans, vol. 27, and in my Abridg. vol.5, pa. 6'2o".

CltAIG (Rev. John), f.r.s, a respectable Scotch di-

vine and mathematician, who it seems settled some time
at Cambridge, but was afterwards vicar of Gillingham, in
Dorsetshire, whence he commonly addressed his letters to
the Royal Society. In 1685 he had a dispute with John
Bernoulli, on the quadrature of curve lines and curvili-
near figures. He had also another difference with that
great mathematician on an algebraic equation ; but after-
wards finding himself wrong, he candidly acknowledged it
in the Philosophical Transactions. He wrote several good
pieces on mathematical subjects, partly published sepa-
rately, and several published in the Philos. Trans, as also
the translation of them from the Latin in my Abridgement,
where a catalogue of them may be seen, in the index at
the end of the iSth or last volume; being, 1. Quadratures
of curves geometrically irrational. — 2. Quadrature of the
logarithmic curve. — 3. On the curve ofquickest descent. —
4. On the solid of least resistance. — 5. Method of deter-
mining the quadrature of figures. — 6. Solution of Ber-
noulli's problem on curves. — 7. On the length of curve
lines. — S. Method of making logarithms. — 9. On the head
of a monstrous calf.

And his separate publications are, 1. Methodus figu-
rarum quadraturas, &c, anno l6S5, 4to. — 2. De quadra-
turis et locis, &c, an. l6j)3, 4to. — 3. De calculo fluenti-
um, et de Optica analytica, &c, an. 17 IS, 4to. — 4. The-
logia; Christianas principia mathematica, l6'99, and after-
wards printed at Leipsic, with a learned preface. The ob-
ject of this curious tract, is to apply mathematical calcu-
lation to the credibility of the Gospel history ; and on this
principle he maintains that the Christian religion must
end, according to the doctrine of chances, annodom.3150,
at which time Christ will make his second appearance.
This work was answered by the abbe Hauteville, in a piece
entitled, The Christian religion proved by facts.

CRAMKR (Gab kiel), an eminent Swiss mathemati-
cian, was born at Geneva in 1704, and died 1752. He
was made professor of mathematics at 19 years of age, and
was author of several good works; as, his Theory of Curve
Lines, 1750, 4to ; he simplified the art of reducing the
equations of a problem to the smallest number possible.
To him the world owes the useful labour of collecting and
publishing the works of James and John Bernoulli. He
possessed a fund of knowledge in physics and the belles
lettres, and rendered himself famous throughout Europe
by his progress and skill in mathematics.

CRANE, a machine used in building, and in commerce,
for raising large stones and other weights.

Perrault, in his notes on Vi.tru.vius, makes the crane the
same with the corvus, or raven, of the ancients.

The modern crane consists of several members or pieces,
the principal of which is a strong upright beam, or arbor,
firmly fixed in the ground, and sustained by eight arms,
coming from the extremities of four pieces of wood laid
across, through the middle of which the foot of the beam
passes. About the middle of the arbor the arms meet, and
are mortised into it : its top ends in an iron pivot, on which
is borne a transverse piece, advancing out to a good di-
stance like a crane's neck; whence the name. The mid-
dle and extremities of this are again sustained by arms
from the middle of the arbor: and over it passes a rope,
or chain, to one end of which the weight is fixed ; She other
is wound about the spindle of a1 wheel, which, turned,
draws the rope, and that heaves up the weight; to be af-
terwards applied to any side or quarter, by the mobility of
the transverse piece on the pivot,

C R E

[ 376 ]

CRE

Several improvements of this useful machine are men-
tioned in Desaguliers's Exper. Philos. pa. 17S & seq. par-
ticularly how to prevent the inconveniences arising from
sudden jerks, as well as to increase its force by using a
double axis in peritrochio, and two handles.

The Crane is of two kinds ; in the first kind, called the
rat-tailed crane, the whole machine, with the load, turns
upon a strong axis: in the second kind, the gibbet alone
moves on its axis. See Desaguliers, as above, for a par-
ticular account of the different cranes, and of the gradual
improvements they have received. See also the Supple-
ment to Ferguson's Lectures, pa. 3, &c; or Philos. Trans,
vol. 54, pa. 24, for a description of a new and safe crane,
with four different powers adapted to different weights.

Crane is the name of a southern constellation. See
Grus.

Crane is also a popular name for a syphon.

CRANK, a contrivance in machines, in manner of an
elbow, only of-a square form ; projecting out from an axis,
or spindle; and serving, by its rotation, to raise and fall
the pistons of engines for raising water, or the like.

CRATER, the Cup, a constellation in the southern he-
misphere; whose stars, in Ptolemy's catalogue, are 7 ; in
Tycho's, 8; in Hcvelius's, 10; and in the Britannic cata-
logue, 31.

CREEK, a part of a haven, where any thing is landed
from the sea. It is also said to be a shore or bank on
which the water beats, running in a small channel from
any part of the sea.

CREPUSCULUM, Twilight; the time from the .first
dawn or appearance of the morning, to the rising of the
sun ; and again, between the setting of the sun, and the
last remains of day.

The Crepusculum, or twilight, it is supposed, usually
begins and ends when the ^un is about 18 degrees below
the horizon ; for then the stars of the 6th magnitude dis-
appear in the morning, and appear in the evening. It is
of longer duration in the solstices than in the equinoxes,
tut it islongerin an oblique sphere, than in a right one; be-
cause, in those cases the sun, by the obliquity of his path,
is longer in ascending through 18 degrees of altitude.

Twilight is occasioned by the sun's rays refracted in our
atmosphere, and reflected from the particles of it to the
eye. For let a be the place
of an observer on the earth
adl, ab the sensible' hori-
zon, meeting in b the circle
cbm bounding that part of
the atmosphere which is ca-
pable of refracting and re-
flecting light to the eye. It
is plain that when the sun is
under this horizon, no direct
rays can come to the eye at
a : but the sun being in the refracted line CG, the parti-
cle c will be illuminated by the direct rays of the sun;
and that particle may reflect those rays to a, where they
enter the eye of the spectator. And thus the sun's light
illuminating an innumerable multitude of particles, may
be all reflected to the. spectator at a. — From B draw bd
touching the circle adl in d; and let the sun be in the
line nn at s : Then the ray SB will be reflected into ba,
and will enter the eye, because the angle of incidence dbe
is equal to the angle of reflection abe : And this will be
the first ray that reaches the eye in the morning, when the

dawning begins; or the last that falls upon the eye at night,
when the twilight ends: for when the sun goes lower down,
the particles at b can be no longer illuminated.

Kepler indeed assigns another cause of the crepusculum,
viz, the luminous matter or atmosphere about the sun ;
which, arising near the horizon, in a circular figure, exhi-
bits the crepusculum; in no wise, he thinks, owing to the
refraction of the atmosphere. — The sun's luminous atmo-
sphere indeed, though neither the sole nor principal cause
of twilight, may lengthen its duration, by illuminating
our air, when the sun is too low to reach it with his own
light. Gregor. Astr. lib. 2, prop. 8.

The depression of the sun beloiv the horizon, at the begin-
ning of the morning, or end of the evening twilight, is deter-
mined in the same manner as the arch of vision ; viz, by
observing the moment when the aii: first begins to shine in
the morning, or ceases to shine in the evening ; then find-
ing the sun's place for that moment, and thence the time
till his rising in the horizon, or from his setting in it in the
evening. It is now generally agreed that this depth is
about 18 degrees upon an average. — Alhazen found it to
be 13°; Tycho, 17°; Rothmann, 24°; Stevenius, 18°;
Cassini,'15°; Riccioli, in the equinox in the morning 16°,
in the evening 20° 30'; in the summer solstice in the morn-
ing 21° 25', in the winter solstice in the morning 17° 25'.

Nor is this difference among the determinations of astro-
nomers to be wondered at; the cause of th« crepusculum
being inconstant : for, if the exhalations in the atmosphere
be either more copious, or higher, than ordinary ; the
morning twilight will begin sooner, and the evening hold
longer than ordinary: for the more copious the exhalations
are, the more rays will they reflect, consequently the more
will they shine; and the higher they are, the sooner will
they be illuminated by the sun. On this account too, the
evening twilight is longer than the morning, at the same
time of the year in the same place. To this it may be
added, that, in a denser air the refraction is greater; and
that not only the brightness of the atmosphere is variable,
but also its height from the earth : and therefore the twi-
light is longer in hot weather than in cold, in summer than
in winter, and also in hot countries than in cold, other cir-
cumstances being the same. But the chief differences are
owing to the different situations of places on the earth, 01
to the difference of the sun's place in the heavens. Thus,
the twilight is longest in a parallel sphere, and shortest in
a right sphere, and longer to places in an oblique sphere, in
proportion as they are nearer to one of the poles; a cir-
cumstance which affords relief to the inhabitants of the
more northern countries, in their long winter nights. And
the twilights are longest in all places of north latitude,
when the sun is in the tropic of cancer; and to those in
south latitude, when he is in the tropic of Capricorn. The
time of the shortest twilight is also different in different la-
titudes; in England, it is about the beginning of October
and of March, when the sun is in the signs t£± and >£.
For the method of determining it by trigonometry, see
Gregor. Astron. lib. 2, prob. 41 . See also Robertson's Na-
vigation, book 5, prob. 12. — Hence, from what has been
said, it appears that when the difference between the sun's
declination and the depth of the equator is less than 18°, so
that the sun does not descend more than 18° below the ho-
rizon ; the crepusculum will continue the whole night, as
is the case in England from about the 22d of May to the
22d of July.

Given the latitude of the place, and the sun's declination ;

CRO

[ 377 ]

CRO

to find the beginning of the morning, and end of the evening
twilight. — In the oblique-angled spherical triangle zps,
arc given zp the colatitude, ps the co.
declination, and zs = 108°, being the
Sum of 90° the quadrant and 18° the
depression at the extremity of the twi-
light. Then, by spherical trigonome-
try-, calculate the angle zps the hour-
angle from noon ; which changed into
time, at the rate of 15° for an hour,
gives the time from noon at the begin-
ning or end of twilight. See Robertson, ubi supra.

Of the Height of the sensible Atmosphere, as determined
from the duration of twilight, see Keil's Astron. Lect.
lect. 20, pa. 235, ed. 1721; or Long's Astron. vol. 1,
pa. 26'0 ; where it is determined that the height at which
the atmosphere is dense enough to reflect the rays of light,
is about,42 miles.

CRESCENT, the newmoon, which, as it begins to recede
from the sun, shows a small rim of light, terminating in
horns or points, which increase till it becomes full, and
round in the opposition.

The term is sometimes also used for the same figure of
the moon in her wane, or decrease, but improperly; both
because the horns are then turned towards the west, and
because the figure is on the decrease ; the crescent pro-
perly signifying increase, from cresco, I grow.

CRONICAL. See Acronical.

CRONOS, a name given to Saturn by some of the old
astronomical writers.

CROSIER, four stars, in form of a cross ; by the help
of which, those that sail in the southern hemisphere find
the antarctic pole.

CROSS, in Surveying, is an instrument consisting of a
brass circle, divided into 4 equal parts, by two lines cross-
ing each other in the centre. At each extremity of these
lines is fixed a perpendicular sight, with small holes below
each slit, for the better discovering of distant objects. The
cross is mounted on a staff, or stand, to fix it in the ground,
and is very useful for measuring small pieces of land, and
taking offsets, &c.

Ex. Suppose it be required to survey the field abcde
with the Cross. Measure along
the diagonal line ac, and ob-
serve, with the Cross, w.hen
you are perpendicularly op-
posite to the corners, as at f,
g, h, and from thence mea-
sure the perpendiculars fe,
gb, hd. When you think you
are nearly opposite a corner, set up the cross, with one
of the bars or cross lines in the direction ac; then look
through the sights of the other cross bar for the corner,
as b; if it be seen through them, the cross is fixed in the
right place ; if not, take it up and move it backward or
forward in the line ac, till the point b be seen through
those sights ; and then you have the true place of the per-
pendicular.

Invention of the Cross, Inventio Ciucis, an ancient feast,
which is still retained in our calendar, and solemnized on
the 3d of May, in memory of the finding of the true cross
of Christ, it is said, deep in the ground, on Mount Calva-
ry, by St. Helena, the mother of Constantine ; where she
erected a church for the preservation of part of it: the

Vol. I.

rest being brought to Rome, and deposited in the church
of the Holy Cross of Jerusalem.

Exaltation of the Cross, an ancient feast, held on the
14th of September, in memory of this, that Heraclitus re-
stored to Mount Calvary the true cross, in 6'42, which had
been carried off, 14 years before, by Cosroes kin" of Per-
sia, on his taking Jerusalem from the emperor Phocas.
This feast is still retained in our calendar, on Sept. 14, un-
der the name of Holy Rood, or Holy Cross.

Cross-Mu.ltiplicatiok, a method t)f. lOinc.
used chiefly by artificers in multiplying feet 6 8
and inches by feet and inches, or the like; "50 q
so called, because the factors are multiplied g g §

crosswise, thus : ' 7. — : — :

60 0 8

Cross-Staff, or Fore-Staff, was a nautical instru-
ment of box, or pear-tree, consisting of a square staff',
of about 3 feet long, having each of its faces divided like a
line of tangents, and having 4 cross pieces of unequal
lengths to fit on to the staff, the halves of these being as the
radii to the tangent lines on the faces of the staff. — The
instrument was used in taking the altitudes of the celestial
bodies at sea. It seems to have been pretty ancient, and was
described by John Werner of Nuremberg, in his Annota-
tions on the 1st book of Ptolemy's Geography, printed in
1514, recommending it for observing the distance between
the moon and some star, for the purpose of determining
helongitude. See Fore-Staff.

CROUSAZ (John Peter de), a learned philosopher
and mathematician, was born at Lausanne in Switzerland,
April 13, 1663 ; where he died in 1748, at 85 years of
age. Having made great progress in mathematics and the
philosophy of Descartes, he travelled into Geneva, Hol-
land, and France. He was successively professor in seve-
ral universities ; and at length was chosen governor to
Prince Frederick of Hesse-Cassel, nephew to the king of
Sweden.

Crousaz was author of many works, in various branches ;
belles-lettres, logic, philosophy, divinity, &c, &c; but the
most esteemed of them are, 1. His Logic; the best edition
of which is that of 1741, in 6* vols. 8vo. — 2. A Treatise on
Beauty. — 3. A Treatise on Education, 2 vols, 12mo. — 4.
A Treatise on the Human Understanding. — 5. Several
Treatises on Philosophy and Mathematics; as a Treatise
on Motion, &c, with several papers inserted in the Memoirs
of the. French Academy of Sciences.

CROW, in Mechanics, an iron lever, made with a sharp
point at one end, and two claws at the other ; being used
in heaving and purchasing great weights, &c. 1

CROWN, in Astronomy, a name given to two constel-
lations, the southern and the northern.

Crown, in Geometry, a plane ring included between
two parallel or concentric peripheries of unequal circles.

The area of this is obtained, by multiplying its breadth
by the length of a middle, periphery, which is an arithme-
tical mean between the two peripheries that bound it ; or
by multiplying half the sum of the circumferences by half
the difference of the diameters; or lastly, by multiplying
the sum of the diameters by the difference of the diameters,
and this last product by '7854. See my Mensuration, pa.
110, 4th. edit.

Crown-Pos*, is a post in some buildings standing up-
right in the middle, between two principal rafters ; and
from which proceed struts or braces to the middle of each
3C

CUB

[ 378 ]

C U B

rafter. It is otherwise called a King-post) or King's-piece,

Of Joggle-piece.

Cs.ows-W/ieel, of a Watch, is the upper wheel next the
balance, or that which drives the balance.

Crow ys -Work, in fortification, is an out-work running
into the field; designed to keep off the enemy, gain some
hill, or advantagecus post, and cover the other works of
the place. It con>ists of two demi-bastions at the ex-
tremities, and an entire bastion in the middle, with cur-
tains.

Crowned Horn-work, is a horn- work with a crown-
work before it.

CRYSTALLINE Humour, is a thick compact humour
of the eye, in form of a flattish convex lens, placed in the
middle of the eye, and serving to make that refraction of
the rays of light which is necessary to have them meet in
the retina, and form an image there, by which vision may
be performed.

Crystalline Heavens, in the Old Astronomy, two
orbs imagined between the primum mobile and the firma-
ment, in the Ptolemaic system, which supposed the hea-
vens solid, and only susceptible of a single motion.

King Alphonsus of Arragon, it is said, introduced the
Crystallines, to explain what was called the motion of
trepidation, or titubation. — The first Crystalline, accord-
ing to Regiornontanus, &c, serves to account for the slow
motion of the fixed stars; by which they advance a de-
gree in about 70 years, according to the order of the
signs, or from west to east ; which occasions a precession
of the equinox. The 2d serves to account for the motion
of libration, or trepidation ; by which the celestial sphere
librates from one pole towards the other, causing a differ-
ence in the sun's greatest declination.

CTESIB1US, orCTESEBES, a mathematician of Alexan-
dria, who was contemporary with Ptolemy king of Egypt,
in the l65th olympiad, about 120 years before Christ. He
was possessed of a very great strength of imagination,
which placed him at once in a situation much above
that which his birth otherwise entitled him toexpect, being
the son of a barber in that city. His memory is particu-
larly cherished as the inventor of the pump. He also in-
vented an hydraulic organ, though the circumstances that
led to the discovery were purely accidental. On lower-
ing a mirror in his father's shop, he observed that the coun-
terpoise, which was included in a cylinder, produced a
sound, by driving the air before it; and on examining
the phenomenon more strictly, he conceived the idea of
making an instrument, in which sounds should be produ-
ced by means of the action of water driving the air before
it, which invention was afterwards carried into effect by
the emperor Nero. Ctesibius was the inventor likewise of
the clepsydra, or water-clock, by means of water falling
upon a wheel, or a train of wheels, which were turned by
it. The wheels communicated their motion to a small
wooden image, which, by being gradually raised, pointed
with an index to the proper hours, that were engraved on
a column near the machine. This invention, it is highly pro-
bable, led to that "I hour-glasses, or sand-glasses, for mea-
suring time, Which seems an imitation of the clepsydra.
Ctesibius was also author of a treatise, " Geodesia, or the
Art of dividing and measuring Bodies," which is said to be
preserved in the library of the Vatican.
Ctesirius' Pump. See Pump.
CUBATURE, or Cuiiation, of a solid, is the measu-

ring the space contained in it, or rinding the solid content
of it, or finding a cube equal to it.

The cubature regards the content of a body, as the qua-
drature does the superficies or area of a figure.

CURE, a regularsolid body, inclosed by six cqualsides
or faces, which are squares. — A die is a small cube. It
is also called a hexaedron, because of its six sides, and is
the 2d of the five Platonic or Regular bodies?

The cube is supposed to be generated by the motion of
a square plane, along a line equal and perpendicular to
one of its sides.

To desci. je a Rete, or Net, for forming a cube, or with
which it may be covered. See Body {regular).

To determine the Surface and Solidity of a Cube. — Mul-
tiply one side by itself, which will give one square or face;
then this multiplied by 6, the number effaces, will give
the whole surface. Also multiply one side, twice by itself,
that is, cube it, which will give the solid content.
Duplication of a Cube. See Duplication.
Cubes, or Cubic Numbers, are formed by multiply-
ing any numbers twice by themselves. So the cubes
of 1, 2, 3, 4, 5, 6, &c.
are 1, 8, 27, 64, 12.5, 2l6, &c.

The third differences of the cubes of the natural num-
bers are all equal to each other, being the constant num-
ber 6". For, let to3, n3, p3, be any three adjacent cubes in
the natural series as above, that, is, whose roots m, n, p,
have the common, difference 1 ; then because
n = to -+- 1, theref. n? = to3 -t- 3mr -+- 3m 4- I,
p =r n ■+■ 1, theref. p3 = ?j3 ■+• 3nz =+■ 3n -+- 1 ; so that
the difference between the 1st and 2d, and between the 2d
and 3d cubes, are

w3 — to3 = 3»r -+■ 3m -+- 1,7 .i_ , . i-o-

, , „ o „ ,'!• the 1st differences ;

p3 — n3 = 3« ■+■ 3» -i- 1 , j
and the dif. of these differences, is

3(n2 — to2) ■+■ 3(n — to) = 3{n ■+- to) . {n — to ) + 3
(n — to), or since n — m = 1, it becomes 3(n •+- to) -+- 3
= 3(« -+■ to -)-!)= 6(to 4- 1), the 2d difference.

In like manner the next 2d dif. is 6(ri -+- 1) : hence the
dif. of these two 2d diffs. is 6(n — m) = 6, which is
therefore the constant 3d difference of all the series of cubes.
And hence the series of cubes will be formed by addition
only, viz, adding always the 3d dif. 6 to find the column
or series of 2d diffs, then these added always for the 1st
dill's, and lastly these always added for the cubes them-
selves, as below :

3d Difs.

2d Difs.

1st Difs.

Cubes

6

6

1

0

6

12

7

I

6

18

19

R

■ 6

24

37

27

6

30

6i

64

6

36'

ftl"

125

6

42

127

^216

6

48

169

343

Peletarius, among various speculations concerning square
and cubic numbers, shows that the continual sums of the
cubic numbers, whose roots are 1, 2, S, &c, form the se-
ries of squares whose roots are 1 , 3, 6, 10, 15, 21, &c.
Thus, 1 = 1 = I'.

l + 8 = 9 = S%
1+ 8 -*- 27 = 36 = 6\
i h- 8 ■+- 27 ■+■ 64 = 100 = 102, &e.

CUB

[ 379 ]

CUB

Or, in general, I3 -4- 23 -t- 33 + 43&c, to»3 = (1 + 2 ■+■ 3 gebra. The rule is this : Put a = ■£/>, and 4 =|?; then

+ 4 . . . ft}2, whose root is •§■». (re -+- 1). by Cardan's method the root is

It is also a pretty property, that any number, and the x =y/[6 -h*/(b* •+- a3)] -t-3/^ — v/(*2 -+- a3)] ; or if there

cube of it, being divided by 6', leave the same remainder; be puts=£/[6-t- -/(&* -+- a3)], and d — z/[b — ^/{lil+ a3)],

the series of remainders being 0, 1,2, 3, 4, 5, continually then s ■+■ d = x, the 1st or Cardan's root,

repeated. Or that the differences between the numbers
and their cubes, divided by 6, leave always 0 remaining;
and the quotients, with their successive differences, form
the several orders of figurate numbers. Thus,

also ■
and •

■ - — -*/ -3 is the 2d root,

s — d .

— tv ■

■3 is the 3d root.

Num.

Cubes.

Difs.

Quot.

lD.f.

■2 Dif.

1

1

0

0

0

0

2

8

6

1

1

1

3

27

24

4

3

2

4

64,

6o

10

6

3

5

125

120

20

10

4

6

216

210

35

15

5

7

343

336

56

21

6"

See a large table- of cubes and squares, with their roots,
&c, in the first volume of my Tracts, at the end.

The Cube of a Binomial, is equal to the cubes of the
two parts or members, together with triple of the two pa-
rallelopipedons under each part and the square of the
other; viz, (a ■+■ b)3 = a3 ■+■ 3a'b ■+■ 3ab2 -+- 63. And hence
the common method of extracting the cube root.

Cubic Equations, are those in which the unknown quan-
tities rise to three dimensions; as i3 = a,
or x3 -t- ax2 = b, or x3 -t- ax"1 ■+■ bx = c, &c.

All cubic equations may be reduced to this form,
x3-hpx = q; viz, by taking away the 2d term, when
there is one; which is done by substituting, in the given
equation z — -J-a for x. /

All cubic equations have three roots ; which are either
all real, or else one only is real, and the other two imagi-
nary ; for all roots become imaginary by pairs.

But the nature of the roots, as to real and imaginary, is
known partly from the sign of the co-efficient p, and partly
from the relation between p and q : for the equation has
always two imaginary roots when p is positive ; it has also
two imaginary roots when/) is negative, provided (ip)3 be
less (kq)1, or 4/r3 less than 27'f ; otherwise the roots are
all real ; namely, whenever p is negative, and ip3 either therefore r = c -t- d =4, Ihe 1st root; and

Now the first of these, or Cardan's root, is always a real
root, though it is not always the greatest root, as it has
been commonly mistaken for. And yet this rule always
exhibits the root in the form of an imaginary quantity
when the equation has no imaginary roots at all ; but in
the form oi a real quantity when the equation has two ima-
ginary roots. See the reason of this explained in my
Tract above cited, art. 47. As to the other two roots, viz,

— i - IT*/ ~ 5, though, in their general form they

have an imaginary appearance ; yet, by substituting cer-
tain particular numbers, they come out in a real form in
all such cases as they ought to be so.

But, after the first root is found by Cardan's rule, the
other two roots may be found, or exhibited, in several
other different ways; some of which are as follow :

Let p denote the 1st, or Cardan's root,

and v and w the other two roots :
then is v -+- w = — r, and vwr = q ;

and the resolution of these two equations will give the
other two roots d and w, by a quadratic equation.

Or resolve the quadratic equation x2 -+- tx h* r2 -+- p = 0,
and its two roots will be those sought.
Or the same two roots will be

either -ir + fv/(-p - -)and-fr- i</(-p-3-l),

or - |r + W{r? - U-) and -\r-l^'{i? -"-?),

or- lr + V(l~ — -) and-!r-JrV(l %-}.

i—pr " y—Pr

Ex. 1. If the equation be x3 — 15r = 4: here/" =

— 15, o = 4, «= - 5, 6=2; hence ,/ (62 -+- a3) =

^(4-125)=^/- 121 = llv/- l,s=^(2 + v'- 121)

= 2 -+V- ]> ancl rf=v/(2-v/~121) =2 —«/ — 1:

equal to, or greater than 27q2.

Every cubic equation of the above form, viz, wanting
the 2d term, has both positive and negative roots, and the
greatest root is always equal to the sum of the two less
roots; viz, either one positive root equal to the sum of
the two negative ones, or else one negative root equal to

— — jj— ± - — -4/ — 3 = — 2 + \/3, the other two roots.

Ex. 2. If x3 - 6x = 4 : here a = — 2, and 6=2;
therefore ^/(b1 -+- a') =^/(4 - 8) = */ — 4 = 2^/- 1,
.5=^(2 4- 2</~ 1) = - 1 -tV-1' andrf=3/(2-2
«J — 1) = — 1 — .*/— 1 : hence then r = s + d = ~ 2,
the sum of two smaller and positive ones. And the sign the first root; and 1 ±^/3 the other two roots,
of the greatest, or single root, is positive or negative, ac- Ex. 3. If a3 -+- 18x = 6: here a = 6, and 6 = 3; then

cording as q is positive or negative when it stands on the \/(b2 ■*■ a3) =\/(9 -H 2l6)=v/225 = I5,s = 3/(3 ■+■ 15)
right-hand side of the equation, thus x3 -h px = q ; and =4^18, and d = 'i/{3 — 15) =1/ — 12 = — ^/12 : there-
the two smaller roots have always the contrary sign to q. fore r =s ■+■ d =^/l8 — £/12 = '331313, the 1st root,
So that, in general, the sign of p determines the nature _.,,i t/ia— y/12 j/ 18+^12 . .

r.i \ . 1 5- - j .l • r and ^ ± ~ V— 3 are the two other

•A the roots, as to real and imaginary ; and the sign of >s 2 v

q determines the affection of the roots, as to positive and roots.

negative. See my Tract on Cubic Equations in the Philos. 2- Another method for the roots of the equation

Trans, for 1780, and in my new Tracts, vol. 2, tract 28. *3 +PX = 9' 's bv means of infinite series, as shown at

art. 68, and seq. of my Tract above cited ; whence it ap-
pears that the roots are exhibited in various forms of scries
as follow : viz,

2J/Jx ( 1 _■ -H£l _ 2-5-8c* &c)

v 3.64* 3.6.9.126* '

for the 1st root, and

To find the Values of the Roots of Cubic Equations. Having
reduced the equation to this form x3+px = q, its root
may be found in various ways : the first of these, is that
which is called Cardan's Rule, by whom it was first pub-
lished, but invented by Ferrcus and Tartalea. Sec Al-

3C2

CUB [ 8801 ] CUB

a , , , 2r- 2.5.8c* , . and secants, in my Course of Mathematics, vol. 3, p. 181,

_ ^A x ( , _____ _ .^ &c) 185j &_

' cs/ ~A (i j. 2 ^ 2.5.8.11c4 „ . 4. Again, the several methods of approach, or approxi-

j/V ^T 3.6.96" a. 6. 9. 12. is/* mation, for the roots of all affected equations, which have

for the two other roots : where a = §p, b = \q, and <■ = been use(| jn various ways by Stevin,Vieta, Newton, Halley,

>/{bl -h a3). And similar series, for the solution of cubic Raphson, and others.

equations, were also employed by .M. Leibnitz of Hanover, f0 these may De a_ded the method of Trial-and-error,
in the year 1677. Various other series for the same or of Double Position, one of the easiest and best of any.
purpose may also be seen in my Tract, so often before Of this method, let ttiere be taken the last example, viz,
cited. x3 ■+- 18x = 6, in which it is evident that x is very nearly
3. A third method for the roots of cubic equations, is equa( to tf Dut a little less; take it therefore x = "33 ;
by angular sections, and the table of sines. It .vas first thenx3 -t- 1 8x = 5*975937, but should be 6", and there-
hinted by Bornbelli, in his Algebra, that angles are trisected fore the error is -024063 in defect.

by the resolution of cubic equations. Afterwards Vieta Again suppose .r = -34; then Xs ■+■ lSx = 6^159304,

gave the resolution of cubics, and the higher equations, by which is -1 59304 in excess,

angular sections. Next, Albeit Girard, in his Invention -34 x -024063 ■+- -33 x -159304 _ 60752

Nouvelle en l'Algebre, shows how to resolve the inedu- therefore -024063 -t- -159104 ~* IS3307 3313,

cible case in cubics by a table of sines : and healso con- the root as before very nearly.

structs the same, or finds the roots, by the intersection of The method of icsolving all the cases of cubic equations
the hyperbola and circle. Halley and Demoivre also by the tables of sines, tangents, and secants, are thus given
gave rules and examples of the same kind of resolutions by Dr. Maskelyne, p. 57, Taylor's Logarithms,
by a table of sines. And, lastly, Mr. Anthony Thacker " The following method is adapted to a Cubic equation,
invented, and Mr. William Brown computed, a large set wanting the second term; therefore, if the equation has the
of similar tables, for resolving affected quadratic and cubic second term, it must be first taken away in the usual man-
equations, with their application to the resolution of bi- ner. There are four forms of cubic equations wanting the
quadratic ones. See Irreducible Case. Also the re- second term, whose roots, according to known rules equi-
solution of these cases, by the tables of sines, tangents, valent to Cardan's are as follow :

lSt.X>+px-q = 0 --x__ ^[|-1-^+g>}-y[-i+^+^)],

2d. ^p+^O x=LVi-'L+y(L +£)] _ y [I t^H.i)],

3d. x'-.px-9 = o x= yt|-.^(f-g)]Vi/c'--/(^-g)J,

4th. x3 -px + q =0 x = -$/[£ -V(i* - £)] - f [f -V(£ T %)■}.

The roots of the first and second forms are negatives of signs belong to the third form, and the lower signs to the

each other ; and those of the third and fourth are also ne- fourth form.

gatives of each other. The first and second forms have _, , . ,

only one root each. The third and fourth forms have also " ° 3

, . ', , ■ „, , .- , ,a' f \- Forms 1st and 2d. Log. - -t- 10 x log.- = log.

only one root each, when the quadratic surd ^/ (J — —)i* * - 3

• , ■ , 4 1 • 2? . loz. tan. (45° — ■§::)+ 20 . _

possible ; but have three roots each when that surd is lm- tan.z; and — ■ = log. tan. tu Ihen log.

possible. 4u

The roots of all the four forms may, in all cases, be * = ! log- j- + log. cot. 2a— 10; and x will be affirma-

easily computed as follows : t;ve ;n tne flrst fornl) an(- negative in the second form.

Forms Ut and 2d. Put q- x (-)* = tang. 2; and Forms 3d -„_ 4,A. t x iog. jj. + 10 - log. £ being less

$/[tan. (45° -42)]= tan. a. Then x = ±>/j * cot. than 10 (which is case first) or log. <L + l0 _ t x log-

2 a; where the upper sign belongs to the first form, and p

the lower sign to the second form. J be-ng less than ]0 (which is case 2d) = log. cos. c.

Forms 3d and Mh. Put - x (?-)$ if less than unity, Caselst. ^S- tan- f45" ~ ^) + 20 _ iogi tan Hi Thcn jog>

otherwise its reciprocal f x (-)* =s cos. z. Then, x — 1 fog. i£ -t- 10- log. sin. 2u; and x will be affirma-

Case 1st. - x f-y < unity. Put ^/[tan. (45° - i 2)] tive in the third form' and negative in the 4th form.

_,. . An „ , Case 2d. Here x has three values.

= tan. u. Then x = ± a/ — x cosec. 2a ; where the up- 4p s

' . ., 3 . c . , . . 1st. Log. + x =ilog. -7- -f- log. cos. - — 10,

per sign belongs to the third lorm, and the lower sign to ° - ° 3 j

the fourth form. 2(]_ Lo„. -r x — « W. ii + fog. cos. (fi0° - -),

Ccisc 2d. - x („-)T> unity. Then x has three values in ap z

9 3 3d. Log. Ti = i l°g- "j" + log- cos. (60° -t- -) - 10;

each form, viz, x ■= ±^ x cos. ~ = Ts/~ x cos. ,vherc the upper signs belong to the third form, and the

/<v>r. x\ -1- /"/> //?„n 2v , . lower signs to the fourth form ; that is, the first value of x

(600 _ _) _ qp ^_ x cos.(6oo + _j . whcre thc uppcr .n the thjrd form .s pos.t.vC) and .te ^^^ and thha

CUB

[ 381 ]

C U N

values negative ; and the first value of x in the fourth form
is negative, and its second and third values affirmative."

See also Irreducible Case.
.. For the construction of cubic equations, see Con-
struction.

Cubic Foot, of any thing, is so much of it as is con-
tained in a cube whose side is one foot.

Cubic Hyperbola, is a figure expressed by the equation
xy'1 = a, having two asymptotes, and consisting of two hy-
perbolas, lying in the adjoining angles of the asymptotes,
and not in the opposite angles, like the Apollonian hy-
perbola^ being otherwise called by Newton, in his Enu-
meratio Linearum Tertii Ordinis, an hyperbolismus of a
parabola ; and is the 65th species of those lines according
to him.

Cubic Numbers. See Cubes.

Cubic Parabola, a curve, as bcd, of the 2d order, hav-
ing two infinite legs cb, cd, tending contrary ways. And
if the absciss, ap or x, touch the curve in c, the relation
between the absciss and ordinate, viz, a P =.r, and pm = y,
is expressed by the equation y = ax3 -t- bx2 -+- ex -+■ d ; or
when a coincides with c, by the equation^ = ax3, which
is the simplest form of the equation of this curvd.

If the right line ap (fig. 2) cut the cubical parabola in
three points a, b, c ; and from any point p there be drawn
the right line or ordinate pm, cutting the curve in one
point m only: then will pm be always as the solid ap x
bp x cp ; which is an essential property of this curve.

And hence it is easy to construct a cubic equation, as
x3 ■+• d'x = b1, by the intersection of this curve and a
right line. See the Construction of a cubic equation by
means of the cubic parabola and a right line, by Dr. Wal-
lis, in his Algebra: As also the Construction of equations
of 6' dimensions, by means of the cubic parabola and a cir-
cle, by Dr. Halley, in a lecture formerly read at Oxford.

Tlie curve of this parabola cannot be rectified, not even
by means of the conic sections. But a circle may be found
equal to the Curve Surface, generated by the rotation
of the curve am about the tangent ap to the principal ver-
tex a. Let mn be an ordinate, and mt a tangent at the
point m; and let pm be parallel to an. Divide mn in
the point o, so that mo be to on as tm to mn. Then a
mean proportional between tm -+- on and | of an will be
the radius of a circle, whose area is equal to the superfi-
cies described by that rotation, viz, of am about ap.

M. —

a —

The Area of a Cubic Parabola is \ of its circumscribi
parallelogram.

Cubic Root, of any number, or quantity, is such a
quantity as being cubed, or twice multiplied by itself,
shall produce that which was given. So, the cubic root
of 8 is 2, because 23 or 2 x 2 x 2 is equal to 8.

The common method of extracting the cube root,
founded on the property given above, viz, (a ■+■ b)3 =o3
■-(- Zuxb -+- 3ab2 -t- b3, is found in every book of common
arithmetic, and is as old at least as Lucas de Burgo, where
it is first met with in print. Other methods for the cube
root may be seen under the article Extraction of Roots,
particularly this one, viz, the cube root of n, or \/n —

x a, very nearly, or 2a3 •+■ n: 2n ■+■ a3

l/n

the cube root of n nearly; where n is any number given
whose cube root is sought, and a3 is the nearest complete
cube to n, whether greater or less.

For example,. Suppose it were proposed to double the
cube, or, which comes to the same thing, to extract the
cube root of the number 2. Here the nearest cube is 1,
whose cube root is \ also, that is, a3 = 1, and a = J, also
n = 2; therefore
2a3 -h n 2b -t a3 a

4 = 5 : : 1 : | = l"25. = ?/2 nearly.

But, for a nearer value, assume now a =-. i., or a3 = U* ;
then is 2a3 -+- n = y° + 2 = ^, and 2n + a3 = 4 ■+■'
3W = W ! hence 378 : 381 : : £ : «££ or 1-25.992 1 =
v/2, or the cube root of 2, which is true in the last place
of decimals— And this is the simplest and easiest method
for the cube root of any number. See its investigation in
my new Tracts, vol. 1, pa. 210. ,

Every number or quantity has three cubic roots, one
that is real, and two imaginary: So, the cube root of 1 is
either I, or - 1+^~3, or - ' ~^~ ' ; and if r be the
real root of any cube r3, the two imaginary cubic roots of
it will be - --+ ^ 3r, and - ?~^ ~3.r. for any one of

these being cubed, gives the same cube r3.

Cubing of a Solid. See Cubature.

Cubo-cube, the 6th power.

Cubo-cubo-cube, the 9th power.

CUBIT, a measure equal to the distance from the elbow
of a man to the extremity of his fingers; and commonly
estimated at half a yard, or 18 inches.

CULMINATION, the passage of a star or planet over
the meridian, or that point of its orbit which it is in at
its greatest altitude. Hence a star is said to culminate,
when it passes the meridian.

To find the time of a Star's culminating. Estimate the
time nearly; and find the right ascension, both of the sun
and star, corrected for this estimated time; then the dif-
ference between these right ascensions, converted into solar
time, at the rate of 15 degrees to the hour, gives the time
of southing. See an example of this calculation every year
in White's Ephemeris, pa. 45.

CULVERIN, was the name of a piece of ordnance; but
is now disused.

CUNTiTTE. See Cuvette.

CUNEUS. See Wedge.

CUN1TIA (Maria), a lady of Silesia, who was famous
for her extensive knowledge in many branches of learning
but more particularly in mathematics and astronomy, o"i
which she wrote several ingenious treatises; particularly
one entitled Urania Propitia, printed in 1650, in Latin
and German, and dedicated to Ferdinand the 3d, emperor

C U R

[ 382 ]

CUR

of Germany. In this work are contained astronomical
tables of great ease ami accuracy, founded on Kapler's hy-
pothesis. But notwithstanding her merit shines with such
peculiar lustre as to reflect honour on her sex, history does
not inform us of the. time of her birth or death.

CURRE.NT, a stream or flux of water in any direction.
The Setting; of the current, is that point of the compass
towards which the waters run; and the Drift of a current
is the rate it runs an hour.

Currents in the sea, are either natural and general, as
arising from the diurnal rotation of the earth on its axis,
or the tides, &c ; or accidental and particular, caused by
the waters being driven against promontories, or into gulphs
and streights; whence they are forced back, and thus dis-
turb the natural Mux of the sea.

The currents are so violent under the equator, where
the motion of the earth is greatest, that they hurry vessels
very speedily from Africa to America, but absolutely pre-
vent their return the same way : so that ships are forced
to run as far as the 40th or 45th degree of north latitude,
to fall into the return of the current again, to bring them
home to Europe. It is shown by Governor Pownal, that
this current performs a continual circulation, setting out
from the. Guinea coast in Africa, for example, thence
crossing straight over the Atlantic Ocean, and so setting
into the Gulph of Mexico by the south side of it ; then
sweeping round by the bottom of the Gulph, it issues out
by the north side of it, and thence takes a direction north-
easterly along the coast of North America, till it arrive
near Newfoundland, where it is turned by a whirling mo-
tion backward across the Atlantic again, upon the coasts
of Europe, and thence southward to the coast of Africa,
from which it commenced.

In the streights of Gibraltar, the currents set in by the
south side, sweep along the coast of Africa to Egypt, by
Palestine, and return by the northern side, or European
coasts, and issue out again by the northern side of the
streights. In St. George's Channel too they usually set
eastward. The great violence and danger of the sea in the
Streights of Magellan, is attributed to two contrary cur-
rents setting in, one from the south and the other from
the north sea.

Currents are of some consideration in the art of navi-
gation, as a ship is either accelerated or retarded by them
in her course, according as the set is with or against the
ship's motion. If the ship sail along the direction of a cur-
rent, it is evident that the velocity of the current must be
added to that of the vessel as observed by the log, because
in this estimation the water is supposed to be at rest; but
if her course be directly against the current, it must for
the same reason be subtracted : and if she sail athwart it,
her motion will be compounded with that of the current;
and her velocity augmented or retarded, according to the
angle of her direction with it; that is, she will proceed in
the diagonal of the parallelogram formed according to the
two lines of direction, and will describe or pass over that
diagonal in the same time in which she would have de-
scribed either of the sides, by the separate velocities.

Tor suppose a rdc be a parallelogram,
the diagonal of which is ad. Now if the.
wind alone would drive the. ship from A
to b, in the same time as the current
alone would drive it from a to c: then,
as the wind neither helps nor hinders the
ship from coming towards the line CD, the

current will bring it there in the same, time as i ftl.e
wind did not act: and as the current neither helps nor
hinders the ship from coming towards the line bd, the wind
will bring it there in the same time as if the current did
not act: therefore the ship must, at the end of tnat time,
be found in both those lines, that is, in their meeting n :
consequently the ship must have passed from a to D in the
diagonal ad. See Comvosition of Forces.

See the Sailing in Currents largely exemplified -in Ro-
bertson's Navigation, vol. 2, book 7, sect. 8.

Currents, Under, are those currents which run at a
certain depth below the surface of the water, and in a
contrary direction. Dr. Smith, in the lull vol. of the
Philosophical Transactions, relates an experiment made
in the Baltic Sound, which was communicated to him by
an able seaman, who was present at the time it was made.
Being at that place in one of his majesty's frigates, they
went with their pinnace into the mid-stream, where they
were carried away violently by the current.' But having
sunk a large basket by means of iron shot placed in it,
and having at the same time a rope affixed to it, the mo-
tion of the boat, which was then drifting with the current,
was checked ; and by sinking the basket still lower and
lower, the motion of the boat was more and more re-
tarded, until it was finally driven in a contrary direction
to that of the upper current, which last did not appear to
extend to a greater depth than that of four or five fa-
thoms ; and as the upper current was stronger, the nearer
it was to the surface; so the under current appeared to
be more rapid, as the depth increased. Dr. Smith derives
another argument in favour of an under current at the
Straits, from the ossing between the north and south
Foreland, where it runs tide and half tide; viz, it is cilhej
ebb or flood in that part of the Downs, three hours before
it is so off at sea; a certain sign that, though the tide of
flood runs above, the tide of ebb must run below, that is,
close to the ground ; and on the contrary, when it is ebb
tide above, it will be flowing below.
Currents of Air. See Wind.

CURSOR, a small piece of brass &c, that slides ; as,
the piece in an equinoctial ring dial that slides to the day
of the mouth ; or the .little ruler or label of brass sliding
in a groove along the middle of another label, representing
the horizon in the analemma; or the point that "slides
along the beam compass ; &c.

CURTATE Distance, is the distance of a planet's place
from the sun, reduced to the ecliptic : or. the interval be-
tween the sun and that point where a perpendicular, let
fall from the planet, meets the ecliptic.

CURTATION, the interval between a planet's distance
from the sun, and the curtate distance. From the fore-
going article it is easy to find the curtate distance; whence
the manner of constructing tables of curtation is obvious ;
the quantity of inclination, reduction, and curtation of a
planet, depending on the argument of latitude. Kepler, in
his Rodolphine Tables, reduces the tables of them all into
one, under the title of Tabula Latitudinal ia\

CURTIN, Curtain, or Courtine, in Fortification,
that part of a wall or rampart that joins two bastions, or
lying between the flank of one and that of another. — The
curtain is usually bordered with a parapet 5 feet high;
behind which the soldiers stand to lire upon the covert-
way, and into the. moat.

CURVATUIU', of a Line, is its bending, or flexure ; by
which it becomes a curve, of any peculiar form and pro-

CUR

[ 383 ]

CUR

perties. Thus, the nature of the curvature of a circle is
such, as that every point in the periphery is equally distant
from a point within, called the centre; and so the curva-
ture of the same circle is every where the same; but the
curvature in all other curves is continually varying. — The
curvature of a circle is so much the more, as its radius
is less, being always reciprocally as the radius, for the
curvature of a circle is its deviation from the tangent in a
very small arc ; and this deviation is equal to the versed
sine of the arc, which is reciprocally as the radius. And,
in like manner, the curvature of other curves is measured
by the reciprocal of the radius of a circle having the same
degree of curvature as itself, at some certain point.

livery curve recedes from its tangent by its curvature,
the measure of which is the same as that of the angle of
contact formed by the curve and tangent. Now the same
tangent ab is common to an
infinite number of circles, or
other curves, all touching it
and each other in the same
point of contact c. So that
any curve dce may be touch-
ed by an infinite number of
different circles at the same
point c; and some of these circles fall wholly within it, be-
ing more curved, or having a greater curvature than that
curve; \yhile others fall without it near the point of con-
tact, or between the curve and tangent at that point, and
being thus less deflected from the tangent than the curve
is, they have a less degree of curvature there. Conse-
quently there is one, of this infinite number of circles;
which neither falls below it nor above it, but, being
equally deflected from the tangent, coincides with it most
intimately of all the circles ; and the radius of this circle
is called the Radius of Curvature of the curve ; also the
circle itself is called the Circle of Curvature, or the
Osculatory circle of that curve, because it touches it so
closely that no other circle can be drawn between it and
the curve.

As a curve is separated from its tangent by its flexure
or curvature, so it is separated from the osculatory circle
by the increase or decrease of its curvature; and as its
curvature is greater or less, according as it is more or less
deflected from the tangent, so the variation of curvature
is greater or less, according as it is more or less separated
from the circle of curvature.

It appears however, from the demonstration of geome-
tricians, that circles may touch curve lines in such a man-
ner, that there may be indefinite degrees of more or less
intimate contact between the curve and its osculatory cir-
cle ; and that a conic section may be described that shall
have the same curvature with a given line at a given point,
and the same variation of curvature, or a contact of the
same kind with the circle of curvature.

If we conceive-the tangent of any proposed curve to be
a base, and that a new line or curve be described, whose
ordinate, on the same base or absciss, is a 3d proportional
to the ordinate and base of the first ; this new curve will
determine the chord of the circle of curvature, by its in-
tersection with the ordinate at the point of contact ; and
it will also measure the variation of curvature, bytneans
of the tangent of the angle in which it cuts that circle :
the less this angle is, the closer is the contact of the curve
and circle of curvature; and of this contact there may be
indefinite degrees.

For example, let emh be any curve, to which ex is a
tangent at the point e ; then let there
be always taken mx : et : : ex : tk,
and through all the points k draw
the curve bkf; then from the point
of contact e draw eb parallel to the
ordinate tk, meeting the last curve
in b; and finally, describe a circle
erqb through the point b and touch-
ing ex in e; and it shall be the oscu-
latory circle to the given curve emu.
And the contact of em and er is al-
ways the closer, the less the angle kbq is. See Maclaurin's
Fluxions, art. 366.

Hence it follows, that the contact of the curve emh
and its osculatory circle, is closest, when the curve bk
touches the arch bq in b, the angle bex being given; and
it is farthest from this, or most open, when bk touches
the right line be in b.

Hence also there may be indefinite degrees of more and
more intimate contact between a circle and a curve. The
first degree is when the same right line touches them both
in the same point; and a contact of this sort may take
place between any circle and any arch of a curve. The
2d is when the curve emh and circle erb have the same
curvature, and the tangents of the curve bkf and circle
bqe intersect each other at b in any assignable angle.
The contact of the curve em and circle of curvature eii
at e, is of the 3d degree, or order, and their osculation is
ot the 2d, when the curve bkf touches the circle bqe at
b, but so as not to have the same curvature with it.
The contact is of the 4th degree, or order, and their oscu-
lation of the 3d, when the curve bkf has the same curva-
ture with the circle bqe at b, but so as that their contact
is only of the 2d degree. And this gradation of more and
more intimate contact, or of approximation towards coin-
cidence, may be continued indefinitely, the contact of em
and er at e being always of an order two degrees closer
than that of bk and bq at b. There is also an indefinite
variety comprehended under each order: thus, when em
and er have the same curvature, the angle formed by the
tangents of bk and bq admits of indefinite variety, and
the contact of em and er is the closer the less that angle
is. And when that angle is of the same magnitude, the
contact of em and er is the closer, the greater the circle
of curvature is. When bk and bq touch at b, they may
touch on the same or on different sides of their common
tangent; and the angle of contact kbq may admit of the
same variety with the angle of contact mer ; but as there
is seldom occasion for considering those higher degrees of
more intimate contact of the curve emh and circle of
curvature erb, Maclaurih calls the contact or osculation
of the same kind, when, the chord eb and angle bex being
given, the angle contained by the tangents of bk and bq
is of the same magnitude.

When the curvature of emh increases from e towards
H, and consequently corresponds to that of a circle gradu-
ally less and less, the arch em falls within er, the arch of
the osculatory circle, and bk is within bq. The con-
trary happens when the curvature of em decreases from e
towards n, and consequently corresponds to that of a
circle which is gradually greater and greater, the arch
em falls without er, and bk is without bq. And ac-
cording as the curvature of em varies more or less, it is
more or less unlike to the uniform curvature of a circle;

CUR

[ 384 ]

CUR

the arch of the curve emu separates more or less from the
arch of the osculatory circle erb, and the angle contained
by the tangents of bkf and bqe at B, is greater or less.
Thus the quality of curvature, as it is called by Newton,
depends on the angle contained by the tangents of bk and
bq at b ; and the measure of the inequability or variation
of curvature, is as the tangent of this angle, the radius be-
ing given, and the angle bet being a right one.

The radii of curvature of similar arcs in similar figures,
are in the same ratio as any homologous lines of these
figures; and the variation of curvature is the same. See
Maclaurin, art. 370.

When the proposed curve emh is a conic section, the
new line bkf is also a conic section ; and it is a right line
when emh is a parabola, to the axis of which the ordi-
nates-TK are parallel, bkf is also a right line when emh
is an hyperbola, to one asymptote of which the ordinate
tk is parallel.

When the ordinate eb, at the point of contact e, in-
stead of meeting the new curve bk, is an asymptote to it,
the curvature of em is less than in any circle ; and this
is the case in which it is said to be infinitely small, or
that the radius of curvature is infinitely great. And of
this kind is the curvature at the points of contrary flexure
in lines of the 3d order.

When the curve bk passes through the point of contact
e; the curvature is greater than in any circle, or the ra-
dius of curvature vanishes; and in this case the curvature
is said to be infinitely great. Of this kind is the curvature
at the cusps of the lines of the 3d order.

As to the degree of curvature in lines of theod and higher
orders, see Maclaurin, art. 379 ; also art. 380, when the
proposed curve is mechanical.

As curves which pass through the same point have the
same tangent when the first fluxions of the ordi nates are
equal, so they have the same curvature when the 2d fluxions
of the ordinate are likewise equal ; and half the chord of
the osculatory circle that is intercepted between the points
where it intersects the ordinate, is a 3d proportional to the
right lines that measure the 2d fluxion of the ordinate and
first fluxion of the curve, the base being supposed to flow
uniformly. When a ray revolving about a given point,
and terminated by the curve, becomes perpendicular to it,
the first fluxion of the radius vanishes ; and if its 2d
fluxion vanish at the same time, that point must be the
centre of curvature. The same may be said, when the
angular motion of the ray about that point is equal to the
angular motion of the tangent of the curve ; as the an-
gular motion of the radius of a circle about its centre is
always equal to the angular motion of the tangent of the
circle. Hence the various properties of the circle may
suggest several theorems for determining the centre of
curvature.

See art. 396 of the said book, also the following, con-
cerning the curvature of lines that are described by means
of right lines revolving about given poles, or of angles
that either revolve about such poles, or are carried along
fixed lines.

It is to be observed that, as when a right line intersects
an arc of a curve in two points, if by varying the position
of that line the two intersections unite in one point, it then
becomes the tangent of the arc; so when a circle touches
a curve in one point, and intersects it in another, if, by va-
rying the centre, this intersection join the point of contact,
the circle has then the closest contact with the arc, and be-

comes the circle of curvature; but it still continues to in-
tersect the curve at the same point where it touches it, that
is, where the same right line is their common tangent, un-
less another intersection join that point at the same time.
In general, the circle of curvature intersects the curve at
the point of osculation, only when the number of the suc-
cessive orders of fluxions of the radius of curvature, that
vanish when this radius comes to the point of osculation, is
an even number.

It has been supposed by some, that two points of con-
tact, or four intersections of the curve and circle of curva-
ture, must join to form an osculation. But Mr. James
Bernoulli justly insisted, that the coalition of one point of
contact and one intersection, or of three intersections, was
sufficient. In which case, and in general., when an odd
number of intersections only join each other, the point
where they coincide continues to be an intersection of the
curve and circle of curvature, as well as a point of their
mutual contact and osculation. See Maclaurin's Flux,
art. 493.

From these principles may be determined the circle ot
curvature at any point of a conic section. Thus, suppose
aemhg be any conic section, to the point e of which the
circle or radius of curvature is to be found. Let et be a
tangent at e, and draw egb and the tangent hi parallel to
the chords of the circle of curvature; then take eb to eg
as El" to hi2; or, when the section has a centre o, as in
the ellipse and hyperbola, as the square of the semi-diame-
ter oa parallel to et, is to the square of the semi-diame-
ter oa parallel to eb ; and a circle eb described on the
chord eb that touches et, will be the circle of curvature
sought.

When bet is a right angle, or eb is the diameter of the
circle of curvature, eg will be the axis of the conic section,
and eb will be the parameter of this axis; also when the
point G, where the conic section cuts eb, and the point
B, are on the same side of e, then emg will be an ellipsis,
and eg the greater or less axis, according as kg is greater
or less than eb.

The propositions relating to the curvature of the conic
sections, commonly given by authors, follow without
much difficulty from this construction.

1. When the chord of curvature, thus found, passes
through the centre of the conic section, it will then be
equal to the parameter of the diameter tjiat passes through
the point of contact.

2. The square of the semi-diameter oa, is to the rect-
angle of half the transverse and hall the conjugate axis, as
the radius of curvature cr. is to oa. And therefore the
cube of the semi-diameter 00, parallel to the tangent et,
is equal to the solid contained by the radius of curvature
ce, and the rectangle of the two semiaxes.

3. The perpendicular to either axis bisects the angle

CUR

[ 385 ]

made by the chord of curvature, and the common tangent
of the conic section and circle of curvature.

4. The chord of the osculatory circle that passes
through the focus, the diameter conjugate to that which
passes through the point of contact, and the transverse
axis of the figure, are in continued proportion.

5. When the section is an ellipse, if the circle of curva-
ture at e meet oa in d, the square of Ed will be equal to
twice the square of oa. Hence Ed : oa : :^/2 : 1. Which
gives an<easy method of determining the circle of curvature
to any point e, when the semi-diameter oa is given in
magnitude and position.

Several other properties of the circle of curvature, and
methods of determining it, when the section is given; or
vice versa, of determining the section when the circle of
curvature is given ; may be seen in Maclaurin's Flux. art.
375. See also the Appendix toMaclaurin'sAlgebra, sect. 1.

To determine the Radius and Circle of Curvature by the
Method of Fluxions. Let AEe be any curve, concave to-
ward its axis ad; draw an ordinate de to the point e
where the curvature is required to be found ; and suppose
EC perpendicular to the curve, and equal to the radius of
the circle BEe of curvature sought ; lastly, draw Ed paral-
lel to ad, and de parallel and indefinitely near to de ;
thereby making Ed the fluxion or increment of the absciss
ad, also de the fluxion of the ordinate de, and Ee that of
the curve ae. Now put x = AX>,y = de, z = ae, and
r = ce the radius of curvature; then is x.d = x, de =y,
and ec = x.
Now, by sim. tri. the 3 lines

CUR

4!/e)i (a

de

, Ee ,

y

> z >

GC

, ce;

X =

ge y;

Ed

are respectively as the three

therefore - - i

and the fluxion of this last equation is

GC.X ■+■ GC .X =z GE ,y -t- GEJ* ,

or because gc = — bg, it is

GC . X — EG . X = GE . y ■+■ GE . y.

But, since the two curves ae and be have the same curva-
ture at the point E, their abscisses and ordinates have
the same fluxions at that point, that is Ed or x is the flux-
ion both of ad and bg, and de ox y is the fluxion both of
DEand ge. In the above equation therefore substitute x
for bg, andi for se, and it becomes

GC . X — XX — GE .y -f- jy ,

or - gc . x — ge . ij = x* + y- or zz.

Now mult, the three terms of this equation respectively

by these three quantities, -^-, ■ — , , which areequal, and

it becomes yi — xj) = — or ; and hence r== - .

JX CE r yi-xf

which is the general value of the radius of curvature for
all curves whatever, in terms of the fluxions of the absciss
and ordinate.

Farther, as in any case either x or y may be supposed
to flow equably, that is, either x oxy constant quantities,
or £ ory = to nothing, by this supposition either of the
terms in the denominator of the value of r may be made
to vanish. So that when * is constant, the value of r is

— 77-) but = ^-^r when y is constant.

-Xj y x

For example, suppose it were required to find the ra-
dius or circle of curvature to any point of a parabola, its
vertex being a, and axis ad. — Now the equation of the
curve is ax = y*; hence ax = 'Jyy, and a'x = 2_r, suppo-
sing constant ; hence r or

Vol. I.

W0I~W~ 1S ~~ or~vi~' the general va-

lue of the radius of curvature for any point E, the ordinate
to which cuts off the absciss ad = x.

■ Hence, when x or the absciss is nothing, the last expres-
sion becomes barely — ya = ia = r, for the radius of cur-
vature at the vertex of the parabola; that is, the diameter
of the circle of curvature at the vertex of a parabola, is
equal to a the parameter of the axis.
See Circle of Curvature.
Variation o/Curvature. See Variation.
Double Curvature, is used for the curvature of a line,
which twists so that all the parts of it do not lie in the
same plane. Such as the rhumb-line, or loxodromic curve.
CURVE, a line whosfrseveral parts proceed bowing, or
tend different ways ; in opposition to a straight line, all
whose parts have the same course or direction.

The doctrine of curves, and of the figures and solids ge-
nerated from them, constitute what is called the higher
geometry.

In a curve, the line ad, which
bisects all the parallel lines mn,
is called a Diameter; and the
point a, where the diameter meets
the curve, is called the Vertex : if
ad bisect all the parallels at right ^
angles, it is called the Axis. The
parallel lines MS are called Ordi-
natesjor Applicates ; and their halves, pm, or pn, Semi-erdi-
nates. The portion of the diameter ap, between the vertex,or
any other fixed point, and an ordinate, is called the Ab*
sciss : also the concourse of all the diameters, if they meet
all in one point, is the Centre. This definition of the dia-
meter, as bisecting the parallel ordinates, respects only the
conic sections, or such curves as are cut only in two points
by the ordinates; but in the lines of the 3d order, which
may be cut in three points by the ordinates, then the dia-
meter is that line which cuts the ordinates so, that the
sum of the two parts that lie on one side of it, shall be
equal to the part on the other side : and so on for curves
of higher orders, the sum of the parts of the ordinates on
one side of the diameter, being always equal to the sum of
the parts on the other side of it.

Curve lines are distinguished into Algebraical or Geo-
metrical, and Transcendental or Mechanical.

Algebraical or Geometrical Curves, are those in which
the relation of the abscisses ap, to the ordinates pm, can
be expressed by a common algebraic equation.

And Transcendental or Mechanical Curves, are such as
cannot be so defined or expressed by an algebraical equa-
tion. See Transcendental Curve.

Thus, suppose, for instance, the curve be the circle *
and that the radius AC = r, the
absciss AP = r, and the ordi-
nate TM=y; then, because
the nature of the circle is such,
that the rectangle ap x pb is
always = p>5H% therefore the
equation is x . (2r — j;) = y*,
or Irx — x~ =y1, defining this
curve, which is therefore an algebraical or geometrical
line. Or, suppose cp = x; then is cm* — cpJ= pm'', that
is r- — x- = y'~ ; which is another form of the equation
of the curve.

3 D

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[ 386 ]

C U 11

The doctrine of curve lines in general, as expressed by
algebraical equations, was first introduced by Descartes,
■who called algebraical curves geometrical ones; as admit-
ting none else into the construction of problems, nor con-
sequently into geometry. But Newton, and after him
Leibnitz and Wolfius, are of another opinion ; and think,
that in the construction of a problem, one curve is not to
be preferred to another for its being defined by a more
simple equation, but for its being more easily described.

Algebraical or geometrical lines are best distinguished
into orders according to the number of dimensions of the
equation expressing the relation between its ordinates and
abscisses, or, which is the same thing, according to the
number of points in which they may be cut by a right line.
And curves of the same kind or order, are those whose equa-
tions rise to the same dimension. Hence, of the first or-
der, there is the right line only ; of the 2d order of lines,
or the first order of curves, are the circle and conic sec-
tions, being 4 species only, viz, dx — x" = y2 the circle,

~~. {dx — x2) =y2 the ellipse, -=- . (dx -+- x1) =y'2 the hy-
perbola, and dx = y'2 the parabola ; the lines of the 3d
order, or curves of the 2d order, are expressed by an equa-
tion of the 3d degree, having three roots; and so on. Of
these lines of the 3d order, Newton wrote an express trea-
tise, under the title of Enumeratio Linearum Tertii Ordi-
nis, showing their distinctive characters and properties, to
the number of 72 different species of curves : «but Mr.
Stirling afterwards added four more to that number; and
Mr. Nic. Bernoulli and Mr. Stone added two more.

Curies of the 2d and other higher feW*,Newton observes,
have parts and properties similar to those of the 1st kind :
Thus, as the conic sections have diameters and axes; the
lines bisected by these are ordinates ; and the intersection
of the curve and diameter, the vertex : so, in curves of the
2d kind, any two parallel right lines being drawn to meet
the curve in 3 points; a right line cutting these parallels
so, as that the sum of the two parts between the secant
and the curve on one side, is equal to the 3d part termina-
ted by the curve on the other side, will cut, in the same
manner, all other right lines parallel to these, and that
meet the curve in three points, that is, so as that the sum
of the two parts on one side, will still be equal to the 3d
part on the other side. These three parts therefore thus
equal, may be called Ordinates, or Applicates ; the cut-
ting line, the Diameter; and where it cuts the ordinates
at right angles, the Axis; the intersection of the diameter
and trie curve, the Vertex; and the concourse of two di-
ameters, the Centre ; also the concourse of all the diame-
ters, the Common or General Centre.

Again, as an hyperbola of the first kind has two asym-
ptotes ; that of the 2d has 3 ; that of the 3d has 4 ; &c :
and as the parts of any right line between the conic hyper-
bola and its two asymptotes, are equal on either side ; so,
in hyperbolas of the 2d kind, any right line cutting the
curve and its three asymptotes in three points; the sum
of the two parts of that right line, extended from any two
asymptotes, the same way, to two' points of the curve, will
be equal to the 3d part extended from the 3d asymptote,
the contrary way, to the 3d point of the curve.

Again, as in the conic sections that are not parabolical,
the square of an ordinate, i.e. the rectangle of the ordi-
nates drawn on the contrary sides of the diameter, is to
the rectangle of the parts of the diameter terminated at the
vertices of an ellipse or hyperbola, in the same proportion

as a given line called the latus rectum, is to that part of
the diameter which lies between the vertices, and called
the latus transversum ; so, in curves of the 2d kind, not
parabolical, the product under three ordinates, is to the
product under the three parts of the diameter cut off at
the ordinates and the three vertices of the figure, in a
given ratio : in which, if there be taken three right lines
situated at the three parts of the diameter between the ver- '
tices of the figure, each to each ; then these three right
lines may be called the Latera Recta of the figure ; and
the parts of the diameter between the vertices, the Latera
Transversa.

And, as in a conic parabola, which has only one vertex
to one and the same diameter, the rectangle under the or-
dinates is equal to the rectangle under the part of the
diameter cut off at the ordinates and vertex, and a given
right line called the latus rectum : so, in curves of the 2d
kind, which have only two vertices to the same diameter,
the product under three ordinates, is equal to the product
under two parts of the diameter cut oft' at the ordinate?
and the two vertices, and a given right line, which may
therefore be called the Latus Transversum.

Further, as in the conic sections, where two parallels,
terminated on each side by a curve, are cut by two other
parallels terminated on each side by a curve, the 1st by
the 3d, and the 2d by the 4th; the rectangle of the parts
of the 1st is to the rectangle of the parts of the 3d, as that
of the 2d is to that of the 4th ; so, when four such
right lines occur in a curve of the 2d kind, each in three
points ; the product of the parts of the 1st, will be to that
of the parts of the 3d, as that of the 2d to that of the4th.

Lastly, the legs of curves, both of the 1st, 2d, and
higher kinds, are either of the parabolic or hyperbolic
kind: an hyperbolic leg being that which approaches in-
finitely towards some asymptote; and a parabolic one, that
which has no asymptote.

These legs are best distinguished by their tangents ;
for, if the point of contact go off to an infinite distance,
the tangent of the hyperbolic leg will coincide with the
asymptote ; and that of the parabolic leg, recede infinitely,
and vanish. Therefore the asymptote of any leg is found,
by seeking the tangent of that leg to a point infinitely di-
stant ; and the direction of an infinite leg is found, by
seeking the position of a right line parallel to the tangent,
where the point of contact is infinitely remote, for this
line tends that way towards which the infinite leg is di-
rected.

Reduction of Curves of the 2d kind.
Newton reduces all curves of the 2d kind to four cases
of equations, expressing the relation between the ordinate
and absciss, viz,

in the 1st case, xy2 -+- ey = ax3 -+■ bx" -+- ex ■+■ d ;
in the 2d, - xy = ax3 -+- bx* -+■ ex + d ;

in the 3d, - yl = ax3 -i- bx" -+- ex -+- d ;

in the 4th, - y = ax3 + bx" ■+■ ex -t- d.

See Newton's Enumeratio, sect. 3; and Stirling's Lineee,
&c, p. S3.

Enumeration of the Curves of the 2d kind.
Under these four cases, the author brings a great num-
ber of different forms of curves, to which he gives different
names. An hyperbola lying wholly within the angle of
the asymptotes, like a conic hyperbola, he calls an In-
scribed Hyperbola; that which cuts the asymptotes, and
contains the parts cut off within its own periphery, a

CUR

t 387 ]

CUR

Circumscribed Hyperbola ; that which has one of its in-
finite legs inscribed and the other circumscribed, he calls
Ambigenal ; "that whose legs tend towards each other, and
are directed the same way, Converging; that where they
tend contrary ways, Diverging ; that where they are con-
vex different ways, Cross-legged ; that applied to its
asymptote with a concave vertex, and diverging legs, Con-
choidal ; that which cuts its asymptote with contrary
flexures, and is produced each way into contrary legs,
Anguineous, or Snake-like ; that which cuts its conjugate
across, Cruciform ; that which returning around cuts
itself, Nodated ; that whose parts concur in the angle of
contact, and there terminate, Cuspidated ; that whose con-
jugate is oval, and infinitely small, i.e. a point, Pointed ;
that which, from the impossibility of two roots, is without
either oval, node, cusp, or point, Pure. And in the same
manner he denominates a parabola Converging, Diverging,
Cruciform, &c. Also when the number of hyperbolic legs
exceeds that of the conic hyperbola, that is more than two,
he calls the hyperbola Redundant.

Under those 4 cases the author enumerates 72 different
curves : of these, 9 are redundant hyperbolas, without di-
ameters, having three asymptotes including a triangle ; the
first consisting of three hyperbolas, one inscribed, another
circumscribed, and the third ambigenal, with an oval ;
the 2d, nodated; the 3d, cuspidated; the 4th, pointed ;
the 5th and 6th, pure ; the 7th and 8th, cruciform ; the
9th or last, anguineal. There are 12 redundant hyper-
bolas, having only one diameter: the 1st, oval; the 2d,
nodated ; the 3d, cuspidated ; the 4th, pointed ; 5th, 6th,
7th, and 8th, pure; the 9th and 10th, cruciform; the
11th and 12th, conchoidal. And to this class Stirling
adds 2 more. There are 2 redundant hyperbolas, with
three diameters. There are 9 redundant hyperbolas, with
three asymptotes converging to a common point; the 1st
being formed of the 5th and 6th redundant parabolas,
whose asymptotes include a triangle; the 2d formed of the
7 th and 8th ; the 3d and 4th, oflhe9th ; the 5this form-
ed of the 5th and 7th of the redundant hyperbolas, with
one diameter; the 6th, of the 6th and 7th ; the 7th, of the
8th and 9th ; the 8th, of the 10th and 11th ; the 9th, of
the 12th and 13th : all which conversions are effected, by
diminishing the triangle comprehended between the asym-
ptotes, till it vanish into a point.

Six are defective parabolas, having no diameters : the
1st, oval; the 2d, nodated ; the 3d, cuspidated ; the 4th,
pointed; the 5th, pure; &c.

Seven are defective hyperbolas, having diameters : the
1st and 2d, conchoidal, with an oval ; the 3d, nodated ;
the 4th, cuspidated, which is the cissoid of the ancients ;
the 5th and 6th, pointed ; the 7th, pure.

Seven are parabolic hyperbolas, having diameters : the
1st, oval ; the 2d, nodated ; the 3d, cuspidated ; the 4th,
pointed; the 5th, pure ; the 6th, cruciform; the 7th, an-
guineous.

Four are parabolic hyperbolas ; four are hyperbolisms
ot the hyperbola: three, hyperbolas of the ellipsis : two,
hyperbolisms of the parabola.

Six are diverging parabolas ; the 1st, a trident; the 2d,
oval; the 3d, nodated; the 4th, pointed ; the 5th, cus-
pidated (which is Neil's parabola, usually called the semi-
cubical parabola); the 6th, pure.

Lastly, one, commonly called the cubical parabola.

Mr. Stirling and Mr. Stone have shown that this enu-

meration is imperfect, the former having added four new
species of curves to the number, and the latter two, or
rather these two were first noticed by Mr. Nic. Bernoulli.
Also Mr. Murdoch and Mr. Geo. Sanderson have found
some new, species ; though some persons dispute the reality
of them. See the Genesis Curvarum per Umbras, and the
Ladies' Diary 1788 and 1789, the prize question.

Organical Description of Curves. — Sir Isaac Newton
shows that curves may be generated by shadows. He says,
if upon an infinite plane, illuminated from a lucid point,
the shadows of figures be projected ; the shadows of the
conic sections will always be conic sections ; those of the
curves of the 2d kind, will always be curves of the 2d
kind ; those of the curves of the 3d kind, will always be
curves of the 3d kind ; and so on ad infinitum.

And, like, as the projected shadow of a circle generates
all the conic sections, so the 5 diverging parabolas, by
their shadows, will generate and exhibit all the rest of the
curves of the 2d kind : and thus some of the most simple
curves of the other kinds may be found, which will form,
by their shadows upon a plane, projected from a lucid
point, all the other curves of the same kind. And in the
French Memoirs may be seen a demonstration of this pro-
jection, with a specimen of a few of the curves of the 2d
order, which may be generated by a plane cutting a solid
formed from the motion of an infinite right line along a
diverging parabola, having an oval, always passing through
a given or fixed point above the plane of that parabola.
The above method of Newton has also been pursued and
illustrated with great elegance by Mr. Murdoch, in his
treatise entitled Newtoni Genesis Curvarum per Umbras,
seu Perspectiva? Universalis Elementa.

Mr. Maclaurin, in his Geometria Organica, shows how
to describe several of the species of curves of the 2d order,
especially those having a double point, by the motion of
right lines and angles: but agood commodious description
by a continued motion of those curves which have no
double point, is ranked by Newton among the most diffi-
cult problems. Newton gives also other methods of de-
scription, by lines or angles revolving above given poles;
and Mr. Brackenridge has given a general method of de-
scribing curves, by the intersection of right lines moving
about points in a given plane. See Philos. Trans. No. 437,
or my Abr. vol. 8, pa. 5; and some particular cases are
demonstrated in his Exerc. Geometrica de Curvarum De-
scriptione.

Curves above the Id Order. The number of species in
the higher orders of curves increase amazingly, those of the
3d order only it is thought amounting to some thousands,
all comprehended under the following ten particular ecpja-
tions,

viz, 1 . y •+- fx-y" -t- gxy3 h- hx2y -+- iy2 ■+• hxy -+• ly,
or 2. y' -n/xy' -+- gxy + hxy2 -+- ixy -t- ky,
or 3. x2y2 -t-fy3 ■+• gxy •+■ hy3 -+- ky,
or 4. x2y2 -t-fy3 -+- gy2 ■+■ hxy + iy,
or 5. y3 +fxy" ■*■ gx-y -+- hy,
or 6. y5 -hfxy2 + gxy -+■ hy

= axA -+- bx3 -+- ex2 ■+• dx h- e\
or 7. / ■+• ex3y + fxy3 -+- gxy" ■+■ hy2 ■+■ ixy ■+- ky,
or 8. x'y -+- exy3 + fx2y -+- gy2 -*- hxy ■+■ iy,
or 9. x'y -hey3 + fxy2 -t- gxy -h hy,
or 10. x3y + <y^ +fy2 -f. gxy ■+■ hy,
= ax* -t- bx2 + ex ■+■ d.

Those who wish to see how far this doctrine has been
3D2

CUR

[ 388 ]

CUR

advanced, with regard to curves of the higher orders, as
well as those of the 1st and 2d orders, may consult Mr.
Maclaurin's Geometria Organica, and Brackenridge's
Exerc. Geom.

All geometrical lines of the odd orders, viz, the 3d, 5th,
7th, &c, have at least one leg running on infinitely ; be-
cause all equations of the odd dimensions have at least one
real root. But vast numbers of the lines of the even orders
are only ovals; among which there are several having very
pretty figures, some being like single hearts, some double
ones, some resembling fiddles, and others again single
knots, double knots, &c.

Two geometrical lines of any order will cut each other
in as many points as are denoted by the product of the
two numbers expressing those orders.

The theory of curves forms' a considerable branch of
the mathematical sciences. Those who are curious of ad-
vancing beyond the knowledge of the circle and the conic
sections, and to consider geometrical curves of a higher
nature, and in a general view, will do well to study Cra-
mer's Introduction a l'Analyse des Lignes Courbes Alge-
braiques, which the learned and ingenious author com-
posed for the use of beginners. There is an excellent
posthumous piece too of Maclaurin's, printed as an Ap-
pendix to his Algebra, and entitled De Linearum Geome-
tricarum Proprietatibus Generalibus. The same author,
at a very early age, gave a remarkable specimen of his ge-
nius and knowledge in his Geometria Organica; and he
carried these speculations farther afterwards, as may be
seen in the theorems he has given in the Philos. Trans. See
Abr. vol. 6. Other writings on this subject, besides the
Treatises on the Conic Sections, are Archimedes de Spi-
ralibus; Descartes Geometria; Dr. Barrow's Lectiones
Geometrical ; Newton's Enumeratio Linearum Tertii Or-
dinis; Stirling's Illustratio Tractatus Newtoni de Lineis
Tertii Ordinis; Maclaurin's Geometria Organica; Brack-
enridge's Descriptio Linearum Curvarum ; M. de Gua's
Usages de l'Analyse de Descartes ; besides many other
Tracts on Curves in the Memoirs of several Academies &c.
Use of Curves in the Construction of Equations. One
great use of curves in Geometry is, by means of their in-
tersections, to give the solution of problems. See Con-
struction.

Suppose, ex. gr. it were required to construct the fol-
lowing equation of 9 dimensions,

a-8 ■+- bx7 -+- cxs -+- dxs -+- ex* -+- (tn +f) x3 -+- gx2 -+- hx ■+■ k = 0 :
assume the equation to a cubic parabola x3 — y; then,
by writing y for x3, the given equation will become
y3 ■+- bxy^ ■+- cy% ■+■ dx^y ■+■ exy -t- my -t-fx3 -+- gx7' -+- hx ■+■ k = 0 ;
an equation to another curve of the 2d kind, where m or
/may be assumed = 0 or any thing else: and by the de-
scriptions and intersections of these curves will be given
the roots of the equation to be constructed. It is suffi-
cient to describe the cubic parabola once. When the
equation to be constructed, by omitting the two last terms
hx aiid k, is reduced to 7 dimensions; the other curve, by
expunging m, will have the double point in the beginning
of the absciss, and may be easily described as above : If
it be reduced to 6 dimensions, by omitting the last three
terms, gx" -t- hx + k; the other Curve, by expunging /,
will become a conic section. And if, by omitting the last
three terms, the equation be reduced to three dimensions,
we shall fall upon Wallis's construction by the cubic pa-
rabola and right line

Rectification, Inflection, Quadrature, Sfc. of Curves.
See the respective terms.

Curve of a Double Curvature, is a curve of which all its
parts are not in the same plane.

A curve which can be traced upon a curve surface, and
not upon a plane, is called a curve of double curvature.
These kinds of curves may be considered as generated by
the motion of a point upon a curve surface, provided the
duration of its motion be not constantly in the same plane:
for, except in the latter case, it is evident that the line so
described is curved in two ways ; that is, it partakes of
the curvature of the surface, and of the continual and
successive deflections of the describing point. Or, we may
consider those curves as generated by the motion of a
variable ordinate along any curve line regarded as an
absciss.

Curves of double curvature may also be formed by two
curve surfaces penetrating each other; as is the case when
a sphere is penetrated by a right cylinder, if the axis of
the latter do not pass through the centre of the former.

In the rectification of those curves, they may be consi-
dered as projected, first, upon some plane, whose position
is known ; by which means a plane curve will be obtained,
which being considered as an abscissr the rectification of
the original curve may be deduced from the usual me-
thods employed for the rectification of plane curves.

Clairaut has published an ingenious treatise on curves
of a double curvature. See his Recherches sur les Courbes
a, Double Courbure. Euler has also treated on this sub-
ject-in the Appendix to his Analysis Infinitorum, vol.2,
pa. 323. As likewise Lacroix in his Calcul Diffcrentiel
&c, and other modern writers.

Family of Curves, is an assemblage of several curves
of different kinds, all defined by the same equation of an
indeterminate degree; but differently, according to the di-
versity of their kind. Fo' example, Suppose an equation
of an indeterminate degree, a"1 ~ x =ym: if m = 2, then
will ax =yx; if m = 3, then will a'x = y3 ; if m = 4, then
is a3x=y\ &c: all which curves are said to be of the
same family or tribe.

The equations by which the families of curves are. de-
fined, are not to be confounded with the transcendental
ones: for though with regard to the whole family, they be
of an indeterminate degree, yet with respect to each several
curve of the family, they are determinate; whereas tran-
scendental equations are of an indefinite degree with re-
spect to the same curve.

All algebraical curves therefore compose a certain fa-
mily, consisting of innumerable others, each of which com-
prehends infinite kinds. For the equations by which
curves are defined involve only products, either of powers
of the abscisses and ordinates by constant coefficients; or
of powers of the abscisses by powers of the ordinates; or
of constant, pure, and simple quantities, by one another.
Moreover, every equation to a curve may have 0 for one
member or side of it; for example ax=yz, by transposi-
tion becomes ax — y* = 0. Theiefoic the equation for all.
algebraic curves will be

. :

•+■ <W J

CatacaUStic, and Diacaustic Cu rves, Sec Catacau-
stic, and Diacaustic.

ay

■ bxy"

■fym'

nx y
kxy

CYC

[ asy ]

CYC

Exponential Curve, is that which is defined by an ex-
ponential equation, as ax*- = y, &c.

Curves by the Light, or Courbes d la Lumiere, a name
given to certain curves by M. Kurdwariowski, a Polish
gentleman. He observed that any line, straight or curved,
exposed to the action of a luminous point, received the
light differently in its different parts, according to their
distance from the light. These different effects of the light
upon each point of the line, may be represented by the or-
dinates of some curve, which will vary precisely with these
effects. Priestley's Hist, of Vision, pa. 752.

Logarithmic Curve. See Logarithmic Curve.

Curve Reflectoire, so called because it is the appear-
ance of the plane bottom of a bason covered with water,
to an eye perpendicularly over it. In this position, the
bottom of the bason will appear to rise upwards, from the
centre outwards; but the curvature will be less and less,
and at last th« surface, of the water will be an asymptote
to it, M. Mai ran, who first conceived this idea from the
phenomena of light, found also several kinds of these
curves; and he gives a geometrical deduction of their pro-
perties, showing their analogy to caustics by refraction.
Mem. Ac. 1740; Priestley's Hist, of Vision, pa. 752.

Radical Curves, a name given by some authors to
curves of the spiral kind, whose ordinates, if they may be
so called, do all terminate in the centre of the including
circle, and appear like so many radii of that circle: whence
the name.

Regular Curves, are such as have their curvature
turning regularly and continually the same way ; in op-
position to such as bend contrary ways, by having points
of contrary flexure, which arc called irregular curves.

-Characteristic Triangle of a Curve, is the differential or
elementary right-angled triangle whose three sides are, the
fluxions of the absciss, ordinate, and curve; the fluxion
of the curve being the hypnthenuse. n £_

So, if p<] be parallel to, and indefinitely
near to the ordinate pq, and q;- pa-
rallel to the absciss ap; then Qr is the
fluxion of the absciss ap, and qr the
fluxion of the ordinate pq, and q^ the AT
fluxion of the curve aq; hence the elementary triangle
Qqr is the characteristic triangle of the curve aq; and the
three sides are *, y, «; in which x- +y* = z'.

CURVILINEAR Angle, Figure, Superficies, frc, are
such as are formed or bounded by curves; in opposition
to rectilinear ones, which are formed by straight lines or
planes.

CUSP, in Astronomy, is used to express the points or
horns of the moon, or other luminary.

Cusp, in. the Higher Geometry, is used for the point or
corner formed by two parts of a curve meeting and termi-
nating there. See Curve.

Cuspidated Hyperbola, he. See Curve.
CUT-Bastion. See Bastion.

CUVETTE, or Cunette, in Fortification, is a kind of
ditch within a ditch, being a pretty deep trench, about
four fathoms broad, sunk and running along the middle
of the great dry ditch, to hold water; serving both to keep
off the enemy, and prevent him from mining.

CYCLE, a certain period or scries of numbers proceed-
ing orderly from first to last, then returning again to the

dies to one another. The apparent revolution of the sua
about the earth has been arbitrarily divided into 24 hours,
which is the basis or foundation of all our mensuration of
time, whether days, years, &c. But neither the annual
motion of the sun, nor that of the other heavenly bodies,
can be measured exactly, and without any remainder, by
hours, or their multiples. That of the sun, for example,
is 365 days 5 hours 49 minutes nearly; that of the moon,
29 days J 2 hours 44 minutes nearly.

Hence, in order to find equivalent expressions for these
fractions in whole numbers, and yet in numbers which
only express days and years, cycles have been invented ;
which, comprehending several revolutionsof the-samebody,.
replace it, after a certain number of years, in the same
points of the heaven whence it first departed ; or, which
is the same thing, in the same place of the civil calendar.

There are various cycles; as, the cycle of indiction, the
cycle of the moon, the cycle of the sun, &c.

Cycle of Indiction, is a series of 15 years, returning
constantly the same as the other cycles ; and it commenced
from the third year before Christ; whence it happens that
if 3 be added to any given year of Christ, and the sum be
divided by 15, what remains is the year of the indiction.
See Indiction.

Cycle of the Moon, or the Lunar Cycle, is a period of
19 years; in which time the new and full moons return
to the same day of the Julian year. See Calippic.

This cycle is also called the Metonic period or cycle,
from its inventor Meton, the Athenian ; and also the
Golden Number, from its excellent use in the calendar:
though, properly speaking, the golden number is rather
the particular number which shows the year of the lunar
cycle, which any given year is in. This cycle of the moon
only holds true for 310-j%- years : for, though the new
moons do return to the same day after 19 years; yet not
to the same time of the day, but near an hour and a half
sooner ; an error which in 310T7^ years amounts to an en-
tire day. Yet those employed in reforming the calendar
made their calculations on a supposition that the lunations
return precisely from 19)ears to 19ycars, forever.

The use of this cycle, in the. ancient calendar, is to show
the new moon of each year, and the time of Easter. In.
the new one, it only serves to find the Epacts ; which show,,
in cither calendar, that the new moon falls 1 1 days too late.
As the Orientals began the use of this cycle at the time
of the Council of Nice in 325, they assumed, that the first
year of the cycle the paschal new moon fell on the 13th of
March; on which account the lunar cycle 3 fell on the
1st of January in the third year.

The Occidentals, on the contrary, placed the number 1
to the 1st of January, which occasioned a considerable dif-
ference in the. time of Easter. Hence, Dionysius Exiguus,.
on framing a new calendar, persuaded the Christians of the
west to salve the difference, and come into the practice of
the church of Alexandria.

To find the Year of the Lunar Cycle, is to find the golden
number. See GoLDEN-iV«/;jitT.

Cycle of the Sun, or Solar Cycle, is a period or revolu-
tion of 28 years ; beginning with I , and ending with 28 ;,
which elapsed, the Dominical or Sunday-letters, and those
that express the other feasts, he, return into their former
place, and proceed in the same order as before. The days
of the month return again to the same days of the week.;.

first, and so circulating perpetually.

Cycles have chiefly arisen from the incommensurability the sun's place to the same signs and degrees of the cclip-
of the periods of revolutions of the earth and celestial bo- tic on the same months and days, so as not to differ one

c y e

I 390 ]

CYC

degree in a hundred years ; and the leap-years begin the cumscribing cylinder, as 5 to S : about the tangent parallel
same course with respect to the days of the week on which to the base, as Z^to S : about the tangent parallel to the
the days of the month fall.

This is called the cycle of the sun, or the solar cycle,
not from any regard to the sun's course, which has no con-
cern in it ; but from Sunday, anciently called dies solis,
the sun's day; as the dominical or Sunday letter is chiefly
sought for from this revolution.

The reformation of the calendar under Pope Gregory the
13th, occasioned a considerable alteration of this cycle :

axis, as 3 to 4: &c.

Honoratus Fabri, in his Synopsis Geom. has a short
treatise on the cycloid, containing demonstrations of the
above, and many other theorems concerning the centres of
gravity of the cycloidal space, &c; which he sayshefound
out before the year J 658.

From the preface to Dr. Wallis's treatise on the Cycloid
e learn, that, in the year 1658, M. Pascal publicly pro-

In the Gregorian calendar, the solar cycle is not constant posed at Paris, under the name of Dettonville, the two

the 9th year before Christ. And therefore, cycloid, cut off by a right line parallel to the base ; also

To find the Cycle of the Sun for my given year: add 9 to the content of the solid generated by the rotation ot the
5 number riven, and divide the sum by 28 ; the remain- same about the axis, and about the base of that segment.

the

der will be the number of the cycle, and the-quotient the
number of revolutions since Christ. If there be no re-
mainder, it will be the 28th or last year of the cycle.

This challenge set the Doctor upon writing that treatise on
the cycloid, which is a much better and more complete
piece than had been given before upon this curve. He
there gives the curve surfaces of the solids generated by the
rotation of the cycloidal space about its axis, and about
its base, with determinations of the centres of gravity, &c.
He here likewise asserts that Sir Christopher Wren, in
165S, was the first who found a right line equal to the
curve of the cycloid ; and Mr. Huygens, in his Horolog.
Oscillat. says that he himself was the first inventor of the
Great Pascal Cycle, is another name for the Victorian segment of a cycloidal space, cut off by a right line parallel
or Dionysian Period. Which see. to the base at the distance of J the axis of the curve from

CYCLOID, or Trochoid, a mechanical or transcen- the centre, being equal to a rectilinear space, viz, to a re-
dental curve, which is thus generated : Suppose a wheel, gular hexagon inscribed in the generating circle; the de-
or a circle, ae, to roll along a straight line ab, beginning monstration of which may be seen in Wallis's treatise,
at the point a, and ending at B, where it has completed Several other authors have treated of the cycloid : as

just one revolution, thus measuring out a right line ab Pascal, in his treatise, under the name of Dettonville J
exactly equal to the circumference of the generating circle Schooten in his Commentary on Descartes's Geometry,

CYCLE of the Sun, and Sunday Letters, from the Year

1800

to the Year 1 900.

l ED 1 5 GF 9 BA

13 DC

17 FE

21 AG

125 CB

2 C 6 E

10 G

14 B

18 D

22 F

26 A

3 B 7 D

11 F

15 A

19 C

23 E

27 G

4 A 1 s C

12 E

16 G

20 B

24 D

J 2S I

ae, while a nail or point a in the circumference of the
wheel, or circle, traces out or describes acurvilineal path
adb ; then this curve adb is the cycloid, or trochoid.
.••■I

A C B

Schooten, in his Commentary on Descartes, says that
Descartes first conceived the notion of this elegant curve,
and after him it was first published by Father Mersenne,
in the year 1(515. But Torricelli, in the Appendix de
Dimensione Cycloidis, at the end of his treatise De Dimcn-
sione Parabola;, published 1644, says that this curve was
considered and named a cycloid, by his predecessors, and
particularly by Galileo about 45 years before, i.e. about
1599. And Dr. Wallis shows that it is of a much older
standing, having been known to Bovilli about the year
1500, and even considered by Cardinal Cusanus much
earlier, viz, before the year 1451. Pliilos. Trans. Abr. vol.
4, pa. 169. It would seem however that Torricclli's was
the first regular treatise on the Cycloid; though several
particular properties of it might bt' known prior to his
work. He first showed, that the cycloidal space is equal
to triple the generating circle, (though Pascal contends
that Roberviil showed this): also that the solid generated
by the rotation of that space about its base, is to the cir-

nearthe end of the 2d book; M. Reinau, in his Analyse
Demontree, torn. 2, pa. 595 : also Newton, Leibnitz, La-
loubere, Roberval, Descartes, Wren, Fabri, the Ber-
noulli's, Lahire, Cotes, Emerson, &c, &c.

Properties of the Cycloid. — The circle ae, by whose
revolution the cycloid is traced out, is called the Genera-
ting Circle; the line ab, which is equal to the circum-
ference of the circle, is the Base of the cycloid; and the
perpendicular DC on the middle of the base, is its Axis.
The properties of the cycloid are among the most beauti-
ful and useful of all curve lines : some of the most remark-
able of which are as follow :

1. The circ. arc dg = gh parallel to ab.

2. The scmicircumf. dgc = the semibase ac

3. The arc dh = double the chord dg.

4. The arc da = double the diameter DC.

5. The tang. Hi is parallel to the chord dg.

6. The space adba = triple the circle ae.

7. The space adgca = the same circle ae, &c.

8. A body falls through any arc kl of a cycloid re-
versed, in the same
time, whether that
arc be great or
small ; that is, from
any point l, to the
lowest point K,
which is the vertex
reversed; and that
time is to the time
of falling pcrpendi-

CYC [

CUlatly through the axis mk, as the semicircumference of
a circle is to its diameter, or as 3'] 41 6" to 2. And hence
st follows that, if a pendulum be made to vibrate in the
arc lkn of a cycloid, all the vibrations will be performed
in the same time. ■■

p. The evolute of a cycloid, is another equal cycloid.
So that if two equal semicycloids or, oq, be joined at o,
so that om be equal to mk the diameter of the generating
circle, and the string of a pendulum hung up at. o, having
its length equal to ok or to the curve op ; then, by plying
the string round the curve op, to which it is equal, and the
ball being let go, it will describe and vibrate in the other
cycloid pkq. We must not however conclude from this,
that a body so vibrating will really perform all its vibra-
tions in equal times, for this is only true in theory, that is,
by supposing the string by which the vibrating body is
suspended to be void of gravity. For in estimating the
centre of oscillation of any body, regard must be had to
this circumstance, and in the final expression the distance
from the point of suspension is found to enter, together with
the distance of the centre of oscillation of the string, and
consequently this last centre of oscillation is constantly
varying; which will prevent the hpdy from performing all
its vibrations in the same time : but the greater the weight
of the body is, when compared with that of the string,
the nearer will these times approach to an equality.

10. The cycloid is the curve of swiftest descent: or a
heavy body will fall from one given point to another,
through the arc of a cycloid passing by those two points, in
a less time, than if it passed along any other curve, or even
along a right line joining those two points.

John Bernoulli was the first who proposed to mathema-
ticians this problem under the title of the Braclij/stodironon,
that is, to find the curve of swiftest descent, or that curve,
along which a heavy body would fall from one point to
another, not in the same \ertical line, in the least time.
And solutions were given to it by his brother James Ber-
noulli, Leibnitz, Hopital, and Newton, bee Gregory's
Mechanics, art. 277, vol. 1 : also the Works of James and
John Bernoulli, for many other curious properties con-
cerning the descents in cycloids, &c.

Cycloids arc also either Curtate or Prolate.

391 ] C Y L

These two curves were both noticed by Torwcelli and
Schooten, and more fully treated of by Wallis, in his Trea-
tise on the Cycloid, printed at Oxford in lGo9; where he
shows that these have properties similar to the first or pri-
mary cycloid ; only the last of them is a curve having a
point of inflection, and the other crossing itself, and form-
ing a node.

By continuing the motion of the wheel, or circle, so as
to describe a right line equal to the generating circum-
ference several times repeated, there will be produced as
many repetitions of the cycloids, which so united together
will appear as in these figures following:
Curtate Cycloid.

R '1'

Cycloid, Curtate, or contracted, is the path described by
some point without the circle, while the circumference
rolls along a straight line; and a

A 11

Cycloid, Prolate, or Inflected, is in like manner the
path of some point taken within the generating circle.

Thus, if, while the circle rolls along the line All, the
point r be taken without the circle, it will describe or
trace out the curtate or contracted cycloid iist; but the
point being taken within the circle, it will describe the
prolate or inflected cycloid nvw.

CYCLOPEDIA. See Encyclopedia.

CYGNUS, the Swan, a constellation of the northern
hemisphere, being one of the 48 old ones, and fabled by
the Greeks to be the swan, under the form of which Jupiter
deceived Leda or Nemesis, from which embrace sprung
the beauteous Helen. The stars in the constellation
Cygnus, in Ptolemy's catalogue arc lg, in Tycho's IS, in
Hevelius's 47, and in the Britannic catalogue 81.

CYLINDER, a solid having two equal circular ends,
and every plane section parallel to the ends a circle equal
to them also. — The cylinder may be conceived to be thus
generated :

Suppose two parallel
circles AB,CD,and a right
line carried continually
round them, always pa-
rallel to itself; this line
will describe the curve
surface of a cylinder,ABDC,
of which the two parallel
circles ab and cd form the two ends. When the line or
side is perpendicular to the ends, the cylinder is a right or
perpendicular one ; otherwise it is oblique.

Or the right cylinder may be conceived to be generated
by the rotation of a rectangle about one of its sides. The
Axis of the cylinder is the line connecting the centres of
its two parallel circular ends; and is equal to the altitude
of the cylinder when this is a right one, but exceeds the
altitude in the oblique cylinder, in the proportion of radius
to the sine of the angle of its inclination to the base.

The convex surface of ,a cylinder is equal to the pro-
duct of the axis and the circumference of its base. — The
solidity of a cylinder is equal t6 the area of its base mul-
tiplied by its perpendicular altitude.

Cylinders of equal bases and altitudes, are equal.

Cylinders are to each other, as the product of their bases
and altitudes. And equal cylinders have their bases rer
ciprocally as their altitudes.

A cylinder is to its inscribed sphere, or spheroid, as 3
to 2 : and to its inscribed cone as 3 to 1,

C Y L

[ 392 ]

C Y S

The oblique plane sections of a cylinder, are ellipses ;
but all the sections parallel to the ends, are circles.

For the surfaces and solidities of the ungulas or oblique
slices of a cylinder, see ray Mensuration, p. l6l, 4th edit.

CYLINDRICAL, pertaining to a cylinder.

CYL1NDROID, a solid resembling the figure of a cy-
linder ; but differing from it as having ellipses for its ends
or bases, instead of circles in the cylinder.

In the cylindroid, the solidity and curve superficies are

found the same way as those of the cylinder ; viz, by mul-
tiplying the circumference of the base by the length or
axis, for the surface ; and the area of the base by the al-
titude, for the solidity.

CYM AT1UM, Cimatium, or Cima, in Architecture, a
member, or moulding of the cornice, whose profile is
waved ; i. e. concave at top, and convex at bottom.

CYPHER, or nought. See Cipher.

CYSSOID. See Cissoid.

D.

DAL

TAADO, that part in the middle of the pedestal of a
■*-^ column &c, between its base and cornice.

DAILY, in Astronomy. See Diurnal.

DALEMBERT, an eminent French mathematician and
philosopher, and one of the brightest ornaments of the
18 th century. He was perpetual secretary to the French
Academy of Sciences, and a member of most of the phi-
losophical academies and societies of Europe.

Dalembcrt was born at Paris, the l6th of November
1717. He derived the name of John le Rond from that
of the church near which, after his birth, he was exposed
as a foundling. But his father, informed of this circum-
stance, listening to the voice of nature and duty, took
measures for the proper education of his child, and for his
future subsistence in a state of ease and independence.
His mother was a lady of rank, the celebrated Made-
moiselle Tencin, sister to Cardinal Tencin, archbishop of
Lyons.

He received his first education among the Jansenists, in
the college of the four nations, where he gave early signs
of genius and capacity. On quitting the college, finding
himself alone, and unconnected in the world, he sought
an asylum in the house of his nurse. He hoped that his
fortune, though not ample, would enlarge the subsistence,
and better the condition of her family, which was the only
one that he could consider as his own. It was here there-
fore that he fixed his residence, resolving to apply himself
entirely to the study of geometry. — And here he lived,
during the space of 40 years, with the greatest simplicity,
discovering the augmentation of his means only by in-
creasing displays of his beneficence, concealing his grow-
ing reputation and celebrity from these honest people,
and making their plain and uncouth manners the subject
of good-natured pleasantry and philosophical observation.
His good nurse perceived his ardent activity ; heard him
mentioned as the writer of many books; but never took
it into her head that he was a great man, and rather be-
held him with a kind of compassion. " You will never,"
said she to him one day, " be any thing but a philosopher —
and what is a philosopher? — a fool, who toils and plagues
himself all his life, that people may talk of him when he
is dead."

In the year 1741 he was admitted a member of the
Academy of Sciences ; for which distinguished literary
promotion, at so early an age (24), he had prepared the
way by correcting the errors of a celebrated work (The
Analyse Demontrec of Reyncau), which was esteemed
classical in France in the line, of analytics. He after-

D A L

wards set himself to examine, with close attention and as-
siduity, what must be the motion and path of a bodv,
which passes from one fluid into another denser fluid, in a
direction oblique to the surface between the two fluids.
Every one knows the phenomenon which happens in this
case, and amuses children, under the denomination of
ducks and drakes ; but it was Dalembert who first ex-
plained it in a satisfactory and philosophical manner.

Two years after his election to a place in the Academy,
he published his Treatise on Dynamics. The new prin-
ciple developed in this treatise, consisted in establishing
an equality, at each instant, between the changes that the
motion of a body has undergone, and the forces or powers
which have been employed to produce them : or, to ex-
press the same thing otherwise, in separating into two
parts the action of the moving powers, and considering the
one as producing alone the motion of the body, in the se-
cond instant, and the other as employed to destroy that
which it had in the first. So early as the year 1744, Da-
lembert had applied this principle to the theory of the
equilibrium, and the motion of fluids : and all the pro-
blems before resolved in physics, became in some measure
its corollaries. The discovery of this new principle was
followed by that of anew calculus, the first essays of which
were published in a Discourse on the General Theory of
the Winds, to which the prize-medal was adjudged by the
Academy of Berlin in the year 1746, which proved a
new and brilliant addition to the fame of Dalembert. This
new calculus of Partial Differences he applied, the year
following, to the problem of vibrating chords, the resolu-
tion of which, as well as the theory of the oscillations of
the air and the propagation of sound, had been but im-
perfectly given by the mathematicians who preceded him;
and these were his masters or his rivals.

In the year 1749 he furnished a method of applying
his principle to the motion of any body of a given figure.
He also resolved the problem of the precession of the equi-
noxes ; determining its quantity, and explaining the phe-
nomenon of the nutation of the terrestrial axis discovered
by Dr. Bradley.

In 1752, Dalembert published a treatise on the Resist-
ance of Fluids, to which he gave the modest title of ai.
Essay; though it contains a multitude of original ideas
and new observations. About the same time he published,
in the Memoirs of the Academy of Berlin, Researches con-
cerning the Integral Calculus, which is greatly indebted
to him for the rapid progress it has since made.

Our author's eulogist ascribes to envy, detraction, &c,

DAL

[ 393 ]

DAL

all the opposition and censure that Dalembert met with
onuccount of the celebrated Encyclopedic, or Dictionary
of Arts and Sciences, in conjunction with Diderot. None
surely will refuse the well-deserved tribute of applause to
the eminent displays of genius, judgment, and true literary
taste, with which Dalembert has enriched that great work.
Among others, the Preliminar}' Discourse he has prefixed
to it, concerning the rise, progress, connexions, and affi-
nities of all the branches of human knowledge, is per-
haps one of the most capital productions the philosophy
of the age can boast of.

Some time after this, Dalembert published his Philoso-
phical, Historical, and Philological Miscellanies. These
were followed by the Memoirs of Christina queen of Swe-
den ; in which he showed that he was acquainted with the
natural rights of mankind, and was bold enough to assert
them. His Essay on the Intercourse of Men of Letters
with Persons high in Rank and Office, wounded the former
to the quick, as it exposed to the eyes of the public the
ignominy of those servile chains, which they feared to
shake off, or were proud to wear. Dalembert gave ele-
gant specimens of his literary abilities in his translations
of some select pieces of Tacitus. But these occupations
did not divert him from his mathematical studies : for
about the same time he enriched the Encyclopedie with a
multitude of excellent articles in that line, and composed
his Researches on several Important Points of the System
of the World, in which he carried to a higher degree of
perfection the solution of the problem concerning the per-
turbations of the planets, that had several years before
been presented to the Academy.

In 1759 he published his Elements of Philosophy: a
work much extolled, as remarkable for its precision and
perspicuity. The resentment that was kindled (and the
disputes that followed it) by the article Geneva, inserted
in the Encyclopedie, are well known. Dalembert did not
leave this field of controversy with flying colours. It was
on this occasion that Frederick king of Prussia offered him
an honourable asylum at his court, and the office of presi-
dent of his Academy: but the king was not offended at
Dalembert's refusal of these distinctions, though he culti-
vated an intimate friendship with him during the rest of
his life. He had refused, some time before this, a propo-
sal made by the empress of Russia to entrust him with the
education of the Grand Duke; — a proposal accompanied
with all the flattering offers that could tempt a man, am-
bitious of titles, or desirous of making an ample for-
tune : but the objects of his ambition were tranquillity and
study.

In the year 1765, he published his Dissertation on the
Destruction of the Jesuits. This piece drew upon him a
swarm of adversaries, who only confirmed the merit of his
work by their manner of attacking it. Besides the works
already mentioned, he published 9 volumes of memoirs
and treatises, under the title of Opuscules ; in which he
has resolved a multitude of problems relating to astro-
nomy, -mathematics, and natural philosophy ; of which
his panegyrist, Condorcet, gives a particular account,
more especially of those which exhibit new subjects, or
new methods of investigation. He published also Ele-
ments of Music ; and rendered, at length, the system of
Rameau intelligible: but he did not think the mathemati-
cal theory of the sonorous body sufficient to account for
the rules of Chat art.

In the year 1772 he was chosen secretary to the French

VpL. I.

Academy of Sciences. He formed, soon after this prefer-
ment, the desjgn of writing the lives of all the deceased
academicians, from 1700 to 1772; and in the space of
three years he executed this design, by.composing 70 eu-
logies. Dalembert died on the 29th of October 17S3,
being nearly 66 years of age.

As it may be curious and useful to have in one view an
entire list of Dalembert's writings, I shall here insert a
catalogue of them, from Rozier'sTNouvelle Table des Ar-
ticles contenus dans les volumes de l'Academie Royale
des Sciences de Paris, &c, as follows :

Traite de Dynamique, in 4fo, Paris, 1743. The 2d ed.
in 1758 — Traite de l'Equilibre et du Mouvement des
Fluides. Paris, 1744; and the 2d edit, in 1770.

Reflexions sur la Cause Generale des Vents; which
gained the prize at Berlin in 1746 ; and was printed at
Paris in 1747, in 4to.— Recherches sur la Precession des
Equinoxes, et sur la Nutation de l'Axe de la Terre dans
le Systeme Newtonien. Paris, 1749, in 4to. — Essais d'une
Nouvelle Theorie du Mouvement des Fluides. Paris, 1752,
in 4to. — Recherches sur differens Points importans du
Systeme du Monde. Paris, 1754 and 1756, 3 vols, in 4to.
— Elemens de Philosophic, 1759.

Opuscules Mathematiques, ou Memoires sur differens
SujetsdeGeometrie, deMechaniques, d'Optiques, etd'As-
tronomie. Paris, 9 vol. in 4to; 176l to 1773.

Elemens de Musique, theorique et pratique, suivant les
Principes de M. Rameau, eclaires, developpes, et simpli-
fies. 1 vol. in 8vo. a Lyon. — De la Destruction des Je-
suites, 1765. — In the Memoirs of the Academy of Paris
are the following pieces, by Dalembert: viz, Precis de
Dynamique, 1743, Hist. 164. — Precis de l'Equilibre et de
Mouvement des Fluides, 1744, Hist. 55. — Methode gene-
rale pour determiner les Orbites et les Mouvements de
toutes les Planetes, en ayant egard a leur action mutuelle,
1745, p. 365. — Precis des Reflexions sur la Cause Gene-
rale des Vents, 1750, Hist. 41. — Precis des Recherches
sur la Precession des Equinoxes, et sur la Nutation de
l'Axe de la Terre dans lc Systeme Newtonien, 1750, Hist.
134. — Essai d'une Nouvelle Theorie sur la Resistance des
Fluides, 1752, Hist. 116. — Precis des Essais d'une Nou-
velle Theorie de la Resistance des Fluides, 1753, Hist.
2S9. — Precis des Recherches sur les differens Points im-
portans du Systeme du Monde, 1754, Hist. 125. — Recher-
ches sur la Precession des Equinoxes, et sur la Nutation
de l'Axe de la Terre, dans l'Hypothese de la Dissimilitude
des Meridiens, 1754, p. 413, Hist. 116. — Reponse a ufi
Article du Memoire de M. l'Abbe de la Caille, sur la
Theorie du Soleil, 1757, p. 145, Hist. 118. — Addition &
ce Memoire, 1757, p. 56'7, Hist. 118.

Precis des Opuscules Mathematiques, 176l, Hist. 86.

Precis du troisieme volume des Opuscules Mathemati-
ques, 1764, H:st. 92. — Nouvelles Recherches sur les Ver-
res Optiqnes, pour servjr de suite a la theorie qui en h ete
donnee dans le volume 3° des Opuscules Mathematiques.
Premier Memoire, 1764, p. 75, Hist. 175. — Nouvelles Re-
cherches sur les Verres Optiques, pour servir de suitea
la theorie qui en a ete donnee dans le troisieme volume
des Opuscules Mathematiques. Second Memoire, 1765,
p. 53, — Observations sur les Lunettes Achromatiques,
1765, p. 53, Hist. 119.— Suite des Recherches sur les
Verres Optiques. Troisieme Memoire, 1767, p. 43, Hist.
153. — Recherches sur le Calcul Integral, 1767, p. 573. —
Accident arrive par l'Explosion d'une Mcillc d'Emouleur,
176S, Hist. 31. — Precis des Opuscules de Mathematiques,
3 E

D A R

[ -3g4 ]

D A V

4e et 5e volumes. Leur Analyse, 1768, Hist. 83. — Re- — His essays, printed in the Memoirs of the Academy of

chc-rches sur les Mouvemens de l'Axe d'une Planete quel- Sciences, are various and very ingenious, and are con-

conque dans l'hypothese de la Dissimilitude des Meridi- tained in the volumes lor the years 1742, 1747 , 174-9,

enes, 176S, p. 1, Hist. 95. — Suite des Recherches sur les 1750, 1751, 1752, 1753, 1754, 1758, 1759, 1760, 1765,

Mouvemens, &c, 1 768, p. 332, Hist. 95. — Recherches sur and in torn. 1 of. the Savans Etrangers.

hCalcul Integral, 1769, p. 73. — Memojre sur les Princi- DARK Chamber. See Camera Obscura.

pes de la Mech. 1769, p. 278. And in the Memoirs of the DARK Ten', a portable camera obscura, made some-

Academy of Berlin, are the following pieces, by our a*i- what like a desk, and fitted with optic glasses, to take

thori'viz, Recherches sur le Calcul Integral, premiere prospects of landscapes, buildings, &c.

partie, 1746. — Solution de quelques Problemes d'Astrono- DATA, in Mathematics, denote certain things or quan-

mie, 1747. — Recherches sur la courbe que forme une
Ccrde Tendue, mise en Vibration, 1747. — Suite des re-
cherches sur le Calcul Integral, 1748. — Lettre a M. de
Maupertuis, 1749. — Addition aux recherches sur la
courbe que forme une Corde Tendue mise en Vibration,
17-50. — Addition aux recherches sur le Calcul Integral,
1750. — Lettre a M. le professeur Formey, 1755.— Lxtr.
de differ, lettres a M. de la Grange, 1763. — Sur lesTauto-

tities, supposed to be given or known, from which other
quantities are discovered that were unknown, or sought.
A problem or question usually consists of two parts, Data
and Qusesita, Given and Sought. Euclid has an express
and excellent treatise of Data ; in which he uses the word
for such spaces, lines, angles, &c, as are given ; or to
which others can be found equal.

Euclid's Data is the first in order of the books that

chrones, 1765. — Extr. de differ, lettres a M. de la Grange, have been written by the ancient geometricians, to facili-

1769. Also in the Memoirs of Turin are, Differentes tare and promote the method of resolution or analysis. In

Lettres a M. de la- Grange, en 1764 et 1765, torn. 3 of general, a tiling is said to be given, which is either actually

these Memoirs.— Recherches sur differens sujets de Math, exhibited, or that can be found, that is, which is either

DARCY (Count), an ingenious philosopher and ma-
thematician, was born in Ireland in 1725 ; but his friends
being attached to the Stuart family, he was sent to France,
at 14 years of age, where he spent the rest of his life. Be

known by hypothesis, or that can be demonstrated to be
known : and the propositions in the book of Euclid's Da-
ta show what things can be found or known, from those,
that by hypothesis are already given : so that in the ana-
lysis or investigation of a problem, from those things that

ing put under the care of the celebrated Clairaut, he im- are laid down as given it is demonstrated, by the help of
proved so rapidly in the mathematics, that at 17 years of these propositions, that other things are given, and from
age he gave a new solution of the problem concerning the these last that others are given, and so on, till i't is de-
curve of equal pressure in a resisting medium. This was monstrated that that which was proposed to be found in
followed the year after by a determination of the curve the problem is given ; and when this is done, the problem
described by a heavy body, sliding by its own weight along a is solved; and its composition is made and derived from
moveable plane, at the same time that the pressure of the the compositions of the Data which were employed in the
body causes a horizontal motion in the plane. Darcy analysis. And thus the Data of Euclid are of the most
served in the war of 1744, and was taken prisoner by the general and necessary use in the solution of problems of
English : and yet, during the course of the war he gave every kind.

two memoirs to the Academy;' the first of these contained Marinus, at the end of his Preface to the Data, is mis-

a general principle in mechanics, that of the preservation taken in asserting that Euclid has not used the syntheti-

of the rotatory motion ; a principle which he again brought cal, but the analytical method in delivering them : for,

forward in 1750, by the name of the principle of the pre- though in the analysis of a theorem, the thing to be de-

servation of action. ■ monstrated is assumed in the analysis ; yet, in the demon-

In 1760, Darcy published' An Essay on Artillery, con- , strations of the Data, the thing to be demonstrated, which

taining some curious experiments on the charges of gun- is, that something is given, is never once assumed in the

powder, &c, &c, and improvements on those of the in- demonstration; from which it is manifest that every one

genious Robins-; which kind of experiments our author of them is demonstrated synthetically : though indeed if a

carried on occasionally to the end of his life. proposition of the Data be turned into a problem, the de-

In 1765, he published his Memoir on the Duration of monstration of the proposition becomes the analysis of the

the Sensation of Sight, the most ingenious of his works: problem. See Simson's edition of Euclid's Data, which

the result of these researches was, that a body may some- is esteemed the best.

times pass by our eyes without being seen, or marking its DAVIS (Capt. John), a celebrated navigator, who
presence, otherwise than by weakening the brightness of flourished in the latter part of the 1 6th century. Hegave
the object it covers. name to the straits which he discovered ; and great ex-
All Darcy's works bear the character which results pectations were formed from his skill and long experience,
from the union of genius and philosophy ; but as he In 1594 he published a small treatise, entitled, The Sea-
measured every thing on the largest scale, and required man's Secrets; a piece written with brevity, and was
extreme accuracy in experiment, neither his time, for- esteemed in its time, an 8th edition being printed in 1657 ;
tune, nor avocations, allowed him to execute more than a so that it seems to have supplanted Cortes'. Davis treats
very small part of what he projected. In his disposition, of plane sailing, calling it Horizontal ; and gives the form
lie was amiable, spirited, lively, and a lover of indepen- for keeping a sea-reckoning. He shows also how to sail
dencc, a passion to which he nobly sacrificed, even in the by the globe, and boasts of what be intended to do ; much
midst of literary society. — He died of a cholera morbus commending great circle sailing, without describing it ; as
in 1779, at 54 years of age. - also what he calls Paradoxal, that is, by a projection on
Darcy was admitted a member of the French Academy the plane of the equator with spiral rhumbs, promising to
in 1749, and was made pensioner-geometrician in 1770. publish a chart for that purpose. But above all he extols

DAY [ 3Q5 ] DEC

the use of calculations in the cases of navigation, and is 23h 56' 4"-!% of mean solar tfme; being the time in

states his intention to treat on that subject. which the earth makes one complete revolution on its

At the end of the book is given the figure of a staff of axis. See Sidereal.
his contrivance, to make a back-observation, which it ap- T)AY's-J¥ork, in Navigation, denotes the reckoning or

pears he was very vain of, and which seems to have been account of the ship's course, during 24 hours, or between

some time in use; but was afterwards supplanted by one noon and noon,
of a more commodious form, called Davis's Quadrant, as DEAD Beat, a peculiar kind of scapement for clock

if it were also of his invention, though perhaps only be-
cause a back-observation was made by both instruments,
like as the quadrant itself was at first called Staff, and
Back-staff.

Davis's Quadrant, the common sea quadrant, or back-

and watch work, invented by Mr. Mudge for correcting
the motion of pendulums &c. See Scapement.

Dead Reckoning, in Navigation, the judgment or
estimation which is made of the place where a ship is,
without any observation of the heavenly bodies; being

staff. See Back-Staff. See also Robertson's Naviga- generally performed by keeping an account of her way by

tion, book 9, sect. 7. the log; 'n knowing the course she has been steered by

DAY, a division of time arising from the appearance the compass, and by combining and rectifying all the. al-
and disappearance of the sun. lowances for drift, lee-way, &c, according to the known

Day is either natural or artificial. trim of the ship. This reckoning however is always to be

Artificial Day is that which is generally meant by the corrected, as often as any good observation can be ob-

word Day, and is the time of its being light, or the time tained.

while the sun is above the horizon. Though sometimes DECAGON, a plane geometrical figure of ten sides and
the twilight is included in the term day-light; in opposi- ten angles. When all the sides and angles are equal, it is
tion to night or darkness, being the time from the end of a regular decagon, and may be inscribed in a circle; other-
twilight to the beginning of day-light. . wise, not.

Natural Day is the portion of time in which the sun If the radius of a circle, or the side of the inscribed hexa-

performs one revolution round the earth ; or rather tin
time in which the earth makes a rotation on its axis. And
this is either astronomical or civil.

Astronomical Day begins at noon, or when the sun's
centre is on the meridian, and is counted 24 hours to the
following noon.

Civil Day is the time allotted for day in civil purposes,
and begins differently in different nations, but still inclu-

gon, be divided in extreme and mean ratio, the greater
segment will be the side of a decagon inscribed in the
same circle. And therefore, as the side of the decagon is
to the radius, so is the radius to the sum of the two.
Whence, if the radius of the circle be r, the side of the
inscribed decagon will be (y' 5 — 1) x ir.

If the side of a regular decagon be 1, its area \Vilr
be 4V(5-t-2v/5) = 7-69420SS; therefore as 1 is to

ding one whole rotation of the earth on its axis ; beginning 7'69420SS, so is the square of the side of any regular de-

either at sun-rise, sun-set, noon, or midnight. cagon, to the area of the same : consequently, if s be the

1st, At sun-rising, amongthe ancient Babylonians, Per- sideof such a decagon, its areawillbeequalto7,69420SS«".

sians, Syrians, and most other eastern nations, with the See Regular Fig-are.

present inhabitants of the Balearic islands, theGreeks,&c To inscribe a decagon in a circle geometrically. Sec

2dly, At sun-setting, among the ancient Athenians and
Jews, with the Austrians, Bohemians, Marcomanni, Sile-
sians, modern Italians, and Chinese. 3dly, At noon, with
astronomers, and the ancient Umbri and Arabians. And
4thly, at midnight, among the ancient Egyptians and Ro-
mans, with the modern English, French, Dutch, Germans,
Spaniards, and Portuguese.

The day is divided into hours; and a certain number of
days makes a week, a month, or a year.

my Mensuration, prob. 35, pa. 20, 4th edit.

DECEMBER, the last month of the year; in which
the sun enters the tropic of Capricorn, making the winter
solstice. In the time of Romulus, December was the
10th month ; whence the name, viz, from decern, ten; for
the Romans began their year ir. March, from which De-
cember is the 10th month. The month of December was
under the protection of Vesta. Romulus assigned it 30
days; Numa reduced it to 29; which Julius Ca?sar in-

The different length of the natural day in different cli- creased to 31. At the latter part of this month they had

mates, has been matter of controversy, viz, whether the the Juveniles Ludi, and the country-people kept the feast

•natural days be all equally long throughout the year; and of the goddess Vacuna in the fields, having then gathered

if not, what their difference is? A professor ofmathema- in their fruits, and sown their corn ; whence it seems is

tics at Seville, in the Philos. Trans, vol. 10, pa. 425, as- derived our popular festival called Harvest-home.
serts, from a continued series of observations for three DECHALES (Claud-Francis-Milliet), an excel-

years, that they are all equal. But Mr. Flamsteed, in the lent mathematician, mechanist, and astronomer, was born

wmc Trans, pa. 429, refutes the opinion ; and shows that at Chambery, the capital of Savoy, in \6\l. He chiefly

a day, when the sun is in the equinoctial, is shorter than excelled in a just knowledge of the mathematical and me-

when he is iti the tropics, by 40 seconds ; and that 14 chanical sciences: not that he was bent upon new dis-

tropical days are longer than so many equinoctial ones, coveries, or happy in making them ; as his talent rather

by 10 minutes. This inequality of the days depends upon lay in explaining those sciences with ease and accuracy;

two different principles : the one, the excenlricity of the which perhaps rendered him equally useful and deserving

earth's orbit; the other, the obliquity of the ecliptic with
regard to the equator, which is the measure of time. As
these two causes happen to be differently combined, the
length of the day is varied. See Equation of time.

of esteem. Indeed it was generally allowed that he made
the best use of the productions of other men, and that he
drew the several parts of the mathematical sciences to-
gether with great judgment and perspicuity. His probity

Day Sidereal, is the time in which any star appears to wasnot inferior to his learning; and these two qualities
revolve from the meridian, to the meridian again, which made him generally admired and beloved at Paris, where
fc 3 E 2

DEC

t 3Q6 ]

DEC

for four years together he read public mathematical lec-
tures in the college of Clermont. — From hence he removed
to Marseilles, where he taught the art of navigation and
the practical'mathematical sciences.— 7He afterwards be-
came professor of mathematics in the university of Turin,
where he died March 28, 16*78, at 67 years of age.

Among other works which do honour to his memory,
are, 1. An edition of Euclid's Elements; in which he has
omitted the less important propositions, and explained the
uses of those he has retained. — 2. A Discourse on Fortifi-
cation; and another on Navigation. — 3. These perfor-
mances, with some others, were collected in 3 volumes
folio, under the title of Mundus Mathematicus, being in-
deed a complete course of mathematics. And the same
was afterward much enlarged, and published at- Lyons,
1690, in 4 large volumes, folio.

DECIMALS, any thing proceeding by tens; as Decimal
arithmetic, Decimal fractions, Decimal scales, &c.

Decimal Arithmetic, in a general sense, may be con-
sidered as the common arithmetical computation in use,
in which the decimal scale of numbers is used, or in .
which the places of the figures change their value in a ten-
fold proportion, being 10 times as much for every place

In setting down a decimal fraction without its denomi-
nator, the numerator must consist of as many places as
there are ciphers in the denominator; and if it has not so
many figures, the defect must be supplied by prefixing
ciphers: thus T3g is "3; and 1?^ is -14; and -j-i§5 is -014;
and -1-0^ is -003; &c.

As ciphers on the right-hand side of integers increase
their value decimally, or in a tenfold proportion, as 2, 20,
200, &c; so, when set on the left-haud of decimal frac-
tions, they decrease the value decimally, or in a tenfold
proportion, as -2, '02, '002, &c. But ciphers set on the
other sides of these numbers, make no alteration in their
value, neither of increase nor decrease, viz, on the left-
hand of integers, or on the right-hand of decimals; so 2,
or 02, or 002, &c, are all the same; as are also -2, or -20,
or -200, &c.

Decimal fractions may be considered as having been in-
troduced by Regiomontanus, about the year 146~4, viz,
when he transformed the tables of sines from a sexagesi-
mal to a decimal scale. They were also used by Ramus,
in his Arithmetic, written in 1550; and before his time
by our countrymen Buckley and Recorde. But it was
Stevinus, who first wrote an express treatise on decimals,

more towards the left hand, or 10 times less for every place viz, about the year 1582, in La Practique d'Arithmetique;

removed towards the right hand ; the places being sup-
posed indefinitely continued, both to the right and left. In
this sense, the word includes both the arithmetic of inte-
gers, and decimal fractions. In a more restrained sense,
however, it means only

Decimal Fractions, which arc fractions whose deno-
minator is always a 1 with some number of ciphers an-
nexed, on the number of which depend the value of the
fraction, but the numerator may be any number whatever;
as -ra> T&S> TErW> &c.

, As the denominator of a, decimal is always one of the
numbers 10, 100, 1000, &c, the inconvenience of writing
these denominators down may be avoided, by placing a
proper distinction before the figures of the numerator only,
to distinguish them from integers, for the value of each
place of figures will be known in decimals, as well as in
integers, by their distance from the 1st or unit's place of
integers, having similar names at equal distances, as ap-
pears by the following scale of places, both in decimals
and integers:

. &C 8888888. 88S888&C

-C _C -C ,C .£ J=
CTT3-OT3 3

V V c 3 3 o

-o $ 5? S3 ~*

C 3 3 3 ^
3 O O O g

The mark of distinction for decimals, called thesepara-
trix, has been various at different times, according to the
fancy of different authors; sometimes a semi parenthesis,
or asemicrotchet, or a perpendicular bar, or the same with
a line drawn under the figures, or simply this line itself,
&C ; but it is usual now to write cither a comma or a full
point near the bottom of the figures; 1 place the point
mar tin- upper part of the figures, in which I follow the
practice of Sir I. Newton; a method which prevents the
eparatrix from being confounded with mere marks of
, ictuation, and which method is now used by the most
1 I mathematicians.

since which time, this has commonly made a part in most
treatises on arithmetic.

To reduce any Vulgar fraction, or parts of any thing, as
suppose |-, to a decimal fraction of the same value; add
ciphers at pleasure to the numerator, and divide by the
denominator : thus,

8 ) 3-000

■375=1;
and therefore -375 or -j-Vtfs- is a decimal of the same value
with the proposed vulgar fraction f.

Some vulgar fractions can never be reduced into deci-
mals without defect ; as ■§•, which by division is -33333 &c
infinitely. Such numbers are very properly called cir-
culating decimals, and repetends, because of the continual
return of the same figures. Sec Repetends and Circu-
lates.

The common arithmetical operations are performed the
same way in decimals, as they are in integers ; regard
being had only to the particular notation, to distinguish
the fractional from the integral part of a sum. Thus, in
Addition and Subtraction of decimals, all figures of the
same place or denomination are set directly under each
other, the separatrix, or decimal points, forming a straight
column. The operation of addition, or subtraction, is
then performed the same as in integers. In Multiplication
of decimals set down the numbers, and multiply them as
in integers ; and point off from the product as many places
for decimals as there arc decimals in both factors; pre-
fixing ciphers if necessary. In Division, set down the
numbers and divide also as in integers; making as
many decimals in the quotient, as those in the dividend
exceed those in the divisor. Examples are numerous and
common in most books of arithmetic. •

Decimal Scales, are any scales divided decimally, or
by tens.

INCLINATION, in Astronomy, is the distance of the
sun, star, planet, Sec, from the equinoctial, either north-
ward or southward; being the same with latitude in geo-
graphy, or distance from the equator.

Declination is either Teal or apparent, according as the

real or apparent place of the point or object is considered.

DEC

[ 397 ]

The declination of any point s is an arch of the meri-
dian se, contained between the given point and the equi-
noctial eq. The declination of a star &c, is found by-
knowing or observing the latitude of y
the place, i. e. the height of the pole,
and then the meridian altitude of the
star, &c ; hence the difference be-
tween the co-latitude and the altitude
of the star &c, is the declination, viz,
the difference between eh the co-la-
titude, and SH the altitude, is ES the
declination. For example, Tycho found at Uranibourg the
meridian altitude of Cauda Leonis, viz, hs = 50° 59' 00"
the co-latitude is ... he = 34 5 45
remains the declin. north - -' es = 16* 53 15

To find the Sim's Declination at any time; having given
his place in the ecliptic ; the rule is, as radius is to the
sine of the sun's longitude, so is the sine of the greatest de-
clination, or obliquity of the ecliptic, to his present decli-
nation sought.

In constructing tables of declination of the sun, planets,
and stars, regard should be had to refraction, aberration,
nutation, and parallax.

By' comparing ancient observations with the modern,
it appears that the declination of the fixed stars is variable;
and that differently in different stars; for in some it increases,
and in others decreases, and that in different quantities.

Circles ©/"Declination., are great circles of the sphere
passing through the poles of the world, on which the de-
clination is measured; and consequently are the same as
meridians in geography.

Parallax of Declination, is an arch of the circle of
declination, by which the parallax in altitude increases or
diminishes the declination of a star.

Parallels of Declination, are lesser circles parallel to
the equinoctial. The tropic of Cancer is a parallel of de-
clination at 23° 28' distance from the equinoctial north-
ward ; and the tropic of Capricorn is the parallel of decli-
nation as far distant southward.

Refraction of the Declination, an arch of the circle of
declination, by which the declination of a star is increased
or diminished by means of the refraction.

Declination of the Compass, or Needle, is its devia-
tion from the true meridian. See Variation.

Declination of a Vertical Plane, or Wall, in Dial-
ling, is an arch of the horizon, comprehended cither be-
tween the plane and the prime vertical-; when it is counted
from the east or west; or between the. plane and the meri-
dian, if it be reckoned from the north or south.

DECLINATOR, or Declinatory, an instrument in
dialling, by which the declination, inclination, and recli-
nation of planes are determined.

DECLINERS, or Declining Dials, are those which
cut obliquely, either the plane of the prime vertical, or the
plane of the horizon.

The use of declining vertical dials is very frequent; be-
cause the erect walls of houses, on which dials arc com-
monly drawn, mostly decline from the cardinal points. But
incliners and recliners are very rare.

DECLIVITY, a sloping or oblique descent.

DECREMENT, Equal, of Life. See Complement
of Life.

Decrements are the small parts by which a variable
and decreasing quantity becomes l< ss and less. The inde-
finitely small decrements are proportional to the fluxions,

DEE

See Fluxions : als

Iw-

which ill this case are negative.
crements.

DECUPLE, a term of relation or proportion in arith-
metic, implying a tenfold change or scale of variation, or
one thing 10 times as much as another.

DECUSSATION, aterm in Geometry and Optics, signi-
fying the crossing of any two lines or rays &c : or the ac-
tion itself of crossing. — The rays of light decussate in the
crystalline, before they reach the retina. — Many of the
lines of the 3d order decussate themselves. See Newton's
Enumeratio, &c.

DEE (John), a famous mathematician and astrologer,
was born at London 1527. In 1542 he was sent to St.
John's-college, Cambridge. After five years close appli-
cation to study, chiefly in the mathematical and astrono-
mical sciences, he went over to Holland, to visit some ma-
thematicians on the Continent ; whence, after a year's ab-
sence, he returned to Cambridge, and was there elected
one of the Fellows of Trinity-college, then first erected by
King Henry the 8th. In 154-S he left England a second
time, his stay at home being rendered uneasy to him, by
the suspicions that were entertained of his being a con-
jurer, arising chiefly from his application to astronomy, and
from some of his mechanical, inventions.

He now visited the university of Louvain ; where he was
much caressed, and visited by several persons of high rank.
After two years he went into France, and read lectures,
in the college of Rheims, on Euclid's Elements. In 1551,
he returned to England, and was introduced to King
Edward, who assigned him a pension of 100 crowns,
which he afterward relinquished for the rectory of Upton-
on-Severn. But soon after the accession of Queen Mary,
holding some correspondence with her sister Elizabeth, he
was accused of practising against the queen's life by en-
chantment : on which account he suffered a tedious con-
finement, and was several times examined ; tdl, in the
year 1555, he obtained his liberty by an order of council.

When Queen Elizabeth ascended the throne, Dee was
consulted concerning a propitious day for the coronation:
on which occasion he was introduced to the queen, who
made him great promises, which were but ill performed.
In 1564, he made another voyage to the continent, to pre-
sent a book which he had dedicated to the Emperor Maxi-
milian, lie returned to England the same year; but in
1571 we find him in Lorrain ; where, being dangerously
ill, the queen sent over two physicians to his relief.
Having once more returned to his native country, he set-
tled at Mortlake in Surrey, where hecontinued hisstudies
with much ardour, and collected agreat library of printed
books and manuscripts, with a number of instruments;
most of which were afterwards destroyed by the mob, as
belonging to one who dealt with the devil.

In 1578, the queen being much indisposed, Mr. Dee
was sent abroad to consult with German physicians and
philosophers (astrologers no doubt) on the occasion;
though some have said she employed him as a spy ; pro-
bably he acted in a double capacity. We next find him
again in England, where he was soon after employed in a
more rational service. The queen, desirous to be informed
concerning her title to those countries which had been
discovered by Englishmen, ordered Dee to consult the an-
cient records, and to furnish her with proper geographical
descriptions. Accordingly, in a short time,*he presented
to the queen at Richmond, two large rolls, in which tlu>
discovered countries were geographically descubed audi

DEF

[ 3y8 ]

D E F

historically illustrated. His next employment was the cover the place, are called the defences, or covers of the
reformation of the calendar, on which subject he wrote a place : and when the cannon have beaten down or ruined
rational and learned treatise, preserved in the Ashrnolean these works, so that the men cannot fight under cover, the

library at Oxford

Hitherto the extravagancies of flur eccentrical philoso-
pher seem to have been tempered with a tolerable propor-
tion of reason and science; but henceforward he is to be
considered as a mere necromancer and credulous alehy-
mist. In the year 1581 he became acquainted with one
Edward Kelly, by whose assistance he performed divers
incantations, and maintained a frequent imaginary inter-
course with spirits and angels ; one of whom made him a
present of a black speculum (a polished piece of cannel-
coal), in which these appeared to him as often as he had
occasion for them, answering his questions, &c. Hence
Butler says,

" Kelly did all his feats upon

The devil's looking-glass, a stone."

In 1583 they became acquainted with a certain Polish
nobleman, then in England, named Albert Laski, a person
equally addicted to the same ridiculous pursuits : he was
so charmed with Dee and Kelly, that he persuaded them
to accompany him to his native country; by whose
means they were introduced to Rodolph king of Bohemia;
who, though a credulous man, was soon disgusted with

defences of the place are said to be demolished.

Line o/"Defence, is that which flanks a bastion, being
drawn from the flank opposite to it. The line of defence
should not exceed a musket shot, i. e. 120 fathoms : in-
deed Melder allows 130, Scheiter 140, Vauban and Pagan
!50.

Line of Defence, greater, or fichant, is a line drawn
from the point of the bastion to the concourse of the op-
posite flank and curtin.

Line of Defence, lesser, or rasant, ovjlanquant, is the
face of the bastion continued to the curtin.

DEFERENT, or Deferens, in the Ancient Astrono-
my, an imaginary circle, which, as it were, carries about
the body of a planet, and is the same with the excentric ;
being invented to account for the excentricity, perigee,
and apogee of the planets.

DEFICIENT Hyperbola, is a curve having only one
asymptote, though two hyperbolic legs running out infi-
nitely by the side of the asymptote, but contrary ways.
See Curve.

This name was given to the. curves by Newton, in his
Enumeratio Linearum tertii Ordinis. There are 6 differ-

their nonsense. They were afterwards introduced to the entspecies ofthem, which have no diameters, expressed by-

king of Poland, but with no better success. Soon after
this they were entertained at the castle of a rich Bohe-
mian nobleman, where they lived for some time in great
affluence ; owing, as they asserted, to their art of transmu-
tation by means of a certain powder in the possession of Kelly.

Dec, now quarrelling with his companion, quitted Bo-
hemia, and returned to England, where he was once more
graciously received by the queen ; who, in 1595, made
him warden of Manchester college, in which town he "re-
sided several years. In l6'04 he returned to his house at
Mortlakc, where he died in 1608, at 81 years of age;
leaving a huge family and many works behind him.

The books that were printed and published by Dee, are,
l.,Propoeilumata Aphoristica, &c. in 155S, in 12mo. —
2. Monas Hieroglyphics ad Regem Romanorum Maximi-
lianum : 1564. — 3. Epistola ad eximium ducis Urbini
mathematicum, Fredericum C^mandinum, prefixa libel-
lo Machomeli Bagdadini de auperlicierum Divisionibus
&c; 1570. — 4. The British Monarchy, otherwise called,
The Petty Navy Royal ; 1576". — 5. Preface Mathemati-
cal to the English Euclid, published by Henry Billingsley,
1570: certainly a very curious and elaborate composi-
tion, and where he says, many more arts are wholly in-
vented by name, definition, property, and use, than either
the Grecian or Roman mathematicians have left to our
knowledge. — 6'. Divers and many annotations and inven-
tions dispersed and added after the 10th book of English
Euclid ; 1570. — 7. Epistola prefixa Ephemcridibus Joannis
Feldi a 1557, cuirationem dcclaraverat Ephemerides con-
scribendi. — 8. ParallaticceCommentationis Paxeosquc Nu-
cleus quidam ; 1573.

This catalogue of Dee's printed and published works is
to be found in his Compendious Rehearsal &c, as well as
in his letter to Abp. Whitgift : and from the same places
might be transcribed more than 40 titles of books unpub-
lished, which he had written.

DEFENCE, in Sieges, is used for any thing that serves
to preserve or screen the soldiers, or the place. So the
parapets, Hanks, cusemates, ravelins, and out-works, that

thecquation xyy -+- ey = — ax3 ■+■ bx2 -+- ex -t- d, the term
axi being negative. When the equation ax1 = bx3 -+- ex2
•+■ dx ■+- -^ee has all its roots real and unequal, the curve has
an oval joined to it. When the two middle roots are equal,
the oval joins to the legs, which then cut one another in
shape of a noose. When three roots are equal, the nodus
is changed into a very acute cusp or point. When, of
three roots with the same sign, the two greatest are equal,
the oval vanishes into a point. When any two roots arc
imaginary, there is only a pure serpentine hyperbola,
without any oval, decussation, cusp, or conjugate point;
and when the terms b and d are wanting, it is of the 6th
species.

There are also 7 different species of these curves, having
each one diameter, expressed by the above equation when
the term ey is wanting : according to the various conditions
of the roots of the equation ax3 = bx* ■+• ex -+- d, as to their
reality, equality, their having the same signs, or two of
them being imaginary.

Deficient Numbers, are those whose aliquot parts
added together, make a sum less than the whole number:
as 8, whose parts 1, 2, 4, make only 7; or the number
16", whose parts 1, 2, 4, S make only 15.

DEFILE, in Fortification, a narrow line or passage
through which troops can pass only in file, making a small
front, so that the enemy may easily stop their maich, and
charge them with the more advantage, as the front and
rear cannot come to the relief of one another.

DEFINITION, an enumeration, or specification of the
chief simple ideas of which a compound idea consists, in
order to ascertain or explain its nature and character.
Definitions are of two kinds; the one nominal, or of the
name; the other real, or of the thing.

Nominal Definition, is an enumeration of such known
characters as are sufficient for distinguishing any proposed
thing from others; as is that of a square, when it is said
that it is a quadrilateral, equilateral, rectangular figure.

Real Definition, a distinct notion, explaining the ge-
nesis of a thing: that is, how the thing is made or done :

D E G

[•399 ]

D E G

is is this definition of -a circle, viz, that it is a figure de-
scribed by the motion of a right iine about a fixed point.

DEFLECTION, the turning any thing aside from its
former course, by some adventitious or external cause. —
The word is often applied to the tendency of a ship from
her true course, by reason of currents, &c, which turn her
out of her right way.

Deflection of the Rays of Light, is a property which
Dr. Hooke observed in l6'75. He found it different both
from reflection and refraction ; and that it was made per-
pendicularly towards the surface of the opacous body.
This is the same property which Newton calls inflection :
and by others it is called diffraction.

DEGREE, in Algebra, is used in speaking of equations,
when they are said to be of such a degree, according to
the highest power of the unknown quantity. If the index
of that power be 2, the equation is of the 2d degree; if 3,
it is of the 3d degree, and so on.

Degree, in Geometry or Trigonometry, is the 3(30th
part of the circumference of any circle; for every circle is
considered as divided into 3(50 parts, called degrees;
which are marked by a small ° near the top of the figure;
thus 45° is 45 degrees. The degree is subdivided into 60
smaller parts, called minutes, meaning first minutes; the
minute into 60 others called seconds; the second into 60
thirds; &c. Thus 45° 12' 20" must be. read 45 degrees,
12 minutes, 20 seconds.

The magnitude or quantity of angles is reckoned in de-
grees; for because of the uniform curvature of a circle in
all its parts, equal angles at the centre are subtended by
equal arcs, and by similar arcs in peripheries of different
diameters ; and an angle is said to be of so many degrees,
as are contained in the arc of any circle comprehended
between the legs of the angle, and having the angular
point for its centre. Thus we say an angle of 90°, or of
45° 24', or of 12° 20' 30". It is also usual to say, such a
star is so many degrees above the horizon, or declines so
many degrees from the equator ; or such a town is situate
in so many degrees of latitude or longitude. — A sign of the
ecliptic, or zodiac, contains 30 degrees.

The division of the circle into 360 degrees is usually
ascribed to the Egyptians, probably from the circle of the
sun's annual course, or according to their number of days
in the year, allotting a degree to each day. It is a conve-
nient number too, as admitting of a great many aliquot
parts, as 2, 3, 4, 5, 6, 8, 9; &c. The sexagesimal subdi-
vision, however, has often been condemned as improper,
by many eminent mathematicians, as Stevinus, Oughtred,
Wallis, Briggs, Gellibrand, Newton, &c; who advisea de-
cimal division instead of it, or else that of centesms ; as
the degree into 100 parts, and each of these into 100 parts
again, and so on. Stevinus even holds, that this division
of the circle which he contends for, obtained in the wise
age, in saeculo sapienti. Stev. Cosmog- lib. 1, def. 6.
And several large tables of sines &c have been constructed
according to that plan, and published, by Briggs, Newton,
and others. And I myself have carried the idea still much
farther, in a memoir published in the Philos. Trans, of
1783, containing a proposal for a new division of the qua-
drant, viz, into equal decimal parts of the radius; by
which means the degrees or divisions of the arch would be
the real lengths of the arcs, in terms of the radius: and I
have since computed those lengths of the arcs, with their
sines, &c, to a great extent and accuracy. See my new
Tracts, vol. 2, pa. 122. The French mathematicians have

lately calculated their new Trigonometrical Tables on an
entire decimal division of arcs; viz, first dividing the qua-
drant into 100 degrees, and these into decimals of degrees.
But it appears by Mr. Crabtree's letter to Mr. Gascoigne
(inventor of the micrometer) Aug. 7, 1640, that both these
gentlemen, as well as their mutual friend Mr. Horrox,
had projected those tables by a complete decimal division,
the whole arch being divided into 100,000,000 parts.
See Philos. Trans, vol. 27, pa. 230 ; or my Abridg. vol.
5, pa. 631.

Degree of Latitude, is the space or distance on the me-
ridian through which an observer must move, to vary his la-
titude by one degree, or to increase or diminish the dis-
tance of a star from the zenith by one degree ; and which,
on the supposition of the perfect sphericity of the earth, is
the 360th part of the meridian.

The quantity of a degree of a meridian, or other great
circle, on the surface of the earth, is variously determined
by different observers; and the methods made use of are
also various.

Eratosthenes, 250 years before Christ, first determined
the measure of a degreeof the meridian, between Alexan-
dria and Syene on the borders of Ethiopia, by measuring
the distance between those places, and comparing it with
the difference of a star's zenith distances in the same
places ; and found it to be 694f stadia.

Posidonius, in the time of Pompey the Great, by means
of the different altitudes of a star near the horizon, taken
at different places under the same meridian, compared in
like manner with the distance between those places, de-
termined the length of a degree to be only 600 stadia.

Ptolemy fixes the degree at 6S| Arabic miles, counting
7\ stadia to a mile. The Arabs themselves, who made a
computation of the diameter of the earth, by measuring
the distance of two places under the same meridian, in the
plains of Sennar, by order of Almamon, make it only 56"
miles. Kepler determined the diameter of the earth by
the distance of two mountains, and makes a degree 13
German miles; but his method is far from being accu-
rate. Snell, by seeking the diameter of the earth from
the distance between two parallels of the equator, finds
the quantity of a degree,

by one method 5706*4 Paris toiscs, or 342384 feet ;
by another tneth. 57057 - toises, or 342342 feet.
The mean between which two numbers, M. Picard found
by mensuration, in 166*9, from Amiens to Malvoisin, to
be the most accurate, and he makes the quantity of a de-
gree 57060 toises, or 342360 feet. However, M.Cassini,
at the king's command, in the year 1700, repeated the same
labour, and measuring the space of 6° 18', from the ob-
servatory at Paris, along the meridian, to the city of Col-
lioure in Roussillon, that the greatness of the interval
might diminish the error, found the. length of the degree
equal to 57292 toises, or 343742 Paris feet, amounting to
365184 English feet.

And this account nearly agrees with that of our coun-
tryman Norwood, who, about the year 16'35, measured
the distance between London and York, and found that
distance to be 905751 English feet; the difference of lati-
tude being 2° 28'; hence, he determined the quantity of
one degree at 367196 English feet, or 57300 Paris toises,
or 69 miles, 288 yards. See Newt. Princ. Phil. prop. 19 ;
and Hist. Acad. Scienc. anno 1700, pa. 153.

M. Cassini, the son, completed the work of measuring
the whole arc of the, meridian through France, in 1718 ;

D E G

[ 400 ]

D E G

to perform which, he divided the meridian of France into
two arcs, which he measured separately.
The one from Paris to Collioure gave him 57097 toises ;
the other from Paris to Dunkirk - - ' 5696O ; and
the whole arc from Dunkirk to Collioure 57060,
the same as Picard's.

M. Muschenbroek, in 1700, resolved to correct the er-
rors of Snell, and found by particular observations, that
the degree between Alcmaer and Bergen-op-zoom contained
57033 toises.

Messieurs Maupertuis, Clairaut, Camus, Monnier, and
Outhier of France, were sent on a northern expedition,
and began their operations, assisted by M. Celsus, an emi-
nent astronomer of Sweden, in .Swedish Lapland, in July
1736, and finished them by the end of May following.
They obtained the measure of that degree, whose middle
point was in lat. 66° 20' north, and found it 57439 toises,
when reduced to the level of the sea. About the same
time another company of philosophers was sent to South
America, viz, Messieurs Godin, Bouguer, and Condamine
of France, with whom were Don Jorge Juan, and Don
Antonio de Ulloa of Spain. They left Europe in 1735,
and began their operations in the province of Quito in
Peru, about October 1736', and finished them, after many
interruptions, about 8 years after. The Spanish gentle-
men published a separate account, and assigned for the
measure of a degree of the meridian at the equator 56768
toises. M. Bouguer makes it 56753 toises, when reduced
to the level of the sea; and M. Condamine states it at
56749 toises.

M. Lacaille, being at the Cape of Good Hope in 1752,
found the length of a degree of the meridian in lat.
33" IS' 30" south, to be 57037 toises; but in 1755, fa-
ther Boscovich found the length of a degree in lat. 43°
north to be 56972 toises, as measured between Rome and
Rimini in Italy. In the year 1740, Messrs. Cassini and
Lacaille again examined the former measures in France,
and, after making all the necessary corrections, found the
measure of a degree, whose middle point is in lat. 49° 22'
north, to be 57074 toises; and in the lat. of 45°, it was
57050 toises.

In 1764, F. Beccaria completed the measurement of a
portion of the meridian near Turin; from which it is de-
duced that the length of a degree, whose middle lat. is
44° 44' north, is 57024 Paris toises.

At Vienna, 3 degrees of the meridian were measured ;
and the medium, for the latitude of 47° 40' north may be
taken at 57091 Paris toises. See an account of this mea-
surement, by father Joseph Liesganig, in the Philos. Trans.
1768, pa. 15.

In the same vol. there is also an account of the mea-
surement of a part of the meridian in Maryland and Pen-
sylvania, North America, 1766, by Messrs. Mason and
Dixon ; from which it appears that the length of a degree
whose middle point is 39° 12' north, is 36376*3 English
feet, or 56904^ Paris toises.

From the trigonometrical survey of England and Wales,
completed in 3 802, under the direction of Lieut. -col.
Mudge, four degrees of the meridian were measured ; the
mean latitude of which was 51° 29' 54|", and the mean
length of a degree 69-1457 English miles.

The rcmeasurcment of a degree in Lapland, as a cor-
rection of the former French operations, was performed
during the years 1801, 1802, and 1803, by the Swedish
mathematicians, Ofvcrboom, Swanberg, llolinquist, and

Polander; and from the account of their operations, pub-
lished by M. Swanberg, the length of the degree in lat.
66° 20' 10", is 69-2689 English miles. From a compa-
rison of this result, with those from the measurements taken
in Peru, the East Indies, and France, M. Swanberg de-
duces a mean of -^ for the elliptic) ty, and 3y63'26 miles
for the equatorial radius of the earth.

In 1803 Colonel Lambton also measured a degree in the
East Indies, north lat. 12° 32', which he made 6S7445
miles. But the latest measurement of this nature is that
of the meridian, comprised between Barcelona in Spain
and the Belearic isles. This meridian had before been
measured by Messrs. Mechain and Delambre, from Dun-
kirk to Barcelona, and the two French mathematicians
Biot and Arago, with the Spanish commissioners Chaix
and Roderiguez, were appointed to continue the line to
the small island of Formentera. The lat. of this island
was found to be 38° 40', that of Dunkirk 51° 5'; and the di-
stance between the two places 137443873 metres: hence
a degree is equal to 6S'769 English miles; and its middle
latitude 44° 52f. '

The following table, collected from the preceding ac-
counts, shows at one view the several lengths of a degree, as
measured in various parts of the earth, with the line of its
measurement, the country in which it was measured, the
names of the measurers, and the lat. of the middle point :

Date.

Latitude.

Length,
miles.

Measurers.

Countries,

1525

490

20'n.

68-763

M. Furnel

France

1620

52

4n.

66-91

Snell

Holland

1685

53

15n.

69-545

Norwood

England

1644

75-066

Riccioli

Italy

1669

49

22n.

68-945

Picard

France

171s

49

22n.

6>li9

Cassini

France

1737

66

20 n.

69'403

Maupertuis

Lapland

1740

49

22n.

69-121

7 Cassini &

France

1740

45

On.

69-092

3 Lacaille

France

1744

0

0

6S-751

Juan &c.

Peru

1744

0

0

6S-732

Bouguer

Peru

1744

0

0

6S-713

Condamine

Peru

1752

33

18s.

69-076

Lacaille

CapeofG.H.

1755

43

On.

6S-998

Boscovich

Italy

1764

44

44n.

69-061

Beccaria

Italy

1766

47

40 n.

69-142

Leisganig

Germany

1768

39

12 N.

6S-S93

Mason & D.

America

1S02

51

30n.

69-146

Mudge

England

1803

66

20n.

69-292

Swanberg

Lapland

1S03

12

32n.

6S743

Lambton

India

1808

44

52n.

68-769

Biot and Ar.

France

The difference between Swanberg's and Maupertuis' re-
sult, taken in the same place, was owing it seems to the
latter having omitted to allow for the difference of level in
the measurement base, above the sea. — On the whole, it
seems the length of the middle degree, or 46° of lat. may
be accounted 69^ miles.

The method of obtaining the length of a degree of the
terrestrial meridian, is to measure a certain distance upon
it by a series of triangles, whose angles may be found by
observation, connected with a base, the length of which
may be taken by an actual survey, or otherwise; and to
observe the different altitudes of some star at the two ex-
tremities of that distance, which gives the difference of la-
titude between them: then, by proportion, as this diffe-
rence, of latitude is to one degree, so is the measured length;

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DEM

to the length of one degree of the meridian sought; which
method was first practised by Eratosthenes, in Egypt.
See Earth, Geography, and the beginning of this ar-
ticle; also Dr. rF. Young's Philos. vol. 2, for a list of wri-
tings on this subject.

Jt appears that the meridional degrees of an ellipsoid in-
crease from the equator to the pole very nearly as the
square of the sine of the latitude. And the length of the
degree at any point, is to the length at the equator, accu-
rately as the cube of a line drawn parallel to the plumb
line, from a point in the axis equidistant from the centre
with the equator, and terminating in a point of the plane
of the equator, to the cube of the line drawn from this
point to the true pole. Or, if e be the ellipticity, and s
the sine of the latitude, the length of the degree will vary

as (1 + (2e -f- ee)xx)^.

Degree of Longitude, is the space between two meri-
dians that make an angle of 1° with each other at the poles;
the quantity or length of which is variable, according to
the latitude, being every where as the cosine of the lati-
tude; viz, as the cosine of one lat. is to the cosine of ano-
ther, so is the length of a degree in the former lat. to that
in the latter; and from this theorem is computed the fol-
lowing Table of the length of a degree of long, in different
latitudes, supposing the earth to be a perfect sphere.

Degr

English

Degr

English

Degr

English

lat.

miles.

lat.

miles.

lat.

miles.

0

69-07

31

59-13

61

33-45

I

69-06

32

58-51

62

3240

2

69 03

33

57-87

63

31-33

3

68-97

34

57-20

64

30-24

4

68-90

35

56-51

65

29-15

5

68-81

36

55-S1

66

2S-06

6

68-62

37

55-10

67

26-96

7

68-48

38

54-37

68

25-85

8

68-31

39

53-62

69

24-73

9

68-15

40

52-85

70

23-60

10

67-95

41

52-07

71

22 47

11

67-73

42

51-27

72

2132

12

67-48

43

50-46

73

20-17

13

67-21

44

49-63

74

19-02

14

66-95

45

48-78

75

17-86

15

06-65

46

47-93

76

16-70

10'

66-31

47

47-06

77

15-52

17

65-98

4S

46-16

78

14-35

18

65-62

49

45-20

79

13-17

19

65-24

50

44-35

80

11 '98

20

64-84

51

43-42

81

10-79

21

64-42

52

42-48

82

9-59

22

63-97

53

41-53

83

8-41

23

63-51

54

40-56

84

721

24

63 03

55

39-58

85

6-00

25

62 53

56

38-58

86

4-81

26

62-02

57

37-58

87

3-fil

27

61-48

58

36-57

88

241

28

60-93

59

35-51

89

121

29

60-35

60

34-50.

90

000

30

59-75

Note, This table is computed on the supposition that
the length of the degrees of llie equator are equal to those
of the meridian at the medium latitude of 45°, which
length is 6.9-i't English miles nearly.

But, taking the earth as a spheroid, the degree of longi-
tude may be found, in any given latitude /, by saying,

Vol.. I.

1st. As the equatorial diameter is to the polar, so is tang.
(90° — I), to tang, of an angle a : then 2d. As radius, to
sine of a, so is the length of a degree of the equator to the
length of a degree on the parallel of the given latitude.

DEINCLINERS, orDEiNciiNOG Dials, are such us
both decline and incline, or recline, at the same time.
Suppose, for instance, a plane cutting the prime vertical
circle at an angle of 30°, and the horizontal plane at an
angle of 24°, the latitude of the place being 52° ; a dial
drawn on this plane, is called a deinclincr.

DELIAC AL Problem, a celebrated problem among the
ancients, concerning the duplication of the cube.

DELISLE. See Lisle (De).

DELPHINUS, the Dolphin,& constellation of the north-
ern hemisphere ; whose stars, in Ptolemy's catalogue, are
10; inTycho's the same; in Hevelius's 14 ; and in Flam-
steed's 18.

DEMETRIUS, a Cynic philosopher, and disciple of
Apollonius Thyaneus, in the age of Caligula. That em-
peror wishing to gain the philosopher to his interest by a
large present, he refused it with indignation, saying, " It-
Caligula wishes to bribe me, let him send ihe his crown."
Vespasian was displeased with his insolence, and banished
him to an island. The cynic derided the punishment, and
bitterly inveighed against the emperor. Demetrius lived
to a very great age. And Seneca observes, that " nature
had brought him forth t'o show mankind that an exalted
genius can live securely without being corrupted by the
vice of the surrounding world."

HEMI-Bastion, in Fortification, that which has only
one face and one flank.

DzMi-Cannon, and Dem i-Culverin, names of certain
species of cannon, now no longer used.

Demi-Ctoss, an instrument used by the Dutch to take
the altitude of the sun or a star at sea; instead of which
we use the cross-staff, or fore-staff.

DEMi-Gorge, is half the gorge or entrance into the
bastion ; not taken directly from angle to angle, where
the bastion joins to the curtin, but from the angle of the
flank to the centre of the bastion ; or the angle the two
curtins would make, were they thus protracted to meet
in the bastion.

DzMi-Lune, or Hulf-moon, an outwork consisting of two
faces, and two little flanks. It is often built before the
angle of a bastion, and sometimes also before the curtin;
though now it is very seldom used.

DEMOCRITUS, one of the greatest philosophers of
antiquity, was born at Abdera, a town of Thrace, about
the 80th olympiad, or about 400 years before Christ.
His father, says Valerius Maximus, was able to entertain
the army of Xerxes; and Diogenes Laertius adds, upon
the testimony of Herodotus, that the king, in requital,
presented him with some Magi and Chaldeans. From
these, it seems, Dcmocritus received the first part of his
education; and from them, while yet a boy, he learned
theology and astronomy. He next applied to Leucippus,
from whom he learned the system of atoms and a vacuum.
His father dying, he and his two brothers divided his ef-
fects. Dcmocritus made choice of that part which con-
sisted in money, as being, though the least share, the most
convenient for travelling ; and it is said that his portion
amounted to more than 100 talents, which is near 20
thousand pounds sterling. His extraordinary inclination
for knowledge and the sciences, induced him to travel into
all parts of the world where he might find learned men.
3F

D E M

[ 402 ]

DEN

He went to visit the priests of Egypt, from whom he learn-
ed geometry : he consulted the Chaldean and Persian
philosophers; and it is said that he penetrated even into
India and Ethiopia, to confer with the Gymnosophists. In
these travels he wasted his substance; after which, at his
return he was obliged for some time to he maintained by
hii brother. Settling himself at Abdera, he there go-
verned in the most absolute manner, by virtue of his con-
summate wisdom. The magistrates of that city made him
a present of 500 talents, and erected statues to him, even
in his lifetime ; but being naturally more inclined to con-
templation than delighted with public honours and em-
ployments, he withdrew into solitude and retirement.

Democritus always laughed at human life, as a conti-
nued farce, which made the people think he was mad ; on
which they sent for Hippocrates to cure him : but that
celebrated" physician having discoursed with the philoso-
pher, told the people that he had a great veneration for
Democritus ; and that, in his opinion, those who esteem-
ed themselves the most healthy, were the most distem-
pered. It is said, though with little probability, that De-
mocritus put out his own eyes, that he might meditate
more profoundly on philosophical subjects. He died, ac-
cording to Diogenes Laertius, in the 36lstyear before
the Christian era, at 109 years of age. He was the author
of many books, which are lost ; from which Epicurus bor-
rowed his philosophy.

DEMOIVRE (Abraham), a celebrated mathemati-
cian, of French original, but who spent most of his life in
England. He was born at Vitri in Champagne, 1667.
On account of the revocation of the edict of Nantes, in
l6'85, he determined, with many others, to take shelter in
England ; where he perfected his mathematical studies,
the foundation of which he had laid in his own country.
A mediocrity of fortune obliged him to employ his talent
in this way in giving lessons, and reading public lectures,
for his better support : in the latter part of his life too, he
chiefly subsisted by giving answers to questions in chances,
play, annuities, &c, and it is said most of these responses
were delivered at a coffee-house in St. Martin's-lane,
where he spent most of his time. The Principia Mathe-
matica of Newton, which chance is said to have, thrown in
his way, soon convinced Demoivre how little he had ad-
vanced in the science he professed. This induced him to
Tedouble his application ; which was attended by a con-
siderable degree of success ; and he soon became con-
nected with, and celebrated among, the first-rate mathe-
maticians. His eminence and abilities in this line, opened
him an entrance into the Royal Society of London, and
into the Academies of Berlin and Paris. By the former
his merit was so well known and esteemed, that they
judged him a fit person to decide the famous contest be-
tween Newton and Leibnitz, concerning the invention of
Fluxions.

Demoivre died at London Nov. 1754, at 87 years of
age. The Philosophical Transactions of London contain
several of his papers, and all of them interesting, viz. in
llir-volumes 1.9, 20, 22, 23, 25, 27, 2£), 30, 32, 40, 41, 43.

lie published also some very respectable works, viz,

1. Miscellanea Analytica, de Seriebus et Quadraturis
&c ; 1730, in 4to. But perhaps he has been more gene-
rally known by his

2. Doctrine of Chances; or, Method of Calculating
the Probabilities of Events at Play. This work was first
printed, 1718, in 4to, and dedicated to Sir Isaac Newton :

it was reprinted in 1738, with great alterations and im-
provements; and a third edition was afterwards printed.

3. Annuities on Lives; first printed 1724, in 8vo. — In
1742 the ingenious Thomas Simpson (then only 33 years
of age) published hisDoctrineof Annuities and Reversions;
in which he paid some handsome compliments to our au-
thor. Notwilhstandingwhich, Demoivre presently brought
out a second edition of his Annuities, in the preface to
which he passed some harsh reflections upon Simpson.
To these the latter gave a handsome and effectual answer,
1743, in An Appendix, containing some Remarks on alate
book on the same subject, with answers to some personal
and malignant misrepresentations, in the preface thereof.
At the end of this answer, Mr. Simpson concludes, "Lastly,
I appeal to all mankind, whether, in his treatment of me,
he has not discovered an air of self-sufficiency, ill-nature
and inveteracy, unbecoming a gentleman." Here it would
seem the controversy dropped : Mr. Demoivre published
the 3d edition of his book in 1750, without any farther no-
tice of Simpson, but omitted the offensive reflections that
had been in the preface.

DEMONSTRATION^ certain or convincing proof of
some proposition: such as the demonstrations of the pro-
positions in Euclid's Elements. The method of demon-
strating in mathematics, is the same with that of drawing
conclusions from principles in logic. Indeed, the demon-
strations of mathematicians are no other than series of en-
thymemes ; every thing is concluded by force of syllogism,
only omitting the premises, which either occurof their own
accord, or are recollected by means of quotations.

DENDROMETER, an instrument lately invented by
Messrs. Duncombe and Whittel; so called, from its use in
measuring trees.

DENEB, an Arabic term, signifying tail; used by astro-
nomers as a name to some of the fixed stars, but especially
for the bright star in the Lion's tail.

DENOMINATOR, of a Fraction, is the number or
quantity placed below the line, which shows the whole in-
teger, or into how many parts it is supposed to be divided
by the fraction; as that which gives denomination or name
to the parts of the fraction. Thus, in the fraction -j?t
(five-twelfths) the number 1 2 is the denominator, and shows
that the integer is here divided into 12 parts, or that it
consists of 12 of those parts of which the numerator con-
tains 5. Also b is the denominator of the fraction -.

b

Denomin ator of a Ratio, is the quotient arising from
the division of the antecedent by the consequent. Thus,
6 is the denominator of the ratio 30 to 5, because 30 di-
vided by 5 gives 6. It is otherwise called the exponent
of the ratio.

DENSITY, that property of bodies by which they con-
tain a certain quantity of matter, under a certain bulk or
magnitude. Accordingly, a body that contains more mat-
ter than another, under the same bulk, is said to be denser
than the other, and that in proportion to the quantity of
matter; or if the quantity of matter be the same, but un-
der a less bulk, it is said to be denser, and so much the more
so as the bulk is less. So that, in general, the density is
directly proportional to the mass or quantity of matter,
and reciprocally or inversely proportional to the bulk or
magnitude under which it is contained.

The quantities of matter in bodies, or at least the pro-
portions of them, are known by their gravity or weight;
every equal particle of matter being endowed with an equal

DEN

[ 403 ]

D E P

gravity, it is inferred that equal masses or quantities of
matter have an equal weight or gravity; and unequal
masses have proportionally unequal weights. So that,
when body, or mass, or quantity of matter is spoken of,
we are to understand their weight or gravity.

From the foregoing general proportion of the density of
bodies, viz, that it is as the mass directly, and as the bulk
inversely, may be inferred the proportion of the masses,
or of the magnitudes; viz, that the mass or quantity of
matter is in the compound ratio of the bulk and density;

ticity of the air, when the thermometer is at 31°, and b be
any other height of the barometer, when the thermometer
is at t degrees ; then in this case, b is the measure of the

elasticity, and — ; b is the measure of the density of

the air.

Density of the Planets. In homogeneous, unequal,
spherical bodies, the gravities on their surfaces are as their
diameters when the densities are equal, or the gravities
are as the densities when the bulks are equal; therefore,

a>id that the bulk or magnitude is as the mass directly, and i» spheres of unequal magnitude and density, the gravity

is in the compound ratio of the diameters and densities, or
the densities are as the gravities divided by the diameters.
Knowing therefore the diameters of the planets by obser-
vation and comparison, and the gravities at their surface by
means of the revolution of the satellites, the relation of their
densities becomes known. And as I have found the mean
density of the earth to be nearly 5 times that of water (my
new Tracts, vol. 2, Tract 27) ; hence the densities of the
planets, with respect to water, become known, and are as
below :

the density inversely. Hence, if b, b be two bodies, or
masses, or weights ;

and d, d their respective densities;

also m, vi their magnitudes, or bulks :
Then the theorems above are thus expressed,

B , B

viz, Da-, and b oc dm, .and m a -;

M D

B b

or d : d : : - : -, and Bib:: dm : dm, &c ;

M m

D Bm . B DM , M Bd

or - = — , and - = — , and - = ,— .
a wm b dm m bD

No body is absolutely or perfectly dense; or no space
is perfectly full of matter, so as to have no vacuity or in-
terstices; on the contrary, it is the opinion of Newton,
that even the densest bodies, as gold, &c, contain but a
small portion of matter, and a very great portion of va-
cuity; or that it contains a great deal more of pores or
empty space, than of real substance.

It has been observed above, that the relative density of
bodies may be known by their weight or gravity; and hence
the most general way of knowing those densities, is by ac-
tually weighing an equal bulk or magnitude of the bodies,

whether solid or fluid ; if solid, by shaping them to the same neal, the densities here determined are supposed to be the
figure and dimensions; if fluid, by filling the same vessel mean densities, or such as the bodies would have if they
with them, and weighing it. were homogeneal, and of the same mass of matter and

For fluids, there are also other methods of finding their magnitude,
density: as 1st, by making an equilibrium between them DENTICLES, or Dentils, are ornaments in a cornice,

in tubes that communicate; for, the diameters of the tubes cut after the manner of teeth. These are mostly affected
being equal, and the weights or quantities of matter also in the Ionic and Corinthian orders ; and of late also in the
equal, the densities will be inversely as the altitudes of the Doric. The square member on which they are cut, is
liquids in them, that is inversely as the bulk. called the Dcnticule.

2dly, The densities of fluids an: also compared together DEPARTURE, in Navigation, is the easting or west-

by immerging a solid in them ; for if the solid be lighter ing of a ship, with regard to the meridian she departed or
than the liquids, the part immerged by its own weight sailed from. Or, it is the difference in longitude, either
will be inversely as the density of the fluid; or if it be east or west, between the present meridian the ship is un-
heavier, and sink in the liquids, by weighing it in them ; der, and that where the last reckoning or observation was
then the weights lost by the body will be directly propor- made. This departure, any where but under the equator,

Water

The Sun

Mercury

Venus

The Earth

Mars

The Moon

Jupiter

Saturn

Herschel

Densities.
1

5

3f
3tV

H

1*

As it is not likely that any of these bodies are homoge-

tional to the densities of the fluids

Density of the Air, is a property that has much en-
gaged the attention of the later philosophers, since the dis-
covery of the Torricellian experiment, and the air-pump.

must be accounted according to the number of miles in a
degree proper to the parallel the ship is in. — The De-
parture, in Plane and Mcrcator's Sailing, is always repre-
sented by the base of a right-angled plane triangle, where

By means of the barometer it is demonstrated, that the air the course is the angle opposite to it, and the distance

is of the same density at all places at the same distance sailed is the hypothenusc, the perpendicular or other leg

from the level of the sea, provided the temperature, or de- being the difference of latitude. And then the theorem

gree of heat, be the same. Also the density of the air al- for finding it, is always this : As radius is to the sine of

ways increases ill proportion to the compression, or the the course, so is the distance sailed, to the departure,
compressing forces. And hence the lower parts of the at- DEPHLOGISTICATED Air. See Oxygen.
inosphere are always denser than the upper : yet the den- DEPRESSION of Equations, in Algebra, is the re-

sity of the lower air is not exactly proportional to the ducing them to lower degrees; as, from biquadratics to

weight of the atmosphere, by reason of heat and cold, cubics, or from cubics to quadratics. It is only in some

which make considerable alterations as to rarity and den- certain cases that this can be performed; as 1st, When

bity; so that the barometer measures the elasticity of the tliere can be found a compound binomial measure, or

air, rather than its density. If the height of the barometer divisor of the given equation, of which one term is the un-

be considered as the measure both of the density and elas- known letter or quantity ; then the other part of the

3F2

D E ¥

t 404 ]

SEP

divisor is one root of the equation, and the whole equation This being the case, it is evident that we shall not alter
bein<* divided by the divisor, the quotient is the equation the value of the equation by writing — x instead of x;

depressed one degree lower. So, for example, if the
equation be x3 — 6xz + 11* — 6 = 0; and it be discovered
that x is = 2, then x — 2 will be the divisor, by which
dividing the given equation, it is depressed to the quadratic
x* _ 4,x -t- 3 == o, the two roots of which are 1 and 3. So
that all the three roots of the given cubic equation, are, 1,
2, 3. — Or, if the given equation were x3 — bxx — arx -+- a"
b=zO, and it be found that b is one root of it, or that x — b
is a divisor; then making the division, the quotient, x° —
a2 =» 0, is the depressed equation, the two roots of
which are x == ■+- a or — a: so that the three roots are
•+■ a, —a, h- b.

2d. If an equation contain equal roots, these may be
found, and the equation reduced as many degrees or dimen-
sions lower, as there are equal roots. For if the original
equation xn — fix11""1 -+- bxr"'2— &c = 0, have m equal roots,
then nx°~' — (n — l)qx?~* ■+-.(« — 2)bxn~3 — &c = 0, has
m — 1 of those roots, as may be readily shown. Hence,
when there are two equal roots, the two equations have a
common measure of this form (x — r)™"1, (r being one of
the roots,) which may be obtained in the usual way, and
thus m roots of the original equation may be known. Then
divide this equation by (x — r)m, and the resulting equa-
tion, of n — m dimensions, contains the other roots.

Thus, let the cubic x3 — ax* -+- bx — c = 0, have two
equal roots ; then 3x2 — lax + b = 0 has one of them ;
and the two equations have a common measure, which is a
simple equation : the quantities 3x3 — Sax1 ■+- 3bx — 3c,
and 3xz — 2ax -t- b, have also the same common measure,
which being found, we have \{6b — 2a'') x — |(6c — ab)
for a divisor of the equation x3 — ax1 -+- bx — c = 0 ; this

, ■ be — ab „.

divisor being also put = 0, gives x = — j. Thus two

° r ' ° 6b — 2a"

roots of the equation are discovered; and, since a is the
sum of all the roots, the 3d root is the difference between
a and the sum of the two equal roots.

If two roots of the equatiun be of the form -+- a, —a,
differing only in their signs; change the signs of the roots,
and the resulting equation has two roots, — a, -+- a ; thus
we have two equations with a common measure, x2 — a2,
which may be found, and the equation depressed, as in the
former case.

To be more particular:— There being only certain cases
in which this depression can be effected, viz, 1st. When there
is any known relation between two or more of the roots;
and, 2dly. When the equation is reciprocal, that is, when
each of the terms of the equation, which are equally distant
from the first and last terms, have equal co-efficients ; under
which form therelationofeach pairof roots becomesknown.

With regard to the first, it seldom happens that any
known relation subsists between the roots of an equation
if the roots themselves are unknown, except it be that of
equality, we shall therefore only consider it under this
point of view; namely, when two or more of the roots are
equal, having either like, or contrary signs ; referring the
reader, for further information on this subject, to Lacroix'
Complement des Elemens d'Algebre, art. 38 ; and lo
Waring's Meditationes Algeb. cap. 3.

1. When an equation has two equal roots, with con-
trary signs, it may be depressed two degrees.

Let there be given the equation
Xs — pr'1 — qx3 ■+- rx2 -i- sx — t = 0, which we are sup-
posed to know contains two equal roots with contrary signs.

which gives us, — xs — px* ■+■ qx3 -t- ri' — sx — t = 0,
Or x5 -I- px* — qx3 — rxx + si + / = 0,

Also Xs — px* — qx3 -+- rxz ■+■ sx — t = 0,
being the given equation. Now these two equations have
a common measure, which, being obtained by the usual
method, will be found to involve the 2d power ofx;
whence the equal roots may be determined, and the whole
equation being divided by this commun measure, will give
us the depressed equation. We may also discover the
value of the equal roots in equations, not exceeding the
5th degree by the addition and subtraction of the above
equations, for by the former we obtain
2x! — 2qx3 -+- 2sx = 0, orx' — qxx ■+■ s = 0, and by the

latter — 2pxi ■+- 2

Therefore — qx
.t - p

x2 — 2t = 0, or x4 x2 h — = 0.

P P

-+• s = 12 ■*■-",. whence we have

P P

x = +*/(- — —), which are the two equal roots, and con-

r-pr

sequently x2 — ( — ) =0 is a divisor of the original equa-
tion, which may thus be depressed to a cube, containin;;
the other three roots.

Suppose, for example, our given equation had been
xs — 3x4 — 17x3 -i- 27x2 ■+• 52x-- 6"0 = 0.

Then p — o,q — \7,r = l27,s = 52, t =z6o -,

And x = ±*/{ _ ) = ± 2; the whole equation
therefore being divided by x2 — 4 = 0, gives x3 — 3x2 —
3x ■+- 15 = 0, which is a cubic equation, in which are
contained the other three roots.

And exactly in the same manner may any other equa-
tion of this kind be depressed two degrees.

2. When two or more of the roots are equal, having
like signs, the equation may be depressed as many degrees
as there are equal roots.

Let x* — pxz + qx — r = 0, be an equation having two
equal roots with like signs, and let a and b represent those
roots; then we shall have the two following identical equa-
tions : a3 — prr ■+- qa — r ==:0, b3 — pbJ -+■ qb — r = 0,
whence we draw (a3 — t3) — p(a" —b") -t- <j(a — b) = 0,
and dividing this by a — b, we obtain (a- -h ab ■+■ b~) — p
(a ■+■ b) -t- q = 0 ; or, since a = b, 3a2 — 2pa ■+■ q = 0 ;
which has a common divisor with the original equation,
and writing x again instead of a, this divisor will he readily
obtained by the rule for finding the greatest common mea-
sure of two quantities, whicli in the present case is

6q - 2/.

= 0, whence x =

9r — 717

6(/ - -2ji-

Thus, two of the roots are discovered, and consequently
the equation may be depressed two degrees as before, by

dividing the whole of it by (x ■ ^-0°.

But in the present case, as the original equation was
only of three dimensions, and p being equal to the sum of
all its roots, from the nature of equations ; we have

x = p — T, ~ ., for the third root.

Suppose, for example, the proposed equation had been
x3 — 4x2 -t- 5x — 2 = 0: here /> = 4, q ss 5, r — 2; and

*"- - VI _

one of the equal roots is

1 ; the third root

«7 - 2/i"

therefore is /> — 2 = 4 — 2 = 2, and the three roots
sought are 1, 1, and 2.

D E P

[ 405 ]

DEE

We should have obtained exactly the same result, by express the side of a regular 17-sided polygon inscribed in

considering that an equation having two equal roots, is a circle, by a quadratic equation; which was before

necessarily divisible by a factor of the form x1 — Zax -+■ or, thought impossible. This curious piece of analysis has

where a represents one of the equal roots: and if three been translated into French under the title of Recherchcs

of the roots are equal, the equation has a divisor of the Arithmetiques, and is highly deserving of an attentive

form x3 — Sax" -+- 3axx — a3, and so on, for any number perusal.

of equal roots : thatis, an equation may always be depressed Depression of the Pole. So many degrees &c as you

as many degrees as there are equal roots contained in it. sail or travel from the pole towards the equator, so many

3 Reciprocal equations may be depressed to others of it is said you depress the pole, because it becomes so much

half the original dimension, when the highest power of lower, or nearer the horizon.

the unknown quantity is even. Let the given reciprocal Depression of a Star, or of the Sun, is its distance be-

equation be x1 ■+■ pr" •+- qxa h- px ■+• 1=0; and dividing low the horizon ; and is measured by an arc of a vertical

this by x\ we obtain I + p1 + a 4- V p ~- + 4 = 0 ; C*rcle' intercePted between the horizon and the place of

J 'i ' r ' r r the star,
whence it is evident, that if a be one of the roots of this

Depression of the Visible Horizon, or Dipofthe Hori-
zon, denotes its sinking or dipping below the true horizon-
tal plane, by the observer's eye being raised above the sur-
face of the sea ; in consequence of which, the observed al-
titude of an object is by so much too
great. Thus, the eye being at E,the
height is ae above the surface of the
earth, whose centre is c ; then eh is
the real horizon, and e/( the visible
one, below the former by the angle
we.Ii, by reason of the elevation 1ae of the eye.

To compute the Depression or Dip of the Horizon.

In the right-angled triangle ce/j, are given c/ttlie earth's
radius = 21000000 feet, and the hypothenuse ce = the
radius increased by the height at. of the eye; to find the
angle c which is = the angle iieA, or the depression
sought; viz, as c/i : ce : : radius : eecant Z. c,
or, as ce : ch : : radius : cosine /_ c.

By either of these theorems are computed the numbers
in the following table, which shows the depression or dip
of the horizon of the sea, for different heights of the eye,
from 1 foot to 100 feet.

equation, - is also a root ; and in the same manner if b be
one of its roots, then - is the other: and so on, for equa-

b '

tions of higher dimensions.

We may therefore consider the given equation as gene-
rated from the product (x -+- a) . (x -t — ) . (x ■+■ b) . (x -t- -),
or from the quadratic factors

(a- + x(a +i) + 1) . (x2 + x(b '■+■ i) + 1),

or making a h — = m, and b -+- - = n, the above becomes
° a b

(r2 -+- mx -+- 1) . (x° ■+• nx ■+- 1) =

x4 -+- {in -+- n)x3 ■+• (2 -+■ mnfx7, -t- (m -+■ n)x -t- 1 = 0,

and equating the coefficients we have m •+■ n =p, mn =

q — 2, whence m — n = \Z{p"~ — 4y 4- 8), m = \p -+-

l</(p" — 4? ■+■ 8), » = |p — !v/0>2 — 4ty + 8).

Thus m and n are found by a quadratic, that is, by an

equation of half the original dimension. As to a and b,

those are now readily determined : for since a h — = m,

we have a = \m +v/(J»<" — 1); and since b -+- - = ??,we

derive b = \n i-s/d"2 — !)■ Suppose, for example, our
equation was x4 -t- 5£x3 -+- lOix2, -i- 5-^-x -i- 1 = 0.

Here/; = 5|, and q = 10|,

whence in = \p -+- \\/ {pL — 4<jr -+- 8) = 3},
and n = \ p — \l/ {p1 — 4j -t- 8) = 1\;
therefore a = \m ±\Z(lm- — 1) = 3 or \,
and b = \n ± y' (|n'2 — l) = 2or|;
consequently, the four roots are 3, 4> 2, and 4..

And in the same manner may any other equation of this
kind be depressed to half its original dimension.

When the highest power of the unknown quantity is un-
even, then -1- 1 or — 1 is a root of the equation, as is evi-
<lant from inspection ; it may therefore be divided by x ±
1=0; which will depress it one degree, still leaving it
reciprocal, alter which, it may be reduced to half that di-
mension, by the method above explained. Let Xs -t-px'' -+-
qx3 ■+-' qx? ■+- px ■+■ 1 =0, be the given equation, then ■+- 1 is See Robertson's Navigation, book 9 appendix; and

one of its roots; dividingit therefore byx— 1 =0,weobtain Tables requisite to be used with the Nautical Ephemeris,
x1 -*-(/>-+- l)x3 + (/j + 5 + \)x- ■+■ {p •+- l)x -t- 1 = 0, pa. 1. See also Levelling.

which is still reciprocal, and may therefore be depressed DEPTH, the opposite of Height, and one of the dimen-

toa quadratic by the method already employed. M.Gauss, sions of bodies, or of space. Sec Heioht, Altitude,
in his Disquisitiones Arithmetical, which is mentioned un- Elevation, &zc.

der the article Algebra, has shown that every equation of DERHAM (Dr. William), an eminent English phi-

two terms, of which the exponent is a prime number, may losopher and divine, was born at Stowton, near Worces-
be rationally decomposed into othci ■equations, the degrees ter, 1G57, and educated in Trinity-college, Oxford. In
of which are indicated by the prim.- factors of the given l6'82, he was presented to the vicarage of Wargrave in
exponent minus 1, by means of which he has been able to Berkshire; and, in 1689, to the valuable rectory of Up-

Height

Dip of

Height

Dip of

Height

Dip of

of the

tlie

of the

the

of the

the

eye

horizon

eye

horizon

eye

horizon

feet

/ </

feet

' 1

feet

/ 11

1

0 57

13

3 26

26

4 52

2

1 21

14

3 34

28

5 3

3

1 39

15

3 42

30

5 14

4

1 55

id

3 49

35

5 39

5

2 8

17

3 56

40

6" 2

6"

2 20

is

4 3

45

6' 24

7

2 31

19

4 10

50

6 44

8

2 42

20

4 16'

60

7 23

9

2 52

21

4 22

70

7 59

10

3 1

22

4 28

80

8 32

11

3 10

23

4 34

90

9 3

12

3 18

24

4 40

100

9 33

D E R

t 406 ]

minster in Essex; which, lying ai a convenient distance
from London, afforded him an opportunity of conversing
and corresponding with the principal literary men of the
nation. Applying himself there with great eagerness to
the pursuit of his studies in natural and experimental phi-
losophy, he soon became a distinguished and useful mem-
ber of the Royal Society, whose Philosophical Transactions
contain a great variety of curious and valuable pieces,
the fruits of his laudable industry, in all or most of the
volumes, from the 20th to the 39th, both inclusive ; the
principal of which are :

1. Experiments on Pendulums in vacuo.

2. Of an instrument for finding the Meridian.

3. Experim. and Observat. on the Motion of Sound.

4. On the Migration of Birds.

5. On the Spots in the Sun, from 1703 to 1711.

6. Observations on the Northern Lights, Oct. S, 1 726,
and Oct. 13, 1728.

7. Tables of the Eclipses of Jupiter's Satellites.

8. Difference of Time in the meridian of diff. places.
9- Of the meteor called Ignis Fatuus.

10. The History of the Death Watch.

1 1 . Meteorological Diaries for several years.

In his younger days he published his Artificial Clock-
maker, a very useful little work, which has gone through
several editions. In 1711, 1712, 1714, he preached
those sermons at Boyle's lecture, which he afterwards
digested under the well-known titles of Physi co-Theology
and Astro-Theology, or Demonstrations of the being and
attributes of God, from his works of creation, and a sur-
vey of the heavens.

In 1716 he was made a canon of Windsor, being at
that time chaplain to the prince of Wales ; and in 1730
he received, from the university of Oxford, the degree of
Doctor of Divinity. He revised the Miscellanea Curiosa,
in 3 vols. 8vo, containing many curious papers of Dr. Hal-
ley and several other ingenious philosophers. To him
also the world isindebted for the publication of the Philo-
sophical Experiments of the late eminent Dr. Hooke, and
other ingenious men of his time; as well as notes and il-
lustrations of several other works.

Dr. Derham was very well skilled in medical as well as
in physical knowledge ; and was constantly a physician to
the bodies as well as the souls of his parishioners. This
great and good man, after spending his life in the most
agreeable and improving study of nature, and the diligent
and pious discharge of his duty, died at Upminster in
1735, at 78 years of age. •

DERIVATIONS {Calcul des), a new species of calcu-
lation, which may be considered as an important discovery
in analysis. — This method was first published by M. Ar-
bogast, professor of mathematics at Strasburgh, in 1800,
and has since beet) translated into French ; but the first
idea of this method appears to have been formed by the
celebrated Dr. Waring, and is alluded to by him in the
conclusion of his Meditationes Analytical.

As the name of this method imports, it depends on the
principle of deriving one term of a series from the foregoing
ones; and by means of a peculiar notation, this derivation
is found to be very simple and commodious. Here, in-
stead of symbols being made use of to represent quanti-
ties, they are employed to represent certain operations
that are performed upon them. Thus let l'(a -+■ x) be any
function of the binomial a -+- x, this function may be ex-
panded into a series proceeding according to the powers

of x, as r (« -t- x) =. a

DES

t- bx H 'X°

£C ;

whence it will be seen that a .-= Fa, that b is derived from
a, or from Fa ; c from b ; d from c, &c, by the same law :
therefore if we conceive the manner in which we derive b
from a, in the same manner we derive c from 6, d from c,
&c. Then representing by d the operation to be per-
formed on Fa, to produce 6, we shall have b = df«,
c = d'2f«, d = d3fo, &c ; so that the above series is

, , , '. DFst , d'fo ..

expressed thus, F(a -+■ x) = ran x -+- x' ■+-

1 v 1 1.2

D3 Fa ,

■nr?* &c-

For farther particulars relating to this important branch
of analysis, the Trader is referred to the work of M. Ar-
bogast, entitled Des Calcul des Derivations; also to Mr.
Woodhouse's Analytical Calculations.

DESAGUL1ERS (John Theophilus), an eminent
experimental philosopher, was the son of the Rev. John
Desaguliers, a French Protestant refugee, and born at
Rochelle in l6S3. His father brought him to England an
infant; and having taught him the classics himself, he sent
him at a proper age to Christ-church college, Oxford ;
where in 1702 he succeeded Dr. Keil in reading lectures
on experimental philosophy at Hart-Hall. In 1712 he
married, and settled in London, when Ire first of any in-
troduced the reading of lectures in experimental philoso-
phy into the metropolis, which he continued during the
rest of his life with the greatest applause, having several
times the honourof reading his lectures before the king and
royal family. In 1714 he was elected f. r.s, and proved
a very useful member, as appears from the great number
of his papers that are printed in their Philos. Trans, on the
subjects of optics, mechanics, and meteorology. The
magnificent Duke of Chandos made Dr. Desaguliers his
chaplain, and presented him to the living of Edgware,
near his seat at Cannons ; and he became afterwards chap-
lain to Frederick prince of Wales; but in the latter part
of his life, he removed to lodgings over the Great Piazza
in Covent Garden, where he carried on his lectures with
great success till the time of his death in 1749>at 66 years
of age.

He was a member of several foreign academies, and cor-
responding member of the Royal Academy of Sciences at
Paris ; from which academy he obtained the prize, pro-
posed by them for the best account of electricity. He
communicated a multitude of curious and valuable papers
to the Royal Society, from the year 1714 to 1743, or from
vol. 29 to vol.42.

Besides those numerous communications, he published
a valuable Course of Experimental Philosophy, 1734, in 2
large vols. 4to. ; and gave an edition of David Gregory's
Elements of Catoptrics and Dioptrics, with an Appendix on
Reflecting Telescopes, 8vo, 1735. This appendix contains
some Original Letters that passed between Sir Isaac New-
ton and* Mr. James Gregory, relating to those telescopes.
He gave also an English translation of Gravesande's Na-
tural Philosophy, in 2 large volumes in 4to, 1747. A son
of Dr. D. was long an officer in the corps of Royal Artil-
lery, was a great favourite with his majesty, and died at
an advanced age, colonel of one of the then four regiments
or battalions of that corps, about the year 1775.

Recollecting all these favourable circumstances; that
Dr. D. was much noticed by tho learned and the great,
and possessed two church livings, wc cannot account for

D E S

[ 407 3

D E S

She circumstance of his dying in obscurity and indigence :
yet such it would seem was the case ; at least, if we may
credit the poet Cawthorne, who, speaking of the neglected
Desagullers, asks,

** How he who taught two gracious kings to view
All Boyle ennobled, and all Bacon knew,
Died in a cell, without a friend to save,
Without a guinea, and without a grave."
DESCARTES. See Cartes.

DESCENDING, a going or moving from above, down-
wards.— There are ascending and descending stars, and
ascending and descending degrees, &c.

Descending Latitude, is the latitude of a planet in its
return from the nodes to the equator.

DESCENSION, in Astronomy, is either Right, or 0-
blique.

Right Descension, is a point, or arch, of the equa-
tor, which descends with a star, or sign, below the hori-
zon, in a right sphere ; and

Oblique Descension, is a point, or arch, of the equa-
tor, which descends at the same time with a star, or sign,
below the horizon, in an oblique sphere.

Descensions, both right and oblique, are counted from
the first point of Aries, or the vernal intersection, accord-
ing to the order of the signs, i.e. from west to east. And,
as they are unequal, when it happens that they answer to
equal arcs of the ecliptic, as for example to the 12 signs of
the zodiac, it follows, that sometimes a greater part of the
equator rises or descends with a sign, in which case the
sign is said to ascend or descend rightly: and sometimes
again, a less part of the equator rises or sets with the same
sign, in which case it is said to ascend or descend ob-
liquely. See Ascension.

Refraction of the Descension. See Refraction.
Descensional Difference, is the difference between the
right and oblique descension of the same star, or point of
the heavens.

Descent, or Fall, in mechanics, &c, is the motion, or
tendency, of a body towards the centre of the earth,
either directly or obliquely. Bodies may be considered
either as freely descending, as in a vacuum, or as resisted
by some external force, as an opposing body, or a fluid
medium, &c. 1st, If the body 6 descend freely, and per-
pendicularly, by the force of gravity; then the motive
force urging it downwards, is equal to its whole weight 6;
and the quantity of matter being b also, the accelerative

force, will be — or I.
b

2dly, If the body b, descending, be opposed by some

mechanical power, suppose a wedge or inclined plane,

that is, instead of pursuing the perpendicular line of

gravity, it is made to descend in a sloping direction down

the inclined plane : then if the natural sine of the angle

the plane makes with the horizon be s, the motive force

urging the body down the plane will be bs; and therefore

the accelerative force is— ors; which is less than in the
b

former case in the proportion of .s to 1.

3dly, A body immersed in a fluid, loses as much of its

weight, as is equal to the weight of a like bulk of the

fluid ; and when descending, it loses the same, besides the

obstruction arising from the cohesion of the parts of the

medium, and the opposing force of the particles struck,

which last produces a greater or less resistance, according

to the velocity of the motion. But, the weight of the
body being b, and that of a like bulk of the fluid medium
m, the motive force urging the body to descend, is only
b — m ; that is, the body only falls by the excess of its-
weight above that of an equal bulk of the fluid Hence

the power that sustains a body in a medium, is equal to
the excess of the absolute weight of the body above an
equal bulk of the medium. Thus, a piece of copper
weighing 47|lb, loses 5 ^lb. of its weight in water: and
therefore a power of 42lb, will sustain it in the water.

4thly, If two bodies have the same specific gravity, the
less the bulk of the descending body is, the more of its
gravity does it lose, and the slower does it descend, in the
same medium. For, though the proportion of the specific
gravity of the body to that of the fluid be still, the same,
whether the bulk be greater or less, yet the smaller the
body, the more the surface is, in proportion to the mass;
and the more the surface, the more the resistance of the
parts of the fluid, in proportion.

5thly, If the specific gravities of two bodies be different;
that which has the greatest specific gravity will descend
with greater velocity in the air, or resisting medium, than
the other body. Thus, a ball of lead descends swifter
than wood or cork, because it loses less of its weight,
though in a vacuum they both fall equally swift.

The cause of this descent, or tendency downwards, has
been greatly controverted. Two opposite hypotheses have
been advanced ; the one, that it proceeds from an inter-
nal principle, and the other from an external one: the first
is maintained by the Peripatetics, Epicureans, and the
Newtonians ; and the latter by the Cartesians and Gassen-
dists. See also Acceleration.

Laws of Descent of Bodies.
1st, Heavy bodies, in an unresisting medium, fall with
an uniformly accelerated motion. For, it is the nature
of all constant and uniform forces, such as that of gra-
vity at the same distance from the centre of the earth, to
generate or produce equal additions of velocity in equal
times. So that, if in one second of time there be pro-
duced a certain velocity, in 2 seconds there will be dou-
ble that velocity, in 3 seconds triple, and so on, the de-
gree or quantity of velocity being always proportional to
the length of the time.

2d, The space descended by an uniform gravity, in any
time, is just the half of the space that might be uniformly
described in the same time by the last acquired velocity
at the end of that time, if uniformly continued. For, as
the velocity increases uniformly in- an arithmetic progres-
sion, the whole space descended by the variable velocity,
will be equal to the space that would be described wilh
the middle velocity uniformly continued for the same
time; and this again will be only half the space thatwould
be described with the last velocity, also uniformly conti-
nued for the same time, because the last velocity is dou- •
ble of the middle velocity, being produced in a double
time.

3d, The spaces descended by an uniform gravity, in
different times, are proportional to the squares of the
times, or to the squares of the velocities. For the whole
space descended in any time, consists of the. sums of all
the particular spaces, or velocities, which are in arithme-
tical progression ; but the sum of such an arithmetical
progression, beginning at 0, and having the last term and
the number of terms the same quantity, is equal to half

D E S

[ 408 ]

D E T

the square of the last term, or of the number of terms ;
therefore the whole sums are as the squares of the times,
or of the velocities.

This theory of the descents by gravity was first disco-

space a heavy body will descend through in one second ;
the very same as before.

4th, For any other constant force, instead of the per-
pendicular free descent by gravity, find by experiment,

vered and tau°ht by Galileo, who afterwards confirmed or otherwise, the space descended in one second by that

the same by experiments; which have often been repeated force, and substitute that instead of 16-jJj. for the value of

in various ways by many other persons since his time, as gin these formula; : or, if the proportion of the force to

Grimaldi, Riccioli, Huyoens, Newton, and many others, - the force of gravity be known, let the value of g be al-

all confirming the same

The experiments of Grimaldi and Riccioli were made
by dropping a number of balls, of half a pound weight,
from the tops of several towers, and measuring the times
of falling by a pendulum. Ricciol. Almag. Nov. torn. 1,
lib. 2, cap. 21, prop. 4. An abstract of their experiments
is exhibited here below:

Vibrations
of the

pendulum

The

time

Space at the
end of the

Sp.ce
descended
each time

5

0

50

Rom. feet
10

Rom. feet
10

10

1

40

40

30

15
20

2
3

30
20

90
lGo

50

70

25

4

10

250

90

b'

1

O

15

15

12

2

0

6'0

45

18

3

0

135

75

24

4

0

240

105

The space descended by a heavy body in any given
time, being determined by experiment, is sufficient, in
connexion with the preceding theorems, for determining
every inquiry concerning the times, velocities, and spaces
descended, depending on a uniform force of gravity. From
many accurate experiments made in England, it has been

lered in the same proportion, and the same formulae will
still hold good. So, if the descent be on an inclined
plane, making, for instance, an angle of 30° with the ho-
rizon; then, the force of descent upon the plane being
always as the sine of the angle it make.s with the horizon,
which in the present case will be as the sine of 30°, that
is, as \ the radius ; therefore the, value of g will be but
half the former, 8JT, in all the foregoing formulae.

Or, if one body descending perpendicularly draw an-
other after it, by means of a cord sliding over a pulley ;
then it will be, as the sum of the two bodies is to the de-
scending body, so is 16^ to the value oPg in this case;
which value of it being used in the said formula?, they
will still hold good. And in like manner for any other
constant forces whatever.

5th, The time of the oblique descent down any chord
of a circle, drawn either from the uppermost point or
lowermost point of the circle, is equal to the perpendicu-
lar descent through the diameter of the circle.

6th, The descent, or vibration, through all arcs of the
same cycloid are equal, whether great or small.

7th, But the descent, or vibration, through unequal
arcs of a circle, are unequal ; the times being greater in
the greater arcs, and less in the less.

8th, For Descents by Forces that are variable, see
Forces, &c. See also Inclined Plane, Cycloid,
Pendulum, &c.

Line of Swiftest Descent, is that which a body, falling

found that a heavy body descends freely through ] 6 feet , t[]e ac[ion of ; describes in the shortest time pos.
1 inch, or lo^ feet, in the first second of time; and con- jjjie from one eiv

sequently, by theorem 2, the velocity gained at the end of
1 second, is 32-J- feet per second. Hence, by the same,
and theorem 3, the velocity gained in any other time t is
32^, and the space descended is l6^C. So that, if v
denote the velocity, and s the space due to the time t,
and there be put g = 16-^; then is

v = 2gt = Zt/gs = — .

= ge =

= ifo.

en point to another. And this line, it
is proved by philosophers, is the arc of a cycloid, when
the one point is not perpendicularly over the other. See
Cycloid.

DESCRIBENT, a term in Geometry, signifying a line
or superficies, by the motion of which a superficies or
solid is described.

DETENTS, in a clock, are those stops which, by be-
ing lifted up or let down, lock and unlock the clock in
striking.

DETENT-W7<ee/, or Hoov-Wheel, that wheel in a clock
which has a hoop almost round it, in which there is a va-

The experiments with pendulums give also the same cancy, where the clock locks,
space for the descent of a heavy body in a second of time. DETERMINATE Number. See Number.
Thus, in" the latitude of London, it is found by experi- Determinate Problem, is that which has but one so-
raent, that the length of a pendulum vibrating seconds is lution, or a. certain limited number of solutions; in con-
nearly 39| inches; and it being known that the time of tradistinction to an indeterminate problem, which admits
one vibration of any pendulum,' is to the time in which a of infinite solutions. Such, for Wstanee, is the problem,
heavy body will fall through half the length of the pen- To form an isosceles triangle on a given line, so that each
dulum, as the circumference of a circle is to its diame- o{ tllt- :111Sll's :lt the bllbc sliaH be. double "' rl>at »t the
i vertex ; which has only one solution: or this, '1 o find an
ter, therefore as 3-1 4lC : 1 : : 1 : ^^ which is the time isosccics t,.jangle whose area and perimeter are given ;
of descending through 1.9T9y inches, or half the length of which admits of two solutions. But if it had only been
the pendulum ; and "the spaces being as the squares of the required to describe an isosceles triangle on a given base,
, l „ „ „ -. then it is evident that an infinite number of such triangles
times, we have as ^^ : I2 : : 10^ : l9r% x 3"14l62 might havc becn con?tructed) am| therefore, the problem

= 193 inches, or l6" feet 1 inch, which therefore is the would havc been an indeterminate one.

D E V

[ 409 _ ]

DEW

Determinate Section, the name of a Tract, or General
Problem, written by the ancient geometrician Apollonius.
None of this work has come down to us, excepting some
extracts, and an account of it by Pappus, in the preface
to the 7th book of his Mathematical Collections. He
there says that the general problem was, " To cut an in-
finite right line in one point so, that, of the segments con-
tained between the point of section sought, and given
points in the said line, either the square on one of them,
or the rectangle contained by two of them, may have a
given ratio, either to the rectangle contained by one of
them and a given line, or to the rectangle contained by
two of them."

Pappus further informs us, that this Tract of Apollo-
nius was divided into two books; that the first book con-
tained 6 problems, and the second 3; that the 6 problems
of the first book contained 16 epitagmas, or cases, respect-
ing the dispositions of the points ; and the second book Q.
Further, that of the epitagmas of the 6 problems of the
first book, 4 were maxima, a.nd one a minimum : that the
maxima are at the 2d epitagma of the 2d problem, at the
3d of the 4th, the 3d of the 5th, and the 3d of the 6th ;
but that the minimum was at the 3d epitagma of the 3d
problem. Also, that the second book contained three de-
terminations ; of which the 3d epitagma of the 1st pro-
blem, and the 3d of the 2d were minima, and the 3d of
the 3d a maximum. Moreover, that the first book had
27 lemmas, and the 2d book 24 ; and lastly, that both
books contained 83 theorems.

From such account of the contents of this Tract, and
the lemmas, also given by Pappus, several persons have
attempted to restore, or recompose what they thought
might be nearly the form of Apollonius's tract, or the sub-
ject of each problem, case, determination, &c ; among
whom are, Snellius, an eminent Dutch mathematician of
the 17th century; a translation of whose work was pub-
lished in English by Mr. John Lawson, in 1772, together
with a new restoration of the whole work by his friend
Mr. William Wales. A more complete restoration is
given in Dr. Simson's posthumous works, in 1776 ; as also
by P. Giannini, at Parma, 1773.

DEVIATION, in Astronomy. See Nutation.

Deviation of a falling body from its perpendicular di-
rection. When the notion of the revolution of the earth
about its axis was revived by Copernicus, various objec-
tions were started against it by the adherents of the old
system ; and among others, it was asserted that, if the
earth did really turn on its 6wn axis, it would have fol-
lowed as a natural consequence, that a stone or other
heavy body, being let fall from an eminence, as the top of
a tower, must have struck the earth at a considerable di-
stance from its foot towards the west; for as the motion of
the earth was from west to east, they conceived that it
must necessarily have passed through a space towards the
east, greater or less, according to the height of the tower,
or to the time the body was in falling, before the latter
reached the ground ; and as, from experiments, this was
tound not to be the case, they considered it as an unan-
swerable objection to the Copernican system. But as
soon as the laws of motion became better known, and the
composition of forces was properly understood, it was
tound that, contrary to the opinions of those objectors,
instead of the stone falling considerably to the westward
of i he tower, it ought to fall a little to the east of it, which
is what experiments have since confirmed to be the case.

Vol. I.

And this is what is now generally understood by the de-
viation of a falling body.

This phenomenon is the natural consequence of the first
law of motion, namely, that every body has a tendency to
preserve its state, whether it be in motion or at rest. Thus,
a bullet let fall from the topmast of a ship, will fall in a
direction parallel to the mast, whatever may be the velo-
city of the vessel, and to those on board it will have the
appearance of a perpendicular descent, notwithstanding its
real path is that of the curve of a parabola, the same as
if it had been projected from a state of rest with a velo-
city equal to that of the ship.

With respect to a body falling from the top of a tower,
the only difference is that, by reason of the revolution of
the earth, the higher a body is elevated, the greater will be
its velocity, because it describes a greater circle in the sainji
time; therefore the body having previously to its fall ac-
quired a greater velocity, from west to east, than the foot
of the tower has, it will naturally be carried a little to-
wards the east. Thus the same circumstance that was
considered as an unanswerable objection to the modern
system of astronomy, may be now regarded as an addi-
tional evidence of its truth.

This fact is confirmed by experiment. M. Guglielmini
was the first who drew the attention of philosophers to
this subject. On causing bodies to fall from a height of
241 feet, he found a deviation of 8 lines to the east from
the vertical. M. Benzenberg, professor of physics and
astronomy at Dusseldorp, made 2S experiments with balls
well turned and polished, which were dropped from a
height of 262 French feet, and which at a medium pro-
duced 5 lines of deviation to the east: the theory gives 4*6
lines. And the last experiments made at Bologna, by Gu-
glielmini, give nearly the same results. Yet so many are
the causes of inaccuracy in the experiments, and so small
the deviation required by the theory, that we hesitate in
laying much stress on the results of these philosophers.
Of the truth of the theory, however, there can be no doubt :
we may therefore here notice M. Laplace's ultimate theo-
rem. Let h denote the height from which the body falls,
g = the space fallen by gravity in the first second, a = the
angle of the earth's rotation in the same time, at the rate
of 360° -4- 0'99727 in a day, and c= cosine of the lati-
tude of the place, also d the deviation towards the east :

then is the deviation d = ^ach^/-. (Bulletin des Scien-
ces, N° 75.) — In this theorem, the value of the angle a is
15"'0411, or nearly 15'; and as a small arc is nearly
equal to its sine or tangent, therefore the value of a is the
sine of 15" to radius 1; that is, a = -000072725, the log.
of which is 5,86l666l, which will be useful in computing
by the above theorem.

DEW, a thin light insensible mist, or rain, ascending
with a slow motion, and falling while the sun is below the
horizon. To us it appears to differ from rain, as less from
more. Its origin and matter are doubtless from the va-
pours and exhalations that rise from the earth and water.
See Exhalation. Some define it a vapour liquefied,
and let fall in drops. M. Huet, in one of his letters,
asserts that, dew does not fall, but rises ; and others have
adopted the same opinion.

M. du Fay made several experiments, first with glasses,

then with pieces of cloth stretched horizontally at different

heights ; and he found that the lower bodies, with their

under surfaces, were wetted before those that were placed

3G

DEW

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D I A

higher, or their upper surfaces. And Du Fay and Mus-
chenbroek both found, that different substances, and even
different colours, receive the dew differently, and some
little or not at all.

From the principles laid down under the article Eva-
poration, the several phenomena of dews are easily ac-
counted for. Such as, for instance, that dews are more
copious in the spring, than in the other seasons of the
year; there being then a greater stock of vapour in readi-
ness, than at other times, by reason of the small expense
of it in the winter's cold and frost. Hence it is too, that
Egypt, and some other hot countries, abound with dews
throughout all the heats of summer ; for the air there be-
ing too hot to constipate the vapours in the day-time, they
never gather into clouds; and hence they have no rain :
lut in climates that are excessively hot, the nights are re-
markably cold ; so that the vapours raised after sun-set,
are readily condensed into dews.

It is natural to conclude, from the different substances
which are combined with dew, that it must be either salu-
tary or injurious, both to plants and animals.

It is not easy to ascertain the quantity of dew that rises
every night, or in the whole year, because of the winds
which disperse it, the rains which carry it down, and other
inconveniences : but it is known that it rises in greater
abundance after rain than after dry weather, and in warm
countries than in cold ones. There are some places in
which dew is observed only to ascend, and not to fall : and
others again in which it is carried upwards in greater plenty
than downwards, being dispersed by the winds.

Dr. Hales made some experiments, to determine the
quantity of dew that falls in the night; and for this pur-
pose, on the 15th of August, at 7 in the evening, he filled
two glazed earthen pans with moist earth ; the dimensions
of the pans being, 3 inches deep, and 12 inches diameter:
and he observes, that the moister the earth, the more dew
falls on it in a night; and that more than a double quan-
tity of dew falls on a surface of water, than on an equal
surface of moist earth. These pans increased in weight,
by the night's dew, ISO grains; and decreased in weight,
by the evaporation of the day, 1 oz 2S2 grs : so that 540
grains more are evaporated from the earth every 24 hours
in summer, than the dew that falls in the night ; i. e. in
21 days near 26 ounces from a circular area of a foot di-
ameter. Now if ISO grains of dew, falling in one night
on such an area, which is equal to 113 square inches, be
equally spread on the surface, its depth will be the 159th
part of an inch. He likewise found that the depth of
dew in a winter's night was the 90th part of an inch. If
therefore we allow 159 nights for the extent of the sum-
mer's dew, it will in that time amount to one inch in
depth ; and reckoning the remaining 206 nights for the
extent of the winter's dew, it will produce 2'28 inches
depth ; and the dew of the whole year will amount to
328 inches depth. But the quantity which evaporated
in a fair summer's day from the same surface, being 1 oz
and 282 grs, gives the 40th part of an inch deep for eva-
poration, which is 4 times as much as fell at night. Dr.
Hales observes, that the evaporation of a winter's day is
nearly the same as in a summer's day ; the earth's greater
moisture in winter compensating for the sun's greater heat
in summer. Ilales's Vegetable Statics, vol.1, p. 52 of
the 4th edit. Sec Evaporation.

Signer Beccaria made several experiments to demon-
strate the existence of the electricity that is produced by

dew. He observes in general, that such electricity took
place in clear and dry weather, during which no strong
wind prevailed ; and that it depends on the quantity of the
dew, as the electricity of the rain depends on the quantity
of the rain. He sometimes found that it began before
sun-set; at other times not till 11 o'clock at night. Arti-
ficial Electricity, Appendix, letter 3.

DE WIT (John), the famous Dutch pensionary, was
born at Dort, in l6"25 ; where he prosecuted his studies so
diligently, that at 23 years of age he published Elements
Curvarum Linearum, one of the deepest books in mathe-
matics at that time. After taking his degrees, and tra-
velling, he, in 1650, became pensionary of Dort, and di-
stinguished himself very early in the management of public
affairs, which soon after raised him to the rank of pen-
sionary of Holland. After rendering the greatest benefits
to his country in many important instances, and serving it
in several high capacities, with the greatest ability, dili-
gence, and integrity, by some intrigues of a party, it is
said, he and his brother were thrown into prison, from
whence they were dragged by the mob, and butchered
with the most cruel and savage barbarity.

D1ACAUSTIC Curve, or the Caustic by Refraction, is a
species of caustic curves, the genesis of which is in the fol-
lowing manner. Imagine an infinite number of rays ba,
em, bi), &c, issuing from the same luminous point B, re-
fracted to or from the perpendicular mc, by the given
curve amd ; and so that ce, the sines of the angles of in-
cidence cme, be always to cg the sines of the refracted
angles cmg, in a given ratio; then the curve hfn that
touches all the refracted rays ah, mf, dn, &c, is called
the Diacaustic, or Caustic by Refraction.

N
5/

DIACOUSTICS, or Diaphonics, the consideration
of the properties of sound refracted in passing through
different mediums ; that is, out of a denser into a more
subtile, or out of a more subtile into a denser medium.
See Sound.

DIAGONAL, is a right line drawn across a figure, from
one angle to another; and is sometimes called a diameter.
It is used chiefly in quadrilateral figures, viz, in parallelo-
grams and trapeziums.

1. Every diagonal, as AC, divides a B_
parallelogram into two equal parts or \
triangles abc, adc.

2. Two diagonals, ac, bd, drawn in
a parallelogram, bisect each Other; as a
ill the point e.

3. Any line, as FG, drawn through the middle of the
diagonal of a parallelogram, is bisected by it at the point
e; and it divides the parallelogram into two equal parts,
Bl'GC and AFCD.

4. The diagonal of a square is incommensurable with its
side.

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t> I A

5. The complements about the diagonal of any paral-
lelogram are equal to each other.

6. In any parallelogram, the sum of the squares of the
four sides is equal to the sum of the squares of the two
diagonals.

7. In any trapezium, the sum of the squares of the four
sides is equal to the sum of the squares of the two dia-
gonals, together with 4 times the square of the distance be-
tween the middle points of the diagonals.

8. In any trapezium, the sum of the squares of the two
diagonals, is double the sum of the squares of two lines bi-
secting the two pairs of opposite sides.

9- In any quadrilateral inscribed in a circle, the rect-
angle of the two diagonals is equal to the sum of the two
rectangles under the two pairs of opposite sides.

Diagonal Scale. See Scales.

DIAGRAM, is a scheme for the explanation or demon-
stration of any figure, or of its properties.

DIAL, or Sun-Dial, an instrument for measuring time
by means of the sun's shadow. Or, it is a draught or de-
scription of certain lines on the surface of a body, so that
the shadow of a style, or ray of the sun through a hole,
should touch certain marks at certain hours.

Sun-Dials are doubtless of great antiquity. But the
first upon record is, it seems, the dial of Ahaz, who began
to reign 400 years before Alexander, and within 12 years
of the building of Rome : it is mentioned in Isaiah, chap.
38, ver.S.

Several of the ancients are spoken of, as the construc-
tors of dials; as Anaximenes Milesius, Thales, &c. Vi-
truvius mentions one made by the ancient Chaldee histo-
rian Berosus, on a reclining plane, almost parallel to the
equator. Aristarchus Samius invented the hemispherical
dial. And there were at the same time some spherical
ones, with a needle for a gnomon. The discus of Aristar-
chus was an horizontal dial, with its rim raised up all
around, to prevent the shadow from stretching too far.

It was late before the Romans became acquainted with
dials. The first sun-dial at Rome was set up by Papyrius
Cursor, about the 460th year of the city; before which
time, Pliny says, there is no mention of any accountof time
but by the sun's rising and setting: the first dial was set
up near the temple of Quirinus ; but being inaccurate,
about 30 years after, another was brought out of Sicily
by the consul M. Valerius -Messala, which he placed on a
pillar near the Rostrum ; but neither did this show time
truly, because not made for that latitude; and, after using
it 99 years, Martius Philippus set up another more exact.

The diversity of sun-dials arises from the different si-
tuation of the planes, and from the different figure of the
surfaces on which they are described ; whence they be-
come denominated Equinoctial, Horizontal, Vertical, Po-
lar, Direct, Erect, Declining, Inclining, Reclining, Cy-
lindrical, &c. For the general principles of their con-
struction, see Dialling.

Dials are distinguished into Primary and Secondary.

Primary Dials, are such as are drawn either on the
plane of the horizon, and thence called horizontal dials ;
or perpendicular to it, and called vertical dials; or else
drawn on the polar and equinoctial planes, though neither
horizontal nor vertical.

Secondary Dials, are all those that are drawn on the
;ilanes of other circles, besides those last mentioned; or
those which either decline, incline, recline, or deincline.

Each of these again is divided into several others, as:

Equinoctial Dial, is that which is described on an
equinoctial plane, or one parallel to it.

Horizontal Dial, is described on an horizontal plane,
or a plane parallel to the horizon. — This dial shows the
hours from sun-rise to sun-set.

South Dial, or an Erect, Direct South Dial, is that de-
scribed on the surface of the prime vertical circle facing
towards the south. — This dial shows the time from 6 in
the morning till 6 at night.

North D14.L, or an "Erect, Direct North Dial, is that
which is described on the surface of the prime vertical
pointing northward. This dial only shows the hours be-
fore 6 in the morning, and after 6 in the evening.

East Dial, or Erect, Direct East Dial, is that drawn on
the plane of the meridian, directly to the east. — This can
only show the hours till 12 o'clock.

West Dial, or Erect, Direct West Dial, is that described
on the western side of the meridian. — This can only show
the hours after noon. Consequently this, and the last pre-
ceding one, will show all the hours of the day between them.
Polar Dial, is that which is described on a plane pass-
ing through 'the poles of the world, and the east and west
points of the horizon. It is of two kinds; the first di-
rected towards the zenith, and called the upper; the lat-
ter, down towards the nadir, called the lower. The polar
dial therefore is inclined to the horizon in an angle equal
to the elevation of the pole. — The upper polar dial shows
the hours from 6 in the morning till 6 at night, and the
lower one shows the hours before 6 in the morning, and
after 6 in the evening, viz, from sun-rise and till sun-set.

Declining Dials, are erect or vertical dials which de-
cline from any of the cardinal points; or they are such as
cut either the plane of the prime vertical, or of the hori-
zon, at oblique angles.

Declining dials are of very frequent use; as the walls of
houses, on which dials are mostly drawn, commonly de-
viate from the cardinal points.

Of declining dials there are several kinds, which are de-
nominated from the cardinal points which they are nearest
to; as decliners from the south, and from the north, and
even from the zenith.

Inclined Dials, are such as arc drawn on planes not
erect, but inclining, or leaning forward towards the south,
or southern side of the horizon, in an angle, either greater
or less than the equinoctial plane.

Reclining Dials, are those drawn on planes not erect,
but reclined, or leaning backwards from the zenith to-
wards the north, in an angle greater or less than the polar
plane.

Dcinclined Dials, are such as both decline and incline,
or recline. — These last three kinds of dials are very rare.

Dials without Centres, are those whose hour lines con-
verge so slowly, that the centre, or point of their con-
course, cannot be expressed on the given plane.
Quadrantal Dial. See Horodictkal Quadrant.
Reflecting Dial. See Reflecting Dial.
Cylindrical Dial, is one drawn on the curve surface of
a cylinder. This may first be drawn on a paper plane,
and then pasted round a cylinder of wood, &c. It will
show the time of the day, the sun's place in the ecliptic,
and his altitude at any time of observation.

There are also Portable Dials, or on a Card, and Uni-
versal Dials on a Plain Cross, &c.

Refracted Dials, are such as show the hour by means
of some refracting transparent fluid.

3G2

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RingDiAL, is a small portable dial, consisting of a
brass ring or rim, about 2 inches in diameter, and one-
third of an inch in breadth. In a point of this rim there
is a hole, through which the sun-beams pass, and form a
bright speck in the concavity of the opposite semi-circle,
which gives the hour of the day in the divisions marked
within it.

When the hole is fixed, the dial only shows true about
the time of the equinox. But to have it perform through-
out the whole year, the hole is made moveable, the signs
of the zodiac, or the days of the month, being marked on
the convex side of the ring ; hence, in using it, the move-
able hole is set to the day of the month, or the degree of
the zodiac the sun is in ; then suspending the dial by the
little ring, turn it towards the sun, and his rays through
the hole will show the hour on the divisions within side.

Universal, or Astronomical Ring-DiA.z., is a ring dial
which shows the hour of the day in any part of the earth ;
whereas the former is confined to a certain latitude. Its
figifre see represented below.

It consists of two rings or flat circles, from 2 to 6 inches
in diameter, and of a proportionable breadth &c. The
outward ring a represents the meridian of any place you
are at, and contains two divisions of go degrees each,
diametrically opposite to each other, the one serving from
the equator to the north pole, the other to the south pole.
The inner ring represents the equator, and turns exactly
within the outer, by means of two pivots in each ring at
the hour of 12.

Across the two circles is placed a thin reglet or bridge,
with a cursor c, sliding along the middle of the bridge, and
having a small hole for the sun to shine through. The
middle of this bridge is conceived as the axis of the world,
and the extremities as the poles: on the one side are
drawn the signs of the zodiac, and on the other the days
of the month. On the edge of the meridian a piece slides,
to which is fitted a small ring to suspend the instrument by.

In this dial, the divisions on the axis are the tangents of
the angles of the sun's declination, adapted to the semi-
diaincter of the equator as radius, and placed on either
side of the centre : but instead of laj ing them down from
a line of tangents, a scale of equal parts may be made, of
which 1000 shall answer exactly to the length of the semi-
axis, from the centre to the inside of the equinoctial ring;
and then 43+ of these parts may be laid down from the
centre towards each end, which will limit all divisions on

the axis, because 434 is the natural tangent of 23° 28r*
And thus, by a nonius fixed to the sliding piece, and ta-
king the sun's declination from an ephemeris, and the tan-
gent of that declination from the table of natural tangents,
the slider may be always set true within 2 minutes of a
degree. This scale of 434 equal parts might be placed
directly against the 23° 28' of the sun's declination, on the
axis, instead of the sun's place, which is there of little use.
By this means the slider might be set in the usual way, to
the day of the month, for common use ; or to the natural
tangent of the declination, when great accuracy is re-
quired.

To use this Dial: Place the line a (on the middle of the
sliding piece) over the degree of latitude of the place, as
for instance 5 If degrees for London: put the line which
crosses the hole of the cursor to the degree of the sign, or
day of the month. Open the instrument so as that the
two rings be at right angles to each other, and suspend it
b}' the ring H, that the axis of the dial, represented by the
middle of the bridge, may be parallel to the axis of the
earth. Then turn the flat side of the bridge towards the
sun, so that his rays, striking through the small hole in
the middle of the cursor, may fall exactly on a line drawn
round the middle of the concave surface of the inner ring ;
in which case the bright spot shows the hour of the day in
the said concave surface of the ring.

Nocturnal or Night-DiAh, is that which shows the hour
of the night, by the light, or shadow projected from the
moon or stars.

Lunar or Moon Dials may be either purposely described
and adapted to the moon's motion ; or the hour may be
found on a sun-dial by the moon shining upon it, thus:
Observe the hour which the shadow of the index points at
by moon-light ; find the days of the moon's age in the ca-
lendar, and take 3-4ths of that number, for the hours to
be added to the time shown by the shadow, to give the
hour of the night. The reason of which is, that the moon
comes to the same horary circle later than the sun by about
three quarters of an hour every day ; and at the time of
new moon the solar and lunar hours coincide.

Dial Planes, are the plane superficies on which the
hour lines of dials are drawn.
Tide Dial. See Tide Dial.

DIALLING, the art of drawing dials on any kind of
surface, whether plane or curved.

Dialling is wholly founded on the first motion of the
heavenly bodies, and chiefly the sun; or rather on the
diurnal rotation of the earth : so that the elements of sphe-
rics, and spherical trigonometry, should be understood,
before a person attempts to enter upon the doctrine of
dialling.

The principles of dialling may be aptly deduced from,
and illustrated by, the phenomena of a hollow or trans-
parent sphere,as of glass. Thus, suppose ttrcp to repre-
sent the earth as transparent; and 'its equator ;is divided
into 24 equal parts, by so many meridian semicircles a,
b,c, d, e, &c, one of which is the geographical meridian
of any given place, as London, which it is supposed is at
the point o; and if the hour of 12 were marked at the
equator, both on that meridian and the opposite one, and
all the rest of the hours in order on the other meridians,
those meridians would be the hour circles of London : he-
, cause, as the sun appears to move round the earth, which
is in the centre of the visible heavens, in 24 hours, he will
pass from one meridian to another in one hour. Then, if

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D I A

the sphere had an opake axis, as PEp, terminating in the
poles p and p, the shadow of the axis, which is in the

same plane with the sun and with each meridian, would
fall upon every particular meridian and hour, when ,the
sun came to the plane of the opposite meridian, and would
consequently show the time at London, and at all other
places on the same meridian. If this sphere were cut
through the middle by a solid plane abcd in the rational
horizon of London, one half of the axis ep would be above
the plane, and the other half below it ; and if straight
lines were drawn from the centre of the plane to those
points where its circumference is cut by the hour circles
of the sphere, those lines would be the hour lines of an
horizontal dial for London; for the shadow of the axis
would fall upon each particular hour line of the dial, when
it fell upon the like hour circle of the sphere.

If the plane which cuts the sphere be upright, as afcg,
touching the given place, for example, London, at F, and

directly opposite to the meridianof London, it will then be-
come the plane of an erect direct south dial; and if right
lines be drawn from its centre E, to those points ol its circum-
fefence where the. hour circles of the sphere cut it, these
will be the hour lines of a vertical or direct south dial for
London, to which the hours are to be set in the figure,
contrary to those on an horizontal dial ; and the lower half
v.p of the axis will cast a shadow on the hour of the day
in this dial, at the same time that it would fall upon the
like hour circle of the sphere, if the dial plane was not in
the way.

If the plane, still facing the meridian, be made to in-
cline, or recline, any number of degrees, the hour circles
of the sphere will continue to cut the edge of the plane in
those points to which the hour lines must be drawn straight
from the centre; and the axis of the sphere will cast a
shadow on these lines at the respective hours. The like
will still hold, if the plane be made to decline by any num-
ber of degrees from the meridian towards the east or west ;
provided the declination be less than QO degrees, or the
inclination be less than the co-latitude of the place; and
the axis of the sphere will be the gnomon: otherwise, the
axis will have no elevation above the plane of the dial, and
cannot be a gnomon.

Thus it appears that the plane of every dial represents
the plane of some great circle on the earth, and the gnomon
the earth's axis ; the vertex of a right gnomon the centre
of the earth or visible heavens; and the plane of the dial
is just as far from this centre as from the vertex of the
stile. The earth itself, compared with its distance from
the sun, is considered as a point ; and therefore, if a small
sphere of glass be placed upon any part of the earth's sur-
face, so that its axis be parallel to the axis of the earth,
and the sphere have such lines upon it, and such planes
within it, as above described ; it will show the hours of the
day as truly as if it were placed at jhe earth's centre, and
the shell of the earth were as transparent as glass. Fergu-
son, lect. 10.

The principal writers on Dials, and Dialling, are the fol-
lowing: Vitruvius, in his Architecture, cap. 4 and 7, lib.
g: Sebastian Munster, his Horolographia: John Dryander
de Horologiorum varia Compositione : Conrade Gesner's
Pandectse: Andrew Schoner's Gnomonicaj; Fred. Com-
mandine de Horologiorum Descriptione : Joan. Bapt. Bc-
nedictus de Gnomonum Umbrarumque Solarium Usu :
Joannes Georgius Schomberg, Exegesis Fundamentorum
Gnomonicorum : Solomon de Caus, Traile des Horologes
Solaires: Joan: Bapt. Trolta, Praxis Horologiorum : De-
sargues, Maniere Universelle pour poser l'Essieu et placer
les Heurcs et autres Choses aux Cadrans Solaires : Ath.
Kircher, Ars magna Lucis et Umbrae: Hallum, Explicatio
Horologii in Horto Regio Londini : Trat:tatus Horologi-
orum Joannis Mark: Clavius, Gnomonices de Horologiis;
in which he demonstrates both the theory and the opera-
tions afterthe rigid manner of the ancient mathematicians:
Dechales, Ozanam, and Schottus, gave much easier trea-
tises on this subject; as did also Wolfius in his Elementa:
M. Picard gave a new method of making large dials, by
calculating the hour lines; and M. Lahire, in his Dial-
ling, printed in l6"83, gave a geometrical method of draw-
ing hour lines from certain points, determined by observa-
tion. Everhard Walper, in 1025, published his Dialling,
in which he lays down a method of drawing the primary
dials on very simple principles; and the same principles
arc also described at length by Sebastian Minister, in his
Rudimenta Mathematica, published in 1 651. In 1672,
Sturmius published a new edition of Walper's Dialling,
with tlie addition of a whole second part, concerning in-
clining and declining dials, &c. In 1708, the same work
was re-published, with Sturmius's additions, and likewise
a 4th part, containing Picard's and Lahire's methods of
drawing large dials, which forms much the best and ablest
work on the subject. Paterson, Michael, and Muller,
have each written on Dialling, in the German language:
Coetsius, in his Iiorologiographia Plana, printed in lGSty:

D I A

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D I A

Gauppen, in his GnomonicaMechanica: Lcybourn, in his
Dialling: Bion, in his Use of Mathematical Instruments:
Wells, in his Art of Shadows. There is also a treatise by
M. Deparceux, 1740. Mr. Ferguson has also written on
the same subject, in his Lectures on Mechanics; besides
Emerson, in his Dialling; and Mr. W. Jones, in his In-
strumental Dialling. See also my Recreations, vol. 3.

Universal Dialling Cylinder, is represented in fig. 2,
plate ix, where abcd is a glass cylindrical tube, closed at
both ends with brass plates, in the centres of which a wire
or axis efg is fixed. The tube is either fixed to an hori-
zontal board n, so that its axis may make an angle with
the board equal to that which the earth's axis makes with
the horizon of any given place, being thus parallel to the
axis of the earth; or it may be made to move on a joint,
and elevated for any particular latitude. The 21 hour
lines are drawn with a diamond on the outside of -the glass,
equidistant from each other, and parallel to the axis. The
xn next b stands for midnight, and the xn next the board
h for noon. When the axis of this instrument is elevated
according to the latitude, and the board set level, with the
line hn in the plane of the meridian, and the end towards
the north ; the axis ErG, when the sun shines on the in-
strument, will serve as a gnomon or stile, and cast a sha-
dow on the hour of the day among the parallel hour lines.
As the plate ad at the top is parallel to the equator, and
the axis efg perpendicular to it, right lines drawn from
the centre to the extremities of the parallels will be the
hour lines of an equinoctial dial,, and the axis will be the
stile. An horizontal plate efput down into the tube, with
lines drawn from the centre to the several parallels, cutting
its edge, will be an horizontal dial for the given latitude;
and a vertical plate gc, fronting the meridian, and touching
the tube with its edge, with lines drawn from its centre to
the parallels, will be a vertical south dial : the axis of the
instrument serving in both cases for the stile of the dial:
and if a plate be placed within the tube, so as to decline,
incline, or recline, by any given number of degrees, and
lines be drawn, as above, a declining, inclining, or recli-
ning dial will be formed for the given latitude. If the axis
with the several plates fixed to it be drawn out of the tube,
and set up in sunshine in the same position as they were in
the tube, ad will be an equinoctial dial, ef an horizontal
dial, and gc a vertical south dial ; and the time of the day
will be shown by the axis efg. If the cylinder were wood,
instead of glass, and the parallel lines drawn upon it in the

square, and f the axis of the cylinder would be its stile.
Thus also inclining, declining, or reclining dials might be
easily constructed, for any given latitude. Ferguson, ubi
supra.

Dialling Globe, is an instrument made of brass, or
wood, with a plane fitted to the horizon, and an index ;
particularly contrived to draw all sorts of dials, and to give
a clear exhibition of the principles of the art.

Dialling Lines, or Scales, are graduated lines, placed
on rules or the edges of quadrants, and other instruments,
to expedite the construction of dials. The principal of
these lines are, 1. A scale of six hours, which is only a
double tangent, or two lines of tangents each of 45 de-
grees, joined together in the middle, and equal to the
whole line of sines, with the declination set against the
meridian altitudes in the latitude of London, suppose, or
any place for which it is made : the radius of which line
of sines is equal to the dialling scale of six hours. 2. A
line of latitudes, which is fitted to the hour scale, and is
made by this canon : as the radius is to the chord of go
degrees; so are the tangents of each respective degree of
the line of latitudes, to the tangents of other arcs: and.
then the natural sines of those arcs are the numbers, which,
taken from a diagonal scale of equal parts, will graduate
the divisions of the line of latitude to any radius. The line
of hours and latitudes is generally for pricking down all
dials with centres. For the method of constructing these
scales, see Scale.

Dialling Sphere, is an instrument made of brass, with
several semicircles sliding over each other, on a moveable
horizon, to demonstrate the nature of the doctrine of
spherical triangles, and to give a true idea of the drawing
of dials on all lands of planes.

DIAMETER, of a Circle, is a right line passing through
the centre, and terminated at thecircumference on both sides.

The diameter divides the circumference, and the area
of the circle, into two equal parts. And half the diameter,
or the semi-diameter, is called the radius.

For the proportion between the diameter and the cir-
cumference of a circle, see Circle and Circumfe hence.

Diameter, of a Conic Section, or Transverse Diameter,
is a right-line passing through the centre of the section, or
the middle of the axis. — The diameter bisects all ordi-
nates, or lines drawn parallel to the tangent at its vertex.
See Conic Sections.

Conjugate Diameter, is a diameter, in Conic Sections,

ame manner, it would serve to facilitate the operation of parallel to the ordinatcs of another diameter, called tin

making these several dials. The upper plate with lines
drawn to the several intersections of the parallels, which
appears obliquely in fig. 2, would be an equinoctial dial as
in fig. 3, and the axis perpendicular to it be its stile. An
horizontal dial for the latitude of the elevation of the axis
might be made, by drawing out the axis and cutting the
cylinder, as at efgh, parallel to the horizontal board h ;
the section would be elliptic as in fig. 4. A circle might
be described on the centre, and lines drawn to the divi-
sions of the ellipse would be the hour lines; and the wire
put in its place again, as e, would be the stile. If ibis cy-
linder were cut by a plane perpendicular to the horizontal
board ii, or to the line sun, beginning at g, the plane of
the section would be elliptical as in fig. 5, and lines drawn
to the points of intersection of the parallels on its edge
would be the hour lines of a vertical direct south dial,
which might be made of any shape, either circular or

transverse; or parallel to the tangent at the vertex of this
other.

Diameter, of any Curve, is a right line which divides
two other parallel right lines, in such manner that, in each
of them, all the segments or ordinatcs on one side, between
the diameter and different points of the curve, are equal to
all those on the other side, according to Newton's defini-
tion of a Diameter.

But some define a diameter to be that line, whether
right or curved, which bisects all the parallels drawn from
one point to another of a curve. So that in this way every
curve will have a diameter; and hence the curves of the
2d order have, all of them, either a right-lined diameter,
oi else the curves of some one of the conic sections for
diameters. And many geometrical curves of the higher
orders, may also have for diameters, curves of more infe-
rior orders.

D I A

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D I F

Diameter of Gravity, is a right line passing through
the centre of gravity.

Diameter, in Astronomy. The diameters of the hea-
venly bodies are either apparent, i. e. such as they appear
to the eye; or real, i. e. such as they are in themselves.

The apparent diameters are best measured with a mi-
crometer, and are estimated by the measure of the angle
they subtend at the eye. These are 'different in different
circumstauccs and parts of the orbits, or according to the
various distances of the luminary; being in the inverse
ratio of the distance.

The sun's vertical diameter is found by taking the
height of the upper and lower edge of his disk, when he is
in the meridian, or near it; correcting the altitude of each
edge on account of refraction and parallax; then the dif-
ference between the true altitudes of the two, is the true
apparent diameter sought. Or the apparent diameter may
be determined by observing, with a good clock, the time
which the sun's disc takes in passing over the meridian:
and here, when the sun is in or near the equator, the fol-
lowing proportion may be used ; viz, as the time between
the sun's leaving the meridian and returning to it again,
is to 360 degrees, so is the time of the sun's passing over
the meridian, to the number of minutes and seconds of a
degree contained in his apparent diameter: but when the
sun is in a parallel at some distance from the equator, his
diameter measures a greater number of minutes and se-
conds in that parallel than it would do in a great circle,
and takes up proportionally more time in passing over the
meridian; in which case we have this proportion: as ra-
dius is to the cosine of the sun's declination, so is the
time of the sun's passing the meridian reduced -to minutes
and seconds of a degree, to the arc of a great circle which
measures the sun's apparent horizontal diameter. See
Transit.

The sun's apparent diameter may also be taken by the
projection of his image in a dark room. There are seve-
ral ways of finding the apparent diameters of the planets;
but the most certain method is that with the micrometer.
In these diameters considerable errors were made by the
more ancient astronomers; but, according to the best of
the modern observations, those diameters are as below :
viz, the sun's diameter is found to vary from 32' 3S"6 in
January, to 3l' 33"8 in July: the apparent diameter of
the moon varies from about 29' 28" to 33' 36"; her real
diameter being about 2180 miles. The apparent diame-
ters of the planets, when at their respective mean distances
from the earth, are as follow :

Mercury, Venus", Mars, Jupiter, Saturn, Herachel,
11" 58" 27" 39" 18" 3" 54."'.

And from these apparent diameters, with the respective
.distances from the Earth, the real diameters of the sun and
planets have been determined in English miles as below:
Mercury, Venn,, Mars, Jupiter, Saturn, Herschcl, Sun,
3224 7687 4189 89170 7904'2 35112 883246.
Observations on the planets Herschel/Saturn, Jupiter,
Mars, and the Earth, prove that there is a sensible diffe-
ence between their equatorial and polar diameters; and
it is probable that there is a like difference between the
diameters of the other planets, but this has not yet been
determined by observation. The mean apparent diameters
• it the planets, as seen from the sun, have been thus given:
Mercury, Venus, Eartli, Mars, Jupiter, Saturn, Herschel,

20" 30" 17" 10" 37" 16" 4".

Diameter of a Column, is its thickness just above the

base. From this the module is taken, which measures all
the other parts of the column.

Diameter of the Diminution, is that taken at the top
of the shaft.

Diameter of the Swelling, is that taken at the height
of one third from the base.

DIAPASON, a musical interval, otherwise called an
octave, or eighth; so called, because it contains all the
possible diversities of sound. If the lengths of two strings
be to each other as 1 to 2, the tensions being equal, their
tones will produce an octave.

DIAPENTE, is- the perfect fifth, or second of the con-
cords, making an octave with theDiatessaron. The lengths
of the chords are as 3 to 2.

DIAPHANOUS Body or Medium, one that is trans-
lucent, or through which the rays of light easily pass ;
as water, air, glass, talc, fine porcelain, &c. See Trans-
parency.

DIAPHONICS, is sometimes used for the science of re-
fracted sound, as it passes through different mediums.

DIASTYLE, a sort of edifice, in which the pillars stand
at such a distance from one another, that three diameters
of their thickness are allowed for the intercolumniation.

DIATESSARON, is the perfect fourth; or a musical
interval, consisting of one greater tone, one lesser, and one
greater semitone. The lengths of strings to sound the di-
atessaron, are as 3 to 4.

DIATONIC, a term signifying the ordinary soit of mu-
sic, which proceeds by tones or degrees, both ascending
and descending. It contains or admits only the greater
and lesser tone, and the greater semitone.

DIESIS, a division of a tone, less than a semitone ; or
an interval consisting of a lesser, or imperfect semitone.
The Diesis is the smallest and softest change, or inflexion,
of the voice imaginable. It is also called a feint, and is
expressed by a St. Andrew's cross, or saltier.

DIFFERENCE, is the excess by which one magnitude
or quantity exceeds another. When a less quantity is
subtracted from a greater, the remainder is otherwise
called difference.

Ascensional Difference. See Ascensional.

Difference of Latitude, in Geography, &c, is an arch
of the meridian included between the parallels of latitude
in which any two places lie. When the latitudes are both
north or both south, the less lat. taken from the greator
leaves the difference of latitude; but when the one lat. is
north, the other south, their sum shows the dif. of lat.

Difference of Longitude, of two places, is an arch of
the equator contained between the meridians of those two
pliices, or the measure of the angle formed by their me-
ridians. When the longitudes are both east or both west,
the one taken from the other leaves the dif. of longitude;
but when the one is east and the other west, they are then
to. be added together for the dif. of longitude.

DIFFERENTIAL, an indefinitely small quantity, part,
or difference. By some, the Differential is considered as
infinitely small, or less than any assignable quantity; and
also as of the same import as fluxion. It is called a Dif-
ferential, or Differential Quantity, because, often consi-
dered as the difference between two quantities ; and as
such it is the foundation of the Differential Calculus.
Newton useil the term Moment in a like sense, as being the
momentary increase or decrease of a variable quantity. M.
Leibnitz and others call it also an infinitesimal.

D I

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Sec

]

D I F

Differential of the 1st, 2d, id, $c, degree.
Different 10 -Differential.

Differential Calculus, is a method of differencing
quantities. See Differential Method, Calculus, and
Fluxions.

Differentio-Differential Calculus, is a method
of differencing differential quantities. As the sign of a dif-
ferential is the letter d prefixed to the quantity, as dx the
differential of x ; so tiiat of a differential of dx is ddx, and
the differential of ddx is dddx, &c ; similar to the fluxions
x, x, x, &c. Thus we have degrees of differentials. The
differential of an ordinary quantity, is a differential of the
first order or degree, as dx; that of the 2d degree is ddx ;
that of the 3d degree is dddx, &c. The rules for diffe-
rentials, are the very same as those for fluxions. See
Fluxions.

Differential, in Logarithms. Kepler calls the loga-
rithms of tangents, Differentiales ; which we usually call
artificial tangents.

Differential Equation, is an equation involving
or containing differential quantities; as the equation
3x2dx — laxdx -+- aydx -+■ axdy = 0. Some mathemati-
cians, as Stirling, &c, have also applied the term differen-
tial equation in another sense, to certain equations de-
fining the nature of series.

Differential Method, a method of finding quanti-
ties by means of their successive differences. This method
is of very general use and application, but especially in
the construction of tables of numbers, and the summation
of series, &c. It was first used, and the rules of it laid
down, by Briggs, in his Construction of Logarithms and
other Numbers, much the same as they were afterwards
taught by Cotes, in his Constructio Tabularum per Diffe-
rentias ; as is shown in the Introduction to my Logs,
pa. 69 et seq. See Briggs's Arithmetica Logarithmica,
cap. 12 and 13, and his Trigonometria Bri'tannica.

The method was next treated in another form by New-
ton, in the 5th Lemma of the 3d book cf\his Principia, and
in his Methodus Differentialis, published by Jones in 1711,
with the other tracts of Newton. This author here treats
it as a method of describing a curve of the parabolic kind,
through any given number of points. He distinguishes
two cases of this problem; the first, when the ordinates
drawn from the given points to any line given in position,
are at equal distances from one another ; and the 2d, when
these ordinates are not at equal distances. He has given
a solution of both cases, at first without demonstration,
which was afterwards supplied by himself and others: see
his Methodus Differentialis above mentioned ; and Stir-
ling's Explanation of the Newtonian Differential Method,
in the Philos. Trans. N° 36"2 ; Cotes, Dc. Methodo Dif-
ferential Newtoniana, published with his Harmonia Men-
surarum ; Herman's Phoronomia; and Lc Scur and Jac-
quier, in their Commentary on Newton's Principia. It
may be observed, that the methods there demonstrated,
by some of these authors, extend to the description of any
algebraic curve through a given number of points, which
Newton, writing to Leibnitz, mentions as a problem of the
greatest use.

By this method, some terms of a series being given, and
conceived as placed at given intervals, any intermediate
term may be found ; which therefore gives a method for
interpolations. Briggs's Arith. Log. ubi supra ; Newton,
Mcth. Differ, prop. 5 ; Stirling, Methodus Differentialis.

Thus also may any curvilinear figure be squared nearly,
having some few of its ordinates. Newton, ibid. prop. 6 ;
Cotes De Method. Differ.; Simpson's Mathematical Dis-
sert, pa. 115. And thus may mathematical tables be con-
structed by interpolations : Briggs, ibid. Cotes Cano-
notechnia. The successive differences of the ordinates of
parabolic curves, becoming ultimately equal, and the in-
termediate ordinate required being determined by these
differences of the ordinates, is the reason for the name
Differential Method.

To be a little more particular.— The first case of New-
ton's problem amounts to this: A series of numbers,
placed at equal intervals, being given, to find any inter-
mediate-number of that series, when its interval or distance
from the first term of the series is given. Subtract each
term of the series from the next following term, and call
the remainders first differences; then subtract, in like man-
ner each of these differences from the next following one,
calling these remainders 2d differences ; again, subtract
each 2d difference from the next following, for the 3d dif-
ferences ; and so on : then if a be the 1st term of the series,
d' the first of the 1st differences,
d" the first of the 2d differences,
d'" the first of the 3d differences, &c;
and if jc be the interval or distance between the first term
of theseries and any term sought, T,thatis, let the number of
terms from a to t, both included, be = x + 1 ; then will
the term sought, T, be =

X „ X X — 1 ,„ X X— 1 X— 2 ,„, -

AH d H . d H . . a , &C.

112 12 3

Hence, if the differences of any order become equal,
that is, if any of the diffs. d", d'", &c, become = 0, the
above series will give a finite expression £or t, the term
sought; it being [evident, that the series must terminate
when any of the diffs. d", d'", &c, become = 0.

It is also evident that the co-efficients - , - . ,

1 1 a '

&c of the differences, are the same a6 the terms of the bi-
nomial theorem.

For Example, Suppose it were required to find the log.
tangent of 5' l" 12'" 24"", or 5' l"/^, or 5' l"'2066, &c.

Take out the log. tangents to several minutes and se-
conds, and take their first and second differences, as
below :

5' 0"

5 1 - . ■- 5

5 2 - - 5

5 3 - - .*!

Here a = 7' 1 6-1 141 7; x =
mean second difference d" = -

Tang.

7-l6'26'96'4
7-1641417
7-1655821
7'l6'70l7S

14453
14404
14357

d"

—49
47

n

xa"

d' = 14404; and the
Hence

- - 71641417

- - - 2977

Theref. the tang, of 5' 1" 12'" 24"" is 7" 1644398

Hence may be deduced a method of finding the sums
of the terms of such a series, calling its terms a, d, cp ,
&c. For, conceive a new series having its 1st term = 0,
its 2d = a, its 3d = a -+■ b, its 4th = a h- b -h C, its
5th = A + B + C+ r>, and so on ; then it is plain that
assigning one term of this series, is finding the sum of all
the terms a, is, c, d, &c. Now since these terms are the
differences of the sums, 0, a, a ■+■ 11, a ■+■ « ■+■ c, &c ;

DIG [4

and as some of the differences of a, b, c, &c, are = 0 by
supposition ; it follows that some of the differences of the

sums will be = 0 ; and since in the series a -i d' -+-

i

— . d" &c, by which a term was assigned, a repre-
sented the 1st term ; d' the 1st of the 1st differences, and
x the interval between the first term and the last; we are
to write 0 instead of a, a instead of d', d! instead of d", d"
instead of d'", &c, also x -+• 1 instead of x ; which being

done, die series expressing the sums will be 0 +

X -+- 1 X + 1 X „ X + 1 X 1-1™ ,

a -i- . —a -+• • — . d , ccc.

1 1 2 12 3

Or, if the real number of terms of the lines be called s,
that is, if z = x -+- 1 , or x = z — 1 , the sum of the series

will be az -+- — .- — d' -+- -. .- — —d" &c. See De

12 12 3

Moivre's Doct. of Chances, pa. 59, 60; or his Miscel.
Analyt. pa. 153; or Simpson's Essays, pa. 95.

For Example. To find the sum of six terms of the se-
ries of squares 1 -t- 4 -t- 9 -t- 16 -t- 25 -H 36, of the natural
numbers.

17 ]

D I G

Terms
1

4

.9

16

25

Here a = 1, d' = 3, d" = 2, d'" &c = 0, and z = 6 ;
therefore the sum is 6 •+• £, £. 3 -+- $,-§, f . 2 =6 "-+■ 45. -^
40 = 91 the sum required, viz, of 1 -+- 4 -1- 9 -+- 16 -+- 25
+ 36.

A variety of examples may be seen in the places above
cited, or in Stirling's Methodus Differentialis, &c.

As to the Differential method, it may be observed, that
though Newton and some others have treated it as a method
of describing an algebraic curve, at least of the parabolic
kind, through any number of given points; yet the con-
sideration of curves is not at all essential to it, though it
may help the imagination. The description of a parabo-
lic curve through given points, is the same problem as the
finding of quantities from their given differences, which
may always be done by algebra, by the resolution of
simple equations. See Stirling's Method. Differ, pa. 97.
This ingenious author has treated very fully of the diffe-
rential method, and shown its use in the solution of some
difficult problems. See also Series.

Differential Scale, in Algebra, is used for the scale
of relation subtracted from unity. See Recurring Series.

DIFFRACTION, a term first used by Grimatdi, to
denote that property of the rays of light, which others
have called Inflection ; the discovery of which is attributed
by some to Grimaldi, and by others to Dr. Hooke.

D1GBY [Sir Ken elm), a celebrated English philoso-
pher, was born at Gothurst in Buckinghamshire, 1603.
He was the descendant of an ancient family : his great-
grandfather, with six of his brothers, fought valiantly
at Bosworth-field on the side of Henry 7th, against
Richard the 3d. His father, Sir Everard, engaged in the
gunpowder plot against James the 1st, for which he was
beheaded ; but the son was restored to his estate, and had
afterwards several appointments under King Charles the
1st, who granted him letters of reprisal against the Vene-
tians, from whom he took several prizes with a small fleet
which he commanded. He fought the Venetians near the

Vol.1.

port of Scanderoon, and bravely made his way through
them with his booty.

In the beginning of the civil wars, he exerted himself
greatly in the king's cause, and was afterwards imprisoned,
by order of the parliament; but was set at liberty in 1643.
He afterwards compounded for his estate ; but being
banished from England, he retired to France, and was
sent on two embassies to Pope Innocent the 10th, from
the queen, widow to Charles the 1st, whose chancellor he
then was. On the restoration of Charles the 2d, he re-
turned to London ; where he died in 1(565, at 6*2 years
of age.

Digby was a great lover of learning, and translated
several authors into English, as well as published several
works of his own; as, 1. Observations upon Dr. Brown's
Religio Medici, 1643. — 2. Observations on part of Spen-
ser's Fairy Queen, 1644.— 3. A Treatise of the Nature of
Bodies, 1644. — 4. A Treatise declaring the Operations
and Nature of Man's Soul, out of which the Immortality
of reasonable souls is evinced : works that discover great
penetration and extensive knowledge.

He applied much to chemistry ; and found out several
useful medicines, which he distributed with a liberal hand.
He particularly distinguished himself by his sympathetic
powder for the cure of wounds at a distance; his discourse
concerning which made great noise for a while. He held
several conferences with Descartes, about the nature of
the soul, and the principles of things. At the beginning
of the Royal Society, he became a distinguished member,
being one of the first council. And he had at his own
house regular levees or meetings of learned men, to im-
prove themselves in knowledge, by conversing with one
another.

This eminent person was, for the early pregnancy of his
talents, and his great proficiency in learning, compared
to the celebrated Picus de Mirandola, who was one of the
wonders of human nature. Yet his knowledge, thouoh
various and extensive, probably appeared greater than it
really was; as he had all the powers of elocution and ad-
dress to recommend it. He knew how to shine in a circle,
either of ladies or philosophers ; and was as much attended
to when he spoke on the most trivial subjects, as when he
spoke on the most important. It has been said that one
of the princes of Italy, who had no child, was desirous
that his princess should bring him a son by Sir Kenelm,
whom he esteemed a just model of perfection.

DIGGES (Leonard), an ingenious mathematician in
the l6th century, was descended from an ancient family,
and born at Digges-court in the parish of Barham in Kent ;
but in what year is not known ; and died about the year
1574. He was educated for some time at Oxford, where
he laid a good foundation of learning ; but retiring from
thence, he prosecuted his studies, and became an excellent
mathematician, a skilful architect, and an expert sur-
veyor of land, &c. He composed several books : as, 1.
Tectonicum: briefly showing the exact Measuring, and
speedy Reckoning of all manner of Lands, Squares, Tim-
ber, Stones, Steeples, &c ; 1 556, 4to. ; this was augmented
and published again by his son Thomas Digges, in 1592 ■
and also reprinted in 1647. — 2. A Geometrical Practical
Treatise, named Puntometria, in three books, which ho
left in manuscript ; but after his death, his son supplied
such parts of it as were obscure and imperfect, and pub-
lished it in 1 5.9 1 , folio; subjoining, " A Discourse Geo-
metrical of the live regular and Platonic bodies, containing
3 11

D I G

[ 418 ]

D I M

sundry theoretical and practical propositions, arising by
mutual conference of these solids, Inscription, Circum-
scription, and Transformation." — 3. Prognostication Ever-
lasting of right good effect: or Choice Rules to judge the
weather by the Sun, Moon, and Stars, &c;in4to, 1555,
1556, and 1564 : corrected and augmented by his' son,
with divers general tables, and many compendious rules,
in4to, 1592.

DIGGES (Thomas), only son of Leonard Digges,
after a liberal education from his tenderest years, went and
studied for some time at Oxford ; and by the improve-
ments he made there, and the subsequent instructions of
his learned father, became one of the best mathematicians
of his age. When Queen Elizabeth sent some forces to
assist the oppressed inhabitants of the Netherlands, Mr.
Digges was appointed muster-master-general of them ; by
which he became well skilled in military affairs; as his
writings afterwards showed. He died in 1595.

Mr. Digges, beside revising, correcting, and enlarging
some pieces of his father's, already mentioned, wrote and
published the following learned works himself: viz, 1. Ala?
sive Seals Mathematical: or Mathematical Wings or
Ladders, 1573, 4to. A book which contains several de-
monstrations for finding the parallaxes of any comet, or
other celestial body, with a correction of the errors in the
use of the radius astronomicus.— 2. An Arithmetical
Military Treatise, containing so much of Arithmetic as is
necessary towards military discipline, 1579i 4to. — 3. A
Geometrical Treatise, named Stratioticos, requisite for
the perfection of Soldiers, 1579, 4to. This was begun by
his father, but finished by himself. They were both re-
printed together in 1590, with several additions and amend-
ments, under this title: " An Arithmetical Warlike
Treatise, named Stratioticos, compendiously teaching the
Science of Numbers, as well in Fractions as Integers, and
so much of the'Rules and Equations- Algebraical, and Art
of Numbers Cossical, as are requisite for the profession of
a souldier. Together with the Moderne militaire disci-
pline, offices, lawes, and orders in every well-governed
campe and armic, inviolably to be observed." At the
end of this work there are two pieces ; the first, " A brieie
and true report of the proceedings of the Earle of Ley-
cester, for the reliefe of the towne of Sluce, from his ar-
rival at Vlishing, about the end of June 1587, untill the
surrendrie thereof 26 Julii next ensuing. Whereby it
shall plainelie appear, his excellencie was not in anie fault
for the losse of that towne :" the second, " A briefe dis-
course what orders were best for repulsing of foraine
forces, if at any time they should invade us by sea in
Kent, or elsewhere." — 4. A perfect Description of the
Celestial Orbs, according to the most ancient doctrine of
the Pythagoreans, &c. This was placed at the end of his
father's " Prognostication Everlasting, &c." printed in
1592, 4to. — 5. A humble Motive for Association to main-
tain the religion established, 1601, 8vo. To which is
added, his Letter to the same purpose to the archbishops
and bishops of England.— 6. England's Defence: or, A
Treatise concerning Invasion. This is a tract of the same
nature with that printed at the end of his Stratioticos,
and called, A briefe Discourse, &c. It was written in
1599, but not published till l686. — 7. A Letter print-
ed before Dr. John Dee's ParallaticaJ Commentationis
praxeosque nucleus quidam, 1573, 4to. — Beside these,
and his Nova Corpora, he left several methematical
treatises ready for the press ; which, by reason of law-

suits and other avocations, he was hindered from pub-
lishing.

If our author was great in himself, he was not less so
in his son, Sir Dudley Digges, so celebrated as a politician
and elegant writer; and also in his grandson Dudley.

DIGIT, in Arithmetic, one of the teti characters 0,
1, 2, 3, 4, 5, 6, 7, 8, 9, by means of which all numbers
are expressed.

Digit, in Astronomy, is the measure by which the
part of the luminaries in eclipses is estimated, being the
12th part of the diameter of the luminary. Thus, an
eclipse is said to be of 10 digits, when 10 parts out of J 2
of the diameter are in the eclipsed part; when the whole
of the luminary is exactly covered, the digits eclipsed are
just 12; and when the luminary is more than covered,
as often happens in lunar eclipses, then more than 12
digits are said to be eclipsed : Thus, if the diameter or
breadth of the earth's shadow, where the moon passes
through, be equal to one diameter and a half of the moon,
then 18° or digits are said to be eclipsed. These digits
are by Wolfius, and some others, called Digiti Ecliptici.

DILATATION, a motion of the parts of a body by
which it expands, or opens itself, so as to occupy a greater
space. — Many authors confound dilatation with rarefac-
tion; but the more accurate writers distinguish between
them ; defining dilatation to be the expansion of a body
into a greater bulk, by its own elastic power; and rare-
faction, the like expansion produced by means of heat.

The moderns have observed, that bodies which, after
being compressed, and again left at liberty, restore them-
selves perfectly, do endeavour to dilate themselves with
the same force by which they are compressed ; and ac-
cordingly they sustain a force, and raise a weight equal to
that with which they are compressed.

Again, bodies, in dilating by their elastic power, exert
a greater foice at the beginning of their dilatation, than
towards the end ; as being at first more compressed; and
the greater the compression, the greater the elastic power
and endeavour to dilate. So that these three, the com-
pressing power, the compression, and the elastic power,
are always equal.

Finally, the motion by which compressed bodies restore
themselves, is usually accelerated : thus, when compress-
ed air begins to restore itself, and dilate into a greater
space, it is still compressed ; and consequently a new im-
petus is still impressed upon it, from the dilatative cause;
and the former remaining, with the increase of the cause,
the effect, that is the motion and velocity, must be increa-i
sed also. Indeed it may happen, that whcre.the com-
pression is only partial, the motion of dilatation shall not
be accelerated, but retarded ; as is evident in the com-
pression of a spunge, soft bread, gauze, &c.

DIMENSION, the extension of a body, considered as
measurable. Hence, as we conceive a body extended,
and measurable in length, breadth, and depth, dimension is
considered as threefold, viz, length, breadth, and thick-
ness. So a line has one dimension only, viz, length ; a
superficies two, length and breadth ; and a body or solid
has three, viz, length, breadth, and thickness.

Dimension is also particularly used with regard to
the powers of quantities in equations. Thus, in a simple
equation x = a ■+■ li, the unknown quantity is only of one
dimension; in a quadratic equation, .ra = «2 ■+- &% it is of
two dimensions ; in a cubic i3 = a3 -t- P, it is of three di-
mensions ; and so on.

D I O [

DIMINISHED Angle, in Fortification. See Angle.

DIMINUTION, in Music, is the abating something of
the fuIL value or quantity of any note.

Diminution, in Architecture, is a contraction of the
upper part of a column, by which its diameter is made less
than that of the lower part.

DINOCRATES, a celebrated ancient architect of Ma-
cedonia, of whom several extraordinary things are related.
He was taken, by Alexander the Great, into Egypt,
where he employed him in marking out and building the
city of Alexandria : here he formed a design, in which
Mount Athos was to be laid out into the form of a man,
in whose left hand were to be represented the walls of a
great city, and all the rivers of the mount flowing into his
right, and from thence into the sea. Another memorable
instance of Dinocrates's architectonic skill, is his restoring
and building, in a more august and magnificent man-
ner than before, the celebrated temple of Diana at Epbe-
sus, after Herostratus, for the sake of immortalizing his
name, had destroyed it by fire. A third instance, more
extraordinary and wonderful than either of the former, is
related by Pliny in his Natural History ; who says he had
formed a scheme for building the dome of the temple of
Arsinoe at Alexandria of loadstone, in order that her image
made entirely of iron should hang in the middle of it, as
if it were in the air; but the king's death, and his own,
prevented the execution or attempt of this project.

DIOCLES, a Greek mathematician of some celebrity,
but more particularly remembered for his invention of the
curve called the cyssoid or the Cissoid of Diodes. See Cis-

SOID.

DIONYSIUS, the Periegetic, an ancient geographer
and poet. Pliny says, he was a native of the Persian
Alexandria, afterwards called Antioch, and at last Char-
rax ; that he was sent by Augustus, to survey the eastern
part of the earth. Dionysius wrote a great number of
pieces, enumerated by Suidas and his commentator Eu-
stathius : but his Periegesis, or Survey of the World, is the
only one now extant; which may be well esteemed one of
the most exact systems of ancient geography, since Pliny
himself proposed it as his pattern.

DIONYSIAN, or Victorian Period. See Period.

DIOPIIANTINE Problems, are certain questions re-
lating to square and cubic numbers, and to right-angled
triangles, &c ; the nature of which were first and chiefly
treated by Diophantus, in his Arithmetic, or rather Algebra.

In these questions, it is chiefly intended to find com-
mensurable numbers to answer indeterminate problems;
which often bring out an infinite number of incommensu-
rable quantities. For example, If it be proposed to find a
right-angled triangle, whose three sides x,y,z are expressed
by rational numbers; from the nature of the figure it is
known that xx ■+■ yx — z*, where z denotes the hypothenuse.
Now it is plain that x and y may also be so taken, that z
shall be irrational ; for ifx = \,a.ndy = 2, then is z —t/6.

Now the art of resolving such problems, consists in or-
dering the unknown quantity or quantities, in such a man-
ner, that the square or higher power may vanish out of the
equation, and then by means of the unknown quantity in
its first dimension, the equation may be resolved without
having recourse toincommensurables. For example, in the
equation above, x7, -+- y2 = I1, suppose z = x ■+- u, then .is
x' H-y2 = .r'J -+- 2xu -+- u2, out of which equation xz vanishes,.

and then it is/ = 2xu ■+ u1, which gives x =y~U : hence,

419 ] D I O

assuming^ and u equal to any numbers at pleasure, the
three sides of the triangle will be v, y ~". and y +— or

■2u ' 2tt '

out of fractions, 2uy, and y2 — uz, and yx + u2, which are
all rational whenever j/and aare rational. For example, if
y = 2, anda=l, then yz — uz — 3, and 2«y = 4, and
y* + «2= 5. It is evident that this problem admits of in-
finite numbers of solutions, as y or u may be assumed any
numbers at pleasure.

But, in questions of this kind, very few directions can
be given, as the method of solution must be left, in a oreat
measure, to the skill and ingenuity employed in the sub-
stitution &c, for unknown quantities. Thus, a person ac-
quainted with this branch of algebra, would have imme-
diately substituted in the above equation,

x = v' — u2, or x2 = -y — 2v2u2 ■+■ uA,

and y = 2vu, or y2 = ±v2u2,

hence z =:v2 ■+■ u2, or z2 =vi -+- 2v2u2 -+- u2.
where v and u may be taken at pleasure.

Or if the equation had been x~ — y2 — z2. Then by
making x2 = (zr -+- u2)2, and y2 = 4s»2 u2, we should have
z- = (vx — u2)2 : where v and u may be taken any num-
bers whatever, as before.

Ex. 2. To find the general values of x and y, such,

that x2 -+- y, and x' — y, maybe both complete squares.

Or, which is the same thing, to find three square numbers

in arithmetical progression.

First, put i! + i) = os) m, „ „ „ ,
r , z ■" ., > Then <2,x- = w- ■+■ z2,
and x — y = z- j '

or 2x2 — w2 = z2. Now make x = w -k-m, or 2x2 =.2zv2

-+- iwm -+- 2m2: hence zx = w2 + 4,wm + 2m2, which must

beacomplete square; therefore substitute z = %v ■+• r, and

we obtain w2 -t- 4tw« + 2m2 = w2 -+■ 2iw -+- ?•'',

or (4ot — 2r) w = r2 — 2m2 ; hence we have w = ;

4m— it '
r* — 2rm + 2m3 — r'+irm— 2n~

•IV •+■ 91 = 1 =

w ■+- r = z = ■

4m — 2r

Or, since the denominators are equal, we may make
w = r2 — 2m2, x =r2 — 2rm -+- 2m2, and z = — rx ■+■ 4m
— 2m2; where r and m may be assumed at pleasure, either
integral or fractional. In the first values ofw, x, and z ;
by assuming r = 6, and m — 4, we have w = 1, x = 5, and
2 = 7; and therefore

1, 25, and 49, the three square numbers required.
Now some of those general expressions being once ob-
tained, they may be frequently introduced to abreviate the
operations in more intricate questions. For an example
of which we will propose the following problem.

Ex. 3. To find the values of x and y, such, that unity
being added to each of them, as also to their sum, and
difference, the four numbers thus arising shall be square
numbers.

r x ■+■ i = r-

r,- , I V ■+■ 1 = n%

l'irst substitute < J , „2

) x + y -t- 1 =/>

Here, we see that the three squares s2, I2, and p2, are in
arithmetical progression, their common difference being y.
The question therefore is reduced to the finding of three
square numbers in arithmetical progression, whose com-
mon difference ■+■ I may be a square ; therefore from the
foregoing example let s2 = (r2 — 2»r)2, l2 = (r2 — 2nn ■+-
2m2)z, p2= ( — r2 ■+- 4n« — 2vir)2, which we know arc in
arithmetical progression, their common difference being
4r% — 12r*m* + 8n»3 tag ; and we have to find 4r*/».
3 112

D I O

[ 420 ]

D 1 O

— 12r-nr ■+• &rm2 -+- 1 a square. Substitute its root = 1
-+- 4rm3, and we thus obtain 1 SrV5 = 4r3m — 12rW; or di-
viding by 4,rzm, we have r = 4?ns ■+■ 3m : where m may be
assumed at pleasure. By taking m = 1, we haver = 7,
alsop2= 529, and '*= 1369; again/1— 1 =1368 = x,
and I2 —p2 = 840 =y, which are the two numbers sought,
r 1368 ■+- i = 372

, . > 840 + 1 = 292

the four squares being ^ 13g8 + 840 + x —472

( 1368 -840+1 = 232.

The operations in the solution of Diophantine problems
may also be frequently simplified, from certain known
properties of numbers, when they are included under pe-
culiar forms, particularly when the answer is required to
be civen in integers ; many ingenious specimens of which
maybe seen in the solution of Diophantine problems in the
New Series of the Mathematical Repository, by Mr. Ley-
bourn, of the Royal Military College.

Another method, which is frequently adopted to consi-
derable advantage, is by means of the development of
certain algebraical functions ; an example of which will
be seen in the following problem,

Ex.4:. To find a number, from which two given squares
being severally subtracted, each of the remainders may
be a square.

Theorem. Let ffla and b7, represent the given squares.
Resolve f a — f b into any two unequal factors m and m' ;
and * a — 4 b into any two unequal factors n and n : then
will {ni1 ■+- nz) . (w/z + n'-) be the number required.

Demonstration. For (m? ■+■ ra1) . {rri~ -+- n1)

f (mni -+- nra')1 -)- (mti — rmi)2

\or{miri — nn')z ■+■ (mn' -+- nm')'2,
as is immediately seen by the development of the above
formula;. And since \a -+- ib = mm', and \a — ±b = nri,
therefore a = mm'+nn'\~c ucntl the s uare of

and b = mm — nn J n - '

each being taking from the above product, will leave a
square remainder.

Suppose, for example, 4 and 36 were the two squares,

then-

4.1, and

= 2.1, therefore (4a ■+- 1*) ,

(22 -i- 1") = S5 the number sought ; for 85 — 4 = 8 1 = 9'i
and S5- 36 = 4.9 = 72.

For further information on this interesting branch of
algebra, the reader may consult Fermat, Bachet, Dzanam,
Kersey, Saunderson, and Euler, particularly the latter,
who in the 2d vol. of his Algebra has carried this subject
to a considerable length, which, with the additions by La
Grange, may be considered as one of the most complete
pieces of Diophantine algebra. But with regard to the
Theory of Numbers, which is intimately connected with
the above subject, the best works are Le Gendre's Essai
sur la Theorie des Nombres ; and the Recherchcs arith-
metiques par M. Gauss.

DIOPHANTUS, a celebrated mathematician of Alex-
andria, who has commonly been reputed to be the inven-
tor of algebra ; but notwithstanding his is the earliest
work extant on that science, there appears to be no reason
for supposing that he was the inventor of it: indeed the
work itself is almost a proof to the contrary ; for though
he does not mention any other authors in the course of
his book, yet he evidently treats the subject as one already
known. It is not certain when Diophantus lived. Some
have placed him before Christ, and some after, in the reigns
of Nero and the Anloinncs ; but all with equal uncertainty.

It seems he is the same Diophantus who wrote the 'Canon
Astronomicus, which Suidas says was commented on by
the celebrated Hypatia, daughter of Theon of Alexandria-
His reputation must have been very high among the an-
cients, since they ranked him with Pythagoras and Euclid
in mathematical learning. Bachet, in his notes upon the
5th book De Arithmeticis, has collected, from Diopliantus's
epitaph in the Anthologia, the following circumstances of
his life; namely, that he was married when he was 33
years old, and had a son born 5 years after ; that this son
died when he was 42 years of age, and that his father did
not survive him above 4 years ; from which it appears,
that Diophantus lived to the age of 84 years.

This epitaph as translated by Bachet is as follows :

Hie Diophantus habet tumulum, qui tempora vita;
Illius rnuta denotat arte tibi

Egit sextantem juvenis, lanugine malas
Vestire hinc ccepit parte duodecimal

Septante uxori post bsec sociatur, et anno
Formosus quinto nascitur inde puer

Sernissem aetatis postquam attigit ille pateruae,
Infelix subita morte peremptus obit

Quatuor asstates genitor lugere superstes
Cogitur; hinc annos illius assequere.

Diophantus wrote 13 books of Arithmetic, or AlgebraT
which Regiomontanus, in his preface to Alfraganus, tells
us, are still preserved in manuscript in the Vatican library :
indeed we are informed by Diophantus himself that his
work consisted of 13 books, viz, at the end of his address
to Dionysius, placed at the beginning of the work, and
hence Regiomontanus might be led to say the 13 books
were in that library; but no more than 6 whole books,
with part of a seventh, have ever been published ; and I
am of opinion there are no more in being; indeed Bom-
belli, in the preface to his Algebra, written 1572, says
there were but 6 of the books then in the library, and that
he and another person were about a translation of them.

Those 6 books, with the imperfect 7th, were first pub-
lished at Basil by Xylander in 1575, but in a Latin version
only, with the Greek scholia of Maximus Planudes on the
first two books, and observations of his own; the same
books were afterwards published in Greek and Latin at
Paris in l621, by Bachet, an ingenious and learned
Frenchman, who made a new Latin translation of the
work, and enriched it with learned commentaries ; and al-
though he did not entirely neglect the notes of Xylander
in his edition, yet he treated the scholiast Planudes with
the utmost contempt. He seems to intimate, in what he
says upon the 2Sth question of the 2d book, that the 6
books which we have of Diophantus, may be nothing more
than a collection made by some novice, of such proposi-
tions as he judged proper, out of the whole 13: butFahri-
cius thinks there is no just ground for such a supposition.
This work was afterwards (l670) revised and republished
by Fermat, who added many excellent notes in addition
to those of Bachet.

DIOPTliR, or Dioptra, the same with the index or
alhidade of an astrolabe, or oilier such instrument.

Diopt'ia was an instrument invented by Hipparchu'S,
which served for several use.s ; ;i s, to level water- courses ;
to take the height of towers, or places at a distance ; to de-
termine the places, magnitudes, and distances of the pla-
nets, &c.

DIOPTRICS, called also Anaclastics, is the doctrine of
refracted vision ; or that part of optics which explains the

D I P

[ «i ]

D I P

effects of light as refracted by passing through different
mediums, as air, water, glass, &c, and especially lenses.

Dioptrics is one of the most useful and pleasant of all
the human sciences ; as it brings the remotest objects
near hand, enlarging the smallest objects so as to show
their minute parts, and even giving sight to the blind ; and
all this by the simple means of the attractive power in
glass and water, causing the rays of light in their passage
through them to alter their course according to the differ-
ent substances of the med.ium ; whence it happens that
the objects seen through them, do, in appearance, alter
their magnitude, distance, and situation.

The ancients -have treated of direct and reflected vi-
sion ; but what we have of refracted vision, is very imper-
fect. J. Baptista Porta wrote a treatise on refraction, in
9 books, but without any great improvement. Kepler was
the first who succeeded in any great degree, on this sub-
ject ; having demonstrated the properties of spherical lenses
very accurately, in a treatise first published anno loll.
After Kepler, Galileo gave somewhat of this doctrine in
his Letters; as also an Examination of the Preface of Jo-
hannes Pena upon Euclid's Oplics, concerning the use of
optics in astronomy. Descartes also wrote a treatise on
Dioptrics, commonly annexed to his Principles of Philoso-
ph}', which is one of his best works ; where the true man-
ner of vision is more distinctly explained than by any for-
mer writer, and in which is contained the true law of re-

self relates, viz, that it being his Custom to finish, and
hang the needles of his compasses, before he touched them,
he always found that, immediately after the touch, the
north point would dip or decline downward, pointing in
a direction under the horizon; so that, to balance the
needle again, he was always obliged to put a piece of wax
on the south end, as a counterpoise. The constancy of
this effect led him, at length, to observe the precise quan-
tity of the dip, or to measure the greatest angle which the
needle would make with the horizon. This, in the year
1576, he found at London was 71° 50'. It is not quite
certain whether the dip varies, as well as the horizontal
direction, in the same place. Mr. Graham made a great
many experiments with the dipping-needle in 1723, and
found the dip between 74 and 75 degrees. Mr. Nairne,
in 1772, found it somewhat above 72°. And by many
observations made since that time at the Royal Society,
the medium quantity is 72°-J. Indeed, in 1812 it is only
stated at 70° 32' at the Royal Society. The trifling dif-
ference between the first observations of Norman, and the
last of Mr. Nairne and the Royal Society, lead to the opi-
nion that the dip is unalterable; and yet it may be diffi-
cult to account for the great difference between these and
Mr. Graham's numbers, considering the well-known accu-
racy of that ingenious gentleman. Philos. Trans, vol. 45,
pa. 279> vol. 6'2, pa. 476', vol. 69, 70, 71. Sec also the
description of a new dipping-needle, by Dr. Lorimer, in

fraction, which was first discovered by Snell, though the the Philosophical Transactions 1775 ; and in the supple-
name of the inventor is suppressed : here are also laid ment to Cavallo's Treatise on Magnetism.

jpp

down the properties of elliptical and hyperbolical lenses,
with the practice of grinding glasses. Dr. Barrow has
treated on Dioptrics in a very elegant manner, though ra-
ther too briefly, in his Optical Lectures, read at Cam-
bridge. Huygens's Dioptrics, is also an excellent work of
its kind. Molyneux's Dioptrics is a work rather heavy
and dull. Besidethese, there are several other treatises on
Dioptrics, such as Hartsoeker's Essai de Dioptrique. Che-
rubiu's Dioptrique Oculaire, et La Vision Parfaite. Da-
vid Gregory's Elements of Dioptrics. Traber's Nervus
Opticus. Zahn's Oculus Artificialis Teledioptricus. Dr.
Smith's Optics, which is a complete work of its kind.
Wollius's Dioptrics, contained in his Elementa Matheseos
Universalis. But especially the Treatise on Optics, and the
Optical Lectures of Newton, in whose experiments are
^contained far more discoveries than in all the former wri-
ters. Lastly, this science was perfected by Dollond's dis-
covery of the achromatic glasses, by which the colours are
obviated in refracting telescopes.

The laws of Dioptrics are delivered under the article
Refuacitox, Lens, &c ; and the application of them in
the construction of telescopes, miscroscopes, and other di-
optrical instruments, under the articles Telescope and
Microscope.

DIP of the Horizon. See Depression.

DippiNG-iVterf/f, or Indinntory Needle, a magnetical
needle, so hung, as that, instead of playing horizontally,
and pointing out north and south, one end dips, or inclines
to the horizon, and the other points to a certain degree of
elevation above it.

The inventor of the dipping-needle was Robert Nor-
man, a compass-maker ;,t Ratcliffe, about the year 15S0 :
this is not only testified by his own account, in his New
Attractive, but also by Dr. Gilbert, Mr. William Bur-
rowes, Mr. Henry Bond, and other writers of that time,
or soon after it. The occasion of the discovery he him-

It is certain however, from many experiments and ob-
servations, that the dip is different in different latitudes,
and that it increases in going northward. It appears from
a table of observations made with a marine dipping-nee-
dle of Mr. Nairne's, in a voyage towards the north pole,
in 1773, that

in latitude 60° IS' the dip was 75° 0',
in latitude 70 45 the dip was 77 52,
in latitude SO 12 the dip was 81 52, and
in latitude 80 27 the dip was 82 2£.
See Phipps's Voyage, pa. 122. See also the Observations
of Mr. Hutchins, made in Hudson's Bay and Straits, Phi-
los. Trans, vol. 65, pa. 129-

Burrowes, Gilbert, Ridley, Bond, &c, endeavoured to
apply this discovery of the dip to the finding of the lati-
tude; and Bond, going still farther, first of an}' proposed
finding the longitude by it; but for want of observations
and experiments, he could not go any length. Mr. Whis-
tou, being furnished with the further observations of Co-
lonel Windham, Ur. Halley, Mr. Pound, Mr. Cunning-
ham, M. Noel, M. Feuille, and his own, made great im-
provements in the doctrine and use of the dipping-needle,
brought it to more certain rules, and endeavoured in good
earnest to find the longitude by it. For this purpose, he
observes, JsJ;, That the true tendency of the north or south
end of every magnetic needle, is not to that point of the hori-
zon, to which the horizontal needle points, but. towards' an-
other, directly under it, in the same vertical, and in differ-
ent degrees under it, in different ages, and at different
places. 2dly, That the power by which the horizontal
needle is governed, and all our navigation usually directed,
it is proved is only one quarter of the power by which the
dipping-needle is moved ; which should render the latter
far the more effectual and accurate instrument. 3dly,
That a dipping-needle of a foot long will plainly show an>
alteration of the angle of inclination, in these parts of the'

n i r

[ 422 ]

D I R

world, in half a quarter of a degree, or 1\ geographical
miles; and a needle of 4 feet, in 2 or 3 miles; i.e. sup-
posing these distances taken along, or near a meridian.
4thlyrA dipping-needle, 4 feet long, in these parts of the
world, will show an equal alteration along a parallel, as
another of a foot long will show along a meridian ; l. e.
that will, with equal exactness, show the longitude, as this
the latitude.

This depends on the position of the lines of equal dip^
in these parts of the world, which are found to lie about
14 or 15 degrees from the parallels. Hence he argues,
that as we can have needles of 5, 6, 7, 8, or more feet
long, which will move with strength sufficient for exact
observation ; and since microscopes may be applied for
viewing the smallest divisions of degrees on the limb of
the instrument, it is evident, that the longitude at laud
may thus be found to less than 4 miles.

And as there have been many observations made at
sea with the same instrument by Noel, Feuille, &c, which
have determined the dip usually within a degree, some-
times within \ or \ of a degree, and this with small nee-
dles, of 5 or 6, or at the most 9 inches long ; it is inferred,
that the longitude may be found, even at sea, to less than
half a quarter of a degree. This premised, the observa-
tion itself follows.

To find the Longitude or Latitude by the Dipping-Needle.
—If the lines of equal dip, below the horizon, be drawn
on maps, or sea-charts, from good observations, it will be
easy, from the longitude known, to find the latitude; and
from the latitude known, to find the longitude either at
sea or land.— Suppose, for example, a person travelling
or sailing along the meridian of London, should find that
the angle of dip, with a needle of one foot, was 75°; the
chart will show that this meridian, and the line of dip,
,meet in the latitude of 53° 1 1'; which is therefore the la-
titude sought.— Or if he be travelling or sailing along the
parallel of London, i. e. in 51° 31' north latitude, and find
the angle of dip 74°; then this parallel, and the line of
this dip, will meet on the map in 1° 46' of east longitude
from London ; which therefore is the longitude sought.

Many other persons have uselessly endeavoured to de-
termine the longitude of places from the dip of the mag-
netic needle. One among this number, was Mandilo, a
Genois pilot, who contrived a small weight for measuring
the degree of inclination of the needle ; and though no-
thing could be much more absurd, he continued in his
opinion to the last, and died under the persuasion that
he had truly discovered a method of determining the lon-
gitude, and that justice had not been done him, nor his
merits fairly appreciated.

DIRECT, in Arithmetic, is when the proportion of any
terms, or quantities, is in the natural or direct order in
which they stand; being the opposite to inverse, which
considers the proportion in the inverted order of the terms.
So, 3 : 4 : : 6' : 8 directly ; or 3 : 4 : : 8 : 6 inversely.

Rule of Three Direct, is when both pairs of terms are
in direct proportion.

Direct, in Astronomy. A planet is said to he direct,
or its motion direct, when it goes forward by its proper
motion in the zodiac, according to the succession or order
of the signs; or when it appears so to do, to an observer
standing on the earth. Whereas it is said to be, or to
move, retrograde, when it appears to go the contrary way,
or backward j and to be stationary, when it seems not to
move either way.

Direct Dials. See Dial and Dialling.
Direct, in Optics. Direct vision is that performed by
direct rays; in contradistinction to vision by refracted, or
reflected rays. Direct vision is the subject of optics,
which prescribes the laws and rules of it.

Direct Rays, are those which pass on in right lines
from the object to the eye, without being turned out of
their rectilinear direction by any intervening body, either
opaque or pellucid ; or without being either reflected or
refracted.

Direct Sphere. See Right Sphere.
DIRECTION, in Astronomy, the motion and other
phenomena of a planet, when direct.

Line of Direction, in Gunnery, is the direct line in
which apiece is pointed. Sometimes a line of direction
is marked on the upper side of the gun, by a small notch
or slit, or knob, in the base and muzzle rings: but, unless
the two wheels of the carriage stand equally high, this
line will be fallacious; for which reason gunners com-
monly find a. new line of direction every time, by means
of a plummet.

Line of Directiost, in Mechanics, denotes the line in
which a body moves, or endeavours to proceed,
i Angle of Direction, is that comprehended between
the lines of direction of two conspiring powers.

Quantity of Direction, is used for the proSuct arising
from multiplying the velocity of the common centre of
gravity, in a system of bodies, by the sum of their masses.
In the collision of bodies, the quantity of direction is the
same both before and after the impulse.

Direction of the Loadstone, that property by which
the magnet, or a needle touched by it, always presents one
of its ends towards one of the poles of the world, and the
opposite end to the other pole. This is also called the po-
larity of the magnet or needle. The attractive property
of the magnet was known long before its directive; and
the directive long before the inclinatory.

Number of Direction, is the number of days that Sep-
tuagesima Sunday falls after the 17th of January. See
Number.

DIRECTLY, in Geometry: We say two lines lie di-
rectly against each other, when they are parts of the same
right line : also quantities are said to be directly propor-
tional, when the proportion is according to the order of
the terms; in contradistinction to inversely, or recipro-
cally proportional, which is taking the proportion contrary
to the order of the terms.

In Mechanics, a body is said to strike or impinge di-
rectly against another body, when the stroke is in a di-
rection perpendicular to the surface at the point of impact.
And a sphere, in particular, strikes directly against an-
other, when the line of direction passes through both
their centres.

DIRECTRIX, a particular right line, perpendicular to
the axis of a conic section, and much referred to by the
authors who demonstrate the properties ot those curves
from the description of them in piano. Thus, the indefi-
nite right line A n is the direc-
trix, and is so placed, that if
any point f be assumed without
it, and while the line in revolves
about F as a centre, a point
d moves in it in such a manner,
that its distance from v shall al-

ways be to en, its distance from the line as, in a constant

D I S

[ 423 ]

D I S

K

F

\

w

B u

\

H\l

ratio ; then the curve vd, described by the point d, is a
conic section. And the curve is an ellipse, or a parabola,
or an hyperbola, according as fd is less than, equal to, or
greater than, CD, or fv than va. The constant ratio of
fd to CD, or of fv to va, is called the determining ratio.
In the ellipse, the distance av of the directrix from the
vertex, is greater than vf the distance of the focus from
the vertex ; in the parabola av is equal to vf; but.in the
hyperbola av is less than vf.

Directrix, in a Parabola, a line perpendicular to
the axis produced, at the distance of the focus without
the vertex ; as shown above.

DIRIGENT, a term expressing the line of motion,
along which a describent line, or surface, is carried in the
genesis of any plane or solid figure.
Thus, if the line ab move parallel
to itself, and along the line ad, so
that the point a always keeps in the
line ad, and the point b in the line
bc ; a parallelogram abcd will be
formed; of which the line ab is
the describent, and the line ad the

dirigent. So also, if the surface abeg be supposed car-
ried along the line ad, in a position always parallel to it-
self at its first situation, the solid af will be formed;
where the surface ae is the describent, and the line ad is
the dirigent.

DISC, or Disk, the body or face of the sun or moon ;
such as it appears to us ; for though they be really sphe-
rical bodies, they are apparently circular planes. — The
diameter of the disc is considered as divided into 12 equal
parts, called digits; by means of which it is, that the
magnitude of an eclipse is measured, or estimated. — In a
total eclipse of either of those luminaries, the whole disc
is obscured, or darkened ; in a partial eclipse, only part
of them.

Illuminated Disc of the Earth. See Circle of Illumi-
nation.

Disc, in Optics, the magnitude of a telescope glass, or
the width of its aperture, whatever its figure be, whether
a plane, convex, meniscus, or the like.

DISCHARGER, or Discharging Rod, in Electri-
city, consists of a handle
of glass or baked wood,
a, and two bent metal
rods,BB, terminating in
points, and capable of be-
in^ screwed into the
knobs dd, which move
by a joint c, fixed to the
handle a. Thus it may be used either with the points or
balls, as occasion requires; and by being made moveable
on a joint, it may be applied to larger or smaller jars at
pleasure. By bringing one of these knobs or points to
one coated side of a charged electric, and the other to
the other side, or to any conductor connected with it, the
communication is completed between the two sides, and
the electric is discharged. — For the description and use
o!' a Universal Discharger, by Mr. Henly, with which
many curious experiments may be performed, see Caval-
lo's Electricity, pa. l64.

DISCORD, the relation of two sounds which are al-
ways, and of themselves disagreeable, whether applied in
succession or consonance

DISCOUNT, or Rebate, is used for an allowance made
on a bill, or any other debt not yet become due, in consi-
deration of making present payment of the bill or debt.

Among merchants and traders, it is usual to allow a sum
for discount that is equal to the interest of the debt, cal-
culated for the time till it becomes due: but this is not
the correct or just sum; for, as the true value of the dis-
count is equal to the difference between the debt and its
present worth, it is equal only to the interest of that pre-
sent worth, instead of the interest on the whole debt. And
therefore the rule for finding the true discount is this;

As the amount of 100/, for the given rate and time :

Is to the given sum or debt : :

So is the interest of 100/, for the given rate and time :

To the discount of the debt.
So that, ifp be the principal or debt, J' the rate of interest
per cent, and t the time ;

VTt

'••ft: ;.. , which is the true dis-

then as 100

100 + rt
prt 1 OOp

count. Hence also p — —JL- js the present

r 100 + rt 100 + rt r

worth, or sum to be received.

For Ex. Suppose it be required to find the discount of
250/, for five months, at the rate of 5 per cent per annum
interest. Here p = 250, r = 5, and t = T52 or 5 months ;

fh prt _ 250 x 5 x fe _ 250 x 25 250 i

100 + rt 100+S x,i 1400 + 25 "*~ "49 49

the discount sought.

A Table of Discounts may be seen in Smart's Tables of
Interest, the use of which makes calculations of discount
very easy.

Discount is also the name of a rule in books of Arith-
metic, by which calculations of Discount are made.

DISCRETE, or Disjunct, Proportion, is that in which
the ratio between two or more pairs of numbers is the same,
and yet the proportion is not continued, so that the ratio
may be the same between the consequent of one pair and
the antecedent of the next pair.

Thus, if the numbers or proportion 6 : 8 : : 3 : 4 be
considered : the ratio of 6 to 8 is the same as that of 3 to
4, and therefore these four numbers are proportional : but
it is only discretely or disjunctly, and nbt continued; for
8 to 3 is not the same ratio as the former; that is, the pro-
portion is broken off between 8 and 3, and not continued
throughout, as it is in these following four numbers, which
are called continual proportionals, viz, 3:6:: 12 : 24.

Discrete Quantity, is such as is not continued and
joined together. Such for instance is any number; for
its parts, being distinct units, cannot be united into one
continuum ; for in a continuum there are no actual deter-
minate parts before division, but they are potentially in-
finite : so that it is usually and truly said that continued
quantity is divisible in infinitum.

DISDIAPASON, in Music, a compound concord, in
the quadruple ratio of 4 to 1, being a fifteenth or double
eighth, and is produced when the voice goes from the first
tone to the fifteenth.

DISJUNCT Proportion. See Discrete Proportion.

DISK. See Disc.

DISPART, a term in Gunnery, used for a mark set
upon the muzzle-ring of a piece of ordnance, of such
height, that a sight-line taken from the top of the base
ring near the vent or touch-hole, to the top of the dispart
near the muzzle, may be parallel to the axis of the con-

D I S

[ 424 ]

D I S

cave cylinder ; for which reason it is evident, that the
height of the dispart is equal to the difference between the
radii of the piece at the base and muzzle-rings, or to half
the difference of the diameters there. Hence comes the
common method of disparting the gun, which is this :
Take, with the calipers, the two diameters, viz, of the
base ring and the place where the dispart is to stand, sub-
tract the less from the greater, and take half the difference,
which will be the length of the dispart ; this is commonly
cut to that length from a small bit of wood, and so fixed
upright in its place with a bit of wax or pitch.

DISPERSION, in Dioptrics, is the divergency of re-
fracted rays of light.

Point of Dispersion, is a point from which refracted
rays begin to diverge, when their refraction renders them
divergent. — It is called point of dispersion, in opposition
to the point of concourse, or point in which converging
rays concur after refraction. But it is more usual to call
the latter the focus, and the former the virtual focus.

Dispersion of Light, occasioned by the refrangibility
of the rays, or the nature of the refracting medium. See
Aberration, and Inflection.

Dr. Wollastons mode of examining refractive and di-
spersive powers is described in Philos. Trans, for 1S02, or
Nicholson's Journ. vol. 4, pa. 89- Extensive tables of re-
fractive and dispersive powers are given in Dr. T. Young's
Nat. Philos. vol. 2, pa. 296, 299-

DISSIPATION, in Physics, a gradual, slow, insensible
loss or consumption of the minute parts of a body ; or,
more properly, the flux by which they fly oft" and are lost.
See Effluvia.

Circle of Dissipation or Aberration, in Optics,
denotes that circular space on the retina of the eye, which
is occupied by the rays of each pencil in indistinct vision :
thus, if the distance of the object, or the constitution of
the eye, be such, that the image falls beyond the retina,
as when objects are too near ; or before the retina, when
the rays have not a sufficient divergency; the rays of a
pencil, instead of being collected into a central point, will
be dissipated over this circular space: and, all other cir-
cumstancs being alike, this circle will be greater or less-,
according to the distances from the retina of the foci of
refracted rays. But this circle causes no perceptible dif-
ference in the distinctness of vision, unless it exceed a
certain magnitude : as soon as that is the case, we begin
to perceive an indistinctness, which increases as that circle
increases, till at length the object is lost in confusion.
This circle is also greater or less, according to the greater
or less magnitude of the visible object; and though it be
not easy to assign the diameter of the said circle, it seems
very probable that vision continues distinct for all such
distances, or so long as these circles, or the pencils of
light from thorn, do not touch one another on the retina ;
and the indistinctness begins when the said circles begin
to interfere. It has been often observed, that a precise
union of the respective rays on ihe retina, is not necessary
to distinct vision ; but the first author who ascertained the
fact beyond all doubt, was Dr. Jurin. See a variety of
observations and experiments on this subject, in his Essay
on Distinct and Indistinct Vision, in Smith's Optics, Ap-
pendix. In the Philos. Trans, for 178,9, pa. 256, is an
excellent paper on this subject by Dr. Maskelyne; in
which he computes the diameter of the circle of dissipation
at '.002667 of an inch, making it answer to an external

angle of 15°, which he shows is very compatible with di-
stinct vision. See also Moon, and Vision.

Radius of Dissipation, is the radius of the circle of
dissipation.

DISSOLVENT, something that dissolves; i.e. divides,
and reduces a body into its smallest parts.

DISSOLUTION, is a separation of the structure of a
body, into small or minute parts. According to Newton,
and others, this is effected by certain powerful attractions.

DISSONANCE, or Discord, is a false consonance or
concord ; being produced by the mixture or meeting of
two sounds which are disagreeable to the ear.

DISTANCE, properly speaking, denotes the shortest
line between two points, objects, &c.

Distance, in Astronomy, as of the sun, planets, co-
mets, &c. The Real Distances are found from the paral-
laxes of the planets, &c. See Parallax, Planet, and
Transit. The distance of the earth from the sun has
been determined at Q5 millions of miles, by the late tran-
sits of Venus : and from this one real distance, and the
several relative distances, by analogy are found all the
other real distances, as in the table below.

The Proportional or Relative Distances of the planets
are very well deduced from the theory of gravity : for
Kepler has long since discovered, and Newton has demon-
strated, that the squares of their periodical times are pro-
portional to the cubes of their distances. Kepler's Epit.
Astrom lib. 4; Newton's Principia, lib. 3, phaen. 4; and
Gregory'sAstron. book 1, prop. 40. If therefore the mean
distance of the earth from the sun be assumed, or sup-
posed 100000, we shall then, from the foregoing analogy,
and the known periodical times, obtain the relative di-
stances of the other planets : thus,

The Periodical Revolutions in Days and Parts.

Mercurv, Venus, Earih, Mars, Jupiter, Saturn, Herschel,

S7f|' 224^. 365* 686'!f 4332* 10759^30445.

Relative Distance of the Planets.

Mercury

Venus

Earth

Mars

Vesta

Juno

Pallas

Ceres

Jupiter

Saturn

Uranus or")

Herschel }

Kepler.

Cass

3S806 3SS06
72413 72340
100000100000
152350 152373

520000 520290
95 1000954 ISO

Halley.

3S710
72333
100000
152369

52009S
954007

Oe la
Lande

"38710

72333
100000
152309
23.5513
26'6'400
276500
276700
520279
954072

1908 ISO

Log. of those
distances.

And from these relative distances, the real distance of any
planet is easily found, that of the earth being known, by
the following proportion, namely; as 100000 : 95000000,
or as 1 to 950, so is the relative distance in the table, to the
real distance required.

Hence we find the real distances expressed in even mil-
lions to be,

Mercury, Venus, Earth, Mars, Vcsia, .Tuno, Pallas,
37 66 95 145 22.'Ve 253* 26if

Ceres, Jupiler, Saturn, Iranus,

26'2-J la!! 903i 1813.

For the distances of the secondary planets from the
centres of their respective primaries, see Satellites*

D I S

[ 425 ]

D I S

As to that of the fixed stars, as having no sensible pa-
rallax, little more can be done tbanJo guess at it.

Distance of the Sun from the Moon's node, or apogee, is
an arch of the ecliptic, intercepted between the sun's true
place and the moon's node, or apogee. See Node.
Curtate Distance. See Curtate.
Distance of the Bastions, in Fortification, is the side
of the exterior polygon.

Accessible Distances, in Geometry, are measured with
the chain, decempeda or ten-foot rod, or the like.

Inaccessible Distances, are found by taking bearings to
them, from the two extremities of a line whose length is
given. Various ways of performing this may be seen in my
Treatise on Mensuration, sect. 3, on Heights and Distances.
Distance, in Geography, is the arch of a great circle
intercepted between two places.

To find the distance of two places, a and b, far remote
from each other. Assume two stations, c and d, from
which both the places a and b may be
seen; and there, with a theodolite, ob-
serve the quantity of the angles acd,
bcd, adc, bdc, and measure any di-
stance as ac. Then, in the triangle
acd, there are given the angles acd,
adc, and the side ac; to find the side
cd. — Next, in the triangle bcd, there are given the angles
bcd, bdc, and the side cd ; to find the side bc. — Lastly,
in the triangle abc, there are given the angle acb, and
the sides ac, cb ; to find the side ab, which is the distance
sought. And in these operations, the triangles may be
computed either as plane triangles, or as spherical ones,
as the case may require, or according to the magnitude of
the distances.

The Distance of a remote object may also be found from
its heights This admits of several cases, according as the
distances are large or small, &c. 1st, Suppose that from
the top of a tower at a, whose height ab is 120feet, there
be taken the angle bac = 33°, and the angle bad = 64°t,
to two trees, or other objects, c, d ; to find the distance
between them cd, and the distance of each from the bot-
tom of the tower at b.

11

c 1>

First, rad.

: tang. Z. bad :

: ab

next, rad.

: tang. Z. bac :

: ab

bd = 251-585,
bc = 77-929,

their difference is the distance cd — 173'650'.

2d, Suppose it be required to find the distance to which
an object can be seen, by knowing its altitude; ex.gr. the
Peak of Teneriffe, whose height is said to be 3 miles above
the level of the sea, supposing the circumference of the
earth 25,000 miles, or the diameter 7958 miles. Let fg
be the radius = 3.979, ef = 3 the height of the mountain,
and ei a tangent to the earth at the point h, which is
the farthest point to which the top of the mountain e can
be seen. Here, in the right-angled triangle egh, are
given the hy pothenuse EG = 39S2, and the leg GH = 3979 ;
to find the other leg 11 e= 154| miles = the distance
sought nearly. Or, rather, as eg : gii : : rad. : cosin.
C r. = 2° 13' 4 ; then as 360° : 2° 13'i : : 25,000 : 154|
miles = the arch of dist. 11 f sought, the same as before.

Vol. I.

3d, If the eye, instead of being in the horizon at H,
were elevated above it at 1, any known height, as suppose
264 feet, or ^th of a mile, as on the top of a high emi-
nence; then the mountain can be seen much farther off
along the line IE, and the distance will be the two tan-
gents ih and he, or rather the two arcs kh and nr.
Hence, as above, as ig : gh : : rad. : cosin. ^igh = 17'f-;
then as 36"0° : 17'f : : 25,000 : 20 miles = the arc kh :
this added to the former arc hf = 1541, makes the whole
arc kf = 174§ miles, for the whole distance to which the
top of the mountain can be seen in this case.

Apparent Distance, in Optics, that distance which we
judge an object is placed at when seen afar off. This is
usually very different from the true distance ; because we
are apt to think that all very remote objects, whose parts
cannot well be distinguished, and which have no other ob-
ject in view near them, are at the same distance from us,
though perhaps the one is thousands of miles nearer than
the other, as is the case with regard to the sun and moon.

M. Cahire enumerates five circumstances, which assist
us in judging of the distance of objects ; viz, their appa-
rent magnitude, the strength of the colouring, the direction
of the two eyes, the parallax of the objects, and the di-
stinctness of their small parts. On the contrary, Dr.
Smith maintains, that we judge of distance principally, or
solely, by the apparent magnitude of objects ; and con-
cludes universally, that the apparent distance of an object
seen in a glass, is to its apparent distance seen by the naked
eye, as the apparent magnitude to the naked eye is to its
apparent magnitude in the glass. But it was long since ob-
served by Alhazen, that we do not judge of distance
merely by the angle under which objects are seen; and
Mr. Robins clearly shows that Dr. Smith's hypothesis is
contrary to fact, in the most common and simple cases.
Thus, if a double convex glass be held upright before some
luminous object, as a candle, there will be seen two images,
one erect, and the other inverted ; the first is made simply
by reflexion from the nearest surface ; the second by re-
flexion from the farther surface, the rays undergoing a re
fraction from the first surface both before and after the
reflexion. If this glass has not too short a focal distance,
when it is held near the object, the inverted image will
appear larger than the other, and also nearer; but if the
glass be carried off from the object, though the eye remain
as near to it as before, the inverted image will be dimi-
nished so much faster than the other, that at length it will
appear much less than it, but still nearer. Here, says Mr.
Robins, two images of the same object are seen under one
view, and their apparent distances immediately compared ;
and it is evident that those distances have no necessary
connexion with the apparent magnitude. This experi-
ment may be made still more convincing, by sticking a
piece of paper on the middle of the lens, and viewing it
through a short tube. He observes further, that the ap-
parent magnitude of very distant objects is neither deter-
mined by the magnitude of the angle only under which
they are seen, nor is the exact proportion of that angle
compared with their true distance, but is compounded
also with a deception concerning that distance; so that if
we had no idea of difference in the distance of objects,
each would appear in magnitude proportional to the angle
under which it was seen ; and if our supposition of the di-
stance were always just, our idea of their magnitude would
be unvaried, in all distances; but in proportion as we err
in our conception of their distance, the greater angle sug-
31

D I S

t 426 ]

DIT

gests a greater magnitude. By not attending to this com-
pound effect, Mr. Robins apprehends that Dr. Smith was
led into his mistake.

Dr. Porterfield has made several remarks on the five
methods of judging concerning the distance of objects
above recited from M. Lahire ; and he has also added
to them one more, viz, the conformation of each eye. See
Circle of Dissipation. This, he says, can be of no use
to us, with respect to objects that are placed without the
limits of distinct vision. But the greater or less confusion
with which the object appears, as it is more or less re-
moved from those limits, will assist the mind in judging of
its distance; and the more confused it appears, the farther
we shall imagine it to be from us. However, this confu-
sion has its limits ; for when an object is placed at a cer-
tain distance from the eye, to which the breadth of the
pupil bears no sensible proportion, the rays proceeding
from a point in the object may be considered as parallel ;
in which case, the picture on the retina will not he sensi-
bly more confused, though the object be removed to a
much greater distance. The most universal, and often the
most sure means of judging of the distance of objects, he
says, is the angle made by the optic axes : our two eyes
are like two different stations, by the assistance of which,
distances are talj.en ; and this is the reason why those per-
sons who have lost the sight of one eye so frequently miss
their mark in pouring liquor into a glass, snuffing a can-
dle, and such other actions as require that the distance be
exactly distinguished. With respect to the method of
judging by the apparent magnitude of objects, he observes
that this can only serve when we are otherwise acquainted
with their real magnitude. Thus he accounts for the de-
ception to which we are liable in estimating distances, by
any extraordinary magnitudes that terminate them; as, in
travelling towards a large city, castle, or cathedral, we
fancy they are nearer than they really are. Hence also,
animals and small objects seen in a valley contiguous
to large mountains, or on the top of a mountain or high
building, appear exceedingly small. Dr. Jurin accounts
for the last-recited phenomenon, by observing that we have
no distinct idea of distance in that oblique direction,
and therefore judge of them merely by their pictures on
the eye.

Dr. Porterfield observes, with respect to the strength of
colouring, that if we are assured they are of a similar co-
lour, and one appears more bright and lively than the
other; we judge that the brighter object is the nearer.
When the small parts of objects appear confused, or do
not appear at all, we judge that they are at a great di-
stance, and vice versa; because the image of any object,
or part of an object, diminishes as the distance of it in-
creases. Finally, we judge of the distance of objects by
the number of intervening bodies, by which it is divided
into separate and distinct parts; and the more this is the
case, the greater will the distance appear. Thus distances
upon uneven surfaces appear less than upon a plane, be-
cause the inequalities do not appear, and the whole appa-
rent distance is diminished by the parts that do not appear
in it; and thus the banks of a river appear contiguous as
viewed at a distance, when the river is low and not seen.
Accidens de la Vue, pa. 358. Smith's Optics, vol. 1,
pa. 52, and Rem. pa. 51. Robins's Tracts; vol. 2, pa. 230,
247, 251. Porterfield on the Eye, vol. 1, pa. 105, vol. 2,
pa. 387- See Priestley's Hist, of Vision, pa. 205, and
pa. 0'»9.

Distance, in Navigation, is the number of miles or
leagues that a ship has sailed from any point or place.
See Sailing.

Line of Distance, in Perspective, is a right line drawn
from the eye to the principal point:
as the line of, drawn between the
eye at o, and the principal point f ;
which being perpendicular to the
plane, pq is therefore the distance
of the eye from that plane.

Point of Distance, in Perspective, is a point in the ho-
rizontal line at the same distance from the principal point,
as the eye is from the same. Such are the points p and
q, in the horizontal line pq, whose distance from the prin-
cipal point f, is equal to that of the eye from the same
point.

DISTINCT Base, in Optics, is that distance from the
pole of a convex glass, at which objects, beheld through
it, appear distinct and well defined : so that the distinct
base is the same with what is otherwise called the focus.
The distinct base is caused by the collection of the rays
proceeding from a single point in the object, into a single
point in the representation: and therefore concave glasses,
which do not unite, but scatter and dissipate the rays, can
have no real distinct base.

Distinct Vision. See Vision.

DITCH, in Fortification, called also Foss, and Moat,
is a trench dug round the rampart, or wall of a fortified
place, between the scarp and counterscarp.

Ditches are either dry, or wet, that is having water in
them ; both of which have their particular advantages.
The earth dug out of the ditch serves to raise the rampart.
The ditch in front should be of such breadth as that tall
trees may not reach over it, being from 12 to 24 falhoms
wide, and 7 or S feet deep. The ditches on the sides are
made smaller ; but the most general rule is perhaps, that
the dimensions of the ditch should be such as that the
earth dug out may be sufficient to build the rampart of a
proper magnitude. The space sometimes left between the
rampart and ditch, being about 6" or 8 feet, is called the
berm, or list, serving to pass and repass, and lo prevent the
earth from rolling into the ditch.

DITONE, in Music, an interval comprehending two
tones, a greater and a less. The ratio of the sounds that
form the ditonc, is 4 to 5 ; and that of the semi-ditonc,
5 tofj.

DITTON (Humphrey), an eminent mathematician,
was born at Salisbury, May 29, l675. Being an only
son, and his father observing in him an extraordinary
good capacity, determined to cultivate it with a good edu-
cation. For this purpose he placed him in a reputable
private academy ; on quitting of which, at the desire of
his father, though against his own inclination, he engaged
in the profession of divinity, and began to exercise his
function at Tunbridgc in the county of Kent, where he
continued to preach some years ; during which time he
married a lady of that place.

But a weak constitution, and the death of his father,
induced Mr. Ditton to quit that profession ; and at the
persuasion of Dr. Harris and Mr. Whiston, both eminent
mathematicians, he engaged in the study of mathematics,
a science to which he had always a strong inclination. In
the prosecution of this science, he was much encouraged
by the. success and applause he received ; being greatly
esteemed by the chief professors of it, and particularly by

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Sir Isaac Newton, by whose interest and recommendation
he was elected master of the new Mathematical School in
Christ's Hospital ; where he continued till his death, which
happened in 1715, in the 40th year of his age, much re-
gretted by the philosophical world, who expected many
useful and ingenious discoveries from his assiduity, learn-
ing, and penetrating genius.

Mr. Ditton published several mathematical and other
tracts, as below. — 1. Of the Tangents of Curves, &c. Philos.
Trans, vol. 23.

2. A Treatise on Spherical Catoptrics, published in the
Philos. Trans, for 1705 ; from whence it was copied an'd
reprinted in the Acta Eruditorum, 1707, and also in the
Memoirs of the Academy of Sciences at Paris.

3. General Laws of Nature and Motion ; 8vo, 1705.
Wolfius mentions this work, and says that it illustrates and
renders easy the writings of Galileo, Huygens, and the
Principia of Newton. It is also noticed by La Roche, in
the Memoires de Literature, vol. 8, pa. 46.

4. An Institution of Fluxions, containing the first Prin-
ciples, Operations, and Applications, of that admirable
Method, as invented by Sir Isaac Newton; 8vo, 1/06.
This work, with additions and alterations, was again pub-
lished by Mr. John Clarke, in the year 1726.

5. In 1709 he published the Synopsis Algebraica of
John Alexander, with many additions and corrections.

6. His Treatise on Perspective was published in 1712.
In this work he explained the principles of that art mathe-
matically ; and besides teaching the methods then gene-
rally practised, gave the first hints of the new method af-
terward enlarged upon and improved by Dr. Brook Taylor;
and which was published in the year 1715.

7. In 1714, Mr. Ditton published several pieces, both
theological and mathematical ; particularly his Discourse
on the Resurrection of Jesus Christ; and The New Law
of Fluids, or a Discourse concerning the Ascent of Liquids,
in exact Geometrical Figures, between two nearly con-
tiguous Surfaces. To this was annexed a tract, to demon-
strate the impossibility of thinking or perception being the
result of any combination of the parts of matter and mo-
tion : a subject which was much agitated about that time.
To this work also was added an advertisement from him
and Mr. Whiston, concerning a method for discovering
the longitude, which it seems they had published about
half a year before. This attempt probably cost our au-
thor his life ; for though it was approved and counte-
nanced by Sir Isaac Newton, before it was presented to the
Board of Longitude, and the method has since been suc-
cessfully put in practice, in finding the longitude between
Paris and Vienna, yet that board determined against it: so
that the disappointment, together with some public ridi-
cule (particularly in some verses written by Dean Swift),
affected his health, that he died the ensuing year, 1715.

In an account of Mr. Ditton, prefixed to the German
translation of his Discourse on the Resurrection, it is
said that he had published, in his own name only, another
method for finding the longitude ; but this Mr. Whiston
denied; However, Raphael Levi, a learned Jew, who had
studied under Leibnitz, informed the German editor, that
he well knew that Ditton and Leibnitz had corresponded
on the subject; and that Ditton had sent to Leibnitz a
delineation of a machine he had invented for that pur-
pose; which was a piece of mechanism constructed with
many wheels like a clock, and which Leibnitz highly ap-
proved of for land use; but doubted whether it would

answer on ship-board, on account of the motion of the
ship.

DIVERGENT Point. See Virtual Focus.
Divergent, or Diverging Lines, in Geometry, are
those whose distance is continually increasing. — Lines
which diverge one way, converge the other.

Divergent, or Diverging, in Optics, is particularly
applied to rays which, issuing from a radiant point, or
having, in their passage, undergone a refraction, or re-
flexion, do continually recede farther from each other.

In this sense the word is opposed to convergent, which
implies that the rays approach each other, or that they
tend to a centre, where they intersect, and, being con-
tinued, go on diverging. Indeed all intersecting rays, or
lines, diverge both ways from the centre, or point of inter-
section.

Concave glasses render the rays diverging; and convex
ones, converging. — Concave mirrors make the rays con-
verge ; and convex ones, diverge.— It is demonstrated in
Optics, that as the diameter of a pretty large pupil does
not exceed \ of a digit; diverging rays, flowing from a
radiant point, will enter the pupil as parallel, to all in-
tents and purposes, if the distance of the radiant from the_
eye amount to 40,000 feet. See Focus, Light, and
Vision.

Diverging Hyperbola, is one whose legs turn their
convexities toward each other, and run out contrary ways.
See Hyperbola.

Diverging Parabola. See Diverging Parabola.
Diverging Series, is a series whose terms always
become larger the farther they are continued.

DIVIDEND, in Arithmetic, is the number given to be
divided by some other number, called the divisor. Or it is
the number given to be divided, or separated, into a certain
number of equal parts, viz. as many as the divisor contains
units; and the number of such equal parts is called the
quotient. Or, more generally, the dividend contains the
divisor, as many times as the quotient contains unity. —
The dividend is the numerator of a fraction, whose deno-
minator is the divisor, and the quotient is the value of the
fraction. Thus, | =4, and -f = '75.

DIVING. The art, or act of descending under water, to
considerable depths, and remaining there for a certain
time. The uses of diving are very considerable, parti-
cularly in the fishing for pearls, corals, sponges, &c.

Various methods have been proposed, and engines con-
trived, to render the business of diving more safe and easy.
The great point in all these, is to furnish the diver with
fresh air, without which he must either make but a short
stay, or perish. Those who dive for sponges in the Medi-
terranean, help themselves by carrying down sponges dipt
in oil in their mouths; but considering the small quantity
of air that can be contained in the pores of a sponge, and
how much that little will be contracted by the pressure of
the incumbent water, it is evident that such a supply can-
not be very useful ; since it is found by experiment, that
a gallon of air included in a bladder, and by a pipe reci-
procally inspired and expired by the lungs, becomes unfit
for respiration in little more than one minute; for though
its elasticity be but little altered in passing the lungs, yet
it loses its vivifying spirit, and is rendered unfit for the
support of animal life. Dr. Halley assures us, a naked
diver, without a sponge, cannot remain above two minutes
inclosed in water; nor much longer with one, without suf-
focating; nor, without long practice, near so long: pcr-
3 I 2

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sons unaccustomed to diving generally begin to be suffo-
cated in about half a minute. Besides, if the depth be
considerable, the pressure of the water on the vessels makes
the eyes blood-shotten, and frequently occasions a spitting
of blood : hence, where there has been occasion to con-
tinue long at the bottom, some have contrived double
flexible pipes, to circulate air down into a cavity inclo-
sing the diver, as with armour, both to furnish air, and to
bear off the pressure of the water, and give leave to his
breast to dilateupon inspiration ; the fresh air. being forced
down one of the pipes with bellows, and returning by the
other, not unlike to an artery and vein.

But this method is impracticable when the depth ex-
ceeds three fathoms ; the water embracing the bare limbs
so closely, as to obstruct the circulation of the blood in
them; and withal pressing so strongly on all the junctures
where the armour is made tight with leather; thatif there
be the least defect in any of them, the water rushes in, and
instantly fills the whole engine, to the great danger of the
diver's life.

Di viNG-BeW, is a machine contrived to remedy all these
inconveniencies. In this the diver is safely conveyed to
any reasonable depth, and may stay more or less time
under the water, as the bell is greater or less ; and it is
most conveniently made in form of a truncated cone, the
smaller base being closed, and the larger open; being
weighed with lead, and so suspended, that it may sink full
of air, with its open basis downward, and as near as may
be in a situation parallel to thehorizon, so as to close with
the surface of the water all at once.

The diver sitting under this machine, sinks down with
the included air to the depth desired ; and if the cavity of
the vessel can contain a ton of water, a single man may
remain a full hour, without much inconvenience, at five
or six fathoms deep ; but the lower he goes, still the more
the included air contracts itself, according to the weight
of the water that compresses it ; so that at thirty-three
feet deep, the bell becomes half-full of water; the pres-
sure of the incumbent fluid being then equal to that of the
atmosphere; and at all other depths, the space occupied
by the compressed air in the upper part of its capacity, is
to the space filled with water, as thirty-three feet to the
depth of the surface of the water in the bell below its com-
mon surface; and this condensed air, being taken in with
the breath, soon insinuates itself into all thecavilies of the
body, and has no ill effect, provided the bell be permitted
to descend so slowly as to allow time for that purpose.

One inconvenience that attends it, is found in the ears,
within which there are cavities which open only outwards,
and that by pores so small, as not to give admission even
to the air itself, unless they be dilated and distended by a
considerable force ; hence, on the first descent of the bell,
a pressure begins to be felt on each ear, which by degrees
becomes painful, till the force overcoming the obstacle,
that constringes these pores, yields to the pressure, and
letting some condensed air slip in, presently ease ensues;
the bell descending lower, the pain is renewed, and after-
wards it is again eased in the same manner. But the
greatest inconvenience of this engine is, that the water en-
tering it, contracts the bulk of air into so small a compass,
that it soon heats, and becomes unfit for respiration: so
that there is a necessity for its being drawn up to recruit
it; besides the uncomfortable situation of the diver, who
is almost covered with water.

To obviate the difficulties of the diving-bell, Dr. Halley,

to whom we owe the preceding account, contrived some
further apparatus, by which not only to recruit and re-
fresh the air from time to time, but also to keep the water
wholly out of it at any depth; which he effected after the
following manner: — His diving-bell (plate ix, fig. 6) was
of wood, 3 feet wide at top, 5 feet at bottom, and 8 feet
high, containing about 63 cubic feet in its concavity, coated
externally with lead so heavy, that it would sink empty ;
a particular weight being distributed about its bottom r,
to make it descend perpendicularly, and no otherwise. In
the top was fixed a meniscus glass d, concave downwards,
like a window, to let in light from above ; with a cock, as
. at b, to let out the hot air ; and a circular seat, as at lm,
for the divers to sit on: and below, about a yard under
the bell, was a stage suspended from it by three ropes, each
charged with a hundred weight, to keep it steady, and for
the divers to stand upon to do their business. The ma-
chine was suspended from the mast of a ship by a sprit,
which was secured by stays to the mast-head, and was di-
rected by braces to carry it overboard clear of the side of
the vessel, and to bring it in again.

To supply air to this bell when underwater, he had two
barrels, as c, holding 36 gallons each, cased with lead, so
as to sink empty, each having a bung-hole at bottom, to
let in the water as they descended, and let it out again as
they were drawn up. In the top of the barrels was an-
other hole, to which was fixed a leathern pipe, or hose,
well prepared with bees wax and oil, long enough to
hang below the bung-hole; being kept down by a weight
for that purpose ; so that the air driven to the upper part
of the barrel by the influx of the water, in the descent,
could not escape up this pipe, unless the lower end were
lifted up.

These air-barrels were fitted with tackle, to make them
rise and fall alternately, like two buckets; being directed
in their descent by lines fastened to the under edge of the
bell: so that they came readily to the hand of a man
placed on the stage, to receive them ; who taking up the
ends of the pipes, as soon as they came above the surface
of the water in the barrels, all the air included in the
upper part of it was blown forcibly into the bell; the water
taking its place.

One barrel thus received, and emptied; on a signal
given, it was drawn up, and at the same time the other
letdown; by which alternate succession, fresh air was
furnished so plentifully, that the learned Doctor himself
was one of five, who were all together in nine or ten fa-
thoms deep of water for above an hour and a half, without
the least inconvenience ; the whole cavity of the bell being
perfectly dry.

All the precaution he observed, was, to be letdown gra-
dually about 12 feet at a time, and then to stop, and drive
out the water that had entered, by taking in three or four
barrels of fresh air, before he descended farther. And, being
arrived at the depth intended, he let out as much of the hot
air that had been breathed, as each barrel would replace
with cold, by means of the cock B, at the top of the bell,
through whose aperture, though very small, the air would
rush with so much violence, as to make the surface of the
sea boil.

Thus he, found any thing could be done that was re-
quired to l>e done underneath; and by taking off the stage,
he could, for a space as wide as the circuit of the bell, lay
the bottom of the sea so far dry as not to be over shoes in,
water. Besides, by the glass vvinirrw so much light was,

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transmitted, that when the sea was clear, and especially
when the sun shone, he could see perfectly well to write
or read, much more to fasten or lay hold of any thing
under him that was to be taken up ; and by the return of
the air-barrel he often sent up orders written with an iron
pen on a plate of lead, directing how he would be moved
from place to place. At other times, when the water was
troubled and thick, it would be as dark as night below ;
but in such cases he was able to keep a candle burning in
the bell.

Dr. Halley observes, that they were subject to one in-
convenience in this bell ; they felt at first a small pain
in their ears, as if the end of a tobacco-pipe were thrust
into them ; but after a little while there was a small puff
of air, with a little noise, and they were easy.

This he supposes to be occasioned by the condensed air
shutting up a valve leading from some cavity in the ear,
full of common air.; but when the condensed air pressed
harder, it forced the valve to yield, and filled every ca-
vity. One of the divers, in order to prevent this pressure,
stopped his ear with a pledget of paper ; which was pushed
in so far, that a surgeon fouqd considerable difficulty in
extracting it.

The same author intimates, that by an additional con-
trivance he has found it practicable for a diver to go out
of the bell to a good distance from it ; the air being con-
veyed to him in a continued stream by small flexible pipes,
which servehimasaclueto direct him back again to the bell.
For this purpose, one end of these pipes, kept open against
the pressure of the sea by a small spiral wire, and made
tight without by painted leather and sheep's guts drawn
over it, being open, was fastened in the bell, as at p, to re-
ceive air, and the other end was fixed to a leaden cap on
the man's head, reaching down below his shoulders, open
at bottom, to serve him as a little bell, full of air, for him
to breathe at his work, which would keep out the water
from him, when at the level of the great bell, because the
density of the air in both was the same : but when he
stooped down lower than the level of the great bell, he
shut the cock f, to cut off the communication between
the two. Phil. Trans, abr. vol. vi. pa. 258, 522.

The air in this bell would serve him for a minute or
two; and he might instantly change it, by raising himself
above the great bell, and opening the cock f. The diver
was furnished with a girdle of large leaden weights, and
clogs of lead for the feet, which, with the weight of the
leaden cap, kept him firm on the ground ; he was also
well clothed with thick flannels, which being first made
wet, and then warmed in the bell by the heat of his body,
kept off the chill of the cold water for a considerable time,
when he was out of the bell.

Mr. Martin Triewald, f. it. s, and military architect to
the king of Sweden, contrived to construct a diving-bell
on a smaller scale, and at a less expense, than that of
Dr. Halley, and yet capable of answering the same pur-
poses. This bell, ab (fig. 7-) sinks with leaden weights
DD, suspended from the bottom of it. It is made of cop-
per, and tinned all over on the inside; and it is illumi-
nated by three strong convex lenses cog, defended by the
copper lids it mi. The iron plate e serves the diver to
stand upon, when he is at work; and this is suspended by
the chains i ft, at such a distance from the bottom of the
bell, that when he stands upright, his head is just above
the water within it, where he has theadvantage of air more
proper for respiration, than when he is much higher up;

but as there is occasion for the diver to be wholly in the
bell, and consequently his head in the upper part of it,
Mr. Triewald also contrived, that even there, after he
has breathed the hot air as long as he can, by means of a
spiral copper tube placed close to the inside of the bell,
he may draw the cooler and fresher air from the lower-
most parts; for which purpose a flexible leather pipe,
about two feet long, is fixed to the upper end of the tube
at b ; and to the other end of the pipe is fastened an ivory
mouth-piece, for the diver to hold in his mouth, by which
to respire the air from below. We shall only remark,
that as air rendered effete by respiration is somewhat hea-
vier than common air, it must naturally subside in the
bell; but it may probably be restored by the agitation
of the sea-water, and thus become more fit for respira-
tion. See Fixed Air, Phil. Trans, abr. vol. viii. pa. 634.
Or Desaguliers's Exper. Phil. vol. ii. pa. 220, &c. Several
other persons have practised with diving-machines, on the
same principles.

The famous Corn. Drcbell had- an expedient in some
respects superior even to the diving-bell, if what is related
of it be true. He contrived not only a vessel to be rowed
under water, but also a liquor to be carried in the vessel,
which supplied the place of fresh air. The vessel was
made for King James the 1st, carrying 12 rowers, besides
the passengers. It was tried in the river Thames ; and one
of the persons in that submarine navigation, then living,
told it one, from whom Mr. Boyle had the relation. As
to the liquor, Mr. Boyle assures us, he discovered by a
physician, who married Drebell's daughter, that it was
used from time to time, when the air in that submarine
boat was clogged by the breath of the company, and ren-
dered unfit for respiration : at which time, by unstopping
the vessel full of this liquor, he could speedily restore to
the troubled air such a proportion of vital parts, as would
make it serve again a good while. The secret of this li-
quor Drebell would never disclose to above one person,,
who himself assured Mr. Boyle what it was. Boyle's Exp.
Phys-. Mech. of the Spring of the Air.

We have had many projects of diving-machines, arfd
diving-ships of various kinds, which have proved abor-
tive.

Divi-NG-Bludi/er, a term used by Borelli for a machine
which he contrived for diving under water to great depths,,
with considerable facility, which he prefers to the com-
mon diving-bell. The vesica, or bladder, as it is usually
called, is of brass or copper, and about two feet in dia-
meter, which is to contain thediver's head; and thisis fixed
to a goat's skin habit, exactly fitted to the shape, of the
body of the person. Within this vesica there are pipes,
by means of which a circulation of air is contrived ; and
the person carries an air-pump by his side, by means of
which he may make himself heavier or lighter, as the fishes
do, by contracting or dilating their air-bladder: by this
means, the objections all other diving-machines are liable
to are obviated, and particularly that of the air;, the
moisture by which it is clogged in respiration, and by which,
it is rendered unfit for the same use again, 'being here
taken from it by its circulation through the pipes, to the-
sides of which it adheres, and leaves the air as free as be-
fore. Borelli Opera Posthuma.

DIVINI (Eus-tachio), an ingenious Italian artist,,
very eminent for his telescopes, though, it seems, inferior
to Campani. He wrote a book (published at the Hague
10'6O, in <tto) against the discovery of Saturn's ring, made'

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D I V

by the celebrated Huygens, contesting the truth of it; to
which the latter wrote a reply. Divini was living in 1663;
though Moreri thinks he died about that time.

DIVISIBILITY, a property in quantity, body, or ex-
tension, by which it becomes separable into parts ; either
actually, or at least mentally. Such divisibility is infinite,
if not actually, at least potentially ; as no part can be
conceived so small, but another may be conceived still
smaller; for every part of matter must have some finite
extension, and that extension may be bisected, or other-
wise divided ; for the same reason, these parts may be di-
vided again, and so on without end.

It is not here contended for the possibility of an actual
division in infinitum : it is only asserted that however
small a body is, it may be still farther divided ; which it
is presumed may be called a division in infinitum, because
what has no limits, is called infinite. The infinite, or in-
' definite divisibility of mathematical quantity is thus
proved, and illustrated by mathema-
ticians: Suppose' a line ad perpen-
dicular to bf ; and another as sh
also perpendicular to the same bf ;
with the centres c, c, c, &c, and di-
stances ca, ca, &c, describe circles
cutting the line gh in the points c,
e, &c. Now, the greater the radius
ac is, the less is the parteG; but the
radius may be augmented in infini-
tum, and therefore the part- ea may be diminished in the
same manner; and yet it can never be reduced to no-
thing, because the circle can never coincide with the right
line bf. Consequently the parts of any magnitude may
be diminished in infinitum.

All that is supposed, in strict geometry, concerning the
divisibility of magnitude, amounts to no more, than that
a given magnitude may be conceived as divided into a
number of parts, equal to any given or proposed number.
It is true, that there are no such things as parts infi-
nitely small ; yet the subtilty of the particles of several
bodies is such, that they far surpass our conception ; and
there are innumerable instances in nature of such parts
actually separated from one another.

Several instances of this are given by Mr. Boyle. He
speaks of a silken thread 300 yards long, that weighed but
two "rains and a half. He measured leaf-gold, and found
by weighing it, that 50 square inches weighed but one
grain i if the length of an inch be divided into 200 parts,
the eye may distinguish them all ; therefore in one square
inch there are 40,000 visible parts; and in one grain of it
there are two millions of such parts; which visible parts
no one will deny are still farther divisible.

Again, an ounce weight of silver may be gilt over with
8 "rains of gold, which may be afterwards drawn into a
wire 13,000 feet long, and still be all covered with the
same gilding.

In odoriferous bodies a still greater subtilty of parts is
perceived, and even such as are actually separated from
one another: several bodies scarce lose any sensible
part of their weight in a long time, and yet continually
fill a very large space with odoriferous particles. Dr. Keil,
in his Vera Physica, Lect. 5, has calculated the mag-
nitude of a particle of assalcetida, which will be the

tli part of a cubic inch. And in the

lououooooouoooooo r

same Lecture, he shows that the particles of the blood in

animalcula;, observed in fluids by means of microscopes,
must be less than that part of a cubic inch which is ex-
pressed by a fraction whose numerator is 8, and denomi-
nator unity with 30 ciphers after it.

The particles of light, if light consist of real particles,
furnish another surprising instance of the minuteness of
some parts of matter. A small lighted candle placed on
a plain, will be visible two miles, and consequently its
light fills a sphere of 4 miles diameter, before it has lost
any sensible part of its weight. Now, as the force of any-
body is directly in proportion to its quantity of matter
multiplied by its velocity; and since it is demonstrated
that the velocity of the particles of light is at least a mil-
lion of times greater than the velocity of a cannon-ball,
it is plain, that if a million of these particles were round,
and of the size of a small grain of sand, we durst no more
open our eyes to the light, than expose them to sand shot
point-blank from a cannon.

By help of microscopes, such objects as would other-
wise escape our sight, appear very large : there are some
small animals scarce visible with the best microscopes;
and yet these have all the parts necessary for life, as
blood, and other fluids. Mow wonderful then must the
subtilty of the parts be, which make up such fluids !
Whence is deducible the following theorem:

Any particle of matter, however small, and any finite
space, however large, being given; it is possible for that
small particle of matter to be diffused through that space,
and to fill it so as that there shall be no pore in it, whose
diameter shall exceed any given line; as is demonstrated
by Dr. Keil, Introduct. ad Ver. Phys.

DIVISIBLE, the faculty or quality of being capable
of being divided.

DIVISION, is one of the four principal rules of arith-
metic, being that by which we find how often one quan-
tity is contained in another; so that division is in reality
only a compendious method of subtraction; its effect be-
ing to take one number from another as often as possible;
that is, as often as it is contained in it. There are there-
fore three numbers concerned in division : 1st, That which
is given to be divided, called the dividend; 2d, That by
which the dividend is to be divided, called the divisor ;
3d, That which expresses how often the divisor is con-
tained in the dividend; or the number resulting from the
division of the dividend by the divisor, called the quotient.

There are various ways of performing division, one
called the English, another the Flemish, another the Ita-
lian, another the Spanish, another the German, and an-
other the Indian way, all equally just, as finding the quo-
tient with the same certainty, and only differing in the
manner of arranging and disposing the numbers.

There is also division in integers, division in fractions,
and division in species, or algebra, &c.

Division is performed by seeking how often the divisor
is contained in the dividend ; and when the latter consists
of a greater number of figures than the former, the divi-
dend must be taken into parts, beginning on the left, and
proceeding to the righ_t, and seeking how often the divisor
is found in each of those parts; utter the manner as taught
in all books of arithmetic, as well as various contractions
adapted to particular cases: such as, 1st, when the divi-
sor has any numbei'of ciphers al the end of it, they are cut
off, with the same number of figures from the end of the
dividend, and then the work is performed without them
both, annexing only the figures last cut off, to the lust re*

D I ' V

[ 431 ]

DIF

mainder; 2d, when the divisor is equal to the product of
several single digits, it is easier to divide successively by
those digits, instead of the divisor at once ; 3d, when it is
required to continue a quotient to a great many places of
figures, as in decimals, a very expeditious method of per-
forming it, is as follows : Suppose it were required to di-
vide 1 by 23, to a great many places of decimals. Adding
ciphers to the 1, first divide 10000 by 29 in the common
way, till the remainder become a single figure, and annex
the fractional supplement to complete the quotient, which
gives -£§ = 0-03448^: next multiply each of these by
the numerator 8, so shall ^ = 0'27584-i-f or rather
0-27586T% ; which figures substituted instead of the frac-
tion -^ in the first value of ■£§, it becomes ■%■§ =
0*0344827 5 86^-: again, multiply both of these by the
last numerator 6, and it will be ■£§ = 0-2068965517^;
which figures substituted for ~ in the last-found value of
■£?, it becomes 4s = 0-03448275862068965517^: and
again, multiplying these by the numerator 7, gives
^ = 0-24137931034482758620^|; which figures sub-
stituted instead of -^ in the last-found value of -£^, it be-
comes ^ =

O-O344827586206S965517241 3793 103448275S6204|,
and so on; where every operation will at least double the
number of figures before found by the last one.

Proof of Division. In every example of division,
unity is always in the same proportion to the divisor, as
the quotient is to the dividend ; and therefore the product
of the divisor and quotient is equal to the product of 1
and the dividend, that is, the dividend itself. Hence, to
prove division, multiply the divisor by the quotient, to the
product add the remainder, and the sum will be equal to
the dividend when the work is right ; if not, there is a
mistake.

Division, in Vulgar Fractions, is performed by divi-
ding the numerators by each other, and the denominators
by each other, if they will exactly divide; butif not, then
the dividend is multiplied by the reciprocal of the divisor,
that is, having its terms inverted ; for, taking the recipro-
cal of any quantity, converts it from a divisor to a multi-
plier,and from a multiplier to a divisor. For Ex. 44 -r by
|- gives 4» by dividing the numerators and denominators;
but \% -f- by •§■ is the same as -J-£ x |., which is = £|.
Where x is the sign of multiplication, and the character
7- is the mark of division. Or division is also denoted
like a vulgar fraction ; so 3 divided by 2, is ■!■•

Division, in Decimal Fractions, is performed the same
way as in integers, regard being had to the number of de-
cimals, viz, making as many in the quotient as those of
the dividend exceed those in the divisor.

But in this case we sometimes make use of a contracted
method, which it may not be amiss to explain. Thus, when
there are many figures in the divisor, or when only a cer-
tain number of decimals are necessary to be retained in
the quotient; then take only as many of the left-hand fi-
gures of the divisor as will exceed the number of figures,
both integers and decimals, to be retained in the quotient
by unity, and find how many times they are contained in
the first figures of the dividend as usual.

Let each remainder form a new dividend ; and for every
such dividend, leave out one figure more on the right-hand
side of the divisor, observing to carry for the increase of
the figures cut off as follow: namely, 1, from 5 to 14;
2, from 15 to 24 ; 3, from 25 to 34, &c.

Hut when there are not so many figures in the divisor as

are required to be in the quotient, then the operation
must be begun with all the figures, continuing it the same
as in the common rule, till the number of figures in the
divisor exceed those remaining to be found in the quotient
by unity, after which begin the contraction. See the fol-
lowing example, where 5 figures are to be retained in the
quotient.

92-4103,5 ) 2508-92806 ( 27-149
660721
13849
4608
912
80
Division, in Algebra, is performed like that of com-
mon numbers, either making a fraction of the dividend
and divisor, and cancelling or dividing by the terms or
parts that are common to both ; or else dividing after the
manner of long division, when the quantities are com-
pound ones. Thus,

ab divided by a, gives b for the quotient :
and I2ab divided by 46, gives 3a for the quotient :
\6abc2 divided by 8ac, gives 2bc :

a divided bv 3b, gives — :

15abc3 divided by 12ic2, gives ^4; = — :

and a2 — b- by a -+- b, gives a — b; thus, '
a ■+■ b ) a2 — b* ( a — b
a2 ■+■ ab
— ab — b2
-ab-b2:
again, to divide x6 — Sx4 — 124x2 — 64 by a:2.— 16 ;
x2- 16 ) xs- Sx'- 124x2-64 ( x1 -+- 8x2h-4

x6-l6x4

Sx4"^- 124x2
Sx4 — 128x*

4x3 — 64
4tx* - 64
In some cases, the quotient will run out to an infinite
series; and then, after continuing it to any certain num-
ber of terms, it is usual to annex, by way of a fraction,
the remainder with the divisor set under it. — It is to be
noted that, in dividing any terms by one another, if the
signs be both alike, either both plus or both minus, the
sign of the quotient will be plus; but when the signs are
different, the one plus and the other minus, the sign of
the quotient is minus.

Division by Logarithms. See Logarithms.
Division of Mathematical Instruments. See Gradua-
tion, and Mural Arc or Quadrant.

Division in Music, is the dividing the interval of an
octave into a number of lesser intervals.

Division by Napier's Bones. See Napier's Bones.
Division of Powers, is performed by subtracting their
exponents. Thus, a6 "-n4 is = o2; and

iah2^ 2a*b$ is = 2a*F.

Division of Proportion, is comparing the difference
between the antecedent and consequent, with either of
them. Thus,

b : :

then by division 1

{a

or -J

2

b
8
~h

6

12:

b :
4 :

d :

d,

2

6,
2.

D I V

[ 432 ]

D I U

Divisor, is the dividing number; or that which shows
how many parts the dividend is to be divided into.

Divisors of a Number, are those numbers by which it
is exactly divisible; thus, 1, 2, 3, 4, 6 and 12, are divi-
sors of the number 12, because 12 may be divided by
each of those numbers, without leaving a remainder. But
if we take a prime number, as for example 13, we shall
find that it has only two divisors ; namely, 1 and 13, or
unity and itself, which is indeed what constitutes a prime
number.

In composite numbers, it is frequently necessary to
know what those numbers are by which they are divisible,
the sum of all those divisors, how many such divisors be-
long to a given number, their forms, and other particu-
lars, which find their application in various branches of
analysis. In order to enter into these particulars, it is
necessary that we know all the prime divisors of a num-
ber, which are found by dividing it by all those prime
numbers, le^s than the square root of itself, by which it is
divisible; repeating the division by each, as often- as it can
be done without leaving a remainder : by which means
ever,)' number n is reduced to the following form, n =
am jr. cp ^q&c; where a, b, c &c are the prime divisors
of n, and m, n, p &c the powers of those divisors, or the
number of times that n is divisible by a, b, c &c.

This reduction being made, it is evident from inspec-
tion only, that by the developement of the formula
(1 -+• a ■+- a2 . . . . am) . (1 -+- b -+- b"...b") . (1 -+- c -+-
c2 . . . . cf) &c, we shall obtain every possible combina-
tion, that can be formed with m-as, n-bs, p-cs &c, and
consequently, every divisor of n is contained in the result
obtained from this developement, and which will there-
fore express the sum of all the divisors of the given num-
ber N.

Suppose, for example, the sum of all the divisors of 360
were required.

Here 360 = 24.32.52, that is a = 2, b = 3, and c = 5;
also m = 4, n = 2, and p = 2 : therefore (1 + 2 + 2*
-+■ 23 ■+- 2') . (1 -+■ 3 -i- 31) . (1 -+• 5 + 5') = 31.13.31
= 12493, which is the sum of all the divisors of 360.
But if, instead of 360, the sum of the divisors of 36*1 had
beeVi required, we should have found it equal only to 362,
because 36l is a prime. From which circumstance we
should be led to conclude, that no order could be ob-
served between the sum of the divisors of the consecutive
numbers; but Euler has proved in his Introduction to
the Analysis Infinitorum, torn. 1, p. 355, that these sums
follow a certain law, after the manner of a recurring
series.

We may also from our first formula find the number of
divisors of any given number, which is evidently expressed
by (m -+- 1) . (n ■+- 1) . {p -+- l) . &c. Suppose, for ex-
ample, it were required to find how many divisors belong
to the number 1800.

First 1800 = 23 . 32 . 5" ; that is, m = 3, n = 2, and
p = 2: therefore (3 + l).(2+ I).(? + l). = 36, the
number of divisors of 1S00. Hence again, we may readliy
find a number that shall have any given number of divi-
sors. Thus, let it be required to find a number that shall
have 20 divisors. Here, it is only necessary to resolve
20 into any number of factors, as 20 = 4.5 ; subtract 1
from each of those factors, and make m = 3, n =4; then
will am.ba be the number sought ; where a and b may be
taken any prime numbers at pleasure. If a = 3, and
b = 2, we have 27. 16 = 432, for the number required.

Various other properties of divisors might be here enu-
merated, but we shall only mention a few of those which
have been most successfully employed in the Theory of
Numbers.

1. If a be a prime number, and x not divisible by a,
then is a a divisor of the 'formula (x"'1 — 1).

2. If a be a prime number, it is a divisor of the for-
mula (1.2.3. — a -l) + l.

3. If a be a prime number, it is a divisor of the for-
mula (18.2*.32. - - (^-—)°) ± 1, the upper sign having

place when a is of the form 4b -1- 1 ; and the under one
when a is of the form An— 1.

4. The divisors of every number comprised under any
of the forms p'z -+- ril, p'1 -+- 2ri2, p- — 2b2, are always of the
same form, as the numbers that they divide. That is, the
sum of two squares can only be divided by numbers that
are also the sum of two squares, &c. It being always un-
derstood that/) and n are prime to each other.

5. Every prime number is a divisor of the formula
p* -i- n'2 -t- 1.

6. Every odd number, except 5 and its multiples, is a
divisor of any repetend digit ; and the number of dicks
necessary to form the dividend, never exceeds the number
expressed by the divisor.

7. If the sum of the digits of any number is divisible by
3 or 9, the number itself is divisible by the same number
3 or 9.

8. If the sum of the digits in the even places, are equal
to the sum of those in the odd places, in any number,
that number is divisible by 1 1.

9. If the n right-hand digits of any number be divisible
by 2", the whole number is divisible by 2".

The reader is referred for further information on this
subject, to Euler's Introduction to the Analysis Infini-
torum, also to the 2d vol. of his Algebra ; to Waring's
Meditationes Algebraica ; Le Gendre's Essai sur la Thco-
rie des Nofhbres ; and to the Recherches Arithmetiques,,
par M. Gauss. See also Numbers in this Dictionary,

DIURNAL, something relating to the day; in opposi-
tion to Nocturnal, relating to the night.

Diurnal Arch, is the arch described by the sun,
moon, or stars, between their rising and setting.

Diurnal Circle, is the apparent circle described by
the sun, moon, or stars, in consequence of the rotation of
the earth.

Diurnal Motion of a Planet, is so many degrees and
minutes &c as any planet moves in 24 hours.

The Diurnal Motion of the Earth, is its rotation round
its axis, the duration of which constitutes the natural day.

Diurnal Parallax. See Parallax.

Diurnal is also used in speaking of what belongs to
the ^cthemeron, or natural day of 24 hours : in which
sense it is opposed to annual, menstrual, &c. The diurnal
phenomena of the heavenly bodies are solved from the
diurnal revolution of the earth; that is, from the rotation
of the earth round its own axis in 24 hours ; which rota-
tion is equable, and from west, to east, about an axis whose
inclination to the ecliptic is now 6*6° 32'. Now since the
earth is an opaque body, that small part of its surface
which comes at the saline time under the confined view of
the spectator, though really spherical, seems to be ex-
tended like a plane: and the eye, taking an entire view of
the heavens, they seem to form a concave spherical su-
perficies, concentric with the earth, or rather with the

D O D

[ 433 ]

D O L

eye, which the said plane of the earth's superficies di-
vides into two equal part3, the one of which is visible,
but the other, because of the earth's opacity, is hid from
the view.

And as the earth revolves ahout its axis, the spectator,
together with the said plane he stands upon, called his ho-
rizon, dividing the visible from the invisible hemisphere
of the heavens, is carried round the same way, viz, to-
wards the east : hence it is, that the sun and stars, placed
towards the east, being before hid, now become visible, the
horizon as it were sinking below them ; and the stars &c
towards the west are covered or hid, and become invisible,
the horizon being elevated above them : so that the former
stars, to the spectator, who reckons the place he stands
on as immoveable, appear to ascend above the horizon, or
rise ; and the latter to descend below the horizon, or set.

Since the earth, with the horizon of the spectator, con-
tinues to move always towards the same parts, and about
the same axis equally; all bodies, and all phenomena,
that do not partake of the said motion, (that is, all such
things as are entirely separate from the earth,) seem to
move in the same time uniformly, but towards the oppo-
site parts, or from east to west: and every one of these
objects, according to sense, describes the circumference of
a circle, whose plane is perpendicular to the axis of the
earth. And because all these circles, together with the
visible objects describing them, appear to be in the con-
cave spherical superficies of the heavens, every visible ob-
ject seems to describe a greater or less circle, according
to its greater or less distance from the poles, or extremi-
ties of the earth's axis produced ; the middle circle be-
tween these poles, called the equator, being consequently
the greatest.

DODECAGON, a regular polygon of 12 equal sides
and angles. If the side of a dodecagon be 1, its area will
be equal to 3 times the tang, of 75° = 3 x (2 +*/3) =
ll'lt)6l524 nearly; and, the areas of plane figures being
as the squares of their sides, therefore 1 I • 1 p6" 1 52-1' mul-
tiplied by the square of the side of any dodecagon, will
give its area. See my Mensuration, pa, 85, 4th edit.

To inscribe a Dodecagon in a given Circle. Carry the
radius 6 times round the circurnference, which will di-
vide it into 6 equal parts, or will make a hexagon; then
bisect each of those parts, which will divide the whole
into 12 parts, for the dodecagon. See also other methods
of describing the same figure in my Mensuration, pa. l6,
&c. See Polygon.

DODECAHEDRON, one of the Platonic bodies, or
five regular solids, being contained under a surface com-
posed of twelve equal and regular pentagons.

To form a Dodecahedron. See Regular Body.

If the side, or linear edge, of a dodecahedron be s, its
surface will be 15sV(! + W5) = 20'64577S8s% and
its solidity £sV(4'7 + 2-1^/5) = 7-6631 1 S.96V.

If the radius of the sphere that circumscribes a dode-
cahedron be r, then is

its side or linear edge = (\/15 — \/3) x |>';
its superficies - = l(>V(2 — |\/5);

:.nd its soliditv - = J .

- 3 v :jo

The side of a dodecahedron inscribed in a sphere, is
equal to the greater part of the side of a cube inscribed
,n the same sphere, cut in extreme and mean propor-
tion.— If a line be cut according to extreme and mean
ratio, and the less segment be taken for the side of a do-

Vol. I.

decahedron, the greater segment will be the side of a cube
inscribed in the same sphere. — The side of the cube is
equal to the right line which subtends the angle of a pen-
tagon of the dodecahedron, inscribed in the same spheie.
See Polyhedron; also my Mensur. pa. 188, &c.

DODSON (James), was an ingenious and very indus-
trious mathematician, as well as author of several useful
books; but we are acquainted with very few particulars
of his life. He was some time master of the Royal Ma-
thematical School in Christ's Hospital, London. His
publications chiefly were, 1. The Antilogarithmic Canon,
folio, 1742. — 2. The Calculator, a collection of useful
tables, large, 8vo, 1747- — 3. Mathematical Repository,
being a collection of analytical questions and solutions,
in 3 vols. 12mo. an. 1748, 1753, 1755.

DOG, a name common to two constellations, called
the Great and Little Dog ; but more usually Canis Major,
and Canis Minor.

DOLLOND (John), the celebrated optician, was born
in Spitalfields, in June, 1706: his parents were French
protestants, who quitted Normandy at the revocation
of the edict of Nantz, in 1685.

The first years of Mr. Dollond's life were employed at
the loom ; but, being of a very studious and philosophic
turn of mind, his leisure hours were engaged in mathe-
matical pursuits ; and though by the death of his father,
which happened in his infancy, his education gave way to
the necessities of his family, yet at the age of fifteen, be-
fore he had an opportunity of seeing works of science or
elementary treatises, he amused himself by constructing
sun-dials, drawing geometrical schemes, and solving
problems. Under- the pressure of a close application to
business for the support of his family, he found time, by
abridging the hours of his rest, to extend his mathemati-
cal knowledge, and made a considerable proficiency in
optics and astronomy, to which he now principally de-
voted his attention, having in the earlier stages of his lite
prepared himself for the higher parts of those subjects, by
a correct knowledge of algebra and geometry. He also
acquired a very respectable knowledge of the Latin and
Greek languages.

He designed his eldest son, Peter Dollond, for the same
business with. himself ; and for several years they carried
on their manufacture together in Spitalfields ; but the em-
ployment neither suited the expectations nor disposition
of the son, who, having received much information on
mathematical and philosophical subjects, from the in-
struction of his father, and observing the great value
which was set upon his father's knowledge in the theory
of optics by professional men, determined to apply that
knowledge to the benefit of himself and his family ; and ac-
cordingly, under the directions of his father, commenced
optician. Success attended every effort; and in the year
1752, John Dollond, embracing the opportunity of pur-
suing a profession congenial with his mind, joined his son,
and in consequence of his theoretical knowledge, soon be-
came a proficient in the practical parts of optics.

His first attention was directed to improve the com-
bination of the eye-glasses of refracting telescopes; and
having succeeded in his system of four eye-glasses, he
proceeded one step further, and produced tele.scopes fur-
nished with five eve-glasses, which considerably surpassed
the former; and of which he gave a particular account
in a paper presented to the Royal Society, and which was
printed in the Philos. Trans, vol. 48, pa. 103.
3K

D O L

[ 434 ]

D O M

Soon after this he made a very useful improvement in
Mr. Saver/ s micrometer: for, instead of employing two
entire object-glasses, as Mr. Savery and M. Bnuguer bad
done, he used only one glass cut into two equal parts,
one of them sliding or moving laterally by the other.
This was considered to be a great improvement, as the
micrometer could now be applied to the reflecting tele-
scope with much advantage, and which Mr. James Short
immediately did. An account of the same was given to
the Royal Society, in a paper, which was afterwards
printed in the Philos. Trans, vol. 48, pa. 178.

Mr. Dbllond's celebrity in optics now became uni-
versal ; and the friendship and protection of the most
eminent men of science flattered and encouraged his pur-
suits. Surrounded by these enlightened men, in a state
of mind prepared for the severest investigation of philo-
sophic truths, and in circumstances favourable to liberal
inquiry, Mr. Dollond engaged in the discussion of a sub-
ject, which at that time not only interested this country,
but all Europe. Sir Isaac Newton had declared, in his
Treatise on Optics, page1 112, " That all refracting sub-
stances diverged the prismatic colours in a constant pro-
portion to their mean refraction ;" and drew this conr
elusion, " that refraction could not be produced without
colour;" and consequently, " that-no improvement could
be expected in the refracting telescope." No one doubted
the accuracy with which Sir Isaac Newton had made the
experiment; yet some men, particularly M. Euler and
others, were of opinion that the conclusion which Newton
had drawn from it, went too far, and maintained, that in
very small angles refraction might be obtained without
colour. Mr. Dollond was not of that opinion, but de-
fended Newton's doctrine with much learning and inge-
nuity, as may be seen by a reference to the letters which
passed between Euler and Dollond on that occasion, and
which were published in the Philos. Trans, vol. 48, pa.
287, and contended that, " If the result of the experi-
ment had been as described by Sir Isaac Newton, there
could not be refraction without colour."

A mind constituted like Mr. Dollond's could not re-
main satisfied with arguing in this manner from an ex-
periment made by another, but determined t6 try it him-
self: and accordingly, in the year 1757, he began the
examination ; and, to use his own words, with " a reso-
lute perseverance," continued during that year, and a
great part of the next, to bestow his whole mind on the
subject, until in the month of June, 1758, he found, after
a complete course of experiments, the result to be very
different from that which he expected, and from that
which Sir Isaac Newton had related. He discovered
" the difference, in the dispersion of the colours of light,
when the mean rays are equally refracted by different
mediums." The discovery was complete, and he imme-
diately drew from it this practical conclusion, " That the
object-glasses of refracting telescopes were capable of
being made without being affected by the different refran-
gibility of the rays of light." His account of this experi-
ment, and of others connected with it, was given to the
Hoyal Society, and printed in their Transactions, vol. .50,
pa. 733, and he was presented, in the same year, by that
learned body, with Sir Godfrey Copley's medal, as a re-
ward of his merit, and a memorial of the discovery,
though not at that time a member of the Society.

This discovery no way affected the points in dispute
between Euler and Dollond, respecting the doctrine ad-

vanced by Sir Isaac Newton. A new principle was in a
manner found out, which had no part in their former
reasonings, and it was reserved for the accuracy of Dol-
lond to have the honour of making a discovery which had
eluded the observation of Newton. This new principle
being now established, he was soon able to construct
object-glasses, in which the different refrangibility of the
rays of light was corrected, and the name of achromatic
given to them by the late Dr. Bevis, on account of their
being free from the prismatic colours.

As usually happens on such occasions/ no sooner was
the achromatic telescope made public, than the rivalship
of foreigners, and the jealousy of philosophers at home,
led them to doubt of its reality ; and Euler himself, in his
paper read before the Academy of Sciences at Berlin, in
the year 1764, says, " I am not ashamed frankly to avow,
that the first accounts which were published of it, ap-
peared so suspicious^ and even so contrary to the best
established principles, that I could not prevail upon my-
self to give credit to them ;" and he adds, " I should
never have submitted to the proofs which Mr. Dollond
produced to support this strange phenomenon, if M.
Clairaut, who must at first have been equally surprised at'
it, had not most positively assured me, that Dollond's
experiments were but too well founded." And when the
fact could no longer be disputed, they endeavoured to find
a prior inventor, to whom it might be ascribed, and several
conjecturers were honoured with the title of discoverers.

In the beginning of the year 176l, Mr. Dollond was
elected Fellow of the Royal Society, and appointed op-
tician to- his majesty, but did not live to enjoy those ho-
nours long; for on the 30th of November, in the same
year, as he was reading a new publication by M. Clairaut,
on the theory of the moon, and on which he had been in-
tently engaged for several hours, he was seized with
apoplexy, which rendered him immediately speechless,
and occasioned his death in a few hours afterwards.

Besides Mr. Peter Dollond, whom we have had occa-
sion to mention in the course of this memoir, his family,
at his death, consisted of three daughters and a son, who,
possessing the name of his father, and, we may add, a
portion of the family abilities, carries on the optical busi-
ness, in partnership with his elder brother. Sec also the
articles Achromatic and Telescope.

DOME, is a round, vaulted, or arched roof, of a
church, hall, pavilion, vestibule, stair-case, &c, by way
of crowning, or acroter. — In plate 7, is represented the
plan and elevation of a dome constructed without cen-
tring, by Mr. S. Bunce ; viz, fig. 1 the plan, and fig. 2
the elevation. The first course consists of the stones
marked 1, 1, 1, &c, of different sizes, the large ones ex-
actly twice the height of the small ones, placed alternately,
and forming intervals to receive the stones marked 2, 2, 2.
The other courses are continued in the same manner, ac-
cording to the order of the figures to the top.

It is evident, from the convergingorwedgc-like form of
the intervals, that the stones they receive can only be
inserted from the outside, and cannot fall through : there-
fore the whole dome may be built without centring or
temporary support. To break the upright joints, the
stones may be cut of the form marked in fig. 3 ; and those
marked 16', 17, &c, near the key-stones, may be enlarged
as at fig. 4. .

For the mathematical theory of Domes, see my new
Tracts, vol. 1, pa. 63, &c.

D O M

[ 436 ]

DOR

DOMINICAL Letter, otherwise called the Sunday
Letter, is one of these first seven letters of the alphabet,
abcdefg, used in almanacs,] &c, to mark or denote
the Sundays throughout the year. The reason for using
seven letters, is because that is the number of days in a
week ; and the method of using them is this : the first
letter a is set opposite the 1st day of the year, the 2d
letter b opposite the 2d day of the year, the 3d letter c
opposite the 3d day of the year, and so on through the
seven letters ; after which they are repeated over and over
again, all the way to the end of the year, the letter a
denoting the Sth day, the letter b the 9th, &c. Then,
whichever of the letters so placed, falls opposite the first
Sunday in the year, the same letter, it is evident, will
fall opposite every future Sunday throughout the year,
because the number of the letters is the same as the num-
ber of days in the week, being both 7 in number ; that is,
in common years ; but in leap-years, an interruption of
the order takes place in them. For, on account of the
intercalary day, either the letters must be thrust out of
their places for the whole year afterwards, so as that the
letter, for example, which answers to the 1st of March,
shall likewise answer to the 2d, &c ; or else the intercalary
day must be denoted by the same letter as the preceding
one. This latter expedient was judged to be the best, and
accordingly all the Sundays in the year after the inter-
calary day .have another Dominical letter.

The Dominical letters were introduced into the ca-
lendar by the primitive Christians, instead of the nundinal
letters in the Roman calendar ; and in this manner were
those seven letters set opposite the days of the year, to
denote the days in the week, in most of our common
almanacs, till the year 1771, when the initial letters of
the days of the week were generally introduced instead of
them, excepting the Sunday letter itself, which is still
retained.

From the foregoing account it follows, that,

1st, As the common year consists of 365 days, or 52
xveeks, and one day over ; the letters go one day back-
wards every common year : so that in such a year, if the
beginning or first day fall on a Sunday, the next year it
will fall on a Saturday, the next on a Friday, and so on.
Consequently, if g be the Dominical letter for the present
year, r will be that for the next year; and so on in a re-
trograde order.

2d, As the bissextile or leap-year consists of 366 days,
or 52 weeks, and 2 days over, the beginning of the year
next after bissextile goes back 2 days. Whence, if in the
beginning of the bissextile year the Dominical letter were
g, that of the following yea^r will be e.

3d, Since in leap-years the intercalary day falls on
the 24th of February, in which case the 24th and 25th
days are considered as one day, and denoted by the same
letter ; after the 24th day of February the Dominical
letter goes back one place: thus, if in the beginning of
the year the Dominical letter be e, it will afterwards
change to the letter F for the remaining Sundays of the
year. With us, however, this day is now added at the
end of February, and from thence it is that the change
takes place.

4th, As every fourth year is bissextile, or leap-year,
and as the number of letters is 7; the same order of
Dominical letters only returns in 4 limes 7, or 28 years ;
which, without the interruption of bissextiles, would re-
turn in 7 years.

5th, Hence the invention of the solar cycle of 28 years ;
on the expiration of which the Dominical letters are
restored successively to the same days of the month, or
the same order of the letters returns.

To find the Dominic a l Letter of any given year. Find
the cycle of the sun for that year, as directed under
Cycle ; and the Dominical letter is found corresponding
to it. When there are two letters, the proposed year is
bissextile ; the former of them serving till the end of
February, and the latter for the rest of the year.

The Dominical letter for any year of the present cen-
tury may be found thus :

To the number of years since the date 1800, add their
fourth part, plus 2; and divide the sum by 7, then sub-
tract the remainder from 7, and the last number will be
that answering to the Dominical letter ; that is, if the
remainder be 1, it answers to a, 2 to b, &c, and 0 to G.
■ Suppose, for example, we wanted the Dominical letter
for the year 1833.

Here 33 + 8 + 2=43, being divided by 7, leaves a re-
mainder 1, which taken from 7, gives 6; answering to F,
which therefore is the Dominical letter for that year.

When the year is bissextile, that is, when the date of
it is divisible by 4, then the letter obtained from this rule
is that used from February to the end of the year, and
the Sunday letter from the beginning of the year, to the
end of February, will be that, next in order in the alphabet,
unless the former is G, in which case the latter will be a.

By the reformation of the calendar under Pope Gre-
gory the 13th, the order of the Dominical letters was
again disturbed in the Gregorian year; for the year 1582,.
which had g for its letter at the beginning, by retrench-
ing 10 days after the 4th of October, came to have c:
by which means the Dominical letter of the ancient Ju-
lian calendar is 4 places before that of the Gregorian,
the letter a in the former answering to d in the latter.

DONJON. See Dungeon.

DONN (Benjamin), a respectable mathematician,
was born at Bideford, in Devonshire, in 1729. He kept
a school in that town for some years, and while there,
made a complete survey and map of the county, for
which he received a premium of 100/. from the society
for promoting arts and commerce. He also published his
Mathematical Essays in Svo, which had a favourable re-
ception, and procured him the office of keeper of the
library at Bristol ; where he also kept a flourishing aca-
demy for some years. In 1771 he printed an Epitome of
Natural and Experimental Philosophy, 12mo; and in
1774, a work, entitled, The British Mariner's Assistant,
being a collection of tables for nautical purposes. In
17.96 he was appointed to the sinecure office of master of
mechanics to the king. Besides the books above men-
tioned, he published also treatises on Arithmetic, Book-
keeping, Geometry, and Trigonometry. Mr. Donn died
in 1798, leaving behind him the character of an ingenious
and worthy man.

DORADO, a southern constellation, not visible in our
latitude; called also Xiphias, or the Sword-fish. Its
stars, in Sharp's catalogue, are 6.

DORIC Order of Architecture, is the second of the
five orders, being placed between the Tuscan and the
Ionic. The Doric seems the most natural, and best pro-
portioned, of all the orders;, all its parts being founded
on the natural position of solid bodies; for which reason
it is most proper to be used in great and massy buildings,
3 K 2

D R A.

[ «6 ]

DUG

as the outside of churches and public places. The Doric
order has no ornaments on its base nor its capital. Its
column is 8 diameters high, and its frieze is divided be-
tween triglyphs and metopes.

DORMER, or Dormant, in Architecture, denotes a
window made in the roof of a building, or above the en-
tablature ; being raised upon the rafters.

DOUBLE Horizontal Dial, one with a double gnomon,
the one pointing out the hour on the outer circle, the
other, on the stereographic projection drawn upon it. This
dial rinds the meridian, the hour, the sun's place, rising,
setting, &c, and many other problems of the globe.

Double Point, in the Higher Geometry, is a point
which is common to two parts or legs, or branches of
some curve of the 2d or higher order : such as, an in-
finitely small oval, or a cusp, or the cruciform intersec-
tion, &c, of such curves. See Newton's Enumeratio
Linearum, &c, de Curvarum Punctis Duplicibus.

Double Stars. See Star.

Double r of Electricity, is an instrument capable of
augmenting a very small quantity of electricity, so as to
render it more than sufficiently manifest, by means of an
electrometer, or even capable of affording sparks.

The invention of this instrument is due to the Rev.
Abraham Bennet; the description of which is contained
in the second part of the 77th. volume of the Philosophical
Transactions; and in the same Transactions, for the year
17S8, Mr. Cavallo has given an account, at considerable
length, of his experiments and observations on this instru-
ment, as improved by himself, and also in its original form;
from which he concludes, " that the invention of the
doubler is very ingenious, but its use cannot be depended
on ;" that is to say, for ascertaining small quantities of
electricity, because though no electricity be communi-
cated to the plates of the instruments, yet by the usual
process of doubling, a certain quantity of it will be pro-
duced. For the description of a very ingenious instru-
ment for this purpose, by Mr. Nicholson, see Philosophi-
cal Transactions for the year 1788.

Doubling a Cape, or Point of Land, in Navigation,
signifies the coming up with it, passing by it, and leaving
it behind the ship. The Portuguese pretend that they
lirst doubled the Cape of Good Hope, under their admiral
Yasquez de Garaa ; but there are accounts in history,
particularly in Herodotus, that the Egyptians, Carthagi-
nians, &c, had done the same long before them.

DOUCINE, in Architecture, is an ornament on the
highest part of the cornice, or a moulding cut in the figure
of a wave, half convex, and half concave.

DRACHM, or Dram, is the name of a small weight
used with us, and is of two kinds, viz, the 8th part of an
ounce in Apothecaries' weight, and the l6th part of an
ounce in Avoirdupois weight.

DRACO, the Dragon, a constellation of the northern
hemisphere, whose origin isvariouslv tabled by the Greeks;
some of them representing it as the Dragon which guarded
the Hesperian fruit, or golden apples, but being killed by
Hercules, Juno, as a reward for its faithful services, took
it up to heaven, and so formed this constellation: while
others say, that in the war of the giants, this Dragon was
brought into the combat, and opposed to Minerva, .when
the goddess taking the Dragon in her hand, threw him,
twisted as he was, up to the skies, and fixed him to the
axis of the heavens, before he had time to unwind his con-
tortions.

The stars in this constellation are, according to Pto-
lemy, 31; according to Tycho, 32 ; Hevelius, 40; Bayer,
33 ; and Flamsteed, SO.

DRAGON, in Astronomy. See Dracq.

Dragon's Head, and Tail, are the nodes of the planets,
but more particularly of the moon, being the points in
which the ecliptic is intersected by her orbit, in an angle
of about 5° 18'. One of these points is northward, the
moon then beginning to have north latitude ; and the other
southward, where she commences south latitude; the for-
mer point being represented by the knot Q for the head,
and the other by the same reversed, or JJ for the tail.
And near these points it is that all eclipses of the sun and
moon happen. These intersections are not always in the
same two point" of the ecliptic, but shift by a retrograde
motion, at the rate of 3' 1 1" per day, and completing their
circle in 18 years 2-25 days.

DRAGOH-Beams, in Architecture, are two strong braces
or struts, standing under a breast-summer, and meeting in
an angle on the shoulder of the king-piece.

DRAM. See Drachm.

DRAUGHT- Conpasses, those provided with several
moveable points, to draw fine draughts in architecture, &c.
See Compass.

DRAUGHT- J/oo&s, are large hooks of iron, fixed on
the cheeks of a gun-carriage, two on each side, one near
the trunnion hole, and the other at the train.

DRAW -Bridge, a bridge made after the manner of a
floor, to be drawn up, or let down, as occasion requires,
before the gate of a town or castle.

DRIFT, Shoot, or Thrust of an Arch, is the push or
force it exerts.

Drift, in Navigation, denotes the angle which the line
of a ship's motion makes with the nearest meridian, when
she drives with her side to the wind and waves, and is not
governed by the power of the helm ; and also the distance
which the ship drives on that line, so called only in a
storm.

DRIP, in Architecture. See Larmier.

DUCTILITY, a property of certain bodies, by which
they are capable of being beaten, pressed, drawn, or
stretched forth, without breaking; or by which they are
capable of great alterations in their figure and dimensions,
and of gaining in one way as they lose in another.

Such are metals, which, being urged by the hammer,
gain in length and breadth what they lose in thickness; or,
being drawn into wire through holes in iron, become longer
as they are made more slender. Such also are gums, glues,
resins, and some other bodies; which, though not malle-
able, may yet be denominated ductile, in as much as, when
softened by water, fire, or some other menstruum, they
may be drawn into threads.

Some bodies are ductile both when they are hot and
cold, and in all circumstances: such are metals, and es-
pecially gold and silver; other bodies are ductile only
when they have a certain degree of heat; such as glass,
and was, and such like substances : Others again are ductile
only when cold, and brittle when hot ; as some kinds of
iron, viz, those called by workmen redshort, as also brass,
and some metallic alloys.

The cause of ductility is very obscure; as depending
much on hardness, a quality whose nature we know very
little about. It is true, it is usual to account for hard-
ness from the force of attraction between the particles of
the hard body ; and for ductility, from the particles of (he

D U C

[ 437 ]

D U C

ductile body being as it were jointed, and entangled with
each other. But without dwelling on any fanciful hypo-
theses about ductility, we may amuse ourselves with some
truly amazing circumstances and phenomena of it, in the
instances of gold, glass, and spiders' webs. Observing,
however, that the ductility of metals decreases in the fol-
lowing order: gold, silver, copper, iron, tin, lead.

Ductility of Gold. One of the properties of gold is,
to be the most ductile of all bodies ; of which the gold-
beaters and gold-wire-drawers, furnish us with abundant
proof. Fa. Mersenne, M. Rohaulf, Dr. Halley, &c, have
made computations of it : but they trusted to the reports
of the workmen. M. Reaumur, in the Memoiresde l'Aca-
demie Royale des Sciences, an. 1713, took a surer way;
he made the experiment himself. A single grain of gold,
he found, even in the common gold leaf, used in most of'
our gildings, is extended into 36' and a half square inches ;
and an ounce of gold, which, in form of a cube, has not
half an inch for its side, is beaten under the hammer into
a surface of 146 and a half square feet; an extent almost
double to what could be done in former times. In Fa.
Mersenne's time, it was deemed prodigious, that an ounce
of gold should form 1600 leaves; which, together, only
made a surface of 105 square feet.

But the distension of gold under the hammer (however
considerable), is nothingto that which it undergoes in the
drawing-iron. There are gold leaveSj in some parts scarce

tne jjtoot Part of an inch thick; but 71

V'

art of an

inch is a considerable thickness, in comparison of that of
the gold spun. on silk in our gold thread. To conceive
this prodigious ductility, it is necessary to have some idea
of the manner in which the wire-drawers proceed. The
wire, and thread that we commonly call gold thread, &c,
(which is only silver wire gilt, or covered over with gold)
is drawn from a large ingot of silver, usually about thirty
pounds weight. This they form into a cylinder, or roll,
about an inch and a half in diameter, and twenty-two
inches long, and cover it with the leaves prepared by the
gold-beater, laying one over another, till the cover is a deal
thicker than that in our ordinary gilding; and yet, even
then, it is very thin ; as will be easily conceived from the
quantity of gold that is used for gilding the thirty pounds
of silver, which seldom exceeds two ounces; and, fre-
quently, little more than one. In effect, the full thick-
ness of the gold on the ingot rarely exceeds ^^ or T^
part ; and sometimes not -j-g'-gs- part of an inch.

The ingot is then successively drawn through the holes
of several irons, each smaller than the other, till it bens
fine as, or finer than, a hair. Every new hole lessens its
diameter; but it gains in length what it loses in thick-
ness; and of course increases in surface: yet the gold still
covers it; it follows the silver in all its extension, and
never leaves the minutest part bare, not even to the mi-
croscope. How inconceivably therefore must it be at-
tenuated while the ingot is drawn into a thread, whose
diameter is 9000 times less than that of the ingot. M.
Reaumur, by exact weighing, and rigorous calculation,
found, that one ounce of the thread was 3232 feet long;
and the whole ingot 1163520 feet, Paris measure, or <J6
French leagues ; equal to 1264400 English feet, or 240
miles English ; an extent which far surpasses what Fa.
Mersenne, Furetiere, Dr. Halley, &c, ever thought of. -

Mersenne says, that half an ounce of the thread is 100
or fathoms long; on which calculation, an ounce

would only be 1200 feet: whereas M. Reaumur finds it
3232. Dr. Halley makes 6 feet of the wire one grain in
weight, and one grain of the gold C)8 vards; and conse-
quently the ten-thousandth part of a grain, above one third
of an inch. The diameter of the wire he found one- 186th
partofaninch; and the thickness of the gold one-154500th
part of an inch. But this too comes short of M. Reaumur;
for, on this principle, the ounce of wire would only be
2680 feet.

But the ingot is not even in this case extended to its full
length. The greatest part of our gold thread is spun, or
wound on silk ; and, before it is spun, they flat it, by pass-
ing it between two" rolls, or wheels of exceedingly well
polished steel; which wheels, in flatting it, lengthen it by
above one-seventh. So that our 240 miles are thus ex-
tended to 274. Now, the breadth of these laminae, or
plates, M. Reaumur finds, is only one-8th of a line, or
one-96'th of an inch; and their thickness one-3072d. The
ounce of gold, then, is here extended to a surface of 1190
square feet ; whereas, the utmost the gold-beaters can do,
we have observed, is to extend it to 146 square feet. From
M. Reaumur's calculus, it is found to be one-175000th of
a line, or one-21 00000th of an inch ; which is scarce one-
13th of the thickness that Dr. Halley supposed.

But he adds, that, from this supposition the thickness
of the gold is every where equal, which is no ways proba-
ble; for in beating the gold leaves, whatever care they can
bestow, it is impossible to extend them equally. This we
easily find, by the greater opacity of some parts than
others ; for, where the leaf is thickest, it willgild the wire
the thickest; and Reaumur, by computing what the thick-
ness of the gold must be where it is thinnest, finds it only
one-3 150000th part of an inch. Yet even this is not the
utmost ductility of gold ; for, instead of two ounces of
gold to the ingot, which we have here computed upon, a
single one might have been used; and then the thickness
of the gold, in the thinnest places, would only have been
the 6300000th part of an inch.

And still by only pressing them more between the flat-
ter's wheels, they are extended to double the breadth and
proportionably in length. So that their thickness, at last,
will be reduced to one-thirteen or fourteen millionth part
of an inch.

Yet, with this amazing thinness of the gold, it is still a
perfect cover for the silver: the best eye, or even the best
microscope, cannot discover the least chasm, or discon-
tinuity. There is not an aperture to admit alcohol of wine, .
the subtilest fluid in nature, or even light itself, unless it
be owing to cracks occasioned by repeated strokes of the
hammer. To which we may add, that if a piece of this
gold thread, or gold plate, be laid to dissolve in aquafortisv
the silver will be all excavated, or eaten out, and the gold
left entire, in little tubules.

It should be observed, that gold, when it has been struck
for some time by a hammer, or violently compressed, as
by gold wire-drawers, becomes more hard, elastic and
stiff, and less ductile, so that it is apt to be cracked or
torn : the same thing happens to the other metals by per-
cussion and compression; but ductility and tractability
may be restored to metals in that state, by annealing them,
or making them red hot. Gold, however, seemslo be
more affected by percussion and annealing, than any other
metals.

As to the Ductility of soft bodies, it.isnotyet carried

DUN

[ 438 ]

D U P

to that pitch. We must not be surprised that, among the works on practical branches- of mathematics ; and he pub-
ductile bodies of this class, the first place is given to the lished a folio Atlas, which has been held in some estima-
most brittle of all others, — glass. tion. Mr. Dunn died in good circumstances, and be-
Ductility of Glass. It is known that, when wellpe- queathed an estate of about 30/. a year, to support a ma-
netrated with the hea( of the fire, the workmen can figure thematical school in his native town, the first master to

and manage glass like soft wax ; but what is most remark-
able, it may be drawn or spun out into threads exceed-
ingly fine and long. Our ordinary spinners do not form
their threads of silk, flax, or the like, with half the ease
and expedition, as the glass-spinners do threads of this

which was appointed in 1793.

DUODECIMALS, a kind of multiplication in arith-
metic, by which artificers square their dimensions, that is,
multiply them together, to obtain the contents of their
worJks, in square feet, inches, and parts. The method is, to

brittle matter. Some of them that are used in plumes multiply the feet and inches in the one dimension, by each
for children's heads, and divers other works, are much of those in the other separately, in so many lines, the like
finer than any hair, and which bend and wave like it with names set under each other, feet under feet, and inches
every wind. under inches, and then adding their like parts together, as
Nothing is more simple and easy than the method of here annexed; first multiplying 14f. 9inc. j^ g'mc
making them : there are two workmen employed ; the first by the 4f. and then by the 6inc. observing
holds one end of apiece of glass over the flame of a lamp; to carry 1 for every 12, from one denomi-
and, when the heat has softened it, a second operator ap- nation to the other. Or, instead of multi-
plies a glass hook to the metal thus in fusion; and, with- plying by the 6' inches, or half a foot, it is
drawing the hook again, it brings with it a thread of glass, better to take half the upper line, or divide
which still adheres to the mass: then, fitting his hook on it by 2. The sum of the products is 66f. 4-^inc. for the
the circumference of a wheel about two feet and a half in content. But it is to be noted, that though the 6'6 are

4 6'

59

7

66 44

diameter, he turns the wheel 'as fast as he pleases ; which,
drawing out the thread, winds it on its rim; till, after a
certain number of revolutions, it is covered with a skein
of glass thread.

The mass in fusion over the lamp diminishes insen-

square feet, yet the 4|inc. are not square inches, but 4£
parts, in 12, of a square foot, or 4-J- twelfth-parts of a
foot, that is, 4^- parallelograms, each of 1 foot long and
1 inch broad. See the article Duodecimal Arithmetic;
as also Cross Multiplication, and most books on Arith-

sibly : being wound out, as it were, like a pelotoon, or metic or Mensuration,

clue of silk, upon the wheel; and the parts, as they recede DUPLE, or Dctuble Ratio, is that in which the ante-

from the flame, cooling, become more coherent to those cedent is double the consequent; or where the exponent

next to them; and this by degrees: the parts nearest the of the ratio is 2. Thus, 6 to 3 is in a duple ratio,

fire are always the least coherent, and of course must give Sub-DuvL'E Ratio, is that in which the consequent is

.way to the effort the rest make to draw them towards the double the antecedent; or in which the exponent of the

wheel. The circumference of these threads is usually a ratio is ■£. As in 3 to 6, which is in subduple ratio,

flat oval, being 3 or 4 times as broad as thick : some of DUPLICATE Ratio, is the square of a ratio, or the

them seem scarce larger than the thread of a silk-worm, ratio of the squares of two quantities. Thus, the dupli-

and are surprisingly flexible. If the two ends of such cate ratio of a to b, is the ratio of a'2 to bz, or of the square

threads be knotted together, they may be drawn and bent, of a to the square of b. — In a series of geometrical propor-

till the aperture, or space in the middle of the knot, doth tiohals, the 1st term is to the 3d, in a duplicate ratio of

not exceed one-4th of a line, or one-4Sth of an inch dia- the 1st to the 2d, or as the square of the first to the square

meter. of the 2d : Thus, in the geometricals 2, 4, 8, 16", &c, the

Hence M.Reaumur advances, that the flexibility of ratio of 2 to 8, is the duplicate of that of 2 to 4, or as the

glass increases in proportion to the fineness of the threads; square of 2 to the' square of 4, that is, as 4 to 16. So that

and that, probably, had we but the art of drawing threads duplicate ratio is the ratio of the squares, as triplicate ra-

as fine as a spider's web, we might weave stuffs and cloths tio is the ratio of the cubes, &c.

of them for wear. Accordingly, he made some experi- DUPLICATION, is the doubling of a quantity, or mul-

ments this way, and found he could make threads fine tiplying it by 2, or adding it to itself,

enough, viz, as fine, in his judgment, as spiders' thread, but Duplication of a Cube, is finding out the side of a

he could never make them long enough to do any thing cube that shall be double in solidity to a given cube: which

with them. is a celebrated problem, much cultivated by the ancient

Ductility of Spider's' Webs. See Web. geometricians, about 2000 years ago.

DUNGEON, Donjon, in Fortification, the highest part It was first proposed by the oracle of Apollo at Del-

of a castle built after the ancient mode; serving as a watch'
tower, or place of observation; and also for the retreat of

phos ; which, being consulted about the manner of stop-
ping a plague then raging at Athens, returned for answer,

a garrison, in case of necessity, so that they may capitulate- that the plague should cease when Apollo's altar, which

with greater advantage.

DUNN (Samuel), was a native of Crediton, in De-
vonshire, where he kept a mathematical school for several
years; but afterwards removed to Chelsea, where be long
kept an academy. He was a person well skilled in nauti-
cal calculations, and was a good practical astronomer,
being employed as mathematical examiner for the East

was cubical, should be doubled. Upon this, they applied
themselves in good earnest, to seek the duplicature of the
cube, which from thence was called the Delian problem.
This problem cannot be effected geometrically, as it re-
quires the solution of a cubic equation, or requires the
finding of two mean proportionals, viz, between the side of
the aiven cube and the double of the same, the first of

India service. Besides several papers of his inserted in the which two mean proportionals is the side of the double
Philos. Trans, be was also the author of some separate cube, as was first observed by Hippocrates of Chios. For,

DUE,

[ 43y ]

D Y P

let a be the side of the given cube, and z the side of the
double cube sought;, then it is z3=2a3, or az : sz : : z :
2a ; so that, if a and 2 be the 1st and 2d terms of a set. of
continued proportionals, then a'2 : zz is the ratio of the
square of the 1st to the square of the 2d, which, it is
known, is the same as the ratio of the 1st term to the 3d,
or of the 2d to the 4th, that is of z to la; therefore, z
being the 2d term, 2a will be the 4th. So that z, the side
of the cube sought, is the 2d of four terms in continued pro-
portion, the 1st and 4th being a and 2«, that is, the side of
the double cube is the first of two mean proportionals be-
tween a and la.

Eutocius, in his Commentaries on Archimedes, gives se-
veral ways of performing this by the mesolabe. In Pappus
too are found three different ways; the first according to
Archimedes, the second according to Hero, and the 3d by
an instrument invented by Pappus, which gives all the
proportions required. The Sieur de Comiers has also pub-
lished a demonstration of the same problem, by means of
a compass with three legs. But all these methods are only
mechanical. See Valerius Maximus, lib. 8 ; also Eu-
tocius's Comm. on lib. 2. Archimedes de Sphasra et Cy-
lindro; and Pappus, lib. 3, prop. 5, and lib. 4, prop. 22.

DURER (Albert), descended of an Hungarian fa-
mily, but born at Nuremberg, in 1471, was one of the best
engravers, painters, and practical geometricians of his age.
He was at the same time a man of letters and a philoso-
pher; and he was an intimate friend of Erasmus, who re-
vised some of the pieces which he published. He was also
a man of business, and for many years the leading magis-
trate of Nuremberg.

Though not the inventor, he was one of the first and
greatest improvers of the art of engraving. He was how-
ever the inventor of cutting in wood, which he devised and
practised in great perfection, using this way for expedi-
tion, as he had a multitude of designs to execute; and as
his work was usually done in the most exquisite manner,
Ij is pieces took him up much time. For in many of those
prints which he executed on copper, the engraving is ele-
gant to a high degree. His Hell-Scene in particular, en-
graven in the year 1513, is as highly finished a print as
ever was engraved, and as happily executed. In his
wooden prints too it is surprising to see so much meaning
in so early a master. In fact, Durer was a man of a very
extensive genius. His pictures were excellent; as well as
his prints, which were very numerous. They were much
admired, from the first, and soon bought up; which put
his wife, who was another Xantippe, on urging him to
spend more time upon engraving than he was inclined to
do : for he was rich ; and chose rather to practise his art
as an amusement, than as a business. He died at Nu-
remberg, in 1528, at 57 years of age.

Albert Durer wrote several books, in the German lan-
guage, which were translated into Latin by other persons,
and published after his death : viz,

1. His book upon the rules of painting, entitled, De
Symmetria Partium in rectis formis Humanorum Corpo-
rum, is one of them: printed in folio, at Nuremberg, in
1532, and at Paris in 1557- An Italian version also was
published at Venice, in 1591.

2. Institutiones Geometrical; Paris 1532.

3. De Urbibus, Arcibus, Castellisque condendis et mu-
niendis; Paris 1531.

4. De Varietate Figurarum, et Flexuris Partium, et
Gestibus Imaginum; Nuremberg 1534.

The figures in these books, which are from wooden
plates, are very numerous, and most admirably well exe-
cuted, indeed far beyond any thing of the kind done in our
own days. Some of them also are of a very large size, as
much as 16 inches in length, and of a proportional breadth,
which being exquisitely worked, must have cost great la-
bour. His geometry is chiefly of the practical kind, con-
sisting of the most curious descriptions, inscriptions, and
circumscriptions of geometrical lines, planes, and solids.
We here meet, for the first time, with the plane figures,
which folded up make the five regular or platonic bodies,
as well as that curious construction of a pentagon, "being
the last method in prob. 23 of my Mensuration.

DYE, in Architecture, the trunk of the pedestal, or that
part between the base and the cornice, being so called, be-
cause it is often made in the form of a dye or cube.

DYNAMICS, is the science of moving powers; more
particularly of the motion of bodies that mutually act on
each other. -See Mechanics, Motion, Communica-
tion of Motion, Oscillation, Percussion, &c; also
my Course of Mathematics, vol. 2; besides the Me-
canique Philosophique of Prony, the Mecanique Analy-
tique of Lagrange, and the Mecanique Celeste of Laplace.

DYNOMETERis the name of an instrument intended
for measuring the muscular strength of men and other
animals. Mr. Graham, many years ago, invented an in-
strument for this purpose, which was afterwards improved
by Dr. Desaguliers; but it was still too bulky and limited
in its operations, to be of much use. The defect of this
machine induced M. Leroy, of the Academy of Sciences
at Paris, to construct another on more' simple principles,
and which was certainly more useful for the purpose it was
designed for, than Mr. Graham's. The principle of this
instrument was not much unlike that of the common
spring steelyard, and perhaps it would have been still
better, had its principles been exactly the same; for with
this last-mentioned instrument, it is evident that the
power of an animal may be as accurately ascertained as
the weight of a body, and nothing more simple can well
be devised.

DYPTERE, or Diptere, was a kind of temple, encom-
passed round with a double row of columns; and the
pseudo-diptere, or false diptere, was the same, only this
was encompassed with a single row of columns, instead of
a double one.

[ 440 ]

E.

EAR

EAGLE, Aquila, is a constellation of the northern he-
misphere, havingits right wing contiguous to the equi-
noctial. For the stars in this constellation, see Aquila.

EARTH, Terra, in Natural Philosophy, one of the four
elements of the ancients; for, according to their system,
all bodies were considered as resolvable into four simple
elements, or constituent parts, namely, air, fire, water,
and earth. The two former were avowedly almost un-
known; most liquids were supposed to be modifications
of the third, and the solid particles of bodies were at-
tributed to the last. Earth in this extended sense was
defined, a simple, dry, and cold substance; and as such,
an ingredient that entered into the composition of all na-
tural bodies. Modern chemists, though they have re-
tained the term in a much more restricted sense, have yet
included under it a sufficiently heterogeneous assemblage
of bodies, many of which may be more properly considered
as belonging to the class of alkalies.

The earths at present known, are nine in number;
namely, Silex, Zircon, Alumine, Glycine, Yttria, Bary
tes, Strontian, Lime, and Magnesia. And when they are
purified by art from all foreign mixture, they agree in the
following properties : 1. They are of a snow-white colour.
2- They are infusible by a Very intense heat. 3. They
are not reducible to the metallic state, by being heated in
contact with combustible matter. Of those nine earths,
however, the last four have all the properties of alkalies;
whence they are sometimes called alkaline earths; and
many chemists are of opinion that the term earth ought
only to be applied to the five former, by which arrange-
ment the. two classes of earths and alkalies would admit
of distinct chemical definitions, which at present is wanted.

Earth, in Geography, the terraqueous globe or ball
which we inhabit, consisting of land and sea.

Figure of the Earth. The ancients had various opi-
nions as to the figure of the earth ; some, as Anaximan-
der and Leucippus, held it cylindrical, or in form of a
drum ; but the principal opinion was, that it was flat;
that the visible horizon was the bounds of the earth, and
the ocean the bounds of the horizon ; that the heavens
and earth above this ocean constituted the universe ; and
that all beneath the ocean was Hades : and of this same
opinion were also some of the Christian fathers ; asLac-
tantius, St. Augustine, &c. See Lactan. lib. 3, cap. 24 ;
St. Aug. lib. l6, de Civitate Dei ; Aristot. de Ccelo, lib.
2, cap.' 13.

Such of the ancients however as understood anything
of astronomy, and especially the doctrine of eclipses,
must have been acquainted with the round figure of the
earth ; as the ancient Babylonian astronomers, who had
calculated eclipses long before the time of Alexander, and
Thalcs the Grecian, who predicted an eclipse of the sun.

That the exterior of the earth is round, or rotund, is
manifest to the most common perception, in the case of a
ship sailing cither from the land, or towards it ; for when
a person stands upon the shore, and sees a ship sail from
the land, out to sea; at first he loses sight of the hull and
lower parts of the ship, next the rigging and middle parts,
and lastly of the tops of the masts themselves, in every

EAR

case the rotundity of the sea between the ship and the
eye being very visible: the contrary happens when a ship
sails towards us; we first see the tops of the masts appear
just over the rotundity of the sea ; next we percei\e the
rigging, and lastly the hull of the ship itself: all which
is well illustrated by the following figure.

The round figure of the earth is also evident from the
eclipses of the sun and moon; for in all eclipses of the
moon, which are caused by the moon passing through
the earth's shadow, that shadow always appears circular
upon the face of the moon, whatever way it be projected,
whether east, west, north, or south, and however its dia-
meter vary, according to the greater or less distance from
the earth. Hence it follows, that the shadow of the
earth, in all situations, is really conical ; and consequently
the body that projects it, i. e. the earth, is at least nearly
spherical.

The spherical figure of the earth is also evinced from
the rising and setting of the sun, moon, and stars; all
which happen sooner to those who live to the east, and
later to those living westwardly; and that more or less so,
according to the distance.

So also, in sailing to the northward, the north pole and
northern stars become more elevated, and the south pole
and southern stars more depressed; the elevation northerly
increasing equally with the depression southerly ; and
each of them proportionably to the distance ; and the
same thing happens in going to the southward ; that is, in
this case the south pole is elevated, and the north pole de-
pressed. Besides, the oblique ascensions, descensions,
emersions, and amplitudes of the rising and setting of the
sun and stars in every latitude, are agreeable to the
supposition of the earth's being of a spherical form: all
which could not happen if it was of any other figure.

But the most convincing proof of the spherical figure of
the earth, and which bids defiance (o every objection, is,
that many celebrated navigators have sailed round it, pro-
ceeding in a certain direction, and by constantly pur-
suing the same course, have finally arrived at the port

EAR

t 441 ]

EAR

whence they departed. The first who succeeded in this
daring attempt, was Ferd. Magellan, a Portuguese, in the
year 1519, and who completed his tour in 1124 days.
In the year 1557 Francis Drake performed the same in
1056 days: in the year 15 S6, Sir Tho. Cavendish made
, the same vo}'age in 777 days; Simon Cordes, of Rotter-
dam, in the year 1590; and in the year 1598, Oliver
, Noort, a Hollander, in 1077 days ; Van Schouten, in the
yearl6l5, in 749 days; Jac. Heremites and Joh. Huy-
gens, in the year l623, in 802 days: and many others
have since performed the same navigation, particularly
Anson Bougainville, and Cook; sometimes sailing round
by the eastward, sometimes to the westward, till at
length they arrived again in Europe, whence they set out ;
and in the course of their voyage, observed that all the
phenomena, both of the heavens and earth, corresponded
to, and evinced this spherical figure.

The globular figure of the earth is also inferred from the
operation of levelling, in which it is found necessary to
make an allowance for the difference between the apparent
and the true level.

The natural cause of this sphericity of the globe is, ac-
cording to Sir Isaac Newton, the great principle of at-
traction, which the Creator has stamped on all the matter
in the universe ; and by which all bodies, and all the
parts of bodies, mutually attract one another. — And the
same is the cause of the sphericity of the drops of rain,
quicksilver, &c.

Whatthe earth loses of its sphericity by mountains and
valleys, is nothing considerable ; the highest eminence
being scarce equivalent to the minutest protuberance on
the surface of an orange. Its difference from a perfect
sphere, however, is more considerable in another respect,
by which it approaches nearly to the shape of an orange,
or to an oblate spheroid, being a little flatted at the poles,
and raised about the equatorial parts, so that the axis
from pole to pole is less than the equatorial diameter.
What gave the first occasion to the discovery of this figure
of the earth, was the observations of some French and
English philosophers in the East Indies, and other parts,
who found that pendulums performed their vibrations
slower, the nearer they came to tlie equator; whence it
follows, that the velocity of the descent of bodies by gra-
vity, is less in countries nearer to the equator ; and con-
sequently that those parts are farther removed from the
centre of the earth, or from the common centre of gra-
vity. Seethe History of the Royal Academy of Sciences,
by Du Hamel, p. 110, 156, 206; and Hist, de l'Acad.
Roy. 1700 and 1701. This circumstance put Huygens and
Newton upon finding out the cause, which they attri-
buted to the revolution of the earth about its axis. If
the earth were in a fluid state, its rotation round- its axis
would necessarily make it put on such a figure, because
the centrifugal force being greatest towards the equator,
the fluid would there rise and swell most; and that its
figure really should be so now, seems necessary, to keep
the sea in the equinoctial regions from overflowing the
earth about those parts. See this curious subject well
handled by Huygens, in his discourse De Causa Gravi-
tatis, pa. 154, where he states the ratio of the polar dia-
meter to that of the equator, as 577 to 578. And New-
ton, in his Principia, first published in 1686, demonstrates,
from the theory of gravity, that the figure of the earth
must be that of an oblate spheroid generated by the ro-
tation of an ellipse about its shortest diameter, provided
Vol. I.

all the parts of the earth were of an uniform density
throughout, and that the proportion of the polar to the
equatorial diameter of the earth, would be that of 689
to 6'92, or nearly that of 229 to 230, or as -9956522 to 1 .
This proportion of the two diameters was calculated by
Newton in the following manner. Having found that the
centrifugal force at the equator is ^th of gravity, he
assumes, as an hypothesis, that the axis of the earth is to
the diameter of the equator as 100 to 101, and thence de-
termines what must be the centrifugal force at the equator
to give the earth such a form, and finds it to be -54Tths of
gravity : then, by the rule of proportion, if a centrifugal
force equal to yg-j-ths of gravity would make the earth
higher at the equator than at the poles by T|^th of the
whole height at the poles, a centrifugal force that is the
^fgth of gravity will make it higher by a proportional ex-
cess, which by calculation is ^th of the height at the
poles ; and thus he discovered that the diameter at the
equator is to the diameter at the poles, or the axis, as
230 to 229. But this computation supposes the earth to
be every where of a uniform density ; whereas if the earth
is more dense near the centre, then bodies at the poles will
be more attracted by this additional matter being nearer ;
and therefore the excess of the semi-diameter of the equa-
tor above the semi-axis, will be different. According to
this proportion between the two diameters, Newton farther
computes, from the different measures of a degree, that
the equatorial diameter will exceed the polar, by 34^- miles.
Nevertheless, Messrs. Cassini, both father and son, the
one in 1701, and the other in 1713, attempted to prove,
in the Memoirs of the Royal Academy of Sciences, that
the earth was an oblong spheroid ; and in 1718, M. Cas-
sini again undertook, from observations, to show that, on
the contrary, the longest diameter passes through the poles;
which gave occasion for John Bernoulli, in his Essai d'une
Nouvelle Physique Celeste, printed at Paris in 1735, to
triumph over the British philosopher, apprehending that
these observations would invalidate what Newton had de-
monstrated. And in 1720, M. De Mairan advanced ar-
guments, supposed to be strengthened by geometrical de-
monstrations, further to confirm the assertions of Cassini.
But in 1735 two companies of mathematicians were em-
ployed, one for a northern, and another for a southern "
expedition, the result of whose observations and measure-
ment plainly proved that the earth was flatted at the poles;
and so it has been considered ever since. See Degree.

The proportion of the equatorial diameter to the polar,
as stated by the gentlemen employed on the northern ex-
pedition for measuring a degree of the meridian, is as 1
to 0'9S91 ; by the Spanish mathematicians as 266 to 265,
or as 1 to 0-99624; by M. Bougueras 179 to 17S, or as
1 to 0-99441.

As to all conclusions however deduced from the length
of pendulums in different places, it is to be observed that
they proceed upon the supposition of the uniform density
of the earth, which is a very improbable circumstance ; as
justly observed by Dr. Horslcy in his letter to Captain
Phipps: "You finish your article," he concludes, "relating
to the pendulum with saying, ' that these observations
give a figure of the earth nearer to Sir Isaac Newton's
computation, than any others that have hitherto been
made ;' and then you state the several figures given, as
you imagine, by former observations, and by your own.
Now it is very true, that if the meridians be ellipses, or
if the figure of the earth be that of a spheroid generated
3 L

E A R

\ 442 ]

E A R

Eiy ihe revolution of an- ellipsis, turning on i is shorter axis, certainty, what relation subsists between the one quantity
the riailTcular figure, or the ellipticity of the generating and the other; our whole theory, except so far as it re-
ellipsis, which your observations give, is nearer to what lates to the homogeneous spheroid, is built upon false as-
Sir Isaac Newton says it should be, if the globe were sumptions, and there is no saying what figure of the earth
homogeneous, than any that can be derived from former any observations of the pendulum give."
observations. But vet it is not what you imagine. Taking He then lays down the following table, which shows

the gain of the pendulum in latitude 79° :}^' exactly as the different results of observations made in different lati-
you state it, the difference between the equatorial and the tudes; in which the first three columns contain the names
polar diameter, is about as much less than the Newtonian of the several observers, the places of observation, and
computation makes it, and the hypothesis of homogeneity the latitude of each ; the 4th column shows the quantity
would require, as you reckon it to be greater. The pro- of P — II in such parts as II is 100000, as deduced from
portion of 212 to 211 -should indeed, according to your comparing the length of the pendulum at each place of
observations, be the proportion of the force that acts upon observation, with the length of the equatorial pendulum
the pendulum at the poles, to the force acting upon it at as determined by M. Bouguer, upon the supposition that
the equator. But this is by no means the same with the the increments and decrements of force, as the latitude is
proportion of the equatorial diameter to the polar. If the increased or lowered, observe the proportion which theory
globe were homogeneous, the equatorial diameter would assigns. Only the 2d and the last value of P — II are con-
exceed the polar by T-J,5- of the length of the latter : and eluded from comparisons with the pendulum at Greenwich
the polar force would also exceed the equatorial by the and at London, not at the equator. 'Ihe5th column shows
like part. But if the difference between the polar and the value of S corresponding to every value of P — II, ac-
equatorial force be greater than -j-J^, (which may be the cording to Clainuit's theorem :
case in an heterogeneous globe, and seems to Le the case
in ours,) then the difference of the diameters should, ac-
cording to theory, be less than -j-J-o, and vice versa.

" I confess this is by no means obvious at first sight ;
so far otherwise, that the mistake, which you have fallen
into, was once very general. Many of the best mathe-
maticians were misled by too implicit a reliance upon the
authority of Newton, who had certainly confined his in-
vestigations to the homogeneous spheroid, and had thought
about the heterogeneous only in a loose and general way.
The late Mr. Clairaut was the first who set the matter
right, in his elegant and subtle treatise on the figure of
the earth. That work has now been many years in the
hands of mathematicians, among whom I imagine there
are none, who have considered the subject attentively,
that do not acquiesce in the author's conclusions.

" In the second part of that treatise, it is proved, that pected ; and they represent the ellipticity of the earth at
putting P for the polar force, II for the equatorial, 8 for about ^ig. But when we come within 10 degrecsof the
the true ellipticity of the earth's figure, and s for the el- equator, it should seem that the force of gravity suddenly
lipticity of the homogeneous spheriod, then becomes much less, and within the like distance of the

f-n _ g - therefore 3* = 2c P~n- Poles much greater than it could be in such a spheroid."

rr ' n ' The following problem, communicated by Dr. Leather-

and therefore, according to your observation, J = ^T. ]anj to Dr. Pemberton, and published by Mr. Robertson,
This is the just conclusion from your observations of the serves for finding the proportion between the axis and the
pendulum, taking it for granted, that the meridians are eqUatorial diameter, from measures taken of a degree of
ellipses: which is an hypothesis, upon which all the tne merjuian in two different latitudes, supposing the earth
reasoningsof theory have hitherto proceeded. Butplausible an 0D]ate spheroid,
as it may seem, I must say, that there is much reason from
experiment to call it in question. If it were true, the
increment of the force which actuates the pendulum,
as we approach the poles, should be as the square of the
sine of the latitude: or, which is the same thing, the de-
crement, as we approach the equator, should be as the
square of the cosine of the latitude. But whoever takes
'the pains to compare together such of the observations of
the pendulum in different latitudes, as seem to have been
made with the greatest care, will find that the increments
and decrements do by no means follow these proportions ;
and in those which 1 have examined, I find a regularity
in the deviation which little resembles the mere error of
observation. The unavoidable conclusion is, that the true
figure of the meridians is not elliptical. If the meridians
are not ellipses, the difference of the diameters may indeed,
or it may not, be proportional to the difference between Let Avap be an ellipse, representing a sect.on of the
the polar and the sunatorial force; but it is quite an un- earth through tbc axis r/> ; the equatorial diameter, or

Observers.

Places.

La,

udes.

p— n.

8.

Bouguer

Equator

0°

0'

Bouguer

Porto Bello

9

34

741-S

Green

Otaheitee

i7

29

563-2

?ia

Bouguer

San Domingo

18

av

5£> J -0

Abbe de la 1
Caille }

Cape of ")
Good Hope J

33

55

7315

- - - -

Paris

48

50

585-1

ihr

TheAcadeO

Pello

66

48

565-9

TST9

micians j

-

Capt. Phipps

....

79

50

47V2

TTX

" By this table it appears, that the observations in the
middle parts of the globe, setting aside the single one at
the Cape, are as consistent as could reasonably be ex-

EAR

[ 443 ]

EAR

the greater axis of the ellipse, being ao; let e and f be
two places where the measure of a degree has been taken ;
these measures are proportional to the radii of curvature
in the ellipse at those places ; andifcQ, cr be conjugates
to the diameters whose vertices are e and f, cq will be to
cr in the subtriplicate ratio of the radius of curvature at
e, to that at f, by Cor. 1, prop. 4. part 6 of Milnes's
Conic Sections, and therefore in a given ratio to one ano-
ther; also the angles qcp, ,rcp are the latitudes of e and
F ; so that, drawing qv parallel to vp, and qxtw to An,
these angles being given, as well as the ratio of cq to cr,
the rectilinear figure cvqxy is given in species ; and the
ratio of vc2 — zc2 (= qx x xw) to rz2 — ■ qvz (= rx x
xs) is given, which is the ratio of CA'tocp2 ; therefore
the ratio of ca to cp is given.

Hence, if the sine and cosine of the greater latitude,
be each augmented in the subtriplicate ratio of the mea-
sure of the degree in the greater latitude, to that in the
lesser, then the difference of the squares of fhe augmented
sine, and the sine of the lesser latitude,- will be to the
difference of the squares of the cosine of the lesser lati-
tude and the augmented cosine, in the duplicate ratio of
the equatorial to the polar diameter. For cq, being taken
in cq equal to cr, and qv drawn parallel to qv, cv and
vq, cz and zk will be the signs and cosines of the respec-
tive latitudes, to the same radius; and cv, vq will be the
augmentations of cv and cq, in the ratio named.

Hence, to find the ratio between the two axes of the
earth, let e denote the greater, and f the lesser of the
two latitudes, m and n the respective measures taken
in each; and let p denote

v2 + sin* e — sin3 f ~ greater axi:

It also appears by the above problem, that when one of
the degrees measured, is at the equator, the cosine of the
latitude of the other being augmented in the subtriplicate
ratio of the degrees, the tangent of the latitude will be
to the tangent answering to the augmented cosine, in the
ratio of the greater axis to the less. For supposing e the
place out of the equator; then if the semi-circle pbnnp be
described, and /c joined, and mo drawn parallel to ac : co
is the cosine of the latitude to the radius cp, and cy that
cosine augmented in the ratio before-named ; yq being to
y/, that is c« to en or cp, as the tangent of the angle ycq,
the latitude of the point e, to the tangent of the angle ycl,
belonging to the augmented cosine. Thus, if m represent the
measure in a latitude denoted by e, and n the measure at
the equator, let a denote an angle whose measure is

, .M . tan. A . lesser axis
cos. e x i/ — . Then is = -.

* N taa, E greateraxis

But m, or the length of a degree, obtained by actual
mensuration in different latitudes, is known from the fol-
lowing table:

Names.

* Eatit.

Value of M.

toises.

Swanberg and assoc.

66°

20'

m= 57209

Cassini and 7

49

22

'm = 57074

LaCaillc j

45

00

m = 57050

Boscovich

43

00

M =5()»72

DelaCaillc -

33

18

M =57037

Bouguer and assoc. \

on

0

M —56757

measur. at equator J

Now, by comparing each of the first five with tfTe 6th
or last, there are obtained 5 results, each showing the

relation of the axes or diameters; the arithmetical means
of all of which gives that ratio as 1 to 0-996747, which
is the ratio of 308 to 307. But if the last of the four re-
sults be omitted, the medium among the other four gives
the ratio of 1 to 0-996876, or of 320 to 319, which is
probably nearest the truth.

Now the magnitude, as well as the figure of the earth,
that is the polar and equatorial diameters, may be de-
duced from the foregoing problem. For, as half the latus
rectum of the greater axis ah is the radius of curvature
at a, it is given in magnitude from the degree measured
there, and thence the axes themselves are given. Thus,
the circular arc whose length is equal to the radius being
57'29578 degrees, if this number be multiplied by 56757
toises, the measure of a degree at the equator, the product,
will be the radius of curvature there, or half the latus
rectum of the greater axis ; and this is to half the lesser
axis in the ratio of the less axis to the greater, that is, as
319 to 320; whence the two axes are 6524252 and
654.4704 toises, or790l and 7926 English miles; and
the difference between the two axes about 25 miles. See
Robertson's Navigation, vol. 2, pa. 206 &c. See also
Suite des Mem. de l'Acad. 1718, pa. 247; and Maclaurin's
Fluxions, vol. 2, book 1, ch. 14. Also the later measure-
ments of the French Institute, of Swanberg, of Colonel
Mudge, &c, &c.

And nearly the same ratio is deduced from the lengths
of pendulums vibrating in the same time, in different la-
titudes; provided it be allowed that the meridians are
real ellipses, or the earth a true spheroid, which however
can only take place in the case of an uniform gravity in
all parts of the: earth.

Thus, in the new Petersburg Acts, for the years, 1788
and 1789, are accounts and calculations of experiments
relative to this subject, by M. Krafft. These experiments
were made at different times, and in various parts of the
Russian empire. This gentleman has collected and com-
pared them, and drawn the proper conclusions from them:
thus he infers that the length x of a pendulum that swings
seconds in any given latitude A, and in a temperature of
10 degreesof Reaumur's thermometer, may be determined
by this equation: 1 = 439-178 -+- 2"32I sine2 A, lines of
a French foot, or x — 39'0045 -+- 0206 sine2 A, in English
inches, in the temperature of 53 of Fahrenheit's thermo-
meter.

This expression nearly agrees, not only with all the
experiments made on the pendulum in Russia, but also
with those of Mr. Graham in England, and those of Mr.
Lyons in 79° 50' north latitude, where he found its length
to be 43T3S lines. It also shows the augmentation of
gravity from the equator to the parallel of a given latitude
A: for, putting g for the gravity under the equator, g for
that under the pole, and y for that under the latitude A,
M. Krafft finds ,y = (1 -1- 00052848 sine1 \)g; and there-
fore G = 1-0052S48£.

From this proportion of gravity under different lati-
tudes, the same author infers that, in case the earth is a
homogeneousellipsoid,itsoblateness must beT-iT; instead
of -jj5, which ought to be the result of this hypothesis:
but on the supposition that the earth is a heterogeneous
ellipsoid, he finds its oblateness, as deduced from these
experiments, to be^-J7; which agrees with that resulting
from the measurement of some of the degrees of the meri-
dian nearly. This confirms an observation of M. Laplace,
that, if the hypothesis of the earth's homogeneity be given
' 3 L 2

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up, then theory, the measurement of degrees of latitude,
and experiments with the pendulum, all agree in their re-
sult with respect to the oblateness of the earth. See Me-
moires de l'Acad. 17S3, pa. 17, and ids later works.

In the Philos. Trans, for 1791, pa. 236, Mr.Dalby has
given some calculations on measured degrees of the meri-
dian; whence he infers, that those degrees measured in
middle latitudes, will answer nearly to an ellipsoid whose
axes are in the ratio assigned by Newton, viz, that of 230
to 229- And as to the deviations of some of the others,
viz, towards the poles and equator, he thinks they are
caused by the errors in the observed celestial arcs.

On the supposition that the earth's figure is that of an
oblate spheroid, it has been shown, 1st, That a degree of
the earth's equator, is the first of two mean proportionals,
between the last and first degrees of latitude. 2d, If the
equatorial diameter be to the polar axis, as ISO to 179;
the lat. in which the degree of lat. is equal to a degree of
the equator, will be 54° 4S' 24". If the proportion be 230
to 229, the lat. comes out 54° 45' 3l". 3d, If the pro-
portion be ISO to 179, the lat. where the degree of the
meridian is equal to a degree of a circle whose diameter is
equal to the earth's axis, will he 35° 20' 30" ; assuming
the proportion 230 to 229, that lat. will be 35° 19' 25".
(Horsley's Tracts, p.394, &c.)

M. Swanberg observes, that occultations of the fixed
stars by the moon is another method by which the figure
of the earth may be determined : and M. Treisnecker, after
comparing a great number of these occultations, concludes
from them that the ellipticity of the earth is T^.

The excellent mathematician and astronomer Laplace,
in his Mecanique Celeste, has calculated the ellipticity of
the earth, from the effect of precession and nutation, to
be tj-$; which is nearly ah arithmetical mean among
those obtained from occultations of the stars, and the
vibrations of the pendulum. The theory of the moon also
gives y^j- for the ellipticity.

The preceding ellipticities of the earth, expressed in
parts of its equatorial diameter, with their authors, and
the principles from which they were derived, may be ex-
hibited in one view, as follows:

Authors.

Huygens -
Newton -

Maupertuis, &c

Swanberg
Clairaut -
Treisnecker
Laplace -

Elliptii

Principle

Theory of gravity.

Mensuration of arcs.

Rotatory motion.
Vibration of pendulum.
Occult, of stars.
Precession and nutation.
Theory of the moon.

The variety in the preceding results presents an in-
teresting circumstance, as it indicates something in the
figure and conformation of the earth, which is not yet un-
derstood, and naturally gives rise to the following im
portant and intricate questions, to which all the accumu-
lated knowledge of the present times has not been able to
obtain complete solutions. Is the earth a spheroid of ro-
tation ? Are the northern ami southern nemispheiesequa-
and similar to each other f What is the ratio of an arc of
the meridian, measured in a given latitude, to the whole

meridian ? It is perhaps no improbable conjecture, that
the effects of precession and nutation, with the vibrations
of the pendulum, will be found among the most accurate
means of determining, at least, the general outlines of
our globe.

Tacquet draws some pleasing inferences, in the form
of paradoxes, from the round figure of the earth ; as, 1st,
That if any part of the surface of the earth were quite
plane, a man could no more walk upright upon it, than
on the side of a mountain. 2d, That the traveller's head
goes a greater space than his feet ; and a horseman than a
footman; as moving in a greater circle. 3d, That a ves-
sel, full of water, being raised perpendicularly, some of
the water will be continually flowing out, yet the vessel
still remain full ; and on the contrary, if a vessel of water
be let perpendicularly down, though nothing flow out,
yet it will cease to be full : consequently there is more
water contained in the same vessel at the foot of a moun-
tain, than on the top ; because the surface ot the water is
formed into a segment of a smaller sphere below than
above. Tacq. Astron. lib. 1, cap. 2.

Changes of the Earth. Mr. Boyle suspects that there
are great, though slow internal changes, in the mass of the.
earth. He draws hfs jirguments from the varieties observed
in the change of the magnetic needle, and from the ob-
served changes in the temperature of climates. But as to
the latter, there is reason to suspect that he could not
have diaries of the weather sufficient to direct his judge-
ment. Boyle's Works abr. vol. 1, pa. 292, &c.

Magnetism of the Earth. The notion of the mag-
netism of the earth was started by Gilbert; and Boyle
supposes magnetic effluvia moving from one pole to the
other. See his Works abr. vol. 1 , pa. 285, 290

Dr. Knight also thinks that the earth may be considered
as a great loadstone, whose magnetical parts are disposed
in a very irregular manner; and that the south pole of the
earth is analogous to the north pole in magnets, that isT
the pole by which the magnetical stream enters. See
Magnet.

He observes also, that all the phenomena attending the
direction of the needle, in different parts of the earth,
in a great measure correspond with what happens to a
needle, when placed on a large terrclla ; if we make al-
lowances for the different dispositions of the magnetical
parts, with respect to each other, and consider the south
pole of the earth as a north pole with regard to magnetism.
The earth might become magnetical by the iron ores it
contains, for all iron ores are capable of magnetism ; and
that a great part of the earth consists of iron is rendered
probable by its great medium, specific gravity, being
nearly 5 times that of water. It is true, the globe might
notwithstanding have remained unmagnetical, unless some
cause had existed capable of making that repellent mat-
ter producing magnetism move in a stream through the
earth.

Now the doctor thinks that such a cause does exist.
For if the earth revolves round the sun in an ellipsis, and
the south pole of the earth is directed towards the sun, at
the time of its descent towards it, a stream of repellent
matter will thence be made to enteral the smith pole, and
issue out at the north. And he suggests, that the earth
being in its perihelion iiv winter may be one reason why
magnetism is stronger in this season than in summer.
The cause, here assigned for the earth's magnetism must
continue, and perhaps improve it, from year to year.

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Hence the doctor thinks it probable, that the earth's mag-
netism has been improving ever since the creation, and
that this may be one reason why the use of the compass
was not discovered sooner. See Dr. Knight's attempt to
demonstrate, that all the phenomena in nature may be
explained by Attraction and Repulsion, prop. 87-

Magnitude and Constitution of the Earth. This has
been variously determined by different authors, both an-
cient and modern. The usual way has been, to measure
the length of one degree ofthe meridian, and multiply it
by 360, for the whole circumference. See Degree.
Diogenes Laertius informs us, that Anaximander, a scholar
of Thales, who lived about 550 years before the birth of
Christ, was the first who gave an account of the circum-
ference of the sea and land ; and it seems his measure
was used by succeeding mathematicians, till the time of
Eratosthenes. Aristotle, at the end of lib. 2 De Coelo,
says, the mathematicians who have attempted to mea-
sure the circuit of the earth, make it 40000 stadiums ;
which it is thought is the number determined by Anaxi-
mander.

Eratosthenes, who lived about 200 years before Christ,
was the next who undertook this business ; which, as
Cleomedes relates, he performed by taking the sun's zenith
distances, and measuring the distance between two places
under the same meridian; by which he deduced for the
whole circuit about 250000 stadiums, which Pliny states
at 31500 Roman miles, reckoning each at 1000 paces.
But this measure was considered to be false by many of
the ancient mathematicians, and particularly by Hippar-
chus, who lived 100 years afterwards, and who added
25000 stadiums to the circuit of Eratosthenes.

Possidonius, in the time of Cicero and Pompey the
Great, next measured the earth, viz, by means ofthe al-
titudes of a star, and measuring a part of a meridian;
from which he concluded that the circumference was
240000 stadiums, according to Cleomedes, but only
180000 according to Strabo.

Ptolemy, in his Geography, says that Marinus, a cele-
brated geographer, attempted something of the same kind ;
and in lib. 1 , cap. 3, he mentions that he himself had tried
to perform the same thing in a manner different from any
others before his time, which was by means of places un-
der different meridians : but he does not say how much he
made the number; for he still made use of the number
180000, which had been before employed.

Snellius relates, from the Arabian geographer Abel-
fedea, who lived about the year 1300, that about the year
S00, Almaimon, an Arabian king, having collected to-
gether some skilful mathematicians, commanded them to
find out the circumference of the earth. Accordingly
they made choice of the fields of Mesopotamia, where
they measured under the same meridian from north to
south, till the pole was depressed one degree lower: which
measure they found equal to 56 miles, or 56f : so that
according to them the circuit of the earth is 20l60or
20340 miles.

It was a long time after this, before any more attempts
of this kind were made. At length, however, Snell,
above mentioned, professor of mathematics at Leyden,
about the year 1620, with great skill and labour, by mea-
suring large distances between two parallels, found one
degree equal to 28500 perches, each of which is 12 Rhin-
land feet, amounting to 19 Dutch miles, and so the whole
periphery 6 8 40 miles; a mile being, according to him,

1500 perches, or 18000 Rhinland feet. See his treatise
called Eratosthenes Batavus.

The next that undertook this measurement, was Richard
Norwood, who in the year 1635, by measuring the dis-
tance from London to York with a chain, and- taking the
sun's meridian altitude, June 11th old style, with a sex-
tant of about 5 feet radius, found a degree contained
367200 feet, or 69 miles and a half and 14 poles; and
thence the circumference of a great circle of the earth is
a little more than 25036 miles, and the diameter a little
more than 7966 miles. See the particulars of this mea-
surement in his Seaman's Practice.

The measurement of the earth by Snell, though very
ingenious and troublesome, and much more accurate than
any of the ancients, being still thought by some French
mathematicians liable to certain small errors, the busi-
ness was renewed, after Snell's manner, by Picard and
other mathematicians, by the king's order; using a qua-
drant of 3f French feet radius; and they thus found a de-
gree contained 342360 French feet. See Picard's treatise,
La Mesure de la Terre.

M. Cassini the younger, in the year 1700, by the king's
command also, renewed the business with a quadrant of
10 feet radius, for taking the latitude, and another of 3f
feet for taking the angles of the triangles; and found a
degree,' from his calculation, contained 57292 toises, or
almost 69$ English miles.

Several other measurements have been taken, the best
of which are the following, with the corresponding mean
semidiameter annexed, in English miles.

Ulloa and Condamine - - 3957"7 miles.
Cassini and Lacaille - - 3957'1
Lacaille .... 3954"7

Boscovich - 3955*0

Liesganig .... 3958*2
General Roy - - - 3956'8

Major Lambton - 39565

The medium among all these is nearly 3957 miles,
which may be taken for the mean semidiameter of th,
earth. And supposing the solar parallactic angle to be
8"-6, it will be, as sin. 8"'6 : radius : : 3957 : 95 millions
of miles nearly, the earth's distance from the sun.

Hence the mean circumference is 2486*2 miles, which
gives 69-rj miles for the medium length of a degree of a
great circle.

See the results of many other measurements under the
article Degree, and from the preceding part of the pre-
sent article. From the mean of all which, the following
dimensions may be taken as near the truth:
the circumference - 4862 miles,

the diameter - - 27914 miles,

the superficies - 196757900 square miles,
the solidity - 25952320OOOO cubic miles.
Also the seas and unknown parts of the earth, by a mea-
surement of the best maps, contain l6t)522026 square
miles, the inhabited parts 38922180; of which Europe
contains 4456065 ; Asia, 10768823; Africa, 9654807;
and America, 14110874.

The terraqueous globe has two motions, besides that
on which the precession of the equinoxes depends ; the
one diurnal around its own axis in the space of 24 hours,
which constitutes the natural day; the other annual,
about the sun, in an elliptical orbit or track, in 365 days
6 hours, constituting the year. From the former arise
the diversities of day and night; and from the latter,

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the vicissitudes of seasons, spring, summer, autumn, and
winter.

The terraqueous globe is distinguished into three parts
or regions, viz, 1st, The external part or crust, being that
from which vegetables spring and animals are nursed.
2d, The middle, or intermediate part, which is possessed
by fossils, extending further than human labour ever yet
penetrated. 3d, The internal or central part, which is
utterly unknown to us, though by many authors supposed
of a magnetic nature ; by others, a mass or sphere of fire ;
by others, an abyss or collection of waters, surrounded by
the strata of earth ; and by others, a hollow, empty space,
inhabited by animals, who have their sun, moon, planets,
and other conveniences within the same. But others di-
vide the body of the globe into two parts, viz, the external
part, called the cortex, including the internal, which they
call the nucleus, being of a different nature from the former,
and. possessed by fire, water, or more likely by a consider-
able portion of metals, and probably mostly iron, as it has
been found, by my calculation, that the mean density of
the whole earth is near double the density of common
stone. See my determination of it, Philos. Trans. 177S,
pa. 7S1 ; or my Tracts, vol. 1, tract 26.

The external part of the globe either exhibits inequali-
ties, as mountains and valleys ; or it is plane andjevel ; or
hollowed into channels, fissures, &c, where rivers, lakes,
and seas, flow in all directions. These inequalities in the
face of the earth most naturalists suppose have arisen from
a rupture or subversion of it, by the force either of the
subterraneous fires or waters. The earth, in its natural
and original state, it has been supposed by Descartes,
and after him, Burnet, Stcno, Woodward, Whiston, and
others, was perfectly round, smooth, and equable ; and
they account for its present rude and irregular form, prin-
cipally from the great deluge. See Deluge.

In the external, or cortical part of the earth, there ap-
pear various strata, supposed the sediments of several
floods; the waters of which, being replete with matters of
divers kinds, as they dried up, or oozed through, deposited
these different matters, which in time hardened into strata
of stone, sand, coal, clay, &c.

Dr. Woodward has considered the circumstances of
these strata with great attention, viz, their order, numher,
and situation with respect to the horizon, depth, inter-
sections, fissures, colour, consistence, &c. He ascribes
the origin and formation of them all, to the great flood,
&c. At that terrible revolution he supposes that all sorts
of terrestrial bodies had been dissolved and mixed with
the waters, forming all together a chaos or confused mass.
This mass of terrestrial particles, intermixed with Water,
he conceives was at length precipitated to the bottom ;
and that generally according to the order of gravity, the
heaviest sinking first, and the lightest afterwards; by
which means the strata were formed of which the earth
consists; which, attaining their solidity and hardness by
degrees, have continued so ever since. These sediments,
he further concludes, were at first all parallel and con-
centrical ; and the surface of the earth formed of tlietn,
perfectly smooth and regular ; but that in course of time,
clivers changes arising, from earthquakes, volcanoes, &C,
the order and regularity of the strata was disturbed and
broken, anil the surface of the earth by such means
brought to the irregular form in which it now appears.

M. Do Bufi'on surmises that the earth, as well as the
other planets, are parts struck oil' from the body of the

sun by the collision of comets ; and that when the earth
assumed its form, it was in a stale of liquefaction by fire.
But this cannot be granted-; for if they were struck off
from the body of the sun, they would move in orbits that
would always pass through the sun, instead of having the
sun for their focus, or centre, as they are now found to
have ; so that having been struck off they would fall down
into the sun again, terminating their career as it were
after one revolution only. >

Besides these, various other hypotheses have been ad-
vanced, all 'of them however more or fess liable to ob-
jections ; as must always be the case with conclusions
drawn from uncertain data. We may however particu-
larize Dr. James Mutton's treatise, on this subject, as one
deserving the most attention, and from which much use-
ful information may be derived.

Earth, in Astronomy, is one of the primary planets,
according to the system of Copernicus, or Pythagoras ;
its astronomical character or mark being © : but ac-
cording to the Ptolemaic hypothesis, it is the centre of
the system. For, whether the earth move or remain at
rest, that is, whether it be fixed in the centre, having the
sun, the heavens and stars moving round it from east to
west ; or, these being at rest, whether the earth only
moves from west to east, is the great article that distin-
guishes the Ptolemaic system from the Copernican.

Motion of the Earth. It has already been observed
that, besides the small motion of the earth which causes
the precession of the equinoxes, the earth has two great
and independent motions; viz, the one by which it turns
round its own axis, in the space of 2-1 hours nearly, which
causes the continual succession of day and night ; and
the other an absolute motion of its whole mass in a large
orbit about the sun, having that luminary for its centre,
so that its axis keeps always parallel to itself, inclined in
the. same angle to its path, and by that means causing the
vicissitudes of seasons, spring, summer, autumn, and winter.

It is indeed true that, as to sense, the earth appears to
be fixed in the centre, with. the sun, stars and heavens
moving round it every day; and such, doubtless would
be considered as the true nature of the motions in the first
early ages of mankind, as they arc still by the rude and
unlearned. But to a thinking and learned mind, the
contrary will soon appear.

Indeed there are traces of the knowledge of these mo-
tions in the earliest age of the sciences. Cicero, in his
Tusc. Quaest. says that Nicetas of Syracuse first disco-
vered that the earth had a diurnal motion, by which it
revolved round ils axis every 24 hours; and Plutarch,
de Placit. Philosoph. informs us that Philolaus discovered
its annual motion round the sun ; also Aristarchns, about
100 years after Philolaus, alludes to the motion of the
earth in stronger and clearer terms, as we are assured by
Archimedes, in his Arenarius. And the same, we are
further assured, was the opinion and doctrine of Pythagoras.

But the religious opinions of the heathen world pre-
vented this doctrine from being more cultivated. For,
Aristnrchus being accused of Bacrilegc l>\ Cleanthes for
moving Vesta and the tutelar deities of the universe out
of, their places, the philosophers were obliged to dissemble,
and seem to relinquish so perilous a position.

Many ages aftewards, Nic. Cusanus revived the ancient
system, in his Doct. de Pignorant. and asserted the mo-
tion of the earth : but the doctrine gained very little
ground till Ihe time of Copernicus, who showed its great

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utility and advantages in astronomy; and who had imme-
diately all the philosophers and astronomers on his side,
that dared to differ from the crowd, and were not afraid
of ecclesiastical censure, which was not less dangerous
under the Christian dispensation, than it, had been under
that of the heathen. For, because certain parts of Scrip-
ture make mention of the stability of the earth, and of the
motion of the sun, as the rising and setting, &c, the
fathers of the church thought their religion required that
they should defend, with all its power, what they con-
ceived to-'be its doctrines, . and to censure and punish
every attempt at innovation on those points. There is
now however no doubt that in such instances the expres-
sions are only to be considered as accommodated to ap-
pearances, and the vulgar notions of things.

-By the diurnal rotation of the earth on its axis, the
same phenomena must take place as if it had no such
motion, that is, as if the sun and stars revolved about it.
For, turning round from west to east, cau=es the sun and
all the visible heavens to have the appearance of moving
the contrary way, or from east to west, as we daily see
them. So that, when in its rotation it has brought the
sun or a star to appear just in the horizon in the east,
they are then said to be rising; and as the earth continues
to revolve more and more towards the east, other stars
seem to rise and advance westwards, passing the meridian
of the observer, when they are due south from him, and
at their greatest altitude above his horizon; after which,
by a continuance of the same motions, viz, of the earth's
rotation eastwards, and the luminaries' apparent counter
motion westwards, these decline from the meridian, or
south point, towards the west, where being arrived, they
are said to set and descend below it; and so on contino
ally from day to day; thus making it day while the sun is
above the horizon, and night while he is below it.

While the earth is thus revolving on its axis, it is at the
same time carried by its-proper motion in its orbit round
the sun, as one of the planet?, namely, between the orbits
of Venus and Mars, having the orbits of Venus and Mercury
within its own, or between it and the sun, in the centre, and
those of Mars, Jupiter, Saturn, &c, without or above it;
which are therefore called superior planets, and the others
the inferior ones; which is called the annual motion of
the earth, because it is performed in a year, of 365 days 6
hours nearly ; or rather 36'5 days 5 hours 49 minutes,
from either equinox or solstice to the same again, which
completes the tropical year; but from any fixed star to
the same again, as seen from the sun, in 365 days 6 hours
p minutes, which is called the sidereal year. The figure
of this orbit is elliptical, having the sun in one focus, the
mean distance being about 95 millions of miles,, which is
upon the supposition that the sun's parallax is about 8"y,
or the angle under which the earth's semi-diameter would
appear to an observer placed in the sun : and the excen-
tricity of the orbit, or distance of the sun, in the focus,
lrom the centre of this elliptic orbit, is about ^th of the
mean distance.

Now this annual motion is performed in such a man-
ner, t hat the earth's axis is every where parallel, or in the
same direction in every part of the orbit; by which means
it happens, that at one time of the year the sun enlightens
more ot the north polar parts, and at the opposite season
of the year more of the southern parts, thus showing all
the varieties of seasons, spring, summer, autumn and win-
ter; which may be illustrated in the following manner:

Let the candle I (fig. 1, plate x) represent the sun, about
which the earth e, or f, &c, is moved in its elliptical or-
bit abcd, or ecliptic, and cutting the equator abed in the
nodes e and g : then, suspending the terrella by its north
pole, and moving it, so suspended, round the ecliptic, its
axis will always be parallel to its first position, and the
various seasons will be represented at the different parts
of the path. Thus, when the earth is at 25 or f, the en-
lightened half of it includes the south pole, and leaves the
north pole in darkness, making our winter; at g it is
spring, and the two poles are equally illuminated, and the
days are every where of the same length ; at it or Y? it is
our summer, the north polar parts being in the illuminated
hemisphere, and the southern in the dark one; lastly, at
e it is autumn, the poles being equally illuminated again,
and the days of equal length every where.

Earth, its Quantity of Mutter, Density, and Attractive
Power. Though the relative densities of the earth and
most of the other planets have been known a considerable
time, it is but very lately that we have come to the know-
ledge of the absolute gravity or density of the whole mass
of the earth. This I have calculated and deduced from
the observations made by Dr.Maskelyne, (late Astronomer
Royal,) at the mountain Schehallien, in the years 1774,-5,
and -6. The attraction of that mountain on a plummet,
being observed on both sides of it, and its mass being com-
puted from a number of sections in all directions, and
consisting of stone; these data being then compared with
the known attraction and magnitude of the earth, nave
by proportion its mean density, which is to that of water
nearly as 5 to 1, and to common stone as 2 to 1 : from
which very considerable mean density, it may be presumed
that the internal parts contain some great quantities of
metals.

From the density, thus found, its quantity of matter
becomes known, being equal to the product of its density
by its magnitude. From various experiments too, we
know that its attractive force, at the surface, is such, that
bodies fall there through a space of l6-JT feet in the first
second of time : whence the force at any other place, ei-
ther within or without it, becomes known ; for the force
at any part within it (supposing it every where uniform)
is directly as its distance from the centre ; but the force
of any part without it, reciprocally as the square of its
distance from the centre.

EAST, one of the cardinal points of the horizon, or of
the compass, being the middle point of it between north
and south, on that side where the sun rises, or the point
in which it is intersected on that side by the prime vertical.

EASTER, a feast of the church, held in memory of
our Saviour's resurrection. This feast has been annually
celebrated ever since the time of the apostles, and is one
of the most considerable festivals in the Christian calendar;
being that which regulates and determines the times of all
the other moveable feasts.

The rule for the celebration of easter, fixed by the
council of Nice, in the year 325, is, that it be held on the
sunday which falls upon, or next after, the full moon
which happens next after the 21st of March ; that is, the
Sunday which falls upon, or next after the first full moon
after the vernal equinox. The reason of which decree
was, that the Christians might avoid celebrating their easter
at the same lime with the Jewish passover, which, ac-
cording to the institution of Moses, was held the very day
of the full moon.

E A S

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e c r-i

To find Easter according
to the Old, or Julian Style.

In the annexed table, find
the golden number, with the
day of the paschal full moon,
and the sunday letter annexed ;
compare this letter with the
dominical letter of the given
year, that it may appear how
many days are to be added to
the day of the paschal full
moon, to give easter-day.

For Example. In the year
1715, the dominical letter is B,
and the golden number is 6,
opposite to which stands April
10 for the day of the paschal
full moon ; opposite to which
is the sunday letter b, which
happening to be the same with
that of the given year, that day
is a sunday ; and therefore
easter will fall 7 days after,
viz, on the 17th of April.

But in this computation, the
vernal equinox is supposed
fixed to the 21st of March;
and the cycle of 19 years, or
golden numbers, is supposed to
point out the places of the new
and full moons exactly ; both
which suppositions are errone-
ous: so that the Julian easter
never happens at its due time,
unless by accident. For instance, in the above example the Sea,

Gold.

Paschal full

Dom.

Num

moon.

letter.

1(5

March 21

C

5

22

D

23

E

13

24

F

2

25

G

26

A

10

27

B

28

C

IS

29

D

7

30

E

31

F

15

April 1

G

4

2

A

3

B

12

4

C

I

5

D

6

E

9

7

F

S

G

17

9

A

6

10

B

11

C

U

12

D

3

13

E

14

F

11

15

G

16

A

19

17

B

8

18

C

For Example. For the year
1813, the golden number is 9;
against which stands April the
15th, and the next Sunday let-
ter, which is c, below that,
stands opposite April 18,' which
is therefore the easter-day for
the year 1813.

Though the Gregorian ca-
lendar be much preferable to
the Julian, it is yet not without
its defects. It cannot, for in-
stance, keep the equinox al-
ways fixed on the2 1st of March,
but it will vary between the
19th and the 23d. To which
we may add, that the full moon
happening on the 20th of March,
might sometimes be paschal ;
yet it is not allowed as such in
the Gregorian computation; as
on the contrary, the full moon
of the 22d of March may be
allowed for paschal, which it is
not. Scaliger and Calvisius
have also pointed out other in-
accuracies in this calendar. An
excellent paper on this subject
by the Earl of Macclesfield,
may be seen in the Philos.
Trans, vol.40, pa. 417; or my
Abridg. vol. x, pa. 33.

Easter Term. See Term.

EBBING and Flowing of
See Tides.

Gold.

Paschal full

Sund.

Num.

moon.

letter.

14

March 21

C

3

22

D

23

E

11

24

F

25'

G

19

26

A

8

27

B

28

C

16

29

D

5

30

£

31

F

13

April 1

G

2

2

A

3

B

10

4

C

5

D

IS

6

E

7

7

F

8

G

15

9

A

4

10

B

11

C

12

12

D

1

13

E

14

F

9

15

G

16

A

17

17

B

6

18

C

the vernal equinox falls on the 10th of March, eleven days ECCENTRIC. See Excentric.

before the rule supposes it; and the paschal full moon on ECCENTRICITY. See Excentricity.

the 7th of April, or 3 days earlier than was supposed : ECHO, a sound reflected, or reverberated from some

and therefore easter-day should be held on the 10th of body, and thence returned or repeated to the ear. — For

April, instead of the 17th. an echo to be heard, the ear must be in the line of reflec-

This error had grown to such a height, that Pope Gre- tion; that the person who made the sound, may hear the

gory the 13th thought it necessary to correct it; and ac- echo, it is necessary he should be in a perpendicular to

cordingly, in the year 1582, by the advice of Aloysius Li- the place which reflects it ; and for a multiple or tautolo-

lius and others, he ordered 10 days to be thrown out of gical echo, it is necessary that there should be a number

October, to bring the vernal equinox back again to the of walls and vaults, rocks and cavities, either placed be-

21st of March : and hence arise the terms Gregorian ca- hind each other, or fronting each other. Those murmurs

lendar, Gregorian year, &c. in the air, that are occasioned by the discharge of great

This correction however did not entirely remove the guns, &c, are a kind of indefinite echoes, and are pro-
error; for the equinoxes and solstices still anticipate 28' duced from the vaporous particles suspended in the atmo-
20" in every 100 Gregorian years; but the difference is so sphere, which resist the undulations of sound, and rever-
inconsiderable as not to amount to a whole day, or 24 berate them to the ear.
hours, in less than 5082 Gregorian years. There can be no echo, unless the direct and reflex

The Gregorian, or New Style, was not introduced into sounds follow one another at a sufficient distance of time;

England till the year 1752, when eleven days were thrown for if the reflex sound arrive at the ear before the imprcs-

out, viz, between the 3d and 14th of September, the error sion of the direct sound ceases, it will not be doubled, but

amounting then to that quantity. only rendered more intense. Now if we allow that 9 or 10

To find EasteK according to the New or Gregorian syllables can be pronounced in a second, in order to pre-

Style, till the Year 1900 exclusive. serve the sounds articulate and distinct, there should be

Look for the golden number of the year in the first co- about the 9th part of a second between the times of their

lumn of the table, against which stands the day of the appulse to the ear ; or, as sound flies about 1 142 feet in a

paschal full moon ; thenlook in the 3d column for the sun- second, the said difference should be £ of 1 142, or 127

day letter, next after the day of the full moon, and the feet ; and therefore every syllable will be reflected to the

day of the month standing against that sunday letter is car at the distance of about 70 feet from the reflecting

easter-day. When the full moon happens on a sunday, body; but as, in the ordinary way of speaking, 3 or 4 syl-

thenthe next sunday after is easter-day. lables only are uttered in a second, the speaker, that he

E C H

[

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may have the echo returned as soon as they are expressed,
should stand about 500 feet (rum the reflecting body ; and
so in proportion for any "other number of syllables.
Mersenne allows for a monosyllable the distance of 69
feet; Morton, £)0 feet; for a dissyllable 105 feet, a trisyl-
lable l6"0 feet, -a tetrasyllable. 182 feet, and a pentasyllable
204 feet. Nat. Hist. Northampton, cap. 5, pa. 35S.

From what has been said, it follows that echoes may be
applied for measuring inaccessible distances. Thus, Mr.
Derham, standing on the banks of the. Thames, opposite to
Woolwich, observed that the echo of a single sound was
reflected back from the houses in 3 seconds; consequently
the sum of the direct and reflex rays must have been 1 142
x 3=3426" feet, and the half of it, 1/13 feet, the breadth
of the river in that place.

It also follows, thai the echoing body being removed far-
ther off, it reflects more of the sound than when nearer;
which is the reason why some echoes repeat but one syl-
lable, or one word, and some many. Of these, some are
tonical, which onty return a voice when modulated into
some particular musical tone; and others polysyllabical.
That fine echo in Woodstock park, Dr. Plot assures us, in
the day-time will return very distinctly 17 syllables, and in
the night 20. Nat. Hist. Oxf. cap. 1, pa. 7-

Echoing bodies may be so contrived, and placed, as that
reflecting the sound from one to the other, a multiple echo,
or many echoes, shall arise. — At Rosneath, near Glasgow,
in Scotland, there is an echo that repeats a tune played
with a trumpet three times completely and distinctly. —
At the sepulchre of Metella, wife of Crassus, there was an
echo, which repeated what a man said five times. — Some
authors mention a tower at Cyzicus, where the echo is
repeated seven times, and one at Brussels, that answers 15
times.

But the finest echo we read of, is that mentioned by
Barthius, in his notes on Statius's Thebais, lib. 6", ver. 30,
which repeated the words a man uttered 17 times. This
was on the banks of the Naha, between Coblentz and Bin-
gen. And whereas, in common echoes, the repetition is
not heard till some time after hearing the words spoken, or
the notes sung ; in this, the person who speaks, or sings, is
scarce heard at all ; but the repetition very clearly, and
always in surprising varieties; the echo seeming some-
times to approach nearer, and sometimes farther off'; some-
times the voice is heard very distinctly, and sometimes
scarce at all: one person hears only one voice, and another
several; one hears the echo on the right, and the other on
the left, &c.

Addison, and other travellers in Italy, mention an echo
at Simonctta palace, near Milan, still more extraordinary,
returning the sound of a pistol 56' times. The echo is
heard behind the house, which has two wings ; the pistol
is discharged from a window in one of these wdngs, the
sound is returned from a dead wall in the other wing, and
heard from a window in the back-front. See Addis. Tra-
vels, pa. 32; Misson. Voyag. d'ltal. torn. 2, pa. 1()6";
Philos. Trans. N° 480, pa. 220.

Again, a multiple echo may be made, by so placing the
echoing bodies, at unequal distances, as that they may re-
flect all oneway, and not one on the other ; by which
means, a manifold successive sound will be heard; one
clap of the hands like many; one ha like a laughter; one.
single word like many of the same tone and accent ; and so
one musical instrument like many of the same kind, imi-
tating each other.
Vol. I.

Lastly, Echoing bodies may be so. contrived, that from
any one given sound, they shall produce many echoes,
different both as to tone and intension. By which means
a musical room may be so contrived, that not only one in-
strument playing in it shall seem many of the same sort and
size, but even a concert of different ones ; this may be
contrived by placing certain echoing bodies so, as that
any note played, shall be returned by them in 3ds, 5ths,
and Sths.

Echo is also used for the place where the repetition of
the sound is produced, or heard. This is either natural or
artificial. The place where the speaker stands is called
the centrum phonicuni; and the object orplace that returns
the voice, the centrum phonocampticum.

Echo, in Architecture, is applied to certain vaults and
arches, mostly of elliptical or parabolical figures; used to
redouble sounds, and produce artificial echoes. — The me-
thod of making them is taught by F. Blancani, in his Echo-
metria, at the end of his book on the Sphere.

One of the finest echoes in England, is that commonly
called the whispering-gallery, in the dome of St. Paul's,
London; and another inGloucester cathedral. See Wiiis-
v erin g- Places.

Vitruvius tells us, that in divers parts of Greece and Italy
there were brazen vessels, artfully ranged under the seats
of the theatres, to render the sound of the actors' voices
more clear, and make a kind of echo; by which- means,
every one of tire prodigious multitude of persons, present at
those spectacles, might hear with ease and pleasure.

ECL1PSAREON, an instrument invented by Mr. Fer-
guson for showing the phenomena of eclfpses; as to their
time, quantity, duration, progress, &c. Ferguson's Astron.,
or Philos. Trans, vol. 4S, pa. 520.

ECLIPSE, a privation of the light of one of the lumi-
naries, by the interposition of some opaque body, either
between it and the eye, or between it and the sun. The
ancients had terrible ideas of eclipses; supposing them pre-
sages of some dreadful events. Plutarch assures us, that
at Rome it was not allowed to talk publicly of any na-
tural causes of eclipses ; the popular opinion running so
strongly in favour of their supernatural production, at
least those of the moon; for as to those of the sun, they
had some idea that they were caused by the interposition
of the moon between us and the sun ; but were at a loss
for a body to interpose between us and the moon, which
they thought must be the way, if the eclipses of the moon
were produced by natural causes. They therefore made
a great noise with brazen instruments, and set tip loud
shouts, during eclipses of the moon ; thinking by that means
to ease her in labour: wdience Juvenal, speaking of a
talkative woman, says, " Una laboranti poterit succurrere
lunse." Others attributed the eclipse of the moon to the
arts of magicians, who, by their enchantments, plucked her
out of heaven, and made her skim over the grass.

The natives of Mexico keep fast during the time of
eclipses; and particularly their women, who beat and abuse
themselves, drawing blood from their arms, &c;' imagi-
ning the moon has been wounded by the sun, in some quar-
rel between them.

The Chinese have an idea that eclipses are occasioned
by great dragons, who are ready to devour the sun and
moon; and therefore when they perceive an eclipse, they
rattle drums and brass kettles, till they think the monster,
terrified by the noise, lets go his prey.
The superstitious notions entertained of eclipses, have
3 M

ECL

[ 450 ]

E C L

been sometimes of considerable advantage, as was the case
■with Christopher Columbus, the discoverer of America,
who, being driven on the island of Jamaica in the year
1493, and" distressed for want of provisions, was refused
relief by the natives; but having threatened them with a
plague, at the same time foretelling an eclipse as a token
of it, which happened according to his prediction, the bar-
barians were so terrified, that they strove who should be
the first in bringing hirn supplies, throwing them at his
feet, and imploring his forgiveness. But the most striking
circumstance of this kind was the prediction of an eclipse
of the sun byThales, the great father of astronomy, which
happened, according to Mayer's calculation, on the 17th ot
May, 603 years before Christ, at the moment that Cyax-
ares king of the Medes, and Aliathos king of the Lydians,
were preparing for battle; but alarmed at the appearance
of the eclipse, they immediately laid down their arms, and
negociated a peace : and thus ignorance, which is often so
injurious to the happiness of man, may, in this instance, be
said to have largely contributed towards it.

Duration of an Eclipse, is the time of its continuance,
or the time between the immersion and emersion.

Immersion, or Incidence, of an Eclipse, is the moment
when the eclipse begins, or when part of the sun, moon, or
planet first begins to be obscured.

Emersion, or Expurgation, of an Eclipse, is the time
when the eclipsed luminary begins to re-appear, or emerge
out of the shadow.

Quantity of an Eclipse, is the part of the luminary
eclipsed. To determine this part, it is usual to divide the
diameter of the luminary into 12 equal portions, called
digits; whence the eclipse is said to be of so many digits,
according to the number of them contained in that part of
the diameter which is eclipsed or obscured.

Eclipses are divided, with respect to the luminary
eclipsed, into Eclipses of the sun, of the moon, and of the sa-
tellites; and with respect to the circumstances, into total,
partial, annular, central, &c.

Annular Eclipse, is when the whole is eclipsed, except
a ring, or annulus, which appears round the border or edge.
Central Eclipse, is one in which the centres of the two
luminaries and the earth come into the same right line.

Partial Eclipse, is when only a part of the luminary is
eclipsed. And a

Total Eclipse, is that in which the whole luminary is
darkened.

Eclipse of the Moon, is a privation of the light of the
moon, occasioned by the interposition of the body ot the
earth directly between the sun and moon, and so inter-
cepting the sun's rays that they cannot arrive at the moon,
to illuminate her. Or, the obscuration of the moon may
be considered as a section of the earth's conical shadow, by
the moon passing through some part of it.

The manner of this eclipse is represented in this figure,
where s is the sun, E the earth, and m or m the moon.

Lunar eclipses only happen at the time of full moon;
because it is only at that lime the earth is between the sun

and moon: nor do they happen every full moon, because
of the -obliquity of the moon's path with respect to the -
sun's, or more properly the earth's; but only in such full
moons as happen either at the intersection of those two
paths, called the moon's nodes, or very near them ; viz,
when the moon's latitude, or distance between the centres
of the earth and moon, is less than the sum of the apparent
semi-diameters of the moon and the earth's shadow.

The chief Circumstances in Lunar Eclipses are the. fol-
lowing:— 1. All lunar eclipses are universal, or visible in
all parts of the earth which have the moon above their ho-
rizon; and are every where of the same magnitude, with
the same beginning and end. — 2. In all lunar eclipses, the
eastern side is what first immerges and emerges again, i.e.
the left-hand side of the moon as we look towards her
from the north ; for the proper motion of the moon being
swifter than that of the earth's shadow, the moon ap-
proaches it from the west, overtakes and passes through it
with the moon's east side foremost, leaving the shadow be-
hind, to the westward. — 3. Total eclipses, and those of the
longest duration, happen in the very nodes of the ecliptic;
because the section of the earth's shadow, then falling on
the moon, is considerably larger than her disc. There
may, however, be total eclipses within a small distance of
the nodes ; but their duration is the less as they are farther
from it; till they become only partial ones, and at last, none
at all. — 4. The moon, even in the middle of an eclipse, has
usually a faint appearance of light, resembling tarnished
copper; which Gassendus, Ricciolus, Kepler, &c, attribute
to the light of the sun, refracted by the earth's atmo-
sphere, and so transmitted thither. — Lastly. She becomes
sensibly more dim and pale, before entering into the real
shadow; owing to a penumbra which surrounds that sha-
dow to some distance.

Jstronomy of Lunar Eclipses, or the method of calcu-
lating their times, places, magnitudes, and other pheno-
mena. The 1st preliminary is to find the length of the
earth's conical shadow; which may be found either from
the distance between the earth and sun, and the proportion
of their diameters, or from the angle of the sun's apparent
magnitude at the time. Thus, suppose the semiaxis of the
earth's orbit $5, 000000 miles, and the excentricity of
the orbit 1,377000 miles, making the greatest distance
96,377000 miles, or 24194 semidiameters of tEe earth;
and the sun's semidiameter being to the earth's, as 112 to
1; then as ad : be : : db : ec, that is, 111 : 1 ::
21194 : 21S semidiameters of the earth = ec the length
of the earth's shadow. Otherwise, suppose the an«lc
aes, or the sun's apparent semidiameter be 15' 56", and
the angle bae, or the sun's parallax 8"6", then is their
difference, or the angle ace = 15'47'4"; hence, as tan<\
15' 47"4": radius : : be or 1 : 21S nearly = ce, the
same distance as before. Hence, as the moon's least dis-
tance from the earth is scarce 50 semidiameters, and the
greatest not more than 64, the moon, when in opposition
to the sun, in or near the nodes, will fall into the earth's
shadow, and will be eclipsed, as the length of the shadow
is almost 4 times the moon's distance.

2. To find the apparent semidiameter of the earth's shadoiv,
in the place where the moon passes through it, at any
given time. Add together the sun and moon's parallaxes,
and from the sum subtract the apparent semidiameter of
the sun; so shall the remainder be the apparent semidia-
meter of the shadow at the place of the moonV passage.
For example, the 28lh of April 1790, at midnight, the

E C L

t 4S1 ]

ECL

moon's parallax is 6l' 9", to which add 8 6" or 9"* for
the sun's parallax, from the sum 6l' IS" take 15' 56", the
sun's apparent semidiameter, and the remainder 43' 22
is the semidiameler of the shadow at the place where
the moon passes through at that time. But some omit
the sun's parallax, as of no consequence; and increase
the apparent semidiameter of the shadow by one. whole
minute, for the shadow of the atmosphere ; which would
give the semidiameter of the shadow, in the case above,
46' 13".

3. There must also be had, the true distance of the
moon from the node, at the mean opposition; and the
true time of the opposition, with the true place of the sun
and moon, reduced to the ecliptic ; also the moon's true
latitude at the time of the true opposition; the angle of
the moon's way with the ecliptic, and the true horary mo-
tions of the sun and moon: from which all the circum-
stances of her eclipse may be computed by common arith-
metic and trigonometry.

To Construct an Eclipse of the Moon. — Lei ew be a
part of the ecliptic, and c the centre »of the earth's
shadow, through which draw perpendicular to ew, the
line cn towards the north, if the moon have north lati-
tude at the time of the eclipse, or cs southward, if she
have south latitude. Make the angle bcd equal to the
angle of the moon's way with the ecliptic, which may be
always taken at 5° 35', on an average, without sensible
error; and bisect this angle by the right linecF; in which
line it is that the true equal time of opposition of the sun
and moon falls, as given by the tables.

From a convenient scale of equal parts, representing
u< minutes of a degree,
take the moon's lati-
tude at the true time
of full moon, and set
it from c to g, on
the line cf; and
through the point G,
at right angles to CD,
draw the right line
hkgli for the path
of the moon's centre.
Then is l the point
in the earth'sshadow,
where the moon's
centre is at the middle of the eclipse ; c. the point where her
centre is at the tabular time of her being full ; and k the
point where her centre is at the instant of her ecliptic op-
position : also i the moon's centre at the moment of im-
mersion, and h her centre at the end of the eclipse.

With the moon's semidiameter as a radius, and the points
i, L, H, as centres, describe circles for the moon at the be-
ginning, middle, and end of the eclipse.

Finally, the length of the line of path in, measured on
the same scale, will serve to determine the duration of the
eclipse, viz, by saying, As the moon's horary motion from
the sun is to ih : : 1 hour or 60 min. to the whole dura-
tion of the eclipse.

To Compute a Lunar Eclipse. This will be very easy
from the foregoing construction. For, 1st, in the triangle
col, right-angled at l, there are given the hypothenuse
co = the moon's latitude at the time of full moon, and
the angle GCL = the half of 5° 35'; to find the legs cl
and LG. — 2d, In the right-angled triangle cm. or CIL, are
given the leg cl, and hypothenuse cn or CI, the sum of

the seniidiameters of the moon and the earth's shadow; to
rind lh or li, half the difference of the sun's and moon's
motions during the time of the eclipse.— 3d, As the dif-
ference of the horary motions of the luminaries is to one
hour, or 60 min. : : hl to the semiduration of the eclipse,
and : : gl to the difference between the opposition and mid-
dle of the eclipse; this last therefore taken from the time
of full moon, gives the time of the middle of the eclipse ;
from which subtracting the time in li, or semiduration
before found, gives the beginning of the eclipse ; or add
the same, and it gives the end of it. — Lastly, from cothe
semidiameter of the shadow, take cl, and there remains
lo ; to which add lp, the moon's semidiameter, when ne-
cessary, the sum of which is op the quantity eclipsed.

Note, When the moon's distance from the node exceeds
12°, there can be no eclipse of the moon ; or, more accu-
rately, the limit is from 104- to 12/^ degrees, according
to the distances of the sun, earth, and moon.

Eclipse of the Sun, is an occupation of the sun's body,
occasioned by the interposition of the moon between the
earth and sun. On which account it is by some consi-
dered as an eclipse of the earth, since the light of the sun
is hid from it by the moon, whose shadow involves a part
of the earth.

The manner of a salar eclipse is represented in this
figure; where s is the sun, m the moon, and CD the earth,

rmso the moon's conical shadow, travelling over a part of
the earth cod, and making a complete eclipse to all the
inhabitants residing in that track, but no wdiere else; ex-
cepting that for a large space around it there is a fainter
shade, included within all the space icds, which is called
the Penumbra.

Hence, solar eclipses happen when the moon is in con-
junction with the sun, or at the new moon, and also in the
nodes or near them, the limit being about 17 degrees on
each side of it; and such eclipses only happen when the
latitude of the moon, viewed from the earth, is less than
the sum of the apparent seniidiameters of the sun and
moon; because the moon's way is oblique to the ecliptic,
or sun's path, making an angle of nearly 5° 35' with it.

In the nodes, when the moon has no visible latitude,
the occultation is total; and with some continuance, when
the disc of the moon in perigee appears greater than that
of the sun in apogee, and its shadow is extended beyond
the surface of the earth ; and without continuance at
small distances when the cusp, or point of the moon's
shadow, barely touches the earth. Lastly, out of thenodes,
but near them, the eclipses are partial.

The other circumstances of solar eclipses are, 1. That
none of them are universal ; that is, none of them are seen
throughout the whole hemisphere which the sun is then
above; the moon's disc being too small and much too
near the earth, to hide the sun from its whole disc. Com-
monly the moon's dark shadow covers only a spot on the
earth's surface, about 180 miles broad when the sun's
3M 2

ECL

[ .«2' ]

ECL

distance is greatest, and the moon's least. But her partial
shadow, or penumbra, may then cover a circular space of
4900 miles in diameter, within which the sun is more or
less eclipsed, as the places are nearer to or farther trom the
centre of this shadow. In this case the axis of the shade
passes through the centre of the earth, or the new moon
happens exactly in the node, and then it is evident that the
section of the shadow is circular ; but in every other case
the conical shadow is cut obliquely by the surface of the
earth, and the section is an oval, and very nearly a true
ellipsis.

2. Nor does the eclipse appear the *ame in all parts of
the earth, where it is seen; but when in one place it is
total, in another it is only partial. Farther, when the
moon appears much less than the sun,- as is chiefly the
case when she is in apogee and the sun in perigee, the
vertex of the lunar shadow is then too short to reach the
earth; for though the moon be in a central conjunction
with the sun at this time, yet she is not large enough to
cover his whole disc, but lets his limb appear like a lucid
ring or bracelet, and so causes an annular eclipse.

3. A solar eclipse does not happen at the same time, in
all places where it is seen ; but appears more early to the
western parts, and later to the eastern ; as the motion of
the moon, and consequently of her shadow, is from west
to east.

4. In most solar eclipses the moon's disc is covered
with a faint light; which is attributed to the reflexion of
the light from the illuminated part of the earth.

Lastly, In total eclipses of the sun, the moon's limb is
seen surrounded by a pale circle of light; which some
astronomers consider as an indication of a lunar atmo-
sphere ; but others as the atmosphere of the sun, because
it has been observed to move equally with the sun, and
not with the moon ; and besides, it is generally believed
that the moon is without any atmosphere, unless it be one
that is very small, and very rare; which latter supposi-
tion may now beconsidered as a fact well established, being
confirmed by a vast number of attentive observations.

To determine the Bounds of a Solar Eclipse. If the
moon's parallax were inconsiderable, the bounds of a solar
eclipse would be determined in the same manner as those
of a lunar; but as it has a- sensible parallax, the method
of proceeding is a little different, viz.

1. Add together the apparent semidiameters of the lu-
minaries, both in apogee and perigee; which gives 33' b"
for the greatest sum of them, and 3U' 31" fur the least sum.

2. Since the parallax diminishes the northern latitude
and augments the southern, therefore let the greatest pa-
rallax in latitude be added to the former sums, and also
subtracted frorn them : thus in each case there will be
had the true latitude, beyond which there can be no
eclipse. This latitude being given, the moon's distance.
from the nudes, beyond which eclipses cannot happen, is
found as for a lunar eclipse. This limit is nearly between
lO'i and IS-j degrees distance from the nodes.

To find the Digits eclipsed. Add the apparent semidia-
meters of the luminaries into one sum ; from which sub-
tract the moon's apparent latitude; the remainder is the
scruples, or parts of the diameter eclipsed. Then say, As
the scmidiameter of the sun is to the scruples eclipsed; so
arc 6' digits reduced into scruples, viz, 30*0 scruples or
minutes, to the digits &c eclipsed.

To determine the Duration of a Solar EcLIPSfiB. Find
the horary motion of the moon from the sun, for one hour

before the conjunction, and another hour after : then say,
As the former horary motion is to the seconds in an hour,
so are the scruples of half duration (found as in a lunar
eclipse') to the timeof ihimersion; and as the latter horary
motion is to the same seconds, so are the same scruples of
half duration to the time of emersion. Lastly, Adding the
times of immersion and emersion together, the aggregate is
the total duration.

The moon's apparent diameter when largest, exceeds
the sun's when least, only by 2' of a degree; and in the
greatest solar eclipse that can happen at any time and'
place, the total darkness cannot continue any longer than
whilst the moon is moving through this 2' from the sun in
her orbit, which is about 4 minutes -of time : for the mo-
tion of the shadow on the earth's disc is equal to the
moon's motion from the sun, which on account of the
earth's rotation on its axis in the same direction, or east-
ward, is about 30f minutes of a degree every hour, at a
mean rate ; but so much of the moon's orbit is equal to
30^- degrees. of a great circle on the earth, because the cir-
cumference of the moon's orbit is about 6*0 times the cir-
cumference of the earth; and therefore the moon's shadow
goes 30-4 degrees, or 1830 geographical miles in an hour,
or 30i. miles in a minute.

To determine the Beginning, Middle, and End, of a Solar
Eclipse. From the moon's latitude, for the time of con-.
junction, find the arch gl (last fig. but one), or the dis-
tance of the greatest obscurity. Then say, As the horary
motion of the moon from the sun, before the conjunction,
is to 1 hour ; so is the distance of the greatest darkness,
to the interval of time between the greatest darkness and
the conjunction. Subtract this interval, in the 1st and 3d M|
quarter of the anomaly, from the time of the conjunc- ^
tion ; and in the other quarters, add it to the same; then
the result will be the time of thegreatest darkness. Lastly,
from the time of the greatest darkness subtract the time of
incidence, and add it to the time of emersion; the diffe-
rence in the first case will be the beginning ; and the sum,
in the latter case, the end of the eclipse.

To Calculate Eclipses of the Sun. 1-. Find the mean
new moon, and thence the true one; with the place of the
luminaries for the apparent time of the true one. — 2. For
the apparent time of the true new moon, compute the ap-
parent time of the new moon observed. — 3. For the ap-
parent time of the new moon seen, compute the latitude
seen. — 4. Thence determine the digits eclipsed. — 5. Find
the times of the greatest darkness, immersion, and emer-
sion.— 6". Thence determine the beginning and ending of
the eclipse.

From the foregoing problems, it is evident that all the
trouble and fatigue of the calculus arises from the paral-
laxes of longitude and latitude, without which, the calcu-
lation of solar eclipses would be the same with that of lunar
ones.

See the Construction and Calculation of Eclipses by
Fhunsteed, in Sit Jonas Moore's System of Mathematics,
and in Ferguson's Astronomy, &c.

In the Philos. Trans. N° 4>6), isa contrivance to repre-
sent solar eclipses, by means of the terrestrial globe, by
M. Segu< r, professor of mathematics at Gottingen. And
Mr. Ferguson titled a terrestrial globe, so as to show
the time, quantity , duration, and progress of solar eclipses,
at any place of the earth where tiny are visible; which
he calls the Eclipsareon. lie has also given a large ca-
talogue of ancient and modem eclipses, including those re-

- E C L [4;

corded in history, from 721 years -before Christ, to A. d.
1485; also computed eclipses from 1485 to 1700, and
all the eclipses visible in Europe from 1700 to 1800. See
his Astron.

The "Number q/Eclipess, of both luminaries, in any
year, cannot be less than two, nor more than seven ; the
most usual number is 4, and it is rare to have more than
6. The reason of which limitation is obvious; because
the sun passes by both the nodes but once in -a year; un-
less he pass by one of them in the beginning of the year;
in which case he will pass by the sameagain a little before
the end of the year; because the nodes move backwards
lg-y degrees every year, so that the sun will come to either
of them 173 days after the other : if either node. be within
17° of the sun at the time of new moon, the sun. will be
eclipsed ; and at the subsequent opposition, the moon will
be eclipsed in the other node, and come round to the next
conjunction before the former node be 17° beyond thesun,
and eclipse him again. When three eclipses happen about
either node, the like number commonly happens about
the opposite one ; as the sun comes to it in 173 days after-
ward, and 6 lunations contain only 4 days more. Thus
there may be two eclipses of the sun, and one of the moon,
about each of the nodes. But when the. moon changes
in either of the nodes, she cannot be near enough the
other node, at the next full, to be eclipsed ; and in 6 lunar
months afterward she will change near the other node; in
which case there cannot be more than two eclipses in a
year, both of the sun.

Period of Eclipses, is the period of time in which the
same eclipses return again ; and as the nodes move back-
wards lQj degrees every year,, they must shift through
every point of the ecliptic in IS years and 225 days ; and
this would 'be the regular period of their return, if any
complete number of lunations were finished without afrac-
tion ; but this is not the case. However, in 223 mean lu-
nations, after thesun, moon, and nodes have been once in
a line of conjunction, they return so nearly to the same
state again, as that the same node which was inconjunction
with the sun and moon at the beginning of the first of these
lunations, will be within 2S' 12" of the line of conjunction
with the sun and moon again, when the last of these luna-
tions is completed ; and in this period there will be a re-
gular return of eclipses for many ages. To the mean time
of any solar or lunar eclipse, by adding this period, or J 8
Julian years 1 1 days 7 hours 43 minutes 20 seconds, when
the last day of February in leap-years is 4 times included,
or a day less when it occurs 5 times, we shall have the
mean time of the return of thesame eclipse. Inan inter-
val ol'O'syu mean lunations, containing 557 years 21 days
18 hours 30 minutes 11 seconds, the sun and node meet
so nearly, as to be distant only 11 seconds.

The Use of Ecu pses. In astronomy, eclipses of the
moon determine the spherical figure of the earth ; they also
show that the sun is larger than the earth, and the earth
than the moon. Eclipses also, that are similar in all cir-
cumstances, and that happen at considerable intervals of
time, serve to ascertain the period of the moon's motion.
In geography, eclipses discover the longitude of different
places ; for which purpose those of the moon are the more
useful, because they are more often visible, and the same
lunar eclipse is of equal magnitude and duration at all
places where it is seen. In chronology, both solar and
lunar eclipses serve i> determine exactly the time of any

past evi n'.

3 ] E G L

The following is a list of all the solar eclipses visible in
this country till the year 2000 ; the times adapted to Lei-
cestershire, the middle of England.

A Table of all the Solar Eclipses that will be visible till
the Year 2000.

Dates.

Beginning.

Middle.

End.

V

gits.

h. m. s.

h. m. s.

ll. III. s.

de£

. m.

5 S 16 Nov. 1 9, m

8 18 28

9 25 28

10 36 28

9"

38

1818 May 5, ill

5 58 21

6 53 21

7 52 51

5

0

1S20 Sept. 7, a

1 3 39

2 36 39

3 58 39

11

20

1823 July 8, m

5 15 30

5 35 30

5 55 50

0

56

1820 Dec. 29, m

10 0 18

116 8

0 13 38

6

41

1833 July 17, m

5 3 10

5 55 10

.6 50 55

0

36

1836 May 15, a

2 1 1

3 29 1

4, 4S 1

11

18

1841 July 18, a

2 36 29

2 43 17

2 50 2

0

of

1S42 JulyS, m

4 51 45

5 44 15

6 40 15

8

54'

1845 May 6, m

8 30 25

9 51 55

11 19 55

6

15

1846 April 25, a

5 44 38

6 21 8

6 56 8

2

' 21

1847 Oct. 9, m

6 22 3S

7 36 38

8 48 38

11

0

1S51 July 28, a

1 57 11

3 5 11

4 10 41

9

43

1858 Mar. 15, m

11 29 30

12 52 2

2 12 0

11

30

1860 July 18, a

1 34 30

2 45 0

3 52 30

9

12

1861 Dec. 31, a

] 50 43

2 46 13

3 37 58

5

0

1863 May 17, a

5 37 50

6 27 20

7 12 35

3

46

1865 Oct. 19, a

3 55 58

5 9 5S

6 17 28

7

36

1866 Sept. 28, a

4 39 14

5 33 14

6 32 44

5

3

I867 March 6,m

8 7 26

9 21 2 6

10 40 26

8

42

1870 Dec. 22, 111

11 15 28

12 32 58

1 48 28

9

36

1873"Maj 26, m

7 53 4li

8 46 46

9 43 16

3

43

1874 Oct'. 10, 111

8 55 7

10 13 37

11 04 37

6

IS

1875 Sept. 20, a

0 6 57

0 32 57

0 58 57

0

33

1880 Dec. 30, a

1 44 46

2 40 16

3 33 16

4

24

1882 May 17, 111

6 15 59

6 54 14

7 31 59

'2

18

1887 Aug. 19,m

3 25 22

4 15 22

5 7 52

11

58

1890 June 17, m

8 20 29

9 23 29

10 30 29

4

39

1891 June 6, a

4 57 56

5 40 26

6 20 26

3

0

1899 June 8, m

4 48 0

5 31 0

6 14 20

3

13

1905 Aug. 30, m

11 54 52

1 8 42

2 19 42

9

30

1908 June 28, a

5 10 0

5 42 0

6 12 30

1

20

1912 April 17, in

10 48 12

12 23 12

1 45 12

11

30

1914 Aug. 2, in

10 38 58

11 52 58

1 6 28

8

9

1916 Fell. 3 a

4 21 10

5 21 40

6 18 10

12

0

1919 Nov. 22, a

3 20 32

4 17 32

5 10 32

3

0

1920 Nov. 10, a

3 53 6

5 1 36

0 3 36

7

3

1921 April s, m

7 39 1

8 52 51

10 9 31

to

48

1922 Mar. 28, a

1 15 25

2 8 55

2 59 25

2

16

1925 Jan. 24, a

2 47 29

3 52 49

4 54 9

10

0

1927 June 29,m

4 29 11

5 24 41

6 25 41

11

6

1928 Nov. 12,m

7 47 4

8 35 .4

9 27 4

2

42

1 929 Nov. 1 , m

10 59 16

11 36 6

12 10 36

0

59

1936 June 1 9, m

3 55 40

4 42 10

5 32 10

6

42

1939 April 19, a

5 32 52

6 24 22

7 12 52

4

42

1942 Sept. 10, a

3 26 6

4 14 36

5 0 6

3

54

1945 July 9, a

0 47 57

2 2 57

3 13 57

7

49

1919 A](il28,ni

6 44 44

7 40 44

8 41 34

4

27

1952 Feb. 25, in

8 38 3

9 11 38

9 46 8

1

29

1954 June 30, m

11 20 2

12 38 32

2 11 2

9

39

1959 Oct. 2, in

11 26 1

12 24 1

1 21 31

4

5

1961 Fell. 15, hi

6 39 10

7 40 40

8 46 40

11

25

1966 May 20, in

8 33 7

9 36 7

10 43 7

f

17

1968 Sept. 22, m

. 9 30 18

10 27 18

11 27 48

4

7

1971 Felj.25, ni

8 37 3S

9 41 38

10 48 38

7

54

1972 July 10, a

7 3 7 52

8 21 22

9 3 2'2

6

4

1973 Dec. 24, a

3 10 6

4 19 36

5 22 36

5

56

1975 May 11, m

5 38 26

6 34 56

7 3i 26

6

48

1976 April 29, m

9 42 0

10 19 0

10 Jl 0

1

2

1982 Dec. 15, m

7 23 28

8 29 28

5 41 28

5

18

1984 May 30, a

5 24 59

6 23 29

7 17 59

6

19

1994 May 10, a

5 52 31

6 50 31 !

7 45 31

5

55

1996 Oct. 12, a

1 9 18

2 23 18 1

3 42 0

7

15

1999 Auir. 1 1,111

9 10 52

10 2" 52 1

11 47 32

1 1

24

Eclipses of the Satellites. Sea Satellites of Jupiter.
— The principal circumstances here observed are, 1. That
the satellites of Jupiter undergo two or three kinds of
eclipses ; the first of which are proper, being such as hap-
pen when Jupiter's body is directly interposed between
them and the sun; which eclipses happen almost every
day. Various authors have given tables for computing

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[ 454 ]

ECL

eclipses of the satellites of Jupiter; as Flamsteed, Cassini,
Wargentin, &c.

The second sort are occultations, rather than observa-
tions ; when the satellites, coming too near the body of
Jupiter, are lost in his light; which Riccioli calls occidere
zeusiace, setting joviall)'. In which case, the nearest or
first satellite exhibits a third kind of eclipse; being ob-
served like a round macula, or dark spot, transiting the
disc of Jupiter, with a motion contrary to that of the sa-
tellite ; like as the moon's shadow, projected on the earth,
will appear to do, to the lunar inhabitants.

The; eclipses of Jupiter's satellites furnish very good
means of finding the longitude at sea. Those especially of
the first satellite are much surer than the eclipses of the
moon, and they also happen much oftener : the manner
of applying them is also very easy. In the Philos. Trans,
for 1786, Mr. Pigott gave a preferable method by the lu-
nar transits. See Longitude.

ECLIPTIC, in Astronomy, a great circle of the sphere
couceived to pass through the middle of the zodiac. It
is sometimes called the via solis, or sun's path, being the
track which he appears to describe among the fixed stars ;
though more properly it is the apparent path of the earth,
as viewed from the sun, and thence called the heliocentric-
circle of the earth. It is called the ecliptic, because all
the eclipses of the sun or moon happen when the moon
crosses it, or is nearly in oneoflhosetwo parts of her orbit
where it crosses the ecliptic, which points are called the
moon's nodes.

Upon the ecliptic are marked and counted the 12 ce-
lestial signs, Aries, Taurus, Gemini, &c; and also the longi-
tude of the planets and stars. Its situation is oblique with
respect to the equator, which it cuts in two opposite points,
viz, the beginning of Aries and Libra, being directly op-
posite to each other, and called the equinoxes, making
one half of the ecliptic to the north, and the other halt'
on the south side of the equator ; the two extreme points
of it, to the north and south, which are opposite to each
other, and at a quadrant distance from the equinoctial
points both ways, arc called the solstices, or solstitial
points, or also the two tropics, which are at the beginning
of Cancer and Capricorn, being at the farthest distance of
any points of it from the equator, which distance is the
measure of the sun's greatest declination, and is the same
with the obliquity of the ecliptic, or (he angle it makes
with the equator.

This obliquity of the ecliptic is not permanent, but is
continually varying, by the ecliptic approaching nearer
and marer to a parallelism with the equator, at the rate
at prcstntof 42"'6 in 100 years, as is deduced from an-
cient and modern observations compared together ; and as
the mean obliquity of the ecliptic was 23° 28' about the
end of the year 1788, or beginning of 1789, by adding
■426" of a secoml for each preceding year, or subtracting
the same for each following year, the mean obliquity will
be found nearly, fot any year either before or since that
period. The quantity however of this change is variously
stajed by different authors. Hipparchus, almost two thou-
sand years since, observed the obliquity of the ecliptic,
and found it about 23° 51'; and all succeeding astronomers,
to the present time, having observed the same, have found
it always less and less ; being now rather under 23° 28' ;
a difference of about 23' in 1950 years ; which gives a me-
dium of 70" in 100 years. There is no doubt, however,
that the diminution is variable.

It is now well known that this change in the obliquity
of the ecliptic, is wholly owing to the actions of the planets
upon the earth, and especially the planets Venus and Ju-
piter, but chiefly the former. See Lagrange's excellent
paper on this subject in the Memoirs of the French Aca-
demy for 1774- ;Cassini's in 177S ; and Lalande's Astron.
vol. 3, art. 2737- According to Lagrange, who proceeds
upon theory, the annual change of obliquity is variable,
and has its limits : about 2000 years ago, he thinks it was
after the rate of 'about 38" in 100 years; that it is now,
and will be for 400 years to come, 56" per century ; but
2000 years hence, 49" per century. According to Cassini,
who computes from observations of the obliquity between
the years 1739 and 1778, the annual change at present is
6'0" or 1' in 100 years. But according to Lalande, the
diminution is at the rate, of 42"'6 per century ; while
Dr. Maskelyne makes it 50" in the same time. But later
observations and the calculations of Laplace, give now
52"' 1 for the mean secular variation.

Besides the diminution of the obliquity of the ecliptic,
at the above rate, which arises from a change of the ecliptic
itself, it is subject to two periodical inequalities, the one
produced by the unequal force of the sun in causing the
precession of the equinoxes, and the other depending on
the nutation of the earth's axis. See the Explanation and
Use of Dr. Maskelyne's Tables and Observations, pa. vi,
where he shows how to calculate those inequalities ; and
also, from his own observations, determines the mean ob-
liquity of the ecliptic to the beginning of 'the year 176~9,
to be 23° 28' 9"-7.

To find the Obliquity of the Ecliptic, or the greatest de-
clination of the sun: — About the time of the summer sol-
stice observe very carefully the sun's zenith distance for
several days together; then the difference between this
distance and the latitude of the place, will be the obliquity
sought, when the sun and equator are botli on one side of
the place of observation ; but their sum will be the obli-
quity when they are on different sides of it. Or, it may
be found by observing the meridian altitude, or zenith
distance, of the sun's centre, on the days of the summer
and winter solstice; then the difference of the two will be
the distance between the tropics, the half of which will
be the obliquity sought.

By the same method too, the declination of the sun
from the equator for any other day may be found ; and
thus a table of his declination for every day in the year
might be constructed. Thus also the declination of the
stars might be found.

The observations of astronomers of all ages, on the ob-
liquity of the ecliptic, have been collected together ; and
although some of them may not be quite accurate, yet
they sufficiently show the gradual and continual decrease
of the obliquity from the times of the earliest observations
down to the present time. The chief of those observations
may be seen in the following table ; where the- firsl co-
lumn contains the name- of the observer, or author ; the
2d, the year before or after Christ; and the 3d, the obliquity
of the ecliptic for the time.

See l'tolem. Aim. lib. 1, cap. 10; [liccioli Aim. vol. i,
lib. 3, cap. 27; Flamsteed, Proleg. Hist. Ccel.vol. 3; Phi-
los. Trans. N° l6S ; ibid. vol. 63, pt. 1 ; Long's Astron.
vol. I, cap. l6j Memoirs of the Acad. an. 1716', 1734,
176"2, 176'7, 177 I, 1778 ; ActaErud. Lipsi;e, 1719; Naut.
Aim. 1779; Maskelyne's Observ. Explan. pa.vi ; &c. Se<
Laplace.

E C L

[

E C L

Table exhibiting the Degree of Obliquity of the Ecliptic at
various Times, as determined from ancient and modern
Observations.

Table of the Obliquity of the Ecliptic for Forty Centuries,
with its Secular Variation.

Authors' Names.

Years before
Christ.

Obliquity.

Pytheas - - - -

324

'is 49 23

Eratosthenes and Hipparchus

230 & 140
^fier Christ

23 51 20

Ptolemy

140

23 48 45

Almahmon -

832

23 35

Albategnius -

880

23 35

Thebat -

911

23 33 30

Abul Wasi and Hamed

999

23 35

Persian Tables in Chrysococea

1004

23 35

Albatrunius -

1007

23 3.5

Arzachel - -

1104

23 33 30

Almaeon - - - -

1140

23 33 30

Choja Nassir Oddin

1290

23 30

Prophatius the Jew

1300

23 32

Ebn Shattir - _ -

1363

23 31

Purbach and Regiomontanus -

1460

23 30

Ulugh Beigh

1463

23 30 17

Walther -

1476

23 30

Ditto corrected by refraction, &c.

23 2y 8

Werner -'-.-'.

1510

23 28 30

Copernicus -

1525

23 28 24

Egnatio Danti ...

1570

23 29

Prince of Hesse ...

1570

23 31

Rothmann and Byrge -

1570

23 30 20

Tycho Brahe - - -

1584

23 31 30

Ditto corrected ...

23 29

Wright -

1594

23 30

Kepler ....

1627

23 30 30

Gassendus - - -

1630

23 31

Ricciolus ....

1646

23 30 20

Ditto corrected - - - .

1655

23 29

Hevelius ....

1653

23 30 20

Ditto corrected - - -

1661

23 28 52

Cassini -

1655

23 29 15

Montons, corrected, &c

1660

23 29 3

Richer corrected

1672

23 28 52

La Hire. ....

1686

23 29

Ditto corrected - - -

,. '

23 29 28

Flamsteed ...

1690

23 29

Bianchini ...

1703

23 28 25

Roemer -

1706

23 28 41

Louville - -

1715

23 28 24

Godin ....

1730

23 28 20

Bradley - #

1750

23 28 18

Mayer ....

1756

'23 28 16

Maskelyne

1769

23 23 10

Hornsby ....

1772

23 28 8

Nautical Almanac

1779

23 23 7

Ditto ....

1800

23 27 50

Dr. Maskelyne's observations

1800

23 27 56-0

Piazzi ....

1800

23 27 56'3

Mr. Pond ... -

1800

23 27 56'5

Delambie with repeating circle"!

ofBorda, mean of many bun- {-

1800

23 27 57

died observations - - J

English and French astronomers

1812

23 27 42

Mean Obliquity.

Secular Variation.

A.C. ' 900

23° 50' 26"-2

SOO

23 49 39 -6

46"-9

700

23 48 52 -7

47 -2

600

23 43 5 -5 .

47-5

500

.23 47 13-0

47 -9

400

23 46 30-1

48 -2

300

23 45 41 -9

48-5

200

23 44 53 -4

48 -9

100

23 44 4'5

49 -1

0

23 43 15-4

49 '3

A.D. 100

23 42 26 • 1

49 '7

200

23 41 36-4

49-9

300

23 40 46 -5

50-1

400

23 39 56 -4

50'3

500

23 39 6-1

50-5

600

23 38 15-6

50-8

700

23 37 24-8

51 -0

800

23 36 33 -8

51 -1

900

23 35 42 -7

51 '3

1000

23 34 51 -4

51 -4

1 100

23 34 0-0

51 -5

1200

23 33 8 -5

51 -7

1300

,23 32 16-8

51 -8

1400

23 31 25-0

51 -9

1500

23 30 33-1

52 -Q

1600

23 29 41 -1

52-0

1700

23 28 49'1

52 '1

1800

23 27 57 -0

52 -1

1900

23 27 5 -0

52-1

2000

23 26 12-8

52-2

2100

23 25 20 -6

52-3

2200

23 24 28 -3

52-2

2300

- 23 23 36 -1

52-2

2400

23 22 43 -9

52-2

2500

23 21 51-7

52-2

2600

23 20 59 -5

52-1

2700

23 20 7 "4

52-1

2800

23 19 15-3

52-1

2900

23 18 23'3

52 '0

3000

23 17 31 3

52'0

According to an ancient tradition of the Egyptians,
mentioned by Herodotus, the ecliptic had formerly been
perpendicular to the equator: they were led into this no-
tion by observing, for a long series of years, that the ob-
liquity was continually diminishing; or, which amounts
to the same thing, that the ecliptic was continually ap-
proaching to the equator. Hence they took occasion to
suspect that those two circles, in the beginning, had been
as far oft' each other as possible, that is, perpendicular to
each other. Diodorus Siculus relates, that the Chaldeans
reckoned 403,000 years from their first observations to
the time of Alexander's entering Babylon. This enormous
account may have some foundation, on the supposition
that the Chaldeans deduced it from the diminution of the
obliquity of the ecliptic at the rate of a minute in 100
years. M. de Louville, taking the obliquity, such as it
must have been at the time of Alexander's entrance into

E L A

[ 45G ]

E L A

Babylon, and going back to the time when the ecliptic,
at that rate, must have been perpendicular to the equator,
actually finds 402,942 Egyptian or Chaldean years;
which is only 58 years short of that epocha. Indeed there
is no way so probable of accounting for the fabulous an-
tiquity of the Egyptians, Chaldeans, &c, as from the sup-
position of long periods of very slow celestial motions, a
small part of which they had observed, and from which
they calculated the beginning of the period, making the
world and their own nation to commence together. Or
perhaps they sometimes counted months or days for years.

Ecliptic, in Geography, a great circle on the terres-
trial globe, in the plane of, or directly under, .the celestial
ecliptic.

Ecliptic, Eclipticus, something belonging to the eclip-
tic, or to eclipses ; as ecliptic conjunction, opposition, &c.

Ecliptic Bounds, or Limits, are the greatest distances
from the nodes at which the sun or moon can be eclipsed ;
namelv, near 18 degrees for the sun, and 12 degrees for
the moon.

Ecliptic Digits, Digiti ecliptici. See Digits.

Poles of the Ecliptic, are the two opposite points of
the sphere which are each every where equally distant
from the ecliptic, viz, 90°. The distance of the poles of
the ecliptic from the poles of the equator, or of the world,
is always equal to the varying distance of the obliquity of
the ecliptic, which for the year 1800 was, according to
Dr. Maskelyne, 23° 27' 56"-f5.

Reduction to the Ecliptic. See Reduction.

EFFECT. The result or consequence of the application
of a cause, or agent, on some subject. It is one of the
great axioms in philosophy, that effects are always pro-
portional to the powers of their adequate causes.

EFFECTION, denotes the geometrical construction of
a proposition. The term is also used in reference to pro-
blems and practices ; which, when they are deducible from,
or founded on some general propositions, are called the
geometrical effection of them.

EFFERVESCENCE, is popularly used for a light
ebullition, or a brisk intestine motion, produced in a liquor
by the first action of heat, with any remarkable separation
of its parts.

EFFICIENT Cause, is that which produces an effect.
See Cause and Effect.

Efficients, in Arithmetic, are the numbers given for
an operation of multiplication, and are otherwise called
the factors. Hence the term coefficients in algebra, which
are the numbers prefixed to, or that multiply the letters
or algebraic quantities.

EFFLUVIUM, a flux or exhalation of minute particles
from any body ; or an emanation of subtile corpuscles
from a mixed, sensible body, by a kind of motion of
transpiration.

ELASTIC, an appellation given to all bodies endow.ed
with the property of elasticity or springiness.

Elastic Body, is that which changes its figure, and
yields to any impulse or pressure, but endeavours by its
own nature and force to restore the same again ; or, it is
a body of such a nature thai, when compressed or con-
densed, it will make an effort to set itself at liberty, and
to repel the body that constrained it: such, for instance,
as a bow, or a sword-blade, ike, which are easily bent, but
presently return to their former figure and extension. All
bodies partake of this property in some degree, though

perhaps none arc perfectly elastic, as none are found to
restore themselves with a force equal to that with which
they are compressed.

The principal phenomena observable in clastic bodies,
are, 1. That an elastic body (i. e. a body perfectly elastic,
if any such there be) endeavours to restore itself with the
same force with which it is pressed or bent. 2. Air elastic
body exerts its force equally towards all sides; though
the effect is chiefly found on that side where the resistance
is weakest; as is evident in the case of a gun exploding a
ball, a bow shooting out an arrow, &c. — 3. Elastic bo-
dies, in whatever manner they are struck or impelled, are
inflected and rebound after the same manner : thus a bell
yields the same musical sound, in whatever manner, or on
whatever side it be struck ; the same of a tense or musical
chord; and a body rebounds from a plane in the same
angle in which it meets or strikes it, making the angle of
incidence equal to the angle of reflection, whether the in-
tensity of the stroke be great or small. — 4. A perfectly
fluid body cannot be elastic, if it be allowed that its parts
cannot be compressed. — 5. A body perfectly solid, can-
not be elastic; because, having no pores, it is incapable
of being convpressed. — ()'. The elastic properties of bodies
seem to differ, according to their greater or less density or
compactness, though not in an equal dearee : thus, metals
are rendered more compact and elastic by being ham-
mered: tempered steel is much more elastic than soft
steel ; and the density of the former is to that of the latter
as 7809 to 7738. Cold condenses solid bodies, and ren-
ders them more elastic ; while heat, that relaxes them,
has the opposite effect : but, on the contrary, air, and
other elastic fluids, are expanded by heat, and rendered
more elastic. — For the laws of motion and percussion in
elastic bodies, see Motion, and Percussion.

Elastic Curve. See Catenaria.

Elastic Fluids. See Air, Electricity, Gas, Elas-
tic Vapours, &c.

Elastic Gum. The same as Caoutchouc, or Indian
Rubber.

Elastic Vapours, or Fluids, are such as may be com-
pressed mechanically into a less space, and which resume
their former state when the compressing force is with-
drawn ; such as atmospheric air, and all the aerial fluids,
with all kinds of fumes raised by means of heat, whether
from solid or fluid bodies.

Of these, some remain elastic only while a considerable
degree of heat is applied to them, or to the substance
which produces them ; while others continue elastic in
every degree of cold that has yet been observed. Of the
former kind, are the vapours of water, spirit of wine, mer-
cury, sal-ammoniac, and all kinds of sublimable salts : of
the latter, those of spirit of salt, mixtures of vitriolic acid
and iron, nitrous acid, and various other metals, and in
short the several species of aerial fluids indiscriminately.

The elastic force with which any one of these fluids is
endowed, has not yet been calculated, as being ultimately
greater than any obstacle we can put in its way. Thus, on
compressing atmospheric air, we find that for some liitle
time at first it easily yields to any force applied ; but at
every succeeding moment the resistance becomes stronger,
and a greater and greater force must be applied, to com-
press it further. As the compression goes on, the vessel
containing the aii becomes hot; hut no power whatever
has yet been able in any degree to destroy the elasticity of

ELA

r 457 ]

E L E

the contained fluid ; for, upon removing the pressure, it is
always found to occupy the very same space that it did
before. The case is the same with the steam of water, to
which a sufficient heat is applied to keep it from condens-
ing into water again.

Elasticity, or Elastic Force, that property of bodies
by which they restore themselves to their former figure,
after any external pressure.

The cause or principle of this important property, elas-
ticity, is variously accounted for. The Cartesians ascribe
it to a subtile matter making an effort to pass through
pores that are too narrow for it. Thus, say they, in bend-
ing or compressing a hard elastic body, as a bow, for in-
stance, its parts recede from each other on the convex
side, and approach on the concave one : consequently the
pores are contracted or straitened on the concave side;
and, if they were before round, are now perhaps oval : so
that the materia subtilis, or matter of the second element,
endeavouring to pass out of the pores thus straitened,
must make an effort, at the same time, to restore the body
to the state it was in when the pores were cylindrical, i.e.
before the bow was bent : and in this consists its elasticity.

Other later philosophers account for elasticity much
after the same manner as the Cartesians ; with this differ-
ence only, that instead of the subtile matter of the Carte-
sians, these substitute ether, or a fine ethereal medium
that pervades all bodies.

Others, setting aside the precarious notion of a materia
subfilis, account for elasticity from the great law of na-
ture, attraction, or the cause of the cohesion of the parts
of solid and firm bodies. Thus, say they, when a hard
body is struck or bent, so that the component parts are
moved a little from each other, but not quite disjointed
or broken off, or separated so far as to be out of the power
of that attracting force by which they cohere ; they must,
on removing the external violence, spring back to their
former natural state.

Others again resolve elasticity into the pressure of the
atmosphere : for a violent tension, or compression, though
not so great as to separate the constituent particles of bo-
dies far enough to let in any foreign matter, must )et occa-
sion many little vacuola between the separated surfaces;
so that on the removal of the force they will close again
by the pressure of the aerial fluid upon the external parts.
Lastly, Others attribute the elasticity of all hard bodies
to the power of resilition in the air included within them;
and so make the elastic force of the air the principle of
elasticity in all other bodies. See Desaguliers's Exper.
Philos. vol. 2, pa. 38, &c.

The Elasticity of Fluids is accounted for from their
particles being all endowed with a centrifugal force ;
whence Sir Isaac Newton demonstrates, prop. 23, lib. 2,
that particles, which naturally avoid or fly off from one
another by such forces as are reciprocally proportional to
the distances of their centres, will compose an clastic fluid,

discovered by Lord Bacon, and farther established by Ga-
lileo. Its existence is thus proved by the latter philoso-
pher: An extraordinary quantity of air being intruded,
by means of a syringe, into a hollow ball or shell of glass
or metal, till such time as the ball, with its accession of
air, weigh considerably more in the balance than it did
before ; then, opening the mouth of the ball, the air
rushes out, till the ball sink to its former weight. Hence
he inferred, that just as much air had issued out, as com-
pressed air had been compressed' in. Air therefore re-
turns to its former degree of expansion, on removing the
force that compressed or resisted its expansion ; and con-
sequently it is endowed with an elastic force. It may be
added, that as the air is found to rush out in every situa-
tion or direction of the orifice, the elastic force acts every
way, or in every direction alike. See Air, and Atmo-
sphere.

The cause of elasticity in air has been usually ascribed
to a repulsion between its particles ; but what is the cause
of that repulsion ? The term repulsion, like that of at-
traction, requires to be defined; and probably it will be
found in most cases to be the effect of the action of some
other fluid. Thus, it is found that the elasticity of the at-
mosphere is very considerably affected by heat. Supposing
a quantity of air heated to such a degree as to raise
Fahrenheit's thermometer to 212, it will then occupy a
considerable space ; but if it be cooled again to such a
degree, as to sink the thermometer to 0, it will shrink up
to less than half the former bulk. The quantity of repul-
sive power therefore acquired by the air, while passing
from one of these states to the other, is evidently owing
to the heat added to it, or taken away from it. Nor does
there seem to be any reason to suppose, that the quantity
of elasticity or repulsive power it still possesses, is owing
to any other cause than the fire contained in it. The sup-
position that repulsion is a primary cause, independent of
all others, has given rise to many erroneous theories, and
very much embarrassed philosophers in accounting for
the phenomena of elasticity.

The elasticity of the air is not only proportional to its
density, but is always equal to the force which compresses
it, because these two forces exactly balance each other.
This elasticity, in the atmospheric air, is measured by the
height of the barometer at any time, allowing for its heat
or temperature, after this rate, viz, the 434th part for each
degree of Fahrenheit's thermometer, above or below some
mean temperature, as 55° ; for by that part of the whole
it is that air expands or contracts, or else increases or de-
creases in its elasticity, for each degree of the thermome-
ter. Sir Geo. Shuckburgh, in the Philos. Trans, for 1777,-
pa. 56 1.

ELECTIONS, or Choice, signify the several different
ways of taking any number of things proposed, either se-
parately, or as combined in pairs, in threes, in fours, &c;
not as to the order, but only as to the number and variety
of them. Thus, of the things a, b, c, d, e, &c, the elec-

whose density is proportional to its compression ; and vice

versa, if any fluid be composed of particles that fly off or' tions of

avoid one another, and have its density proportional to its one thing are (a,) 1 = 2* — 1,

compression, then the centrifugal forces of those particles

will be reciprocally proportional to the distances of their

centres.

Elasticity of the Air is the force with which that
element endeavours to expand, and with which it does
actually dilate itself, on removing the force that com-
presses it. The elasticity or spring of the air was first

Vol. I.

two things are (a, b, ab) 3 = 22 — 1,
three things are (a, b, c, ab, ac, be, abc) 7 = 23 — 1, &c ;
and of any number, n, all the elections are 2" — 1 ; that
is, one less than the power of 2 whose exponent is n, the
number of single things to be chosen, either separately or
in combination. See Combination.

ELECTRIC, in Physics, is a term applied to those sub-
3 N

E L E

[ 458 ]

E L E

stances in which the electric fluid can be excited, and
accumulated, without transmitting it; and which are
therefore called non-conductors. They are also called
original Electrics, and Electrics per se. The word is de-
rived from yteTtrpov, amber, one of the most observable
non-conductors, and the only one known to the ancients.
To this class alsp belong glass, and all vitrifications, even
of metals ; all precious stones, of which the most trans-
parent are the best ; resins, and resinous compositions ;
sulphur, and baked wood; all bituminous substances, wax,
silk, and cotton; all dry animal substances, as feathers,
wool, hair, &c ; also paper, white sugar, and sugar-candy;
likewise air, oils, chocolate, calces of metals and semi-
metals, the ashes of animal and vegetable substances, the
rust of metals, all dry vegetable substances, and stones, of
which the hardest are the best.

Substances of this kind may be excited, so as to exhi-
bit the electric appearances of attracting and repelling light
bodies, emitting a spark of light, attended with a snapping
noise, and yielding a current of air, the sensation of which
resembles that of a spider's web drawn over the face, &c,
and a smell like that of phosphorus ; and this exciting
may be either produced by friction, or by heating and
cooling, or by melting, and pouring one melted substance
into another. — The term is peculiarly applied to the elec-
tric, viz, the globe, or cylinder, Sec, used in electrical
machines, for collecting the fluid by friction.

Electric a I. Air Thermometer, an instrument contrived
by Mr. Kinnersley of Philadelphia, and used in determin-
ing the effects of the electrical explosion upon air ; a de-
scription of which may be seen in Franklin's Letters, &c,
pa. 3S9, 4to, 176"9.

Electrical Apparatus, consists of glass tubes, about
3 feet long, and an inch and a half in diameter, one of
which should be closed at one end, and furnished at the
other end with a brass cap and stop-cock, to rarefy or
condense the inclosed air ; sticks of sealing-wax, or tubes
of rough glass, or glass tubes covered with sealing-wax, or
cylinders of baked wood for producing the negative elec-
tricity; with proper rubbers, as black oiled silk, with
amalgam upon it for the former, and soft new flannel, or
hare-skins, or cat-skins, tanned with the hair on, for the
latter; coated jars, or plates of glass, either single, or
combined in a battery, for accumulating electricity ; me-
- tal rods, as dischargers; an electrical machine; electro-
meters, and insulated stools, supported by pillars of glass,
covered with sealing-wax, or baked wood, varnished or
boiled in linseed oil.

Electrical Atmosphere, is a stream or mass of the
electric fluid which surrounds an excited or electrified
body, to a certain distance.

Electrical Balls. See Balls, and Electrometer.
Electrical Batten/, consists of a number of coated
jars, placed near each other in a convenient manner.
These being charged, or electrified, and connected with
each other, are then suddenly exploded or discharged,
with a prodigious effect. Sec Battery.

Electrical Bells, are a set of bells mounted in a pe-
culiar manner, so as to be rung by means of the action of
the electric fluid. Fig. 10, plate vii, represents a set of
bells of this kind; when the chipper is suspended from
the fly bed, the axis of which rests in a small hole on the
top of the glass pillar ef; the upper part of the axis moves
freely on, and is supported by a hole in the brass piece gs,
and bells of different tones arc placed round the board

/iik. This apparatus being placed near the cylinder, the
prime conductor being removed, will, when the machine is
in action, be put in motion, the clapper of which striking
the bells in succession, produces an agreeable harmony.

Electric Experiments, are certain experiments made
by means of an electric machine and apparatus; some-
times in order to discover the nature and properties of the
electric fluid, and at others for the amusement of the
operator or by-standers. But, in a work like the present,
it is impossible to give any thing like a complete set of
experiments : we have however selected the following ;
and must refer the reader, for others, to Adams's, Caval-
lo's, and Cuthbertson's works on this subject.

Experiment 1. Put the machine in action, connect the
cushion by a chain with the ground, and those bodies
which communicate with the positive conductor will be
electrified positively. Connect the positive conductor
with the earth by a chain, take off the chain from the
cushion, and those bodies which communicate with the
cushion, or negative conductor, will be electrilied nega-
tively.

Exper. 2- Connect the positive conductor by a chain
with the table; turn the cylinder, and the cushion will
be found to be negatively electrified. Take the chain off
from the positive conductor, and both will exhibit signs of
electricity ; but any electrified body, which is attracted
by the one, will be repelled by the other. If the)' are
brought sufficiently near to each other, sparks will pass
between them, and they will act on each other stronger
than on any other bodies. If they are connected toge-
ther, the electricity of the one will destroy that of the
other; for though the fluid seems to proceed from the
cushion to the conductor, the two, when thus conjoined,
will exhibit no signs of electricity, because the fluid is
continually circulating from one to the other, and is there-
fore kept always, in the. same state.

We see, by this experiment, that electric appearances
are produced both in the electric which is excited, and
the substance by which it is excited, provided that sub-
stance be insulated ; but their electric powers are directly
reverse of each other, and may be distinguished by oppo-
site effects.

Expcr. 3. If the cushion and the conductor are both in-
sulated, it is observed, that the less electric fluid is ob-
tained, the more perfect the insulation is made.

■ The moisture, which is at all times floating in the
air, together' with the small points, from which it is im-
possible totally to free the cushion, do not permit it So be
perfectly insulated, so as to afford no supply of electric
matter to the cushion. If the air, and other parts of the
apparatus, are very dry, little or no electricity will be
produced under the above-mentioned circumstances.
From this experiment it is inferred, that the electric
powers do not exist in the electrics themselves, but are
produced from the earth by the excitation of electrics;
or that the electric matter on the prime conductor is not
produced by the friction of the cylinder against the
cushion, but is collected by that operation from it, and
from those bodieswhich arc connected with il.

Exper. 4. Touch a pair of insulated pith balls with an
excited glass tube, they will become electrilied, and will
separate from each other; the balls are electrified posi-
tively, and are therefore attracted by excited wax, and
repelled by excited glass. As those light substances,
which possess the same electric power, repel each other ;.

E L E

[ 459 ]

ELE

we can easily discover whether they are electrified posi-
tively or negatively, by presenting an excited stick of
sealing-wax or glass to them. If they are attracted by the
glass, they are negatively, if repelled by it, they ai-e po-
sitively electrified; on the contrary, if repelled by the
excited wax, they are negative, if attracted, positive.

In ascertaining the nature of the electric powers, we
must avoid bringing the bodies to be tried near each other
suddenly ; or one with a strong electricity near another
which is weakly so; as it may render the experiment
doubtful, by attracting, and not repelling the light body.

Exper. 5. Insulate a long metallic rod, suspend a pair
of pith balls from each end of it, place one of the ends at
about two inches from the prime conductor, the other
end as far from it as possible, electrify the conductor,
and the electric fluid in the rod will be driven to that end
which is furthest from the conductor; so that one end
will be electrified negatively, the other end positively, as
will be seen by the balls.

Exper. 6. Fix the end of a wire, having feathers at-
tached to it, in the small hole which is at the end of the
prime conductor; turn the cylinder, and the feathers,
which are connected with the wire by linen threads, will
separate from each other ; the fibrous and downy parts
will become turgid, and expand in a pleasing manner,
and in a variety of directions.

Present a metallic point, or the finger, or any other
conducting substance, to the feathers, the downy parts
thereof will immediately collapse, the divergence of the
feathers will cease, and they will approach each other,
and cling round the non-electric body.

The feathers separate from each other, and tend to-
wards unelectrified bodies, from the effort made by the
electricity which is communicated to them to diffuse itself,
and the resistance it meets with from the air.

Exper. 7. Fix the end of the wire into the hole at the
end of the conductor, as before, having now two pith
balls attached, instead of the feathers ; put the machine
in action, and the two small balls will recede from each
other. Bring a conducting substance within the sphere of
their action, and they will fly towards it ; touch the con-
ductor with a non-electric, and they will immediately
come together.

The balls do not always diverge so much as might be
expected, from the action of their atmospheres, because
they are influenced by that of the conductor.

'i lie balls, or feathers, will separate, &c, in the same
manner, if they are annexed to a negative conductor.

Exper. 8. Present a fine thread towards an electrified
conductor ; when it is at a proper distance, it will fly to-
wards, and stick to the conductor, and convey the electric
fluid from it to the hand ; remove the thread to a small
distance from the conductor, and it will fly backwards
and forwards with great velocity and in a very pleasing
manner : present the same thread towards one that hangs
from the conductor, they will attract and join each other,
llring a nonelectric body, as a brass ball, near these
threads, the ball will repel that held by the hand, and
attract that which is fixed to the conductor : the upper
thread renders the brass ball negative, and therefore goes
towards it ; while the under thread, which is also nega-
tive, is repelled. Let the ball be brought near the lower
part of the under one, audit will be attracted by it. The
junction of the threads arises from the effort the electric
fluid makes to diffuse itself through them.

Exper. 9. Place a square piece of leaf brass or silver on
the under plate, hold this parallel to the upper one, at
about 5 or 6 inches from it ; turn the machine, and the
leaf will then rise up into a vertical situation, and remain
between the two plates, without touching either of them.
Present a metal point towards the leaf, and it will imme-
diately fall down. Place a brass ball at the end of the
conductor, and when the- leaf of brass is suspended be-
tween the plate and ball, moye the plate round the ball,
and the leaf will also move round without touching
either plate or ball.

Exper. 10. Place two wires directly under, and parallel,
to each other; suspend one from the conductor, let the
other communicate with the table ; then a light image
placed between these, will, when the conductor is elec-
trified, appear like a kind of electrical rope-dancer.

Exper. 11. Put a pointed wire into one of the holes
which are at the end of the conductor, hold a glass
tumbler over the point, covering it with your hands, which
serve as a temporary coating; then electrify the con?
ductor, and turn the tumbler round, that the whole in-
terior surface may receive the fluid from the point ; place
a few pith balls on the table, and cover them with this
glass tumbler, the balls will immediately begin to leap up
and down as if they were animated, and will continue to
move for a considerable time.

Exper. 12. Connect one end of a chain with the outside
of a charged jar, and let the other end lie on the table,
place the end of another piece of chain at about one
quarter of an inch distant from the former; then set a
decanter of water on these separated ends, and on making
the discharge through the chain, the water will appear
perfectly and beautifully luminous.

Exper. 13. Let any person stand on an insulated stool,
and connect himself by a wire or chain with the prime
conductor, arrd he will then exhibit the same appearances
which are observed in the conductor, and will attract
light bodies, give the spark, &c, and thus afford a pleasing
mode of diversifying every experiment. It is however
absolutely necessary, to the complete success of this ex-
periment, that no part of the clothes touch the floor,
table, &c, and that the glass feet be carefully dried; a
sheet of dry brown paper placed under the stool will be
found of considerable service, by rendering the insulation
more complete.

If the insulated person lav his hands on the clothes of
one that is not so, especially if they are woollen, they
will both feel as it were many pins pricking them, as long
as the cylinder is in motion.

Exper. 14. Place the brass ball of a coated jar in con-
tact with the prime conductor, while the outside com-
municates with the table, turn the cylinder, and the
bottle will in a little time be charged. Now to discbarge
the jar, that is to restore it to its natural state, bring one
end of a conducting substance in contact with the outside
coating, and let the other be brought near the knob of
the jar which communicates with the inside coating, and
a strong explosion will take place, the electric light will
be visible, and the report very loud. 'Charge the. jar
again, then touch the outside coating with one hand, and
the knob with the other, by which means the bottle will
be discharged, and a sudden peculiar sensation will be
perceived, which is called the electric shock. The shock,
when it is taken in this manner, generally affects the
wrists, elbows) and breast; and when it is strong it re-
3 N 2

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ambles a universal blow. This peculiar sensation is
probably owing to the two-fold and instantaneous action
of the electric fluid, which enters and goes out of the
body, and the various parts through which it passes, at
one "and the same instant. It has been also observed,
that nature has appointed a certain modification of the
electric fluid in all terrestrial bodies, which we violate
in our experiments; when this violation is small, the
powers of nature operate in a gentle manner, to rectify
the disorder we have introduced; but when the deviation
is considerable, the natural powers restore the original
constitution with extreme violence.

If several persons join hands, and the first touches the
outside of a charged jar, and the last the knob, the bottle
will be discharged, and they will all feel the shock at the
same instant ; but the greater the number of persons are
that take a shock, the weaker are its effects.

The force of the shock is in proportion to the quantity
of coated surfaces, the thinness of the glass, and the power
of the machine ; or the effect of the Leyden phial is in-
creased in proportion as we destroy the equilibrium on the
surfaces.

Exper. 15. A cork ball, or an artificial spider, made
of burnt cork with legs of linen thread, suspended by
silk, will play between the knobs of two bottles, one of
which is charged positively, the other negatively, and
will in a little time discharge them.

Exper. l6. A ball suspended on silk, and placed be-
tween two brass balls, one proceeding from the outside,
and the other from the inside of a Leyden jar, when the
bottle is charged, will fly from one knob to the other ;and
by thus conveying the fluid from the inside to the outside
of the bottle, will soon discharge it.

Exper. 17. Take a glass tube, round which, at small,
but equal distances from each other, pieces of tin-foil are
pasted in a spiral form from end to end; this tube is in-
closed in a larger one, fitted with brass caps at each end,
which are connected with the tin-foil of the inner tube.
Hold one end in the hand, and apply the other near
enough to the prime conductor to take sparks from it, a
beautiful and lucid spot will then be seen at each separa-
tion of the tin-foil ; which multiply, as it were, the spark
taken from the conductor ; for if there was no break in
the tin-foil, the electric fluid would pass off unperceived.

Exper. 18. To take the electric spark with a metal point,
screw a pointed brass wire into one end of a spiral tube,
and present it to the conductor while the machine is in
action ; then a strong spark will pass between the con-
ductor and the point.

Exper. 19. Take a clean dry glass tube, of about a
quarter of an inch bore, and insert a pointed wire in this
tube, keep the pointed end at some distance from the end
of the tube, at the same time let the other end be con-
nected with the ground, bring the former towards the
prime conductor, and strong zig-zag sparks, attended
with a peculiar noise, will pass between the conductor
and the point.

Exper. 20. The Luminous Word. This experiment is
exactly on the same principles as Experiment 17. The
word is formed by the small separations made in the tin-
foil, which is pasted on a piece of glass, that is fixed in a
frame of baked wood, having a brass ball at one end.
To make the experiment, hold the frame in the hand,
mid present the ball to the conductor, the spark received
on this will be communicated to the tin-foil, and follow

it in all its windings, till it arrives at the hook, whence it
is conveyed to the ground by a chain : and thus the lucid
appearance at each break exhibits a word in characters
of fire.

Electrical Fluid, is a fine rare fluid which issues
from, and surrounds, electrified bodies.

Electrical Kite, an instrument contrived by Dr.
Franklin, to verify his hypothesis of the identity of elec-
tricity and lightning. It consisted of a large thin silk
handkerchief, extended and fastened at the four corners
to two slender strips of cedar, and accommodated with a
tail, loop, and string, so as to rise in the air like a paper
kite. To the top of the upright stick of the cross^yas
fixed a very sharp-pointed wire, rising a foot or more
above the wood ; and to the end of the twine, next the
hand, a silk ribband was tied. From a key suspended at
the junction of the twine and silk, when the kite is raised
during a thunder-storm, a phial may be charged, anxl
electric fire collected, as is usually done by means of an
excited glass tube or globe. Philos. Trans, vol.47, pa.565,
or Franklin's Letters, pa. Ill and 112. — Kites made of
paper, covered with varnish, or with well boiled linseed
oil, to preserve them from the rain, with a stick and cane
bow, like the common ones used by boys, will answer the
purpose extremely well, and are very useful in deter-
mining the electricity of the atmosphere. See Con-
ductor.

Electrical Machine, a piece of mechanism in which
an electric body, such as glass, sulphur, resin, &c, is
subjected to friction, in order to excite in it that power
which is commonly known under the name of electricity.

It would be useless to trace this machine through all
its various forms and improvements, which the ingenuity
and industry of philosophers have given to it; we shall
therefore limit ourselves to describing and exhibiting some
of the most approved constructions, which may be enu-
merated as follows.

Description of the most useful Electrical Machines.

Fig. 1, plate xi, represents Dr. Priestley's machine,
which is a very extensively useful one, described in his
History of Electricity ; in which g is the globe, or electric ;
/the rubber; in the two pillars d, d, of baked wood, are
several Uioles to receive the spindles of different globes or
cylinders, several of which may be put on together, to in-
crease the electricity : Mm is the prime conductor, being a
copper tube, supported on a stand of glass or baked wood.

Dr. Watson's machine, for using several globes at once,
to accumulate a great quantity of electricity, is represented
fig. 2.

Fig. 3 represents a very portable electrical machine
invented by Mr. Read, and improved by Mr. Lane, a is-
the glass cylinder, moved vertically by means of the pulley
at the lower end of the axis, the pulley being turned by
the large wheel B parallel to the table : there are also se-
veral pulleys, of different sizes, eitherof which may be used,
according as the motion is required to be quicker or slower.
The conductor c is furnished with points to collect the
fluid, and is screwed to the wire of a coated jar d. The
figure shows also the manner of applying Mr. Lane's elec-
trometer to this machine.

Electrical machines have of late years undergone some
very essential alterations and improvements; both from
the suggestions of private electricians, and the inventions
of Messrs. Adams, Nairne, and Jones, instrument-makers
in London; some of which are as follow:

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Fig. 4 represents a very convenient machine for practice, some general and practical directions necessary to be ob-
Thc frame of this machine consists of the bottom board served for the preservation and proper management of
abcd; which, when the machine is used, must be fasten- them, in making experiments.

ed to the table by two metal cramps, ef are two round
pillars, of baked wood, which support the cylinder g by
the axles of the brass or wooden caps n, turned sometimes
by a simple winch I, and sometimes by a pulley and wheel,
as in the next fig. The rubber is fixed to a glass pillar k,
which is fastened to a wooden basis l at the bottom. The
conductor s is usually made of brass or tin japanned, and
is insulated by a glass pillar, screwed into a wooden basis
or foot, most conveniently placed parallel to the cylinder.

Fig. 5 represents an electrical machine, with a con-
ductor in the shape of a t ; and an improved medical ap-
paratus, where it is necessary to give the shock in the arms,
or any other particular part of the body.

Fig. 6 shows Mr. Nairne's patent machine for medical
purposes. Its glass cylinder is about 7 inches in diame-
ter, and 12 long, with two conductors parallel to it. The
rubber is fastened to the conductor r; and consists of a
cushion of leather stuffed, having a piece of silk glued to
its under part. The conductors are made of tin, and co-
vered with black lacker, each of them containing a large
coated glass jar, and likewise a smaller one, or a coated
tube, which are visible when the caps nn are removed.
To each conductor is fixed a knob o, for the occasional
suspension of a chain to produce positive or negative elec-
tricity. That part of the winch c which acts as a lever in
turning the cylinder, is of glass. Thus every part of the
machine is insulated, the cylinder itself and its brass caps
not excepted; by which means very little of the electri-
city is dissipated, and hence of course the effects are likely
to be the more powerful : the inventor has also adapted to
this machine some flexible conducting joints, a discharging
electrometer, &c, for the practice of medical electricity.

The large electrical machine placed in Teyler's Museum
at Harlem, was constructed by Mr. John Cuthbertson, an
English instrument-maker; and it has, for the electric, two
glass plates of 65 inches diameter, made of French glass,
as this is found to produce the most electricity next to
English flint glass, which could not be made of a sufficient
size : these plates are set on the same horizontal axis, at the
distance of 74- inches, and are excited by 8 rubbers, each
15£ inches long; and both sides of the plates are covered
witfl a resinous substance to the distance of l6| inches
from the centre, both to strengthen the plates, and to pre-
vent any electricity from being carried off by the axis. Its
battery of jars contains 225 square feet of coated surface,
and its effects are astonishingly great; as a proof of which
we mention the following particulars: The sensation com-
monly called the spider's web, when this machine is in
action, is felt by the by-standers at the distance of 8
feet from the prime conductor. A thread 6 feet long,
suspended perpendicularly, was sensibly attracted by the
prime conductor at the distance of 38 feet, and a pointed
wire presented to the prime conductor appeared luminous
even at the distance of 28 feet. A single spark from the
prime conductor melted a considerable length of gold leaf.
And a Leyden phial, containing about a square foot of
coated surface, was fully charged by about half a revolu-
tion of the plates, so as to discharge itself; and by repeated
trials it was found that, in one minute's time, this phial
discharged itself 76, 78, and frequently even 80 times in a
minute. Having thus described the electrical machines of
the most approved construction, it only remains to add

Moisture and dust, but particularly the former, being
detrimental to the power of an electrical machine, it be-
comes necessary lo guard against both, as much as may be
practicable ; hence when not actually in use, the machine
should be kept in a dry and clean place, and at least the
glass part of it should not be suffered to remain dirty and
soiled. If it has been long neglected, the operator, in or-
der to render it ready for use, must, in the first place, re-
move the rubber; he must then place the machine at a
moderate distance from the fire, so as to render every part
ot it perfectly dry, but not too warm. This done, and the
dust removed, the glass part must be repeatedly rubbed
with a clean and warm handkerchief, or towel; the rubber
likewise must be cleaned, removing all the old amalgam
that may ha\e adhered to it. The glass plate, or cylin-
der, in its rotation frequently contracts some dark spots or
concretions on its surface, which tend to diminish its
power, and which must therefore be removed. Previously
to replacing the rubber, the following operation generally
contributes towards increasing the excitation. It consists
in touching the cylinder with the bottom of a tallow can-
dle, in streaks parallel to the axis of the cylinder, then
rub the cylinder again with a dry warm linen cloth, taking
care that this cloth be not very old, for in that case it is
apt to leave filaments about the glass and about the rest of
the machine. This done, the rubber is fixed in its place,
and its supporter being properly adjusted as to pressure, as
also a chain suspended from the rubber, connecting it with
the floor of the room, then the machine is ready for opera-
tion, except the application of the amalgam, in which the
following directions must be observed.

Formerly the amalgam was spread upon the rubber; but
experience has shown that it is much better for it to be
fixed clean in its place, and then to apply the amalgam
upon a piece of leather, to the surface of the cylinder,
while this is revolving in its usual direction; for by this
means the revolution of the plate or cylinder will carry
from the leather a sufficient quantity of the amalgam, and
will deposit it upon the rubber. The leather with the
amalgam need not be applied longer than the cylinder
makes 10 or 12 revolutions, moving the leather at the same
time, backward and forward, from one end of the cylinder
to the other, in order that the amalgam may be equally dis-
tributed. Now if the cylinder be turned, and a hand or
the ends of the fingers be presented to it, a crackling noise,
which is accompanied with luminous brushes and dense
sparks of the electric fire, will be observed, particularly in
a dark room, which appearances indicate that the ma-
chine is in good action; then, the prime conductor being
situated in its proper place, you may proceed to perform
the experiments. During the performance of which, the
electrified part of the machine is apt to attract dust from
all quarters, which the operator must guard against as
much as possible, by frequently wiping those parts with a
clean and perfectly dry cloth, or silk handkerchief.

The amalgam which we have mentioned above is gene-
rally composed of 1 part of zinc, and 4 or 5 parts of
quicksilver,whicharcamalgamated in thefollowing manner;

Let the quicksilver be heated to about the degree of boil-
ing water; then, the zinc being already melted in a cru-
cible, pour the quicksilver first into a wooden box, and
immediately after the melted zinc. Then shut the bos,

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F. L F.

and shake it lor about half a minute. After this you must
wait till the amalgam is quite cold, or nearly so, when
yau may mix some grease with it by trituration. It the
melted rinc be poured into the quicksilver when cold, a
very small portion of the former will be amalgamated, the
iv>t remaining in lumps of different sizes.

The above is the method recommended by M. Cavallo,
but Mr. Cuthbertson gives the following directions for pre-
paring amalgam. Take one part of tin and zinc, melt them
in a crucible, and pour them on two parts of mercury,
which put into a wooden box made for the purpose;
and shake the box till the metals are cold. The amalgam
is then to be pulverised in a metal mortar to a very fine
powder, and afterwards mixed with a sufficient quantity of
hogs-lard, to make it into a paste.

Electrical Phial. See Leyden Phial.
Electrical Rubber. See Electrical Apparatus, and
Electrical Machine.

Electhical Shock, is the sudden explosion between the
opposite sides of a charged electric ; so called because if
the discharge be made through the body of an animal, it
occasions a sudden motion by the contraction of the mus-
cles through which it passes, accompanied with a disagree-
able sensation. The force of this shock is proportioned to
the quantity of coated surface, the thinness of the glass,
and the power of the machine by which it is charged. Its
velocity is almost instantaneous, and it has not been found
to take up the least sensible time in passing to the greatest
distances.

It has however been observed that the electrical shock
is weakened by being communicated through several per-
sons in contact witli,one another. Indeed it is obstructed
in its passage, even through the best conductors, as it will
prefer a short passage through the air to a long one through
the most perfect conductors; and if the circuit be inter-
rupted, either by electrics, or very imperfect conductors
of a moderate thickness, the shock will rend them in its
passage, disperse them in every direction, and exhibit the
appearance of a sudden c>>pansion of the air about the cen-
tre of the shock. A strong shock made to pass through or
over the. belly of a muscle, forces it to contract; and sent
through a small animal hody, deprives it instantly of life,
and hastens putrefaction. It gives polarity to magnetic
needles, reverses their poles, and produces effects precisely
similar, though inferior in degree, to those of lightning.
Electrical Star. See Star.

ELECTRICITY, or Electrical Force, is that power
or property, which was first observed in amber, the lyneu-
rium, or tourmalin, and which sealing-wax, glass, and a
variety of other substances, called electrics, are now known
to possess, of attracting light bodies, when excited by heat
or friction; and which is also capable of being communi-
cated in particular circumstances to other bodies.

Electricity also denotes the science, or that part of
natural philosophy, which proposes to investigate the nature
and effects of this power. From vjAsxrf ov, the Greek name
for amber, is derived the term electricity, which is now
very extensively applied, not only to the power of attracting
light bodies inherent in amber, but to other similar powers,
and their various effects, in whatever bodies they reside, or
to whatever bodies they may be communicated. Muschcn-
broek and jEpinus have observed a considerable analogy,
in a variety of particulars, between the powers of electri-
city and magnetism; and they have also pointed out many
instances in which they differ.

History of Electricity. — The property which am-
ber possesses of attracting light bodies, was observed bv
Thales of Miletus, 600 years before Christ, who concluded
from hence that it was animated. But the first person who
expressly mentioned this substance, was Theophrastus,
about 300 years before Christ. The attractive property
of amber is also occasionally noticed by Plinv, and other
naturalists, particularly Gassendus, Sir Kenelm Digby, and
Sir Thos. Brown. But it was generally apprehended that
this quality was peculiar to amber and jet, and perhaps
agate, till Dr. Gilbert, a native of Colchester, and a phy-
sician in London, published his treatise De Magnete, in the
year 1600. Dr. Gilbert made many considerable experi-
ments and discoveries, considering the then infant state of
the science. He enlarged the list both of electrics, and
of the bodies on which they act; and remarked, that a
dry air was most favourable to electrical appearances, while
a moist air almost annihilates the electric virtue : he also
observed the conical figure assumed by electrilied drops of
water. He considered electrical attraction separately from
repulsion, which he thought had no place in electricity, as
a phenomenon similar to the attraction of cohesion, and he
imagined, that electrics were brought into contact with the
bodies on which they act by their effluvia, excited by fric-
tion.

The ingenious Mr. Boyle added to the catalogue of elec-
tric substances; but he thought that glass possessed this
power in a very low degree: he found; that the electricity
of all bodies, in which it might be excited, was increased
by wiping and warming them before they were rubbed ;
that an excited electric was acted on by other bodies as
strongly as it acted upon them : that diamonds rubbed
against any kind of stuff, emitted light in the dark; and
that feathers would cling to the fingers, and to other sub-
stances, after they had been attracted by electrics. He ac-
counted for electrical attraction, by supposing a glutinous
effluvia emitted from electrics, which laid hold of small
bodies, in its way, and carried them back to the body from
which it proceeded.

Otto Guericke, the celebrated inventor of the air-pump,
lived about the same time. This ingenious philosopher
discovered, by means of a globe of sulphur, that a body
once attracted by an electric, was next repelled, and that
it continued in this state of repulsion till it was touched
by some other body; he also observed the sound and
light produced by the excitation of his globe; and that
bodies immerged in electrical atmospheres tire themselves
electrilied, with an electricity opposite to that of the at-
mosphere.

The light emitted by electrical bodies was, soon after,
observed to much greater advantage by Dr. Wall, who
ascribes the light to the electrical property which they
possess ; and he suggests a similarity between the effects of
electricity and lightning.

Sir Isaac Newton was not inattentive to this subject: he
observed that excited glass attracted light bodies on the
side opposite to that on which it was rubbed; and he
ascribed the action of electric bodies to an elastic fluid,
which freely penetrates glass, and the emission of it to the
vibratory motions of the parts of excited bodies.

Mr. flawksbee wrote on this subject in the year 1709,
when a new tcra commenced in the history of this science,
lie first took notice of the great electrical power of glass,
and the light proceeding from it ; though others had be-
fore observed the light proceeding from other electrilied

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substances : he also noted the noise occasioned by it, with
a variety of phenomena relating to electrical attraction
and repulsion. He first introduced a glass globe into the
electrical apparatus, to which circumstance many of his
important discoveries may be attributed.

After his time there was an interval of near 20 years in
the progress of this science, till Mr. Stephen Grey estab-
lished a new eera in the history of electricity. To him we
owe the important discovery of communicating the power
of native electrics to other bodies, in which it cannot be
excited, by supporting them on silken lines, hair lines,
cakes of resin or glass; and also a more accurate distinc-
tion than had hitherto been made between electrics and
non-electrics: he likewise showed ithe effect of electricity
on water much more obviously than Gilbert had done in
the infancy of this science.

The experiments of Mr. Grey were repeated by M. du
Fay, member of the Academy of Sciences at Paris, to which
he added many new experiments and discoveries of his
own. He observed, that electrical operations are obstruct-
ed by great heat, as well as by a moist air; that all bodies,
both solid and fluid, would receive electricity, when placed
on warm or dry glass, or sealing-wax; that those bodies
which are naturally the least electric, have the greatest
degree of electricity communicated to them by the ap-
proach of the excited tube. He transmitted the electric
virtue through a distance of 1256 feet; and first observed
the electric spark from a living body, suspended on silken
lines, and noted several circumstances attending it. M.
dn Fay also established a principle, first suggested by Otto
Guericke, that electric bodies attract all those that are

stances: which was first done by Dr. Ludolf, in the be-
ginning of the year 1744, who, with sparks excited by the
friction of a glass tube, kindled the ethereal spirit of Fro-
benius. Winkler did the same by a spark from his own
finger, by which he kindled French brandy, and other
spirits, after previously healing them. Mr. Gralath fired
the smoke of a candle just blown out, and so lighted it
again; and Mr. Boze fired gun-powder, by means ofitsin»
flammable vapour. It was also at this time Ludolf the
younger demonstrated, that the luminous barometer was
made perfectly electrical by the motion of the quicksilver.
The electrical star and electrical bells were also of Ger-
man invention.

From this period Dr. Watson considerably distinguished
himself in the pursuit of this science ; he fired a variety of
substances by the electrical spark, and first discovered that
they are capable of being fired by the repulsive power of
electricity. In the year 1745, the accumulation of the
electrical power in glass, by means of the Leyden phial,
was first discovered. See Leyden Phial : and for the me-
thod practised about this time, of measuring the distance
to which the electrical shock may be conveyed, see Elec-
trical Circuit. Dr. Watson discovered that the glass
tubes and globes do not contain the electric matter in
themselves, but only serve as first-movers or determiners,
as he expresses it, of that power ; which was also con-
firmed towards the end of 1/46, by Mr. Benjamin Wilson,
who made the same discovery, that the electric fluid does
not come from the globe, but from the earth, and other non-
electric bodies about the apparatus. Dr. Watson also dis-
covered what Dr. Franklin observed about the same time

not so, and repel them as soon as they are become elec- in America, and called the plus and minus elecfricity

trie, by the vicinity or contact of the electric body. He
likewise inferred from other experiments, that there were
two kinds of electricity; one of which he called the vitreous,
belonging to glass, rock crystal, &c; and the other resinous,
as that of amber, gum-lac, &c, which are distinguished by
their repelling those of the same kind, and attracting each
other. He farther observed, that communicated electri-
city had the same property as the excited ; and that elec-
tric substances attract the dew more than conductors.

Mr. Grey, resuming his experiments in 1734, suspended
several pieces of metal on silken lines, and found that by
electrifying them they gave sparks; which was the origin
of metallic conductors: and on this occasion he discovered
a cone or pencil of electric light, such as is now known
to issue from an electrified point. And from other ex-
periments he concluded that the electric power seems to
be of the same nature with that of thunder and lightning.

Dr. Desaguliers succeeded Mr. Grey in the prosecution
of this branch of philosophy. The account of his first
experiments is dated in 1739, and to him we owe those
technical terms of conductors or non-electrics, and electrics
perse; he also first ranked pure air among the electrics per
se, and supposed its electricity to be of the vitreous kind.

After the year 1742, in which Dr. Desaguliers con-
cluded his experiments, the subject was taken up and pur-
sued in Germany : the globe was substituted for the tube,
which had been used ever since the time of Hawksbec,

He likewise showed that theclectricmatter passed through
the substance of the metal of communication, and not
merely over the surface. The history of medical electri-
city commenced in the year 1747 ; but we are under the
necessity of omitting many experiments, and conclusions
drawn from them, by Mr. Wilson, Mr. Smeaton, and Dr.
Miles in England, and by the Abbe Nollet, with regard
to the effect of electricity on the evaporation of fluids, on
solids, and on animal and other organized bodies, in
France.

While the philosophers of Europe were busily employed
in electrical experiments and pursuits, those of America,
and Dr. Franklin in particular, were equally industrious,
and no less successful. His discoveries and observations
in electricity were communicated in several letters to a
friend; the first of which is dated in 1747, and the last
in 1754; the particulars of his system may be seen under
the articles, Theory of Electricity, Leyden Phial,
Points, Charging, Conductors, Electrics, &c.

The similarity between electricity and lightning had
been suggested by several writers: and Dr. Franklin first
proposed a method of bringing the matter to the test of
experiment, by raisins; an electrical kite; by which means
he succeeded in collecting electrical fire from the clouds,
in 1752, one month after the same theory had had been
verified in France, and without knowing what had been
done there : and to him we owe the practical application

and a cushion was soon after used as a rubber, instead of of this discovery, in securing buildings from the damage

the hand ; it was also about this time that cylinders be-
gan to be used. instead of the globes; and some of the
German electricians made use of many globes at the same
time. By thus increasing the electrical power, they were
the first who succeeded in setting fire to inflammable sub-

of lightning, by erecting metallic conductors. Si-e Con-
ductors, and Lightning.

In the subsequent period of the history of electricity,
Mr. Canton in England, and Signior Beccaria in Italy,
acquired distinguished reputation. They both discovered,

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£L>E

independently of each.other, that air is capable of receiv- their several treatises on the subject of electricity, any of
iiiff electricity by communication, and of retaining it when
received. Mr. Canton also, towards the latter end of the
year 1753, pursued a series of experiments, which proved
that the appearances of positive and negative electricity,
which had hitherto been deemed essential and unchange-
able properties of different substances, as of glass and seal-
ing-wax for instance, depend on the surface of the electrics,
and that of the rubber. This hypothesis, verified by nu-
merous experiments, occasioned a controversy between
Mr. Canton and Mr. Delaval, who still maintained that
these different powers depended entirely on the substances
themselves. About this time also some curious experi-
ments were performed by four of the principal electricians
of that period, viz. Dr. Franklin, and Messrs. Canton,
Wilcke, and jEpinus, to ascertain the nature of electric at-
mospheres ; the resultof which see under that article. The
theory of two electric fluids, always co-existent and coun-
teracting each other, though not absolutely independent,
was maintained by a course of experiments on silk stock-
ings of different colours, communicated to the Royal So-
ciety by Mr. Symmer, in the year 1759, and which were
farther pursued by Mr. Cigna of Turin, who published an
account of them in the Memoirs of the Academy at Turin
for the year 1765.

Many instances occur in the history of the science
about this time, of the astonishing force of the electric
shock, in melting wires, and producing other similar ef-
fects: but the most remarkable is an experiment of S.
Beccaria,in which he thus revivified metals. Several ex-
periments were also made by Dr. Watson, Mr. Smeaton,
Mr. Canton, and others, on the passage of the electric fluid
through a vacuum, and its luminous appearance, and on
the power possessed by certain substances of retaining the
light communicated to them by an electric explosion. Mr.
Canton, S. Beccaria, and others, made many experiments
to identify electricity and lightning, to ascertain the state
of the atmosphere at different times, and to explain the
various phenomena of the aurora borealis, water-spouts,
hurricanes, &c. on the principles of this science.

Those who are desirous of farther information with re-
spect to the history of electrical experiments and disco-
veries, may consult Dr. Priestley's History and Present
State of Electricity. This author however, is not merely
an historian : his work contains many original experiments
and discoveries made by himself. He ascertained the
conducting power of charcoal and of hot glass, the elec-
tricity of fixed and inflammable air, and of oil; the dif-
ference between new and old glass, with respect to the
diffusion of electricity over its surface; the lateral explo-
sion in electrical discharges; a new method of fixing cir-
cular-coloured spots on the surfaces of metals, and the
most probable difference between electrics and conductors,
&.C. The science is also greatly indebted to many other
persons, either for their experiments and improvements
in it, or for treatises and other writings upon it ; as Mr.
Henley, to whom we owe several curious experiments and
observations on the electrical and conducting quality of
different substances, as chocolate, vapour, &c, with the
reason of the difference between them ; the fusion of pla-
tina ; the nature of the electric fluid, and its course in a
discharge; the method of estimating the quantity of it in
electrical bodies by an electrometer ; the influence of
points ; &c, &c. Also Messrs. Van Marum, Van Swinden,
Ferguson, Cavallo, Lord Mahon, Nairnc, &c, &c, for

which may be consulted with advantage for the experi-
ments and principles of the science.

Medical Electricity. It is natural to imagine that
a power of such efficacy as that of electricity would soon
be applied to medical purposes; especially, since it has
been found invariably to increase the sensible perspira-
tion, to quicken the circulation of the blood, and to pro-
mote the glandularsecretion : accordingly, many instances
occur in the latter period of the history of this science, in
which it has been applied with considerable advantage
and success. And among the variety of cases in which it
has been tried, there are none where it has been found pre-
judicial, except those of pregnancy and the venereal dis-
ease. In most disorders, in which it has been used with
perseverance, it has given at least a temporary and partial
relief, and in many it has effected a complete cure. Nu-
merous instances of this kind may be seen in the Philos.
Trans., and the writings on this science by Messrs. Lovet,
Westley, Ferguson, Cavallo, &c, &c.

Theory of Electricity. It is hardly necessary to re-
cite the ancient hypotheses on this subject ; such as that
of the sympathetic powder of the Peripatetics; that of
unctuous effluvia emitted by excited bodies, and return-
ing to them again, adopted by Gilbert, Gassendus, Sir
Kenelm Digby, &c; or that of the Cartesians, who ascribed
electricity to the globules of the first elements, discharged
through the pores of the rubbed substance, and in their
return carrying with them those light bodies, in whose
pores they were entangled: these hypotheses were framed
in the infancy, not only of this science, but of philosophy
in general, and have long since been deservedly exploded.
In the more advanced state of electricity there have been
two principal theories, each of which has had its advo-
cates. The one, is that of two distinct electric fluids, re-
pulsive with respect to themselves, and attractive of one
another, adopted by M. du Fay, on discovering the two
opposite species of electricity, viz, the vitreous and re-
sinous, and since new-modelled by Mr. Symmer. It is
supposed that these two fluids are equally attracted by all
bodies, and exist in intimate union in their pores; and that
in this state they exhibit no mark of their existence. But
that the friction of an electric by a rubber separates these
fluids, and causes the vitreous electricity of the rubber to
pass to the e'lectric, and then to the prime conductor of a
machine, while the resinous electricity of the conductor
and electric is conveyed to the rubber: and thus the
quality of the electric fluid, possessed by the conductor
and the rubber, is changed, while the quantity remains the
same in each. In this state of separation, the two electric
fluids will exert their respective powers; and any num-
ber of bodies charged with cither of them will repel each
other, attract those bodies that have less of each parti-
cular fluid than themselves, and be still more attracted by
bodies that are wholly destitute of it, or that are over-
charged with the contrary. According to this theory, the
electric spark makes a double current ; one fluid passing
to an electrified conductor from any substance presented
to it, while the same quantity of the other fluid passes
from it; and when each body receives its natural quantity
of both fluids, the balance of the two powers is restored,
and both bodies are unelectrified. For a farther account
of the explication of some of the principal phenomena of
electricity by this theory, sec Dr. Priestley's History,
vol. 2, § 3.

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The other theory is commonly distinguished under the
denomination of positive and negative electricity, being first
suggested by Dr. Watson, but digested, illustrated, and
confirmed by Dr. Franklin; and since that it has been
known by the appellation of the Franklinian hypothesis.
It is here supposed that all the phenomena of electricity
depend on one fluid, sui generis, extremely subtile and
elastic, dispersed through the pores of all bodies, by which
the particles of it are as strongly attracted as they are re-
pelled by one another. When bodies possess their natu-
ral share of this fluid, or such a quantity as they can re-
tain by their non-attraction, it is then said they are in an
unelectrified state ; but when the equilibrium is disturbed,
and they either acquire an additional quantity from other
bodies, or lose part of their own natural share by commu-
nication to other bodies, they exhibit electrical appear-
ances. In the former case it is said they are electrified
positively, or plus ; and in the other negatively, or minus.
This electric fluid, it is supposed, moves with great ease
in those bodies that are called conductors, but with ex-
treme difficulty and slowness in the pores of electrics ;
whence it happens, that ail electrics are impermeable to
it. It is further supposed that electrics contain always an
equal quantity of tins fluid, so that there can be no sur-
charge or increase on one side, without a proportionate de-
crease or loss on the other, and vice versa; and as the
electric does not admit the passage of the fluid through
its pores, there will be an accumulation on one side, and a
corresponding deficiency on the other. Thus when both
sides are connected together by proper conductors, the
equilibrium will be restored by the rushing of the redun-
dant fluid from the overcharged surface to the exhausted
one. Thus also, if an electric be rubbed by a conduct-
ing substance, the electricity is only conveyed from one to
the other, the one giving what the other receives ; and if
one be electrified positively, the other will be electrified
negatively, unless the loss be supplied by other bodies
connected with it, as in the case of the electric and insu-
lated rubber of a machine. This theory serves likewise
to illustrate the other phenomena and operations in the
science of electricity. Thus, bodies differently electrified
will naturally attract each other, till they mutually give
and receive an equal quantity of the electric fluid, and the
equilibrium is restored between them. Beccaria sup-
poses, that this effect is produced by the electric matter
making a vacuum in its passage, and the contiguous air
afterwards collapsing, and so pushing the bodies toge-
ther.

The influence of points, in drawing or throwing off the
electric fluid, depends on the less resistance it finds to en-
ter or pass off through fewer particles than through a
greater number, whose resistance is united in fiat or round
surfaces. The electric light is supposed to be part of the
electric fluid, which appears when it is properly agitated ;
and the sound of an explosion is produced by vibrations,
occasioned by the air's being displaced by the electric
fluid, and again suddenly collapsing.

As to the nature of the electric fluid, philosophers have
entertained very different sentiments: some, and among
them Mr. Wihou, have supposed that it is the same with
the ether of Sir Isaac Newton, to which the phenomena
of attraction and repulsion are ascribed; while the light,
smell, and other sensible qualities of it, are referred to the
grosser particles of bodies, driven from them by the forci-
ble action of this ether; and other appearances are ex-

V'OL. I.

plained by means of a subtile medium diffused over the
surfaces of all bodies, and resisting the entrance, and exit
of the ether; which medium, it is supposed, is the same
with the electric fluid, and is more rare on the surfaces of
conductors, and more dense and resisting on those of elec-
trics : but Dr. Priestley remarks that, though they may
possess some common properties, they have others essen-
tially distinct; the ether is repelled by all other matter,
whereas the electric fluid is strongly attracted by it.
•Others have had recourse to the element of fire ; and
from the supposed identity of fire and the electric fluid,
as well as from the similarity of some of their effects, the.
latter has been usually called the electric fire: but most
electricians have supposed that it is a fluid sui generis.
Mr. Cavendish has published an attempt to deduce and
explain some of the principal phenomena of electricity in
a mathematical and systematic manner, from the nature
of this fluid, considered as composed of particles that re-
pel each other, and attract the particles of all other mat-
ter, with a force inversely as some less power of the di-
stance than the cube, while the particles of all other mat-
ter repel each other, and attract those of the electric fluid,
according to the same law. Philos. Trans, vol. 61, pa. 5S4
— 667- And a similar hypothesis and method of reason-
ing was also proposed by M. iEpinus, in his Tentamen
Theoria: Electricitatis et Magnetism!.

Dr. Priestley concludes, from experiments which he
made, that the electric matter either is phlogiston, or con-
tains it, since he found that both produced similar effects.
And Mr. Henley also apprehends, that the electric fluid
is a modification of that element, which, in its quiescent
state, is called phlogiston; in its first active state, electri-
city; and when violently agitated, fire. Perhaps we may
be allowed to enlarge our views, and consider the sun as
the fountain of the electric fluid, and the zodiacal light,
the tails of comets, the aurora borealis, lightning, and ar-
tificial electricity," as its various and not very dissimilar
modifications. On this subject, see Priestley's Hist, of
Electr. vol. 2, part 3, § 1, 2, 3 ; Wilson's Essay towards
an Explication of the Phenomena of Electricity, &c ; Wil-
son and Hoadley's Obs. ccc, pa, 55, 1753 ; Freke's Essay
on the Cause of Electricity, 17-16' ; Priestley on Air, vol. I,
pa. 1S6\274, &p; Philos. Trans, vol. 67, pa. 129; Eales's
Letters, on the same subject; and Cavallo's Electricity.

EEECTllOMETER, is an instrument that measures the
quantity, and determines the quality of electricity, in any
electrified body.. Previous to the invention of instruments
of this kind, Mr. Canton estimated the quantity of elec-
tricity in a charged phial, by presenting the phial with
one band to an insulated conductor, and giving it a spark,
which he took off with the other ; proceeding in U.is man-
ner till the phial was discharged, when he determined tj.ie
height of tlie charge by the number ol spaiks. Electro-
meters are of various kinds: as 1, the single thread ; 2,
the cork or pith balls; .;, the quadrant ; and 4, the dis-
charging electrometer, <\c

The 1st, or most simple electrometer, is a linen thread,
called by Dr. Desaguliers, the thread of trial; which, be-
ing brought near an electrified body, is attracted by it :
but this does little more than determine whether the body
is in any degree electrified or not; without determining
with aii)' precision its quantity, much less the quality of
it. The Abbe Nollct used two threads, showing tin de-
gree of electricity by the angle of their di.vergencj exhi-
bited in their shadow on a board placed behind them.
3 O

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Mr. Canton's electrometer consisted of two balls of
cork, or pith of elder, about the size of a small pea, sus-
pended by fine linen threads, about 6 inches long, which
may be wetted in a weak solution of salt. (See fig. 7.) If
the box containing these balls be insulated, by placing it
on a drinking- glass, &c, and an excited smooth glass tube
be brought near them, they will first be attracted by it,
and then be both repelled, from the glass, and from each
other; but if excited wax be applied to them, they will
gradually approach and come together; and vice versa.
This apparatus will also serve to determine the electricity
of the clouds and air, by holding them at a sufficient di-
stance from buildings, trees, &c ; for if the electricity of
the clouds or air be positive, their mutual repulsion will
increase by the approach of excited glass, or decrease by
the approach of amber or sealing-wax ; and on the con-
trary, if it be negative, their repulsion will be diminished
by the former, and increased by the latter. See Philos.
Transact, vol. 48, part 1 and 2, for an account of Mr.
Canton's curious experiments with this apparatus. It
two balls of this kind be annexed to a prime conductor,
they will serve to determine both the degree and quality
of its electrification, by their mutual repulsion and diver-
gency.

The Discharging Electrometer (fig. 3, plate xi.) was in-
vented by Mr. Lane. It consists of brass work g, the
lower part of which is enclosed in the pillar f, made of
baked wood, and boiled in linseed oil, and bored cylindri-
cally about two-thirds of its length; the brass work is
fixed to the pillar by the screw h, moveable in the groove
i ; and through the same is made to pass a steel screw
1, to the end of which, and opposite to k, which is a po-
lished hemispherical piece of brass, attached to the prime
conductor, is fixed a ball of brass m, well polished ; and to this
screw is annexed,a circular plate o, divided into 12 equal
parts. The use of this electrometer is to discharge a jar
d, or any battery connected with the conductor, without
a discharging rod, and to give shocks successively of the
same degree of strength ; on which account it is very fit
for medical purposes. Now, if a person holds a wire fast-
ened to the screw h in one hand, and another wire fixed
to e, which is a loop of brass wire passing from the frame
of the machine to a tin-plate, on which the phial d stands,
he will perceive no shock, while k and m are in contact;
and the degree of the explosion, as well as the quantity of
electricity accumulated in the phial, will be regulated by
the distance between K and M. Philos. Trans, vol. 57,
pa. 451. — Mr. Henley much improved Mr. Lane's electro-
meter, by taking away the screw, the double-milled nut,
and the sharp-edged graduated plate, and adding other
things in their stead. Mr. Henley's discharger of this kind
has two tubes, one sliding within the other, to lengthen
and accommodate it to larger apparatus.

The Quadrant Electrometer of Mr. Henley, consists of
a stem, terminating at its lower end with a brass ferrule
and screw, for fastening it on any occasion; and its up-
ber part ends in a ball. Near the top is fixed a graduated
semicircle of ivory, on the centre of which the index, be-
ing a very light rod with a cork ball at its extremity,
reaching to the brass ferrule of the stem, is made to turn
on a pin in the brass piece, so as to keep near the gra-
duated limb of the semicircle. When the electrometer is
not electrified, the, index hangs parallel to the stem ; but
as soon as it begins to be electrified, the index, repelled
by the stem, begins to move along the graduated edge of

the semicircle, and so mark the degree to which the con-
ductor is electrified, or the height to which the charge of
any jar or battery is advanced.1 (See fig. S.)

M. Cavallo also contrived several ingenious electrome-
ters, for different uses ; as may be seen in his Treatise on
Electricity, pa. 370, &c, and in the Philos, Trans, vol. 67,
pa.^48 and 399.

ELECTROPHOR, or Electropmorus, an instru-
ment for showing perpetual electricity ; which was in-
vented by M. Volta, of Como, near Milan, in Italy. The
machine consists of two plates, fig. 9, one of which r is a
circular plate of glass, covered on either side with some
resinous electric, and the other a is a plate of brass, or a
circular board, coated with tin-foil, and furnished with a
glass handle I, which may be screwed into its centre by
means of a socket. Now, if the plate r be excited by rub-
bing it with new white flannel, and the plate a be applied
to its coated side, a finger, or any other conductor, will
receive a spark on touching this plate; and if the plate a
be then separated, by means of the handle I, it will be
found strongly electrified, with an electricity contrary to
that of the plate r. Then replacing the plate a, touching
it with the finger, and separating it again, it will be found
electrified as before, and give a spark to any conductor,
attended with a snapping noise; and by this means a
coated phial may be charged. The same phenomena may
repeatedly be exhibited, without any renewed excitation
of the electric plate r ; the electric power of r having
continued for several days, and even weeks, after excita-
tion; though there is no reason to imagine that it is per-
petual.

M. Cavallo prepares this machine by coating the glass
plate with sealing-wax ; and the late Mr. Adams, philoso-
phical-instrument-maker, prepared them with plates form-
ed from a composition of two parts of shell-lac, and one
of Venice turpentine, without any glass. The action of
this plate depends on a principle discovered and illustrated
by the experiments of Franklin, Canton, Wilcke, and iEpi-
nus, viz, that an excited electric repels the electricity of
another body, brought within its sphere of action, and
gives it a contrary electricity. Thus the plate a, touched
by a conductor, while in contact with the plate r, which
is negatively electrified, it will acquire an additional quan-
tity of the electric fluid from the conductor; but if.it
were in contact with a plate electrified positively, it would
part with its electricity to the conductor connected with
it. See an account of several curious experiments with
this machine, by Mr. Henley, M. Cavallo, and Dr. In-
genhousz, in the Philos. Trans, vol. 66", pa. 513 ; vol. 6*7,
pa. 116 and 389; and vol. 6'S, pa. 1027 and 1049.

ELEMENTARY, something that relates to the princi-
ples or elements of bodies, or sciences ; as Elementary Air,
Fire, Geometry, Music, &c.

ELEMENTS, the first principles of which all bodies
and things are composed. These are supposed few in
number, unchangeable, and by their combinations produ-
cing that extensive variety of objects to be met with in the
works of nature. Democritus stands at the head of the
elementary philosophers, in which he is followed by Epi-
curus, and many others after them, of the Epicurean and
corpuscular philosophers.

Among those who consider matter, or the elements, as
corruptible, some assert that there is only one element in
nature, while others maintain that there are several. Of
the former, the principalarcHeraclilus, who considers fire

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as this element; Anaximenes asserts that it is air; Thales
Milesius says water, and Hesiod earth. Hesiod is fol-
lowed by Bernardin, Telesius; and Thales by many of
the chemists.

Among those who admit several corruptible elements,
the principal are the Peripatetics; who, after their leader
Aristotle, contend for four elements, viz, lire, air, water,
and earth. Aristotle took the notion from Hippocrates;
Hippocrates- from Pythagoras ; and Pythagoras from
Ocellus Lucanus, who- it seems was the first author of it,
or from the orientals.

The modern chemists, however, do not acknowledge
any particular number of elements, but freely admit into
this class all substances that have not been decomposed.
An element is therefore merely one of the last results of
chemical analysis ; and in denominating substances ele-
mentary or simple, we consider them as such only with
regard to the present state of chemical science. It is
not probableany one of the simplest bodies with which we
are acquainted, is really and essentially elementary, and
it is at best needless to enter into an enquiry in which there
are no facts to rest on for support.

Elements, a term also used for the first grounds and
principles of arts and sciences ; as the Elements of geome-
try, Elements of mathematics, &c. So Euclid's Elements,
or simply the Elements, as they were anciently and pecu-
liarly named, denotes the treatise on the chief properties
of geometrical figures by that author.

The Elements of mathematics have been delivered
by several authors in their courses, systems, &c. The
first work of this kind is that of Herigon, in Latin and
French, and published in 1634, in 5 tomes; which con-
tains Euclid's Elements and Data, Apollonius, Theodo-
sius, &c ; with the modern elements of arithmetic, alge-
bra, trigonometry, architecture, geography, navigation,
optics, spherics, astronomy, music, perspective, &c.
The work is remarkable for this, that a kind of real and
universal characters are used throughout; so that the
demonstrations may be understood by such as only re-
member the characters, without any dependence on lan-
guage or words at all.

Since Herigon, the elements of the several parts of
mathematics have been also delivered by many others ;
particularly the Jesuit Schottus, in his Cursus Mathema-
ticus, in l6"74; De Chales, in his Cursus, l6"74; Sir Jonas
Moore, in his New System of Mathematics, in lCSl;
Ozanam, in his Cours de Mathematique, in 1 6*99; Jones,
in his Synopsis Palmariorum Matheseos, in 170(5 ; and
many others, but above all, Christ. Wolfius, or Wolf,
in his Elementa Matheseos Universal, in 2 vols -ilo, the
1st published in 1713, and the 2d in 1713 ; a very excel-
lent work of the kind. Another edition of the work was
published at Geneva, in 5 vols 4to, of the several dates
1732, 1733, 1735, 173-8, and 1741-

The Elements of Euclid, as they were the first, so
they still continue to be the best system of geometry yet
extant. They are contained in 15 books, of which nume-
rous editions and commentaries have been published.
Proclus wrote a commentary on this work, and Orontius
Fineus gave a printed edition of the first 6 books, in 1 530,
with notes, to explain Fuclid's sense. Peletarius did the
same in 1557- Nic.Tartaglia also, about the same time,
made a comment on all the 15 books, with the addition
of many things of his own. And the same was also done
by Billingsley in 1570; and by Flussates Caudalla, a

French nobleman, in the year 157S, with -considerable
additions as to the comparison and inscriptions of solid
bodies; which work was afterwards republished with a
prolix commentary, by Clavius. But the first who trans-
lated them into Latin, while they were yet unknown to the
European nations, was Adhelard, or Adelard, a learned
Englishman, who performed this task in the 12th century.
Commandine also gave a good edition of this work, and
in 1703 Dr. Gregory published an edition of the whole
works of Euclid, in Greek and Latin. But it would be
endless to notice all the editions of this celebrated per-
formance that have been given ; some of the best of these
however are those of Tacquet, Ozanam, Whiston, Stone,
Playfair, Ingram, and more particularly that of Dr. Rob.
Simson, of Glasgow.

Beside these there are several other writers on geome-
try, who have not followed Euclid strictly either in form
or arrangement ; as it is thought by some, that in the
present state of mathematical knowledge, a system of
geometry less tedious and prolix than Euclid's would be
a considerable acquisition.

Elements, in the Higher or Sublime Geometry, are
the infinitely small parts, or differentials, of a right line,
curve, surface, or solid.

Elements, in Astronomy, are those principles de-
duced from astronomical observations and calculations,
and such fundamental numbers, as are employed in the
construction of tables exhibiting the planetary motions.
Thus, the elements of the theory of the sun, or rather of
the earth, show its mean motion and excentricity, with
the motion- of the aphelia. And the elements of the
theory of the moon, relate to her mean motion, that of
the node and apogee, the excentricity, the inclination of
her orbit to the plane of the ecliptic; &c. For the par-
ticulars of which see the respective names above men-
tioned.

ELEVATION, the height or altitude of any object.

Elevation, in Architecture, denotes' a draught or
description of the principal front or side of a building;
called also its upright or orthography.

Elevation, in Astronomy and Geography, is various ;
as elevation of the equator, of the pole, of a star, &c.

Elevation of the Equator, is the height of the equator
above the horizon ; or an arc of the meridian intercepted
between the equator and the horizon of the place. — The
elevation of the equator and of the pole together always
make up a quadrant ; the one being the complement of
the other. Therefore, the elevation of the pole being
found, and subtracted from 90°, leaves the elevation of
the equator.

Elevation of the Pole, is its height above the ho-
rizon ; or that arc of the meridian which is comprehended
between the equator and the horizon of the place. The
elevation of the pole is always equal to the latitude of the
place; that is, the arc of the meridian intercepted be-
tween the pole and the horizon, is every where equal to
the arc of the same meridian intercepted between the
equator and the zenith. Thus the north pole is elevated
51° 3l' above the horizon of London ; and the distance,
or number of degrees, is the same between London and
the equator; so that London is also in 51°3l' of north
latitude.

Elevation of a Star, or of any other point in the
sphere, is the angular height above the horizon ; or an
arc of the vertical circle intercepted between the star and
3 0 2

ELL

C 4fiS ]

ELL

the horizon. The meridian altitude of any such point, or its
altitude when in the meridian, is the greatest possible.

Elevation of a Cannon, or Mortar, is the angle which
the bore or the axis of the piece makes with the hori-
zontal plane.

Angle o/Eleva.tiqn, is the angle which any line of
direction makes above a horizontal line.

Elevation is also used by some writers on Perspec-
tive, for the scenography, or perspective representation
iii the whole body or building.

ELIMINATION, in Algebra, is that operation by
which any number >i of equations, containing n unknown
quantities, being given, we find one equation which in-
volves only one unknown quantity ; so that by resolving
this equation, we obtain the value of that unknown
quantity, and retracing our steps back again, we obtain
lJso the values of all the other quantities, which, in the
course of the operation, are found to have certain rela-
tions with, or dependence on each other. See Exter-
mination.

ELLIPSE, or Ellipsis, is one of the conic sections,
popularly called an oval : it was first called an Ellipse or
Ellipsis by Apollonius, the first and principal author on
the conic sections, because in this figure the squares of
the ordinates are less than, or defective of, the rectangles
under the parameters and abscisses.

This figure is differently defined by different authors;
cither from some of its properties, or from mechanical
construction, or from the section of a cone, which is the
best and most natural way. Thus;

Fig. 1. Fig. 2.

And, secondly, that the rectangle AG x gb (fig. 3) of the
abscisses, tending contrary wavs, is to gii° the square of
the ordinate, as AB^to ik°, the square of the transverse-
axis to the square of the conjugate, or, which is the same
thing, as the transverse axis is to the parameter; beside
various other properties.

3. The ellipse is also variously described from mecha-
nical constructions, which also depend on some of its
peculiar properties. 1st, If in the axis ae, there be taken
any point I (fig. 4) ; and with the radii ai, bi, and centres
f and f, the two foci, arcs be described, these arcs will
intersect in certain points e, e, e, e, which will be in the
curve or circumference of the figure : and thus several
points i being taken in the axis ab, as many more points,
e, e, &c, will be found ; then the curve line drawn
through all these points e,' e, will be an ellipse. And if
there be taken a thread of the exact length of the trans-
verse axis ab, and the ends of the thread be fixed by
pins in the two foci f and / (fig. 5) ; then moving a pen
or pencil within the thread, so as to keep it always
stretched, it will describe an ellipse. Or it may be de-
scribed from the property of the directrix.

Fig. 5. , Fig. 6.

1. An ellipse is a plane figure made by cutting a cone
by a plane obliquely through the opposite sides of it; or
so as that the plane makes a less angle with the base than
the side of the cone makes with it; as abd fig. 1. The
line ab connecting the upper and lower points of the
section, is the transverse axis ; the middle of it c, is the
centre; and the perpendicular to it dce, through the
centre, is the conjugate axis. The parameter, or latus
rectum, is a 3d proportional to the transverse and con-
jugate axes; and the foci are two points in the transverse
axis, at such equal distances from the centre, that the
double ordinates passing through those points, and per-
pendicular to the transverse, are equal to the parameter.

2. The ellipse is also variously described from some of
its properties ; the principal of which are as follow: That
if two lines be drawn from two certain points c and d in
the axis, fig. 2, to any point e in the circumference, the
sum of those two lines ce and de will be every where
equal to the same constant quantity, viz, to the axis ab.

Fig. 3.

Fig. 4.

To construct an Ellipse. There are many other
ways of describing or constructing an ellipse, besides those
already mentioned.

1st. If upon the given transverse axis there be described
a circle agb (fig.6), to which there be drawn any ordi-
nate dg, and de be taken a 4th proportional to the trans-
verse, the conjugate, and the ordinate dg ; then e is a
point in the curve. Or if the circle agb be described on
the conjugate axis ab, to which any ordinate <%is drawn,
in which taking du in like manner a 4th proportional to
the conjugate, the transverse, and ordinate dg, then shall
e be a point in the curve. Or, having described the two
circles, and drawn the common radius cgG cutting them
in g and g ; and then dgE being drawn parallel to the
transverse, and dge parallel to the conjugate, the inter-
section e of these two lines will be in the curve of the
ellipse. And thus several points e being found, the cm ve
may be drawn through them all with a steady hand.
Fig. 7- Fig. 8.

2. Or take three rulers, of which the two Gil and fi
(fia. 7) arc of the length of the transverse axis lk, and
the third FG equal to in the distance between the foci;
then connecting these, rulers so as to be moveable about
the foci h and I, and about the points f and g, their in-

ELL

[

tersection e will always be in the curve of the ellipse; so
that by moving the rulers about the joints, with a pencil
passed through the slits made in them, it will trace out
the ellipse lk

Fig. 9-

Fig. 10.

3. If one end a of any two equal rulers ab, db, (fig. 9
and 10), which are moveable about the point b, like a
carpenter's joint-rule, be fastened to the ruler lk, so as
to be moveable about the point a ; and if the end d of
the ruler db be drawn along the side of the ruler lk, then
any point e, taken in the side of the ruler db, will describe
an ellipse, whose centre is a, conjugate axis 2de, and
transverse axis 2ab -+- 2be.

Another method of description is by the elliptical com-
pass. See that article, below.

The most Remarkable Properties of the Ellipse, are as
follow: — 1. The rectangles under the abscisses are pro-
portional to the squares of their ordinates ; or as the
square of any axis, or any diameter, is to the square of
its conjugate, i0 is the rectangle under two abscisses of
the former, to the square of their ordinate parallel to the
latter ; or again, as any diameter is to iis parameter, so is
the said rectangle under two abscisses of that diameter, to
the square of their ordinate. So that if d be any diame-
ter, c its conjugate, p its parameter, so that p — ca-f- d, x
the one absciss, d — x the other, and y the ordinate; then,
as d° : cz : : x(d — x) : \f~, or d*yl = c*x(d — x) ;
or as d : p : : x(d — .r) : yl, or dyl = px[d — x) .
From either of which equations, called the equation of
the curve, any one of the quantities may be found, when
the other three are given.

2. The sum of two lines drawn from the foci to meet
in any point of the curve, is always equal to the trans-
verse axis; that is, ce ■+- de = ab, in the 2d fig. Con-
sequently the line cg drawn from the focus to the end of
the conjugate axis, is equal to ai the semi-transverse.

Fig. 11.

3. If from any
point of the curve,
there be drawn an
ordinate to either
axis, and also a tan-
gent meeting the axis
produced; then half
that axis will be a
mean proportional
between the distances from the centre to the two points of
intersection; viz, ca a mean proportional between CD
and ct ; and consequently all the tangents te, te, meet
in the same point of the axis produced, which are drawn
from the extremities E, e, of the common ordinates de,
d-e, of all ellipses described on the same axis ab.

4. Two lines drawn from the foci to any point of the
curve, make equal angles with the tangent at that point :
that is, the ^.fet = £-fv,t. And hence a line drawn
to bisect the angle vvf, will be perpendicular to the tan-
gent and curve at the point e.

5. All parallelograms that are formed by tangents drawn

4Co ] ELL

through the extremities of any diameter and its conjugate,
and thus circumscribed about an ellipse, are equal to
each other; and every such parallelogram is equal to the
rectangle of the two axes.

6'. The sum of the squares of every pair of conjugate
diameters, is equal to the same constant quantity, viz, the
sum of the squares of the two axes.

7. If a circle be described upon either axis, and from
any point in that axis an ordinate be drawn both to the
circle and to the ellipsis; then will the ordinate of the
circle be to the ordinate, of the ellipse, as that axis is to
the other axis : viz,

ab : ab : : dg : de, (in the 6th fig.)
and ab : ab : : dg : dn.

And in the same proportion is the area of the circle to the
area of the ellipse, or any corresponding segments adg,
ade. Also the area of the ellipse is a mean proportional
between the areas of the inscribed and circumscribed cir-
cles. Hence therefore,

S. To find the area of an Ellipse. Multiply the two
axes together, and that product by 78 54, for the area. Or,

9. To find the area of any segment ade. Find the area
of the corresponding segment adg of a circle on the same
diameter ab ; then say, as the axis ab : its conj. ab : : circ.
seg. adg : elliptic seg. ade.

10. To find the length of the whole circumference of the'
Ellipse. Multiply the circumference of the circumscribing
circle by the sum of the series

3(P

3*.StP

1 — . —3 , ,. „ , &c, lor the circum-

23 2"'.42 2".4_.6 2_.4".6".8

ference; where rfis the difference between an unit and the
square of the less axis divided by the square of the greater.

See another series, p. 178, vol. 4, Edinb. Philos. Trans.

Or, for a near approximation, take the circumference
of the circle whose diameter is an arithmetical mean be-
tween the two axes, or half their sum : that is,
i(t + c~) x 3' 141 6 = the perimeter nearly; being about
the 200th part too little ; where t denotes the transverse,
and c the conjugate axis.

Or, again, take the circumference of the- circle, the
square of whose diameter is half the sum of, or an
arithmetical mean between, the squares of the two axes:
that is, \/\{t- ■+-' c-) x 3-1416 = the perimeter nearly;
being about the 200th part too great.

Hence combining these two approximating rules to-
gether, the periphery of the ellipse will be very nearly
equal to half their sum, or equal to
.*[£.(/■+ c) ■+- t/U1' ■+■ c*)l x 3-1416, within about the
30000th part of the truth.

For the length of any particular arc, and many other
parts about the ellipse, sec my Mensuration, pa. 212, 6cc,
4th edit. See also my Conic Sections, for many other
properties of the ellipse, especially such as are common
to the hyperbola also, or to the conic sections in'general.

Infinite Ellipses. See Elliptoid.

ELLIPSOID, is an elliptical spheroid, being the solid
generated by the revolution of an ellipse about either axis.
See Spheroid.

ELLIPTIC or Elliftical, something relating to an
ellipse.

Elliptic Compasses, or Elliptical Compass, is an
instrument for describing ellipses at one revolution of the
index. It consists of a cross abgh (fig. 8.) with grooves
in it, and an index ce, sliding in dovetail grooves; by
which motion the end E describes the curve of an ellipse.

E M B

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M E

Elliptical Conoid, is sometimes used for the spheroid,
or ellipsoid.

Elliptical Dial, an instrument usually made of brass,
with a joint to fold together, and the gnomon to fall flat,
for the convenience of the pocket. By this instrument
are found the meridian, the hour of the day, the rising
and setting of the sun, &c.

ELLIPTICITY, of the terrestrial spheroid, is the ratio,
or the difference, between the two semiaxes. It is usually
expressed in terms of the greater, or radius of the equator,
that is, by what fractional part this exceeds the axis or
polar diameter. The quantity of this ellipticity has been
variously assigned by different mathematicians. Sir Isaac
Newton, supposing the earth of uniform density, gave T| ■&
for the ellipticity ; Boscovich, from a mean of several ad-
measurements, stated it at ^i¥ ; Lalande at TfT ; Laplace,
tkr'> Sajour, TJT; Carouge, j^; Kraft, T|,-; and Play-
fair, from a theorem of Clairaut, applied to the hetero-
geneous spheroid, states it at T§^. Setting aside those
which are deduced from the hypothesis of uniform density,
-j-lo may be admitted as the most probable value of the
ellipticity. Under the article Earth, see other propor-
tions, and the manner of deducing them.

Elliptoid, arj infinite or indefinite ellipse, defined by
the indefinite equation aym + n = bxm(a — x)n, where m
or n are greater than I : for when they are each I, it de-
notes the common ellipse. There are several kinds or
degrees of elliptoids, which receive their denominations
from the exponent m -+- n of the ordinate y. As the cu-
bical elliptoid, expressed by «y3 = bx1 {a — x) ; the bi-
quadratic, or sursolid ay* = bx-(q, — x)%; &c.

ELONGATION, in Astronomy, the distance of a
planet from the sun, with respect to the earth ; or the
angle formed by two lines drawn from the earth, the one
to the sun, and the other to the planet ; or the arc mea-
suring that angle: or it is the difference between the sun's
place and the geocentric place of the planet.

The greatest elongation, is the greatest distance to
which the planets recede from the sun, on either side.
This is chiefly considered in the inferior planets, Venus
and Mercury; the greatest elongation of Venus being
about 47 -J degrees, and of Mercury only about 27-j de-
grees ; which is the reason that this planet is so rarely
seen, being usually lost in the sun's beams.

EMBER-Days, are certain days observed by the church
at four different seasons of the year; viz, the Wednesday,
friday, and Saturday next after Quadragesima Sunday, or
the 1st Sunday in Lent; after Whit-sunday ; after Holy-
rood, or Holycross, the 14th day of September; and after
St. Lucy, the 13th day of December. The name, it
seems, is derived from embers, or ashes, which it is sup-
posed the primitive Christians strewed on their heads at
these solemn fasts.

Ember- Weeks, are those weeks in which the ember-
days fall. These ember-weeks are now chiefly noticed on
account of the ordination of priests and deacons ; because
the canon appoints the Sundays next alter the ember-weeks
for the solemn times of ordination ; though the bishops, if
they please, may ordain on any Sunday or holiday.

EMBOL1M/EAN, and Embolismic, Intercalary, is
chiefly used in speaking of the additional months inserted
by chronologists to form the lunar cycle of 1 9 years. The
19 solar years, consist of 6939 days and 18 hours, and
the 19 lunar years of only 0726' days, so that it was found
necessary to intercalate or insert 7 lunar months, contain-

ing 209 days; which, with the 4 bissextile days happen-
ing in the lunar cycle, make 213 days, and the whole
6939 days, the same as the 19 solar years, which make
the lunar cycle.

In the course of 19 years there are 22S common moons,
and 7 embolismic moons, which are distributed in this
manner, viz, the 3d, 6'th, 9th, 11th, 14th, 17th, and 19th
years, are embolismic, and so contain 384 days each.
And this was the method of computing time among the
Greeks; though they did not strictly observe it, as it
seems the Jews did. And the method of the Greeks was
followed by the Romans till the time of Julius Cassar.
The embolismic months, like other lunar months, some-
times contain 30 days, and sometimes only 29 days.

The Embolismic Epacts are those between 19 and 29;
which are so called, because, with the addition of the
epact 11, they exceed the number 30: or rather, because
the years which have these epacts, are embolismic ; having"
13 moons each, the 13th being the embolismic.

EMBOL1SMUS, in Chronology, signifies intercalation.
As the Greeks used the lunar year, which contains only
354 days; in order that they might bring it to the solar
year, of 365 days, they had an embolism every two or
three. years, when they added a 13th lunar month.

EMBRASURE, in Architecture, an enlargement of the
aperture or opening of a door, or window, within-side the
wall, sloping back inwards, to give the greater play for the
opening of the door, casement, &c, or to take in the more
light. V

Embrasures, in Fortification, are the apertures or
holes through which the cannon are pointed, whether in
casemates, batteries, or in the parapets of walls. In the
navy, these are called port-holes. The embrasures are
placed 12 or 15 feet apart from each other; being made
sloping or opening outwards, from 6 to 9 feet wide on the
outside of the wall, and from 2 to 3 within, to allow the
gun to traverse from side to side. Their base is about 2f
or 3 feet above the platform on the inside of the wall, but
sloping down outwards, so as to be only about 1 \ above
it on the outside; in order that the muzzle may be occa-
sionally depressed, when it is necessary to point the gun
in that direction. -

EMERGENT Year, in Chronology, is the epoch, from
which any people begin to compute their time or dates.
So, our emergent year is sometimes the year of the crea-
tion, but more usually the year of the birth of Christ.
The Jews used that of the Deluge, or the Exodus, &c.
The emergent year of the Greeks, was the beginning of the
Olympic games ; while that of the Romans was the date
of the Building of their City.

EMERSION, in Astronomy, is the re-appearance of
thesun, moon, or other planet, after having been eclipsed',
or hid by the interposition of theraoon, earth, or other body.
The immersions and emersions of Jupiter's first satellite,
are particularly useful for finding the longitudes of places;
the immersions being observed from Jupiter's conjunction
with thesun, till his opposition ; and the emersions fruni
the opposition till the conjunction. But within 15 days
of the conjunction, both before and after it, they cannot
be observed, the planet and his satellites being then lost
in the sun's light.

EMERSION is also used when B star, after being hid by

the sun, begins to re-appear, and emerges out of his rays.

Minutes or Scruples of Emeus ion, an ate of the moon's

orbit, which her centre passes over, from the tjinc she

EME

[ 471 ]

E N C

begins to emerge out of the earth's shadow, to the end of
the eclipse.

Emersion, in Physics, the rising of any solid above
the surface of a fluid that is specifically heavier than the
solid, into which it had been violently immerged,orpushed.
It is one of the known laws of hydrostatics, that a lighter
solid, being forced down into a heavier fluid, immediately
endeavours (p emerge; and that with a force equal to the
excess of the weight of a quantity of the fluid above that ot
an equal bulk of the solid. Thus, if the body be immerged
in a fluid of double its specific gravity, it will emerge
again, till half its bulk be above the surface of the fluid.

EMERSON (William), a lute eminent mathemati-
cian, was born in June 1701, at Hurworth, a village
about three miles south of Darlington, on the borders of
the county of Durham ; at least it is certain that he re-
sided here from his childhood. His father Dudley Emer-
son was a schoolmaster, and a tolerable proficient in the
mathematics; and without his books and instructions,
probably his son's genius, though eminently fitted for ma-
thematical studies, might never have been unfolded : but
besides his father's instructions, our author was assisted in
the learned languages by a young clergyman, then curate
of Hurworth, who was boarded at his father's house.
Being satisfied with a moderate competence left him by his
parents, he devoted himself to a studious retirement,
which he thus closely pursued, in the same place, through
the course of along life, being mostly very healthy, till to-
wards the latter part of his days, when he was much afflicted
withthestone. Towards the close, of the year 1781, being
sensible of his approaching dissolution, he disposed of the
whole of his mathematical library to a bookseller at York ;
and on May the 20th, 1782, his lingering and painful dis-
order put an end to his life at his native village, being near
8 1 years of age.

Mr. Emerson, in his person, was rather short, but strong
and well made, with an open countenance and ruddy com-
plexion. He was of a singular turn in his behaviour,
dress, and conversation; his manner and appearance being
rather plain and uncourtly: he had strong natural mental
parts, and could discourse sensibly on any subject, but
was always positive and impatient of contradiction. He
spent his whole life in close study, and writing books,
from the profits of which, he redeemed his little patrimony
from some original incumbrance. In his dress he was as
singular as in every thing else. He possessed commonly
but one suit of clothes at a time, and those very old in
their appearance. He seldom used a waistcoat; and his
coat he wore open before, except the lower button ; and
his shirt quite the reverse of one in common use, the hind
side turned foremost, to cover his breast, and buttoned
close at the collar behind. He also had a kind of rusty-
coloured wig, without a crooked hair in it, which pro-
bably had never been tortured with a comb from the time
of its being made ; and the hat lie wore had served him the
greater part of his life, of which he gradually lessened the
flaps, bit by bit, as it lost its elasticity, till little or no-
thing but the crown remained.

He sometimes walked up to London when he had any
book to be published, revising sheet by sheet himself: — for
trusting no eye but his own, was always a favourite maxim
with him. In mechanical subjects, he always tried the
propositions practically, making all the different parts
himself on a small scale ; so that his house was filled with
all kinds of mechanical instruments, together or disjointed.

He would frequently stand up to his middle in water while
fishing; a diversion he was very fond of. He used to
study incessantly for some time, and then for relaxation
take a ramble to some pot-alehouse, where he could getany
body to drink with and talk to. He was a married man,
but without children; and his wife used to spin on an old-
fashioned wheel, of his own making, a drawing of which
is given in his Mechanics.

Mr. Emerson, from his strong vigorous mind and close
application, had acquired a deep knowledge of all the
branches of mathematics and physics, on every part of
which he wrote good treatises, though in a rather anti-
quated style and manner. He was not remarkable, how-
ever, for genius or discoveries of his own, as his works
show hardly any traces of original invention. He was
well skilled in the science of music, the theory of sounds,
and the various scales both ancient and modern; but he
was a very poor performer, though he could make and re-
pair some instruments, and sometimes went. about the
country tuning harpsichords, as well as cleaning clocks.

The following is a .list of Mr. Emerson's works ; all of
them printed in Svo, excepting his Mechanics and his In-
crements in 4to, and his Navigation in 12mo. 1. The Doc-
trine of Fluxions. — 2. The Projection of the Sphere, or-
thographic, stereographic, and gnomonical. — 3. The Ele-
ments of Trigonometry. — 4. The Principles of Mechanics.
— 5. A Treatise of Navigation. — 6". A Treatise on Arith-
metic.— 7. A Treatise on Geometry. — 8 A Treatise of
Algebra, in 2 books. — 9. The Method of Increments. —
10. Arithmetic of Infinites, and the Conic Sections, with
other Curve Lines.— 11. Elements of Optics and Perspec-
tive.— 12. Astronomy. — 13. Mechanics, with Centripetal
and Centrifugal Forces. — 14. Mathematical Principles of
Geography, Navigation, and Dialling. — 15. Commentary
on the Principia, with the Defence of Newton. — 16. Mis-
cellanies.

Besides these, Mr. Emerson was a frequent mathema-
tical correspondent in the Ladies' Diary, the Palladium,
&c, in the fictitious names of Merones, Nichol Dixon,
and other fanciful appellations. In the same works, some-
times a riddle or a question in rhyme, showed that he was
not destitute of poetical talents.

EMINENT1AL £</;<«/«>«, a term used by some algebra-
ists, in the investigation of the areas of curvilineal figures, for
a kind of assumed equation that contains another equation
eminently, the latter being a particular case of the former.
Hayes's Flux. pa. 97-

ENCEINTE, a French term, in Fortification, signify-
ing the whole inclosure, circumference, or compass, of a
fortified place, whether built with stone or brick, or only
made of earth, and whether with or without bastions, &c.
ENCYCLOPAEDIA, the circle or chain of arts and
sciences ; sometimes also written Cyclopasdia.

Numerous are the encyclopaedias, or general dictionaries
that have been published ; but our business is only with
those which treat chiefly on the mathematical or philoso-
phical sciences. The first dictionary of this kind, it ap-
pears, was that of Dasypodius, printed in 1573, under
the title of Asfyxov, seu Dictionarium mathematicarum,
in quodefinitionesetdivisionescontinenturscientiarum ma-
thematicarum, arithmeticae,&c,M.CouradoDasypodio au-
thore; a very extraordinary work for the time of its pub-
lication, and of which a new edition was printed at Stras-
burg, in 1579. — I" l632 came out the work J. H. Alslcdii
Encyclopedia, in 2 vols, folio : an elaborate performance;

E N C

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E N C

which was followed in 16\57, by Erhardi Wegalii Idea En-
cyclopaedia Mathematico-philosophicae; a work not su-
perlatively interesting, even for that period. Of a similar
kind is the Lexicon Math. Astron. Geomet. of Vatalis,
printed at Paris in 16'6"S, 8vo ; as also the Dictionnaire
Mathematique of Ozanam, printed in 1 vol. 4to, 169I-

As to the works of Ger. J. Vossius (16'50), and of
Beughem (l6<)4), they are Jittle more than catalogues of
mathematical books.

The next important step was made by Dr. Harris, in
his Lexicon Technic'um, published, the 1st vol. in 170-4, the
2d in 1710.' Thisisthe earliest English work that assumes
to good purpose the systematic form of a general dic-
tionary, and attempts to allot to each article its compa-
rative portion in the scale of science. The author pos-
sessed very considerable general knowledge ; but his at-
tainments as a mathematician and philosopher were most
conspicuous. To the 5th edit, in 1736", a supplement was
added by the booksellers, but of minor importance. — Va-
rious dictionaries, comprising more or less of scientific ma-
terials, were published between the times of Harris's Lex-
icon and Chambers's Cyclopaedia ; such, for instance, as
the Great Dictionary of the French Academy, the Dic-
tionary of the Jesuits of Trevoux, the Chemical Dictionary
of Johnson, the Medical ones ofBlanchard and Castellas,
the Mathematical Dictionaries of Stone and Wolfius, the
Sea Dictionary of Mainwaring, the Dictionary of the Bible
by Calmet, the Lexicon Philosophicum of Chauvin, a work
of importance, especially in the mathematics and philo-
sophy of the ancients ; the Lexicon of J. Burkard Monkens,
in, 1715; Jablonski's Lexicon in 1-721; and Collier's great
Historical Dictionary, begun in 1694, and finished in 1727.

The first edition of Chambers's Cyclopaedia came out in
1727, 2 vols, folio ; of which the public demand occa-
sioned a 2d edit, in 1738, a 3d in 173y, a 4th in 1741,
and a 5th in 1746. This unprecedented success induced
the proprietors, at a great expense, to engage G. L. Scott
and Dr. Hill to prepare a Supplement to the 6th edition,
which was accordingly published in 2 additional volumes.
The7th edition, completed in 17S6 in'four thick folio vo-
lumes, removed the disadvantage of the double alphabets,
by incorporating them into one, by the labour and skill
of that able editor Dr. Ahr. Rees.

From this period, dictionaries devoted to arts and sciences,
as well as general dictionaries, increased very rapidly, both
in number and importance; of which a few of the chief
may be noticed : as, the .Medical Dictionaries of Motherby,
Quincy, James, Turton, and the Edinburgh Dictionary ;
the Chemical Dictionaries of Macquer, Nicholson, and
the Aikins ; the Dictionaries of Gusseme and Rasches, on
Numismatology ; those of Miller, Martyn, and Dickson,
on Gardening ; those of Burn, Cunningham, and Jacob,
011 Law : the Marine Dictionaries of Chapman and Fal-
coner; the Builder's Magazine, and Felihien's Dictionary
of Architecture ; Jom belt's Dictionnaire de l'Engcnieur et
de I'Arfllleur, and James's Military Dictionary; Pilking-
ton's Dictionary of Painters ; Mortimer's and Postle-
thwaite's Dictionariesof Trade a ml Commerce; Rousseau's,
1 lo\ Ic's, and Busby's Dictionaries ol Mumc ; our own Ma-
thematical and Philosophical Dictionary; and the Diction-
naire de Physique of M. Lihes. The diligence exerted by
the authors of these, ami a few other dictionaries appro-
priate 'I to separate branches of science, and by the editors
of encyclopaedias, has operated reciprocally to improve
both ; and hence it has happened, that many of such dic-

tionaries, during the last 50 years, have greatly contributed
to the improvement and diffusion of human knowledge.

The labours of the continental encyclopaedists, during
this period, are also of much importance. Among the
works of the Germans may be noticed, the Universal Lex-
icon of Ludwig, in 1732 — 1750, in 64 volumes ; Oekono-
mische Encyclopedia, by Kriinitz, in 1773; the Encyclo-
pedia der Historischen, Philosophiscen, und Mathematis-
chen Wissenchaften, by Biisch,in 1775and 1795; Kliigel's
Encyclopedic aller Mathematischen Wissenchaften ihre
Geschichte und Litteratur, by Rosenthal, in 170,0. To
these may be added, the Swedish Encyclopaedia, by Gior-
well, in 1785; and the Encyclopedia Italiana ovvero Bi-
bliotheca universale della umane cognizioni, at Naples in
17S8.

Our neighbours the French also have the Dictionnaire
universel de Mathematique et de Physique, by M. Save-
rien, in 1753; the Encyclopedic, ou Dictionnaire rai-
sonne des Sciences, des Arts, et des Metiers, by Diderot,
Dalembert, &c, in 1754 — 1757; the Dictionnaire Porta-
tif, in 1760; the Dictionnaire de Physique Portatif, in
176'3; the new edition of the Encyclopedic, by Diderot,
&c, in 3j) vols, in 1778, 1779; and the Ency clopedie Me-
thodique, by Dalembert, Bossut, Condorcet, Lalande, &c,
which commenced in 1785, and consists of separate al-
phabets, or dictionaries, for the several arts and sciences.

We must now glance rapidly at the labours of British
encyclopaedists since the time of Chambers ; passing over
the productions of Owen, Procter, Castieau, Hall, Howard,
and Kendal, with a mere notice of their names ; the only-
one of which that we do not feel desirous to forget is
Owen. We must however mention, with commendation,
the Dictionary of Arts and Sciences, published in 1766",
by Crowder, in 3 vols. 4to ; the editors of which were,
the Rev. J. Scott, Trinity-college, Cambridge; Mr. Chas.
Green, assistant at the royal observatory ; Mr. J. Meadcs;
and Mr. Falconer, the unfortunate author of the Ship-
wreck.

The work last mentioned, though respectable, is far in-
ferior to Chambers, who has however now had some for-
midable rivals in Britain; viz, the Encyclopaedia Britan-
nica, published in Edinburgh, first we believe in 176S, in
4 vols. 4to, and afterwards in 10 vols. This work seems
also to have been the first that attempted the innovation
of incorporating systems or treatises, along with the usual
articles in the alphabetical arrangement. The 3d edition
of this Encyclopaedia, superintended by Dr. Gleig, was
finished in 1800; the whole with a supplement, making
20 large volumes. It contains, besides the general mat-
ters treated in Chambers's work, the additional subjects of
biography, history, and geography. It is a work, in many
respects, of considerable excellence: it commonly ox-
plains the principles and practices, in the various arts and
sciences, with great correctness and perspicuity. Many
of the treatises it contains, were drawn up by some of the
most eminent Scotch professors, which do them great cre-
dit.— The English Encyclopaedia was completed in 10 vols.
4t0, in 1803. The names of its conductors are not men-
tioned; but they are commonly understood Id have been
Dr. Aikin and Mr. Houlston. Its general plan is much
like that of the Encyclopaedia Britannica, but much less
imbued with original masterly compositions.

Such is the general diffusion of knowledge in this coun-
try, and so prevalent is the desire to possess a library as
it were in one work, that besides two general dictionaries

E N T

[ 473 ]

EPA

just finished, no less than six others are now in course of order of a column which is over the capital, comprehend

publication, viz, The Encyclopaedia Londinensis; anew
edition of the Encyclopeedia Perthensis ; Brewster's Edin-
burgh Encyclopaedia ; the Pantologia, or New Cyclopae-
dia, by Mr. Good, Dr. O. Gregory, and Mr. Bosworth, in-
cluding all English words, besides all sciences and human

knowledge, to be comprised in 10 or 12 volumes; also throw the water off.

ing the architrave, frize, and cornice.

Entablature, or Entablement, is sometimes also
used for the last row of stones on the top of the wall of a
building, on which the timber and covering rest ; some-
times also called the drip, because it projects a little, to

a new edition of the Encyclopaedia Britannica, and Dr.
Rees's New Cyclopaedia, both to be extended to several
more volumes.

EN DECAGON, a plane geometrical figure of eleven
sides and angles, otherwise called undecagon. If each side
of this figure be 1, its area will be g-3656399 = V of the
tang, of 73^j. degrees, to the radius 1. See my Mensura-
tion, pa. S5,&c, 4th edit. See also Regular Figure.

ENFILADE, a French term in Fortification, applied to
those trenches, and other lines, that are ranged in a right
line, and so may be scoured or swept by the cannon length-
ways, or in the direction of the line.

To Enfilade, is to sweep lengthways by the firing of
cannon, &c.

ABatttry '^'Enfilade, is that where the cannon sweep
a right line.

ENVELOPE, in Fortification, is a mound of earth,
sometimes raised in the ditch of a place, and sometimes
beyond it, being either in form of a single parapet, or of a
small parapet bordered with a parapet. These envelopes
are made only to cover weak parts with single lines, with-
out advancing towards the field, which cannot be done
without works that require a great deal of room, such as
horn-works, half-moons, &c. Envelopes are sometimes
called Sillons, Contregards, Conserves, Lunettes, &c.

ENUMERATION, numbering or counting. Sir Isaac
Newton wrote a very ingenious treatise on the enumeration
of the lines of the 3d order.

EOLIP1LE. See jEolipile.

EPACT, in Chronology, the excess of the solar month
above the lunar synodical month; or of the solar year
above the lunar year of 12 synodical months; or of several

A Post, or Command ^'Enfilade, is a height from solarmonths above as manysynodical months; or of several

which a whole line may be swept at once.

ENGINE, in Mechanics, a compound machine, consist-
ing of several simple ones, as wheels, screws, levers, or the

like, combined together, in order to lift, cast, or sustain a

weight, or produce some other considerable effect, so as to

save either force or time. There are numberless kinds of
engines; of which some are for war, as the Balista, Cata-

pulta, Scorpion, Aries or Ram, &c; others for the arts of And the lunar month

peace, as mills, cranes, presses, clocks, watches, &c, &c. The menstrual epact is

ENGINEER is applied to a contriver or maker of any

kind of useful engines or machines; or who is particularly

skilled or employed in them. And he is denominated

either a civil or military engineer, according as the objects

of his profession respect civil or military purposes.

A militaiy engineer should be an expert mathematician

and draughtsman, and particularly versed in fortification

and gunnery, being the person officially employed to di-
rect the operations both for attacking and defending works.

When at a siege the engineers have narrowly surveyed the

place, they are to make their report to the general, or

commander, by acquainting him which part they judge

the weakest, and where approaches may be made with

most success. It is their business also to draw the lines

of circumvallation and contravallation ; also to mark out

the trenches, places of arms, batteries, and lodgments,

and in general to direct the workmen in all such opera-
tions.

ENHARMONIC, the last of the three kinds of music.
It abounds in dieses, or the least sensible divisions of a tone.
See Philos. Trans. No. 481; also Wallis's Appendix to
Ptolem.pa. l6"5, l66\

' ENNEADECATERIS, in Chronology, a cycle or pe-
riod of 19 solar years, being the same as the golden num-

'ber and lunar cycle, or cycle of the moon; which see; as
also Embolismic.

ENNEAGON, a plane geometrical figure of 9 sides and Again, as the new moons are the same, or fall on the

angles ; and is otherwise called a nonagon. If each side same day, every 19 years, so the difference between the so-
of this figure be 1, its area will be 6"-1818242 = -J of the lar and lunar years is the same every 19 years. And be-
fang. of 70 degrees, to the radius 1 . See my Mensuration, cause the said difference is' always to beadded to thelunar
ji;i. 8 ">, -1 1 li edit. See also the article Reg ula u Figure. year, to adjust or make it equal to the solar year ; there-
F.NTABLATURE, in Architecture, is that part of the fore the said difference respectively belonging to each year
Vol. I. • s 3 P "

solar years above as many periods each consisting of 12
synodical months.

The epacts therefore are either annual or menstrual.

Menstrual Epacts, are the excesses of the civil calen-
dar month above the lunar month. Suppose, for example,
it were new moon on the 1st day of January: then since
the month of January contains 31 days,

29ds 12" 44m 3';
1 11 15 57

Annual Epacts, arc the excesses of the solar year above
the lunar. Hence,

As the Julian solar year is - 365d" 61' 0m 0",
And the Julian lunar year - 354 8 48 3S,
The annual epact will be - 10 21 11 22,

that is, almost 11 days. Consequently the epact of 2
years, is 22 days; of 3 years, 33 days; or rather 3, since
30 days make an embolismic, or intercalary month ;
therefore, adding still 11, the epact of 4 years is 14 days;
and so of the rest as in the following table, where it may
be observed, that they do not become 30, or 0 again, till
the 19th year; so that at the 20th year the epact is 11
again ; and hence the cycle of epacts expires with the gol-
den number, or lunar cycle of 19 years, and begins with
the same again.

Table of Julian Epacts.

Golden
Numb.

Epacts.

Golden
Numb.

Epacts.

Golden
Numb.

Epacts.

I

11

VIII

2S

XV

15

II

00

IX

9

XVI

26"

III

3

X

20

XVII

8

IV

14

XI

1

XVIII

19

V

25

XII

12

XIX

30")

VI

6

XIII

23

or Oj

VII

17

XIV

4

E P H

[ 474 ]

E P I

of the moon's cycle, is called the epact of the said year,
that is, the number to be added to the said year, to make
it equal to the solar year ; and upon this mutual relation
between the cycle of the moon and the cycle of theepacts,
is founded this

Rule for finding the Julian Epact, belonging to any year
of the Moon's Cycle. — Multiply the golden number, or the
given year of the moon's cycle, by 1 1, and the product will
be the epact, if it be less than 30 ; but if it exceed 30, then
reject as many 30's as the product contains, and the re-
mainder will be the epact.

Rule to find the Gregorian Epact. — 1st, The difference
between the Julian and Gregorian years being equal to the
difference between the solar and lunar year, or 11 days,
therefore the Gregorian epact for any year is the same with
the Julian epact for the preceding year ; and hence the
Gregorian epact will be found, by subtracting .1 from the
golden number, multiplying the remainder by 11, and re-
jecting the 30's ; which rule will serve till the year 1900;
but after that year, the Gregorian epact will be found by
this rule: Divide the centuries of the given year by 4;
multiplythe remainder by 17; and to this product add 43
times the quotient, as also the number 86, and divide the
whole sum by 25, reserving the quotient: next multiply
the golden number by 11, and from the product subtract
the reserved quotient; so shall the remainder, after reject-
ing all the 30's contained in it, be the epact sought.

The following table contains the golden numbers, with
their corresponding epacts, till the year 1900.

Table of Gregorian Epacts.

Golden
Numb.

Epacls.

Golden
Numb.

Epacts.

Golden
Numb.

Epacts..

I

0

VIII

17

XV

4

II

11

IX

28

xvt

15

III

22

X

9

XVII

20

IV

3

XI

20

XVIII

7

V

14

XII

1

XIX

IS

VI

25

XIII

12

I

0

VII

6

XIV

23

On the subject of epacts, see Wolfius's Elementa Chro-
nologia? ; apud Opera, torn. 4, pa. 133; also Philos. Trans,
vol. 46, pa. 417, or No. 495, art. 5.

EPAULE, or Espaule, in Fortification, the shoulder
of the bastion, or the angle made by the face and flank,
otherwise called the angle of the epaule.

EPAULEMENT, in Fortification, a side-work hastily
raised, to cover the cannon or the men ; and is made either
of earth thrown up, or bags filled with earth or sand, or of
gabions, or fascines, &c; of which latter sort are commonly
constructed the epaulements of the places of arms for the
cavalry behind the trenches.

Epaulement, is also used for a demi-bastion, consist-
ing of a face and flank, placed at the point of a horn-
work or crown-work : and a little (lank added to the sides
of horn-works, to defend them when they are too long: also
for the redoubts made on a right line, to fortify it. And
lastly, for an orillon, or mass of earth almost square, faced
and lined with a wall, and designed to cover the cannon of
a casemate.

EP11EMERIS, EniEMEiiiDES, tables calculated by
astronomers, showing the present stale of the heavens for
(very day at noon; that is, the places in which all the
planets are found at that time ; differing but little from an

astronomical almanac. It is from such tables as these that
the eclipses, conjunctions, and aspects of the planets are
computed; as also horoscopes, or celestial schemes, con-
structed, &c.

The following list of authors of published epheraeride*
has been chiefly collected from Weidler's Hislory of As-
tronomy, in 4to. 74 1.

Kegiomontanus

,fiom\4T5 to

1506

Beaulieu

from

1703 to 1714

Angelas

—

1494, —

1500

Manfredi

—

1715 — 1725

Stoefler

—

149* —

1551

—

1725 — 1750

Scboner

—

1533

Gbisler

—

1721 — 1740

Gauricus

—

1534 —

1551

—

1739 — 1756

Rheticus

—

1550

Connoissance

des

Simi

—

1554 —

1568

Temps, by

Mizald

—

1555 —

1557

Picard

—

1679 — 1635

Pitatus

—

1552 —

1562

Lefebvre

—

16S6 — 170L

Stadias

—

1554 —

1576

Lieutaud

—

1701 — 1729

Corelli

—

155S —

1577

Godin

—

1729 — 1735

Leovitius

—

1556 —

1606

Maraldi

—

1/35 — 1760

Moletius

—

1564 —

1584

Lalande

—

1760 —. 1775

Maginus

—

1581 —

1630

Jeaurat

—

1775 — 178*

Scala

158S —

1600

Mechain

—

1788 — 1794

Everhart

1590 —

1615

Commission

—

1795

Origan
Kepler

—

1595 —
1617 —

1655
1620

Board of Lon-
gitude

}-

1796, &C, &C.

'■ —

—

1630 —

1636

Ephemerides

Argoli

1621 —

1640

des Mouve

Durret

—

1637 —

1649

mensCeles

—

1648 —

1700

tes, by

Montebrun

—

1641 —

1660

Desplaces

—

1715 — 1725

Wing

. —

1652 —

1672

—

1725 — 1735.

Malvasia

—

166* —

1667

—

1735 — 1744

Palati

—,

1664 —

1670

Lacaille

—

1745 — 1755

Hecker

—

1666 —

1680

—

1755 — 1765

Gadbury

—

1672 —

1701

—

1765 — 17*5

Mozzavachis

—

"1675 —

1684

Lalande

—

1775 — 1784

1702 —
16S1 —

• 1720
1692

1785 — 1792

Kircbius

Nautical Alma-

Lieutaud

—

1693 —

1729

nac, by

Junius

—

1701 —

1703

Maskelyne

—

1767 — 1814

Lahire

—

1701 —

1702

|

There are now published such ephemerides by the Aca-
demies of several other nations; but that which is in most
esteem for its accuracy and use in finding the longitude, is,
the Nautical Almanac, or Astronomical Ephemeris, pub-
lished in England by the Board of Longitude, under the
direction of the astronomer royal, which commenced with.
the year 1767, by Dr. Maskelyne : besides many other,
smaller annual ephemerides, by White, Partridge, Parker,
and many more.

EPICHARMUS, an ancient poet and philosopher,
born in Sicily, was a scholar of Pythagoras, and flourished
in the time of Hiero, in whose reign it is said he introduced
comedy at Syracuse. He also wrote treatises on philoso-
phy and medicine; but none of his works have been pre-
served. He died at 90 years of age, according to Laertius,
who has preserved four verses inscribed on his statue.

EPICURUS, a celebrated ancient philosopher, was
born at Gargettium in Attica, v in the 109th Olympiad, ofi
about 340 years before Christ. He settled at Athens in a
fine garden he had bought ; where he lived with liis friends
in much tranquillity, and educated a great number of dis-
ciples ; who lived all in common with their master. His
school was never divided, but his doctrine was followed as
an oracle; and the respect which his disciples paid to his
memory is admirable; his birth-day being kept even in
Pliny's time, and the month he was born in was ob-
served as a continued festival; his picture was also dis-
played in numberless places. He wrote a great many
books, and valued himself upon making no quotations. He
raised the atomical system to great reputation, though he

E P I

[ 475 1

E P 1

was not the inventor of it, but only made some change in
that of Democritus. As to his doctrine concerning the
supreme good or happiness, it was very liable to be misre-
presented, and some ill effects proceeded from thence,
which discredited his sect, though undeservedly. He was
charged with perverting the worship of the gods, and in-
citing men to debauchery. But he did not forget himself
on this occasion : he published his opinions to the whole
world; wrote some books of devotion; recommended the
veneration of the gods, sobriety, and chastity, living in an
exemplary manner, and conformably to the rules of philo-
sophical wisdom and frugality. Gassendus has given us
all that he could collect from the ancients concerning the
person and doctrine of this philosopher; by whom we are
informed that he died of a suppression of urine, at the age
of 72 years.

Epicurean Philosophy, the doctrine, or system of phi-
losophy, maintained by Epicurus and his followers; which
consisted of three parts; Canonical, physical, and ethical.
The first respected the canons or rules of judging; in which
soundness and simplicity of sense, assisted by some natural
reflections, chiefly formed his art. His search after truth
proceeded only by the senses ; to the evidence of which he
gave so great a certainty, that he considered them as an
infallible rule of truth, and termed them the first natural
light of mankind. In the 2d part of his philosophy he
laid down atoms, space, and gravity, as the first principles
of all things. He asserted the existence of a God, whom
he-accounted a blessed immortal being, but who did not
concern himself with human affairs.

As to his ethics, he made the supreme good of man to
consist in pleasure, and consequently supreme evil in pain.
Nature itself, says he, teaches us this truth ; and prompts
us from our birth to procure whatever gives us pleasure,
and to avoid that which gives us pain. To this end he
proposes a remedy against the sharpness of pain, which is
to divert the mind from it, by turning our whole attention
upon the p easures we have formerly enjoyed. He held
that the wise man must be happy, as long as he is wise:
pain, by not depriving him of his wisdom, cannot deprive
him of his happiness : from which it would seem that his
pleasure consisted rather in intellectual than in sensual
enjoyments: though this is a point strongly contested : pro-
bably in both.

EPICUREANS, the sect of philosophers holding or
following the principles and doctrine of Epicurus. As
the nature of the pleasure, in which the chief happiness
of man is supposed to be seated, is a great problem in the
morals of Epicurus, there hence arise two kinds of Epicu-
reans, the rigid and the remiss : the first were those who
understood Epicurus's notion of pleasure in the best sense,
and placed all their happiness in the pure pleasures of the
mind, arising from the practice of virtue: while the loose
•or remiss Epicureans, taking the words of that philoso-
pher in a gross sense, placed all their nappiness in bodily
pleasures or debauchery.

EPICYCLE, in the ancient Astronomy, a little circle
having its centre in the circumference of a greater one:
or a small orb or sphere, which being fixed in the de-
ferent of a planet, is carried along with it, and yet, by its
own peculiar motion, carries the planet fastened to it
round its proper centre. — It was by means of epicycles
that Ptolemy and his followers solved the various pheno-
mena of the planets, but more especially their stations and
retrogradations.

EPICYCLOID, is a curve generated by the revolution
of a point of the periphery of a circle, which rolls along
or upon the circumference of another circle, either on
the convex or concave side of it. When a circle rolls
along a straight line, a point in its circumference describes
the curve called a cycloid. But if, instead of the right
line, the circle roll along the circumference of another
circle, either equal to the former or not, then the curve
described by any point in its circumference is what is
called the epicycloid.

If the generating circle roll along the convexity of the
circumference, the curve is called an upper, or exterior
epicycloid ; but if along the concavity, it is called a
lower, or interior epicycloid. Also the circle that re-
volves is called the generant ; and the arc of that circle
along which it revolves, is called the base of the epicy-
cloid.

Fig. 1.

Fis-2.

Thus, in the above figure, geif is the exterior epicy-
cloid, and ge¥ the interior epicycloid, to the former of
which eb is the generant, and gbf the base, and to the
latter the circle fh is the generant, having the arc gf for
its base.

The honour of the invention of this curve is due to M.
Roemer, a celebrated Danish astronomer, who did not
treat of it as a purely geometrical speculation ; but as one
of great utility, being, as he thought, the form which
ought to be given to the teeth of wheels, in machinery,
in order to diminish the friction of them against each
other, and to render the action of the powers more equal.
This idea occurred to M. Roemer while he was in Paris
about the year l6"74, notwithstanding M. Lahire, i n
1694, published a work upon this subject, and seems to
lay claim to the indention; but he was justly confuted
by Leibnitz, who was in Paris during the visit of the
Danish mathematician Roemer, and knew both of the
invention and its application to mechanical purposes.
Sir Isaac Newton and John Bernoulli have each consi-
dered the nature and properties of this curve, which are
both numerous and interesting ; but our limits will only
admit the insertion of some of those which are more par- •
ticularly deserving of attention. It is a remarkable pro-
perty of circular epicycloids, that they are sometimes
geometrical, though the common cycloid, which is much
more simple in appearance, as a right is more simple than
a curve, is always mechanical, or transcendental. The
cases where the epicycloid is geometrical, are those in
which there is a numerical ratio between the circum-
ferences of the circle which forms the base, and the gene-
rant ; for where there is no rational ratio between them,
the curve becomes mechanical.

It is also a remarkable property of epicycloids, whether
they be geometrical or transcendental, that they are al-
ways rectifiable, at least in the case where the describing
point is taken in the circumference of the generating
3 P 2

E P I

[ 476 ]

E P O

circle. Thus, the curve of the epicycloid gef, is to 4
times the diameter of the generating circle be, as the sum
of the diameters of the two circles, is to that of the base.
But if the epicycloid is inferior, as Teg, then instead of the
sum of the diameters we must take their difference.

It may also be observed, as a curious circumstance,
that when the diameter of the generant is equal to half
that of the circle forming the base, as in fig. 2; then the
epicycloid degenerates into a right line, describing the
diameter of the latter circle. We may also further re-
mark, that when the describing point is taken either
within or without the generant, the length of the epicy-
cloid is equal to the circumference of an ellipse, which is
easily constructed. '"

As to the areas of epicycloids, they are determined
from the following proportion. As the radius of the
circle forming the base, is to 3 times that radius plus
twice that of the generant ; so is the circular segment bi, to
the epicycloidical space bvs ; or, so is the whole area of
the generant, to the entire area of the epicycloid fegb.

Dr. Halley has also given a general proposition for the
measuring of all cycloids and epicycloids : which is this,
the area of a cycloid, or epicycloid, either primary, or
contracted, or prolate, is to the area of the generating
circle; and also the areas of the parts generated in those
curves, to the areas of analogous segments of the circle;
as the sum of double the velocity of the centre and the
velocity of the circular motion, is to this velocity of the
circular motion. See the Demonstr. in the Philos. Trans.
No. 218, or my Abridg. vol.4, pa. 47.

Spherical Epicycloids are formed by a point of the
revolving circle, when its plane makes a constant angle
with the plane of the circle on which it revolves. Messrs.
Bernoulli, Maupertuis, Nicole, and Clairaut, have de-
monstrated several properties of these epicycloids, in Hist.
Acad. Sci. for 1732.

Epicycloids. Parabolic, Elliptic, &c.

If a parabola roll upon another equal to it; its focus
will describe a right line perpendicular to the axis of the
quiescent parabola : also the vertex of the rolling para-
bola will describe the cissoid of Diocles; and any other
point of it will describe some one of Newton's defective
hyperbolas, having a double point in the like point of the
quiescent parabola.

In like manner, if an ellipse revolve upon another el-
lipse, equal and similar to it, its focus will describe a'
circle, whose centre is in the other focus, and consequently
the radius is equal to the axis of the ellipsis; and any
other point in the plane of the ellipse will describe a line
of the 4th order.

The same may be said also of an hyperbola, revolving
upon another, equal and similar to it; for one of the foci
will describe a circle, having its centre in the other focus,
and the radius will be the principal axis of the hyperbola;

and any other point of the hyperbola will describe a line,
of the 4th order.

Concerning these lines, see Newton's Principia, lib. 1 ;
also Lahire's Memoires de Mathematique &c, where he
shows the nature of this line, and its use in mechanics ;
see also Maclaurin's Geometria Organica; the Philos.
Trans. No. 217, pa. 113 ; and No. 218, pa. 125.

EPIPHANY, a Christian festival, otherwise called the
Manifestation of Christ to the Gentiles, observed on the
6th of January, in honour of the appearance of Christ to
the three magi or wise men, who came to adore him and
bring him presents.

EPISTYLE, in the ancient Architecture, a term used
by the Greeks for what we call the architrave, viz, a massive
stone, or a piece of wood, laid immediately over the
capital of a column.

EPOCH, or Epochs, a term or fixed point of time,
whence the succeeding years are numbered or reckoned.
Different nations make use of different epochs. The
Christians chiefly use the epoch of the nativity or incarna-
tion of Jesus Christ ; the Mahometans, that of the He-
gira; the Jews, that of the Creation of the World, or that
of the Deluge; the ancient Greeks, that of the Olym-
piads; the Romans, that of the Building of their City ; the
ancient Persians and Assyrians, that of Nabonassar ; &c.

The doctrine and use of epochs is of very great extent
in chronology. To reduce the years of one epoch to those
of another, i. e. to find what year of one corresponds to
a given year of another ; a period of years has been in-
vented, which, commencing before all the known epochs,
is, as it were, a common receptacle of them all, called
the Julian Period. To this period all the epochs are re-
duced ; that is, the year of this period when each epoch
commences, is so determined; that, adding the given
year of one epoch to the year of the period corresponding
with its rise, and from the sum subtracting the year of the
same period corresponding to the other epoch, the re-
mainder is the year of the required epoch.

Epoch of Christ, is the common epoch throughout
Europe, commencing at the supposed, time of his nativity,
December 25 ; or rather, according to the usual ac-
count, from his circumcision, or the 1st of January.
The author of this epoch' was an abbot of Rome, one
Dionysius Exiguus, a Scythian, about the year 507 or
527- Dionysius began his account from the conception
or incarnation, usually called the Annunciation, or Lady-
Day; which method obtained in the dominions ofGreal
Britain till the year 1752, before which time the Diony-
sian was the same, as the English epoch: but in that year
the Gregorian calendar having been admitted by act ol
parliament, they now reckon from the first of January,.
as in the other parts of Europe, except in the court of
Rome, where the epoch of the Incarnation still obtains
for the date of their bulls.

EOU

[ 477 1

EOU

A Table of the Years of the most remarkable Epochs or Eras and Events.

N. B. The years before Christ, are those before the reputed year of his birth, and not
reckoned back from the first year of his age, as is generally done in such tables.

The Creation of the World --------

The Deluge, or Noah's flood - -•

Assyrian monarchy founded by Nimrod - - - - -

The birth of Abraham --------

Kingdom of Athens founded by- Cecrops - - - - -

Entrance of the Israelites into Canaan ------

The destruction of Troy --------

Solomon's temple founded --------

The Argonautic expedition --------

Lycurgus formed his laws --------

Arbaces, first king of the Medes -------

Olympiads of the Greeks began -------

Rome built, or Roman Era --------

Era of Nabonassar ---------

First Babylonish captivity, by Nebuchadnezzar - - - -

The 2d ditto, and birth of Cyrus -------

Solomon's temple destroyed - - - -

Cyrus began to reign in Babylon -------

Peloponnesian war began ------

Alexander the Great died - - - - "

Captivity of 100,000 Jews by Ptolemy- - - - , -

Archimedes killed at Syracuse -------

Julius Caesar invaded Britain . - - - -

He corrected the calendar - - - " .

The true year of Christ's birth - - - - -

The Christian Era begins here.
Dionysian, or vulgar era of Christ's birth . . - - -

Christ crucified, Friday April 3d ------ -

Jerusalem destroyed ---------

Adrian's wall built in Britain -------

Dioclesian A*.poch, or that of Martyrs - - - - . -

The council of Nice ---------

Constantino the Great died --------

The Saxons invited into Britain --------

Hegira, or flight of Mohammed -------

Death of Mohammed --------

The Persian Yesdegird --------

Sun, Moon, and Planets h, %> 6* , S , g '» =£=> SL'cn fl'om tue eartn

Art of printing discovered - -.-

The Reformation begun by Martin Luther - -

The Calendar corrected by Pope Gregory . - - - -

Oliver Cromwell died - - - - -.-

Sir Isaac Newton born, Dec. 25 -------

Made President of the Royal Society, ------

Died, March 20th

New Planet discovered byHcrschel

Julian

Year of the

Years before

Period.

World.

Christ.

706

0

4007

2362

1656

2351

2537

1831

2167

27J4

2008

1999

3157

2451

1556.

3262

2556

1451

3529

2823

1184

3701

2995

1012

3776

3070

937

3829

3103

884

383S

3132

875

393 S

3232

775

3901

3255

752

396'7

320'1

746

410/

3401

606 ~

4114

3408

599

4125

3419

588

4177

3471

536

42S2

3576

- 431

4390

3684

323

4393

3687

320

4506

3800

207

4659

3953

54

466'7

396l

46

4709

4003

4

Years since
Christ.

4717

4007

0

4746

4040

33

4783

4077

70

4S33

4127

120

4997

4291

284

5038

4332

325

5050

4344

337

5158

4452

445

5335

4629

622

5343

4637

630

5344

463S

631

5899

5193

11S6

6153

5447

1440

6230

5524

1517

6295

5589

1582'

6371

566*5

165S

6355

5619

1642

6416

5710

1703

6440

5734

1727

6494

5788

17SI

EPROUVETTE, in Gunnery, is a machine for showing
the strength or quality of gunpowder. Eprouvcttes have
been made of various kinds and shapes. The best is that
described in my Tracts, vol. 3, pa. 153.

EQUABLE Motion, Celerity, Velocity, &C, is that which
is uniform, or without alteration, or by which equal spaces
are passed over in equal times. Hence, thespaces, passed
over in equable motions, are proportional to the times. So
that if a body pass over 20 feet in 1 second of time, it
will pass over 40 feet in 2 seconds, and so on.

Equably Accelerated or Retarded, &c, is when the mo-

tion or change is increased or decreased by equal quan-
tities or degrees in equal times.

EQUAL, a term of relation between different things,
but of the same kind, magnitude, quantity, or quality.—
Wolfius defines equals to be those things that may be substi-
tuted for each other, without any alteration of their quan-
tity.— It is an axiom in mathematics &c, that two things
which are equal to the same thing, arc also equal to each
other. And if equals be equally altered, by equal ud-
dition, subtraction, multiplication, division, &c, the re-
sults will be also equal.

E Q U [ 47S ]

Equal Circles, are those whose diameters are equal.
Equal Angles, are those whose sides are equally in-
clined, or which are measured by similar arcs of circles.
Equal Lines, are lines of the same length.
Equal Plane Figures, are those whose areas are equal;
whether the figures be of the same form or not.

Equal Solids, are such as are of the same space, capa-
city, or solid content; whether they be of the same kind
or not.

Equal Curvatures, are such as have the same, or equal
radii of curvature.

Equal Ratios, are those whose terms are in the same
proportion.

Equal, in Optics, is said of things that are seen under
equal angles.

EQUALITY, the exact agreement of two things in re-
spect to their quantity. Those figures are equal which
may occupy the same space, or ttiay be conceived to pos-
sess the same space, by the flexion or transposition of their
parts. See a learned discourse upon this subject, by Dr.
Barrow, in his 11th and 12th Mathematical Lectures.

Equality, in Algebra, the relation or comparison be-
tween two quantities that are really or effectually equal.
See Equation. — Equality, in Algebra, is usually denoted
by two equal parallel lines, as = : thus 2 ■+■ 3 = 5, i. e.
2 plus 3, are equal to 5. This character = was first in-
troduced by Robert Recorde. Descartes, and some others
after him, use the mark oe instead of it: as 2 -+- 3 oc 5.

Equality, in Astronomy. Circle of Equality, or
the Equant. See Circle, and Equant.

Ratio or Proportion of Equality, is that between two
equal numbers or, quantities.

Proportion of Equality evenly ranged, or ex cequo or-
dinata, is that in which two terms, in a rank or series, are
proportional to as many terms in another series, compared
to each other in the same order, i. e. the first of one rank
to the first of another, the 2d to the 2d, &c.

Proportion of Equality evenly disturbed, called also ex
aquo perturbata, is that in which more than two terms of
one rank, are proportional to as many terms of another,
compared to each other in a different and interrupted or-
der; viz, the 1st of one rank to the 2d of another, the 2d
to the 3d, &c.

EQUANT, or Equant, in Astronomy, a circle for-
merly conceived by astronomers, in the plane of the de-
ferent, or excentric; for regulating and adjusting certain
motionsofthe planets, and reducing them more easily to a
calculus : but in the modern astronomy they are not used.
' EQUATED Anomaly. See Anomaly.
Equated Bodies. On Gunter's Sector are sometimes
placed two lines, answering to one another, and called the
lines of equated bodies ; being situated between the lines
of superficies and solids, and are marked with the letters
D, i, c, s, o, t, to signify dodecahedron, icosahedron,
cube, sphere, octahedron, and tetrahedron.

The uses of these lines arc, 1st, When the diameter of
the sphere is given, to find the sides of the five regular
•bodies, each equal to that sphere ; 2d, From the side of
any one of those bodies being given, to find the diameter
of the sphere, and the sides of the other bodies, which
shall be each" equal to the first given body. So that when
the sphere is given, take its diameter, and apply it on the
(sector in the points s, s; but when one of the other five
bodies is given, apply its side over in its proper points;

E Q U

then the parallels taken from between the points of the
other bodies, or sphere, will be the sides or diameter, equal
severally to the sphere or body first given.

EQUATION, in Algebra, an expression of equality
between two different quantities ; or two quantities, whe-
ther simple or compound, with the mark of equality be-
tween them : as 2 + 3 = 7 — 2; or 2 x 3 = 6; or 5 x 3
= 30-r2; or a -t- b =zc; or x- -+■ ax = b ; &c. When
the two sides of the equation are the same, the expression
becomes an identity, as 5 = 5, ora = a, &c. Sometimes
the quantities are placed all on one side, and made equal
to 0, or nothing, on the other side ; as 5 — 5 = 0, or a ^- b
= 0 : which is no more than setting down the difference
of two equal quantities equal to nothing, — The character
or sign usually employed to denote an equation, is = ,
which is placed between the two equal quantities, called
the two sides of the equation.

The Terms of an Equation, are the several quantities
or parts of which it is composed. Thus, of the equation
a -+■ b = c, the terms are a, b, and c : and the tenor or im-
port of the expression is, that some quantity represented
by c, is equal to two others represented by a and b.
Equations are either simple or affected or compound.
A Simple Equation is that which has only one power
of the unknown quantity : as a -+- x = 36, or axz = bc, or
a3 -+- 2x3 = 5b, &c; where x denotes the unknown quan-
tity, and the other letters known ones.

An Affected, or Adfected, or Compound Equation, con-
tains two or more different powers: as x2 -+- ax = b, or
x3 — 4x2 -+- 3x = 25, &c.

Again, equations are denominated from the highest
power contained in them ; as quadratic, cubic, biquadratic,
&c. Thus,

A Quadratic Equation, is that in which the unknown
quantity rises to two dimensions, or to the square or 2d
power : as x1 ■+■ 20x = 200, or xx — ax = b.

A Cubic Equation, is that in which the unknown
quantity is of three dimensions, or rises to the 3d power :
as x3 = 25, or x3 — 2xa = 27, or x3 — ax2, -+- bx = c.

A Biquadratic Equation, is that in which the unknown
quantity is of 4 dimensions, or rises to the 4th or biqua-
dratic power : as x4 = 25, or x* — 20x = 10, or x* -I- ax3
-t- ftx2, ■+- ex = d.

And so on for other higher orders of equations.
The Root of an Equation, is the value of the un-
known letter or quantity contained in it. And this value
being substituted in the terms of the equation instead of
that letter or quantity, will cause both sides to vanish, or
will make the one side exactly equal to the other. So the
root of the equation 3x -+• IS = 24, is 2; for by using
2 for x, it becomes 3 x 2 ■+- IS = 0 -+■ 18 = 24.

Every equation has as many roots as it has dimensions,
or as it contains units in the index of the highest power,
when the powers are all reduced to integral exponents.
So the simple equation of the 1st power, has only one
root; but the quadratic has 2, the cubic 3, the biqua-
dratic 4, &c. Thus the two roots of this equation
x- — 4x= — 3 are 1 and 3 ; for either of these substi-
tuted for x makes x" — 4x = — 3. Also the three roots
of x3 — 4x3 — 1 lx = - 30, or x3 — 4.r2 -l]r + 30= 0,
arc 2, 5, and — 3; as will appear by substituting each of
these instead of x in the equation, by which means it will
always be found to be equal to 0. And so of others.
On the Relation between the Roots of Equations, and the

EQ U

L 479 ]

EQ u

Coefficients of their Terms. — In every equation, when the
terms are ranged in order according to the order of the
powers, the greater before the less ; .the first term or
highest power being freed of its coefficient, by dividing all (
the terms of the equation by it, and all brought to one
side, and made equal to nothing on the other side, then
it will appear in this form,

n n — 1 » a — 2 n — 3 „

x ■+■ ax -t- bx -+- ex . . . . = 0 ;

and the relations between the roots and coefficients, are
as follow :

1st, The coefficient a of the 2d term, is equal to the
sum of all the roots.

2d, The coefficient b of the 3d term, is equal to the
sum of all the products of the roots that can be made by
multiplying every two of them together.

3d, The coefficients c, d, e, &c, of the following terms,
are respectively equal to the sum of the products of the
roots made by multiplying every three together, or every
four together, or every five together, &c, the signs of all
the roots being changed, as will be seen when we treat of
the Generation of Equations.

The roots of equations are positive or negative, and
real or imaginary. Thus, the two roots of the equation
xz — 4x = — 3, are 1 and 3, real and both positive ; but
the roots of the equation xz — 4x2 — 1 lx =r — 30, are 2,
5, and — 3, viz, real, two positive and one negative; and the
roots of the equation x3 -+- 9x = 10, are 1 and — -J i -r
tj — 39, one real and positive, and two imaginary.

On the Generation of Equations. — This is only the mul-
tiplying of certain assumed simple equations together, to
produce compound ones, with intent to show the nature
of these ; a method which was invented by Harriot, and is
as follows: Suppose x to denote the unknown quantity
of any equation, and let the roots of that equation, or the
values of x, be, a, b, c, d, &c ; that is x =a, and x=b,
and x = c, &c ; or x — a = 0, and x — 6=0, andx — c = 0,
&c; then multiply these last equations together, thus,
x — a =0
x - b =0

'-*}

x -+- ab = 0
. . . =0

a 1 ab 1

1 — 6 > x2 + ac > x — abc = 0

c) be)
— d =0

i*-

abed = 0

ab~\

a~\ ac I abc "1

b I 3 be I 2 abd [
'fx5— , ^x2 — , \:
ad J acd [

dj be | dbcj

dej

Now the roots of these equations are a, b, c, d, &c;
and it is obvious that the sum of all the roots is the co-
efficient of the 2d term, the sum of all the products of
every two is the coefficient of the 3d term, the sum of all
the products of every three that of the 4th term, and so
on, to the last term, which is the continual product of all
the roots; and the same law may always be observed,
whatever be the dimension <ir degree of t lie equation.

On the Reduction of Equations. — This is to transform
or change them to their simplest and most commodious
form, in order to prepare them for finding or extracting
their roots. , It is always most convenient to arrange them

according to the powers of the unknown fetter, the highest
power being placed first on the left hand, and that term
to have only -+- ] for its coefficient; also all the terms
containing the unknown letter to be on one side of the
equation, and the absolute known term only on the
other side.

Now this reduction chiefly respects the first term, or
that which contains the highest power of the unknown
quantity; and the general rule for reducing it is, to con-
sider in what manner it is involved or connected with other
quantities, and then perform the counter or opposite re- ,
lation or operation ; for every operation is undone or
counteracted by the reverse of it ; as addition by sub-
traction, multiplication by division, involution by evolu-
tion, &c : then bring all the unknown terms to one side,
and the known term to the other side, changing the signs,
from ■+- to — , or from — to -i-, of those terms which are
changed from one side to the other; and lastly divide by
the coefficient of the first term, with its sign.

Thus, for example, if 5x — 12 = 3x -+- 4 :

then is 5x ■— 3x = 12 -t- 4, or 2x =16";
and so x =z\- = 8.
See other examples in my Course of Mathematics, vol. 1.

On Extracting or Finding the Roots of Equations.

This is finding the value or values of the unknown letter
in an equation, the rules for which are various, according
to the degree of the equation.

1. For the Root of a Simple Equation.

Having reduced the equation as above, by bringing the
unknown terms to one side, and the known ones to the
other,' freeing the former from radicals and fractions, by
their counter operations, and lastly dividing by the co-
efficients of the unknown quantity, the value of it is then
found : as in the example of reduction above given..
2. For the Roots of Quadratic Equations.

These are usually found by what is called completing
the square; which consists in squaring half the coefficient
of the 2d term, and adding it to both sides of the equa-
tion ; for then the unknown sfde is a complete square of
a binomial, and the other side consists only of known
quantities. Therefore, extracting the root on both sides, the
root of the first side will be a binomial, one part of which
is the unknown letter, and the other a known quantity,
and the mot of the other side is taken either -t- or — ,
since the square of either of these is the same given quan-
tity : lastly, bringing over the known part of the binomial
root to the other side, with a contrary sign, it will give
the two roots or values of the unknown letter sought. See
examples in my-Course, vol. 1.

3. For the Roots of Cubic Equations.

A Cubic Equation is that in which the unknown letter
ascends to the 3d power ; as x3 -t- a? =: S3, or x3 -+■ ax2 -+-
bx = c. The 2d term of every cubic equation being taken.
away, (see Transformation,) those equations may all'
be reduced to this form, vX3 -+- ax = b ; and the general
value of one root is x = V[i^ ■+• s/ii^ + tV^)] +,
\/[-r& — \/(ib° -+- x,-a3)]. This rule is usually called
Cardan's, because first published by him, but it was in-
vented both by Scipio Ferrous, and Nich. Tartalea, by
the latter of whom it was communicated to Cardan. See
the article Aloebra.

When the 2d term is negative, or the equation of this
form, x3 — ax = ± b, the radical ^/{\ bl -+• -rja^) be-
comes \/{\b2 — ^a3), which will be imaginary or impos-
sible when ,'Ta3 is greater than ^b2, for \/{$bl — tt"3)

E Q U [ 480 ]

will then be the square root of a negative quantity, which
is impossible: and yet, in this case, the root x is a real
quantity ; though algebraists have never been able to find
a real finite general expression for it. And this is what
is generally called the Irreducible Case.

This case may indeed be resolved by the trisecfion of
an arc or angle; or by any of the usual methods of con-.
verging; or by general expressions in infinite series. See
Saunderson's Algebra, pa. 713; Philos. Trans, vol. 18,
pa. 136, and vof. 70, pa. 415 , or my Abridg. vol. 3, pa.
396; vol. 5, pa. 334, and vol. 15, pa. 139- See also the
article Cubic Equations.

Mr. Cotes observes, in his Logometria, p. 29, that the
solution of all cubic equations depends either on the tri-
section of a ratio, or of an angle : which method is ex-
plained in Saunderson's Algeb. pa. 718.

Biquadratic Equations, or those that are of 4 dimen-
sions, are resolved after various methods. The first rule
was given by Lewis Ferrari, the companion of Cardan,
which is one, of the best. A second method was given by
Descartes, and another by Mr. Simpson and Dr. Waring.
For the explanation of which, see Biqu avratic Equations.
Equations of the Higher Degrees or Orders.
There is no general rule to express algebraically the
roots of equations above those of the 4th degree ; and
therefore methods of approximation are here made use of,
which, though not accurately, are yet practically true.

Some of these excel in ease and simplicity, and others
in quickness of converging. Among these may be reckoned
first, Double Position, or Trial-and-Error, both in respect
of ease and universality, as it applies in the simplest man-
ner to. all kinds of equations whatever, not excepting even
exponential ones, radical expressions of ever so complex
a form, expressions of logarithms, of arches by the sines
or tangents, of arcs of curves by the abscisses, or any
other fluents, or roots of fluxional equations. For an ex-
planation of this and other methods of converging to the
roots of equations, by Hall'ey, Newton, Raphson, &c, &c,
see Approximation, and Converging; and my Course
of Mathematics, vol. 1.

Besides the methods above adverted to, there have been
some others, given in the Memoirs of several Academics,
and elsewhere. As, by M. Daniel Bernoulli, in the Acta
Petropolitana, torn. 3, pa. 92 ; by Euler, in the same, vol.
6, New Scries, and torn. 5, pa. 6'3 and 82 ; by Mr. Thos.
Simpson, in his Essays, pa. 82 ; in his Dissertations, pa.
102 ; in his Algebra, pa. 158; and in his Select Exercises,
pa. 215.

Construction of Equations. See Construction.
Depression of Equations. See Depression.
Elimination in Equations. See Elimination.
Transformation of Equations. See Transforma-
tion.

Absolute Equation. See Absolute.
Adjected, or Effected Equation. See Affected.
Differential Equation, is the equation of differences or
fluxions.

EQUATIONS of Condition. See Condition, Equa-
tions of.

Eminential Equation. Sec Eminent! al.
Exponential EQUATION, one in which the exponents of
the powers are variable or unknown quantities. See Ex-
PON V.V\ i a 1..

Fluentml Equation, is the equation of the fluents.
Fluxional Equation, is the equation of the fluxions.

EOU

Limits of the Roots of an Equ at ion. See Limits.

Literal Equation, is a general equation expressed in
letters, as contradistinguished from a

Numeral Equation, one expressed in numbers.

Reciprocal Equation. See Reciprocal.

Transcendental Equation. See Transcendental,

Equation of Payments, in Arithmetic, is the finding a
time when, if a sum of money be paid, which is equal to
the sum of several others due at different times, no loss
will be sustained by either party. — The rule usually given
for this purpose, in the common books of arithmetic, is
this : Multiply each payment by the time it is due at,
then divide the sum of the products by the sum of the
payments, for the equated time. For example, If 20/. be
due at 2 months, and 40/. at 4 months, to find, as an
equivalent, when the whole 601. may be paid at once.
Here then 20 x 2 -t- 40 x 4 = 40 -1- l6'0 = 200, is the
sum of the products; and 20 h- 40 = 60, is the sum of
the payments ; hence 200 -f- 60 = 3i is the equated time,
for making the whole payment at once.

This method, it is obvious, cannot be quite correct
when the two payments, as here, are unequal, because
no account of the interest on the two sums is considered.
When this is allowed for, Mr. Malcolm, in his Arithmetic,
has given a correct rule for two payments at simple in-
terest, which is this : If jt> denote the first payment, and
t the time till it is due ; also p = another payment, and
t = its time; and if r denote one year's interest of it;

next, putting a = t

, andc = Ti ■

pt + pt

pr pr

then \a — \<sj (ar — 4c) is the equated time for the two
payments. And here if the former example be taken, it
gives p = 20, t = 2, p = 40, t = 4, and r = ^, for a
rate of 5 per cent. Hence a = 66, and c ="208; then §
a — i;*J {a? — 4c) = 3"3436 is the true equated time, al-
lowing simple interest, being very nearly equal to the for-
mer, which was 3'3333 &c. And if compound interest
were allowed, the result would be a small matter different
still.

When three or more payments are concerned, then Mal-
colm's method recommended, of first equating for two pay-
ments, and next for the result and a 3d payment, and so
on, is not strictly just. But in all such cases, to obtain a
true answer, Malcolm's general principle ought to be used,
viz, making the interests of the sums that are kept later
than they are due, equal to the discounts of those that
are paid before they are due. The resolution of the result
ing equation will, indeed, require some knowledge in al-
gebra. But for ordinary use, the common rule will bring
out answers sufficiently near the truth.

Equation of a Curve, is an equation showing the
nature of a curve by expressing the relation between any
absciss and its corresponding ordinate, or else the relation
of their fluxions, ecc. Thus, the equation to the circle,
is ax — xz =y':, where Wis its diameter, 1 any absciss,
or part of that diameter, and y the ordinate at that point
of the diameter; that is, whatever absciss is denoted by
x, the square of its corresponding ordinate will be ax— x\
In like manner (he equation

- *8) = r,

of the ellipse is - (ax ■
of the hyperbola is (ax

*■) = y%

of the parabola is- - - px = jA
Where a is an axis, and p the parameter.

EQU

[ 481 ]

EQU

And in like manner for any other curves. See Absciss, same, between the mean and equated anomaly. The
This method of expressing the nature of curves by alge- greatest equation of the centre may be obtained by find-

braical equations, was first introduced by Descartes
who, by thus connecting together the two sciences of al-
gebra and geometry, made them mutually assist each,
other, and so laid the foundation of the great improve-
ments that have been made in every branch of them since
that time. See Descartes's Geometry; also Newton's
Lines of the 3d Order, and many other similar works on
curve lines, by several authors. See Curve.

Equation, in Astronom\', as Annual Equation, is
either of the mean motion of the sun and moon, or of the
moon's apogee and nodes.

The annual equation of the sun's centre being given, the
other three correspondent annual equations will be also
given, and therefore a table of the first will serve for them
all. Thus, if the annual equation of the sun's centre,
taken from such a table for any time, be called s ; and if
-rIs-s = a, and-£s = b ; then will the other annual equations
for that time be as follow :

a •+■ -j-1^ a = m, that of the moon's mean motion ;
and b -(- -j^b = a, that of the moon's apogee ;
and f b — i-^b = n, that of her nodes.

And here it must be observed, that when s, or the equa-
tion of the sun's centre, is additive; thenm is negative, a
is positive, and n is negative. But on the contrary, when s
is negative or subductive ; then dispositive, a negative,
and n positive.

There is also an Equation of the moon's mean motion, de-
pending on the situation of her apogee in respect of the
sun ; which is greatest when the moon's apogee is in an
octant with the sun, and is nothing at all when it is in the
quadratures or syzygies. This equation when greatest,
and the sun in perigee, is 3' 56". But it is never above
3' 34" when the sun is in apogee. At other distances of
the sun from the earth, this equation when greatest, is re-
ciprocally as the cube of that distance. But when the
moon's apogee is any where out of the octants, this equa-
tion becomes commonly less, and is at the same distance
between the earth and sun, as the sine of double the
distance of the moon's apogee from the next ouadrature
or syzygy, is to radius. This is to be added to the moon's
motion while her apogee passes from a quadrature with
the sun to a syzygy ; but is to be subtracted from it, while
the apogee moves from the syzygy to the quadrature.

There is another Equation of the moon's motion, which
depends on the aspect of the nodes of the moon's orbit
with respect to the sun : and this is greatest when her nodes
are in octants to the sun, and quite vanishes when they
come to their quadratures or syzygies. This equation is
proportional to the sine of double the distance of the node
from the next syzygy or quadrature ; and at the greatest
is only 47". This must be added to the moon's mean mo-
tion while the nodes are passing from the syzygies with the
sun to their quadratures; but subtracted while they pass
from the quadratures to the syzygies. From the sun's true
place subtract the equated mean motion of the lunar apo-
gee, as was shown above, and the remainder will be the
annual argument of the said apogee ; from which the ex-
centricity of the moon and the 2d equation of her apogee
may be compared. See Theory of the Moon's motions, &c.

Equation of the Centre, called also Prosthapheresis, and
Total Prosthapheresis, is the difference between the true
and mean place of a planet, or the angle made by the lines
of the true and mean place ; or, which amounts to the

Vol. I.

ing the sun's longitude at the times when he is near his
mean distances, for then the difference will give the true
motion for that interval of time : next find the sun's mean
motion for the same interval of time; and then half the
difference between the true and mean motions will show
the greatest equation of the centre.

For example, by observations made at, the Royal Ob-
servatory at Greenwich, it appears that at the following
mean times the sun's longitudes were as expressed below, viz,

Mean times. Sun's longitudes.

1769 Oct. 1 at 23h4Qm12s - - 6"s 9° 32' 0-6"

1770 Mar. 29 at 0 4 50 . - - 0 8 50 27"5
dif. of time 17Sd 0 15 38 ; true dif. Ion. 5 29 18 27
tropical year = 365d 5h 48m 42s = 365-2421527;
observed interval =178 0 15 38 = 17S'0108565 :
then 365-2421527: 17S-01085648 : : 360° : 175"455948
or 175° 27' 21" the mean motion.

Therefore - 175° 27' 21" of mean motion,

answers to - 179 18 27 of true motion ;

their difference is 3 51 6

and its half - 1 55 33

is the greatest equation of the centre according to these

observations.

To find the Equation of the Centre, which is Kepler's
problem, and requires a very troublesome operation, es>-
pecially in the more excentric orbits. The method of
performing this, has been shown by Newton, Gregory,
Keil, Machin, Lacaille, and others, by methods little dif-
fering from one another; which consist principally in find-
ing a certain intermediate angle, called the excentric ano-
maly; having known the mean anamoly, and the dimen-
sions of the sun's orbit. The mean anomaly is easily
found, by determining the exact time when the sun is in
the aphelion, and using the following proportion, viz,
As the time of a tropical revolution, or solar year,
Is to the interval between the aphelion and given time,
So is 360 degrees, to the degrees of the mean anomaly.
Or it may be found by taking the sun's mean motion at
the given time out of tables.

To find the Excentric Anomaly, say,
As the aphelion distance,
Is to the perihelion distance ;
So is the tangent of half the mean anomaly,
To the tangent of an arc.

Which arc added to half the mean anomaly, gives the
excentric anomaly. Then,

To find the True Anomaly, say,
As the square root of the aphelion distance,
Is to the square root of the perihelion distance ;
So is the tangent of half the excentric anomaly,
To the tangent of half the true anomaly.

Then, the difference between the true and mean ano-
maly, gives the equation of the centre, sought. Which is
subtractive, from the aphelion to the perihelion, or in the
first 6 signs of anomaly ; and additive, from the perihelion
to the aphelion, or in the last 6 signs of anomaly; and
hence called Prosthapheresis.

By this problem a table may easily be formed. When
the equations of the centre for every degree of the first 6
signs of mean anomaly are found, they will serve also for
the degrees of the last 6 signs, because equal anomalies
are at equal distances on both sides of their apses. There-
fore set these equations orderly to theirsigns and degiees
3 Q

EQU

[ 482 ]

EQU

of anomaly ; the first 6 being reekoned from the top of
the table downwards, and signed subtract; the last 6, for
which the same equations serve, in a contrary order, be-
in" reckoned from the bottom upwards, and marked add.
Let also the difference between every adjacent two equa-
tions, called tabular differences, be set in another column.
Hence, from these equations of the centre, augmented or
diminished by the proportional parts of their respective
tabular differences, for any given minutes and seconds,
may easily be deduced equations of the centre to any
mean anomaly proposed. See Robertson's Elem. of Na-
vig. book 5, pa. 286, 290, 295, and 308, where such a ta-
ble of equations is given.

Euler has particularly considered this subject, in the
Mem. de l'Acad. de Berlin, torn. 2, pa. 225 et seq. where
he resolves the following problems :

1. To find the true and mean anomaly corresponding
to the planet's mean distance from the sun ; that is, where
the planet is in the extremity of the conjugate axis of its
orbit. — 2. The excentricity of a planet being given, to
find the escentric anomaly corresponding to the greatest
equation.— 3. The excentricity being given, to find the
mean anomaly corresponding to the greatest equation. —
4. From the same data, to find the true anomaly corre-
sponding to this equation. — 5. From the same data, to
find the greatest equation. — 6*. The greatest equation be-
ing given, to find the excentricity.

Euler observes, that this problem is very difficult, and
that it can only be resolved by approximation and tenta-
tively, in the manner he mentions : but if the excentricity
be not great, it may then be found directly from the
greatest equation. Thus, if the greatest equation be = m,
and the excentricity = n ; then is

m = 2n -) — — n -+- —

24.3 26.3.5.7

n = \tn — tif — t

2".3 2'".3.5

equation m must be expressed in parts of the radius,
which- may be done by reducing the angle m into seconds,
aud adding 4-6855749 to the log. of the resulting num-
ber, which will be the log. of the number m. The mean
anomaly to which this greatest equation corresponds, will

be x = 90° -+- -I- tn -Trtn3 x-m% — &c. Whence,

* 2. 3 2.95

if 90° be added to £ of the greatest equation, the sum will
be the mean anomaly sufficiently exact.

Euler subjoins a table, by which may be found the
greatest equations, with the mean and excentric anoma-
lies corresponding to these greatest equations for every
100th part of unity, which he supposes equal to the great-
est excentricity, or when the transverse and distance of
the foci become infinite. The last column of the table
gives also the logarithm of that distance of the planet
from the sun where its equation is greatest. By means of
this table, any excentricity being given, the corresponding
greatest equation will be found by interpolation. But the
principal use of the table is to determine the excentricity
when the greatest equation is known ; and without this
help Euler thinks the problem cannot be resolved.

Equation of Time-, denotes the difference between
mean and apparent time, or the reduction of the apparent
unequal time, or motion of the sun or a planet, to equal
and mean lime, or motion; or the equation of time is the
difference between the sun's mean motion, and his right
ascension. Apparent time is that which takes its begin-
ning Tom the passage of the sun's centre over the raeri-

■ &c ; and by reversion
&c. Where the greatest

dian of any place ; and had the sun no motion in the
ecliptic, or was his motion reduced to the equator or ii*
right ascension uniform, he would always return to the
meridian after equal intervals of time. But his apparent
motion in the ecliptic being continually varying, and his
motion in right ascension being rendered further unequal
on account of the obliquity of the ecliptic to the equator,
it follows from these causes that the intervals of his return
to the meridian become unequal, and the sun will gradu-
ally come too slow or too soon to the meridian for an
equable motion, such as that of clocks and watches ought
to be ; and this retardation or acceleration of the sun's
coming to the meridian, is called the equation of time.

Now, computing the celestial motions according to
equal time, it is necessary to turn that, time back again
into apparent time, that they may correspond to observa-
tion : on the contrary, any phenomenon being observed,
the apparent time of it must be converted into equal time,
to have it correspond with the times marked in the astro-
nomical tables.

The equation of time is nothing at four different times
in the year, at which time the whole mean and unequal
motions exactly agree; viz, about the 15th of April, the
loth of June, the 31st of August, and the 24-th of Decem-
ber : but at all other times the sun is cither too fast or
too slow for mean, equal, or clock time, by a certain num-
ber of minutes and seconds, which at the greatest is 16''
14", and happens about the 1st of November ; every other
day throughout the year having a certain quantity of this
difference belonging to it ; which however is not exactly
the same every year, but only every 4th year ; for which
reason it is necessary to have 4 tables of this equation, viz,
one for each of the four years in the period of leap-years.
Instead of these, we shall insert here, one general equation
of time, according to the place of the sun, in every point
of the ecliptic: where it is to be observed, that the sign
of the ecliptic is placed at the tops of the columns, and
the particular degree of the sun's place, in each sign, in
the first and last columns ; and in the angle of meeting in.
all the other columns, is the equation of time, in minutes
and seconds, when the sun has any particular longitude:
supposing the obliquity of the ecliptic 23° 2S', and the
sun's apogee in 9° of 25.

The equations with the sign -h, are to be added to,
and those with the sign — are to be subtracted from, the
apparent time, to give the mean time. The preceding
sign, whether it be -h or — , at the top of any column,
belongs to all the numbers or equations in that column
till the sign changes; after which, the remainder of the
column takes the contrary sign.

The equation answering to any point of longitude be-
tween one degree and another, or any number ol minutes
or parts of a degree, is to be found by proportion in the
usual way, viz, as 1° or 6*0', is to that lumber of minutes,
so is the whole difference in the equation, from the given
whole degree of longitude to the next degree, to the pro-
portional part of it answering to the given number of
minutes.

See Tables of the Equation of Time computed for
every year, in the Nautical Almanac, by a method pro-
posed and illustrated by Dr. Maskelyne, late astronomer
royal, viz, by taking the difference between the sun's
true right ascension and his mean longitude, corrected
by the equation of the equinoxes in right ascension, and
turning it into time at the rate of 1 minute of time to 15.
of light ascension. Philos. Trans. vol.54, pa. 330".

E Q U L 483 ] EQU

Table of the Equation of Time, for every Degree of the Sun's Longitude.

'b

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13

43

a

6

5

42

2

22

3

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33

5

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0

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9

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6

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1

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34

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7

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2

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3

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13

12

13

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7

8

5

5

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42

3

4

2

58

5

54

0

1

10

20

16

11

11

1

2

43

13

24

13

14

8

9

4

47

2

51

2

56

3

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0-

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10

39

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13

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3

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10

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9

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15

16

1:3

9

53

4

7

13

54

12

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11

12

3

50

3

16

2

29

3

46

5

37

1

17

1 1

33

16

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9

29

4

35

14

2

12

27

12

13

3

32

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23

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19

3

58

5

31

1

38

11

51

16

10

9

5

5

2

14

9

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13

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13

3

30

2

8

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3

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4

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14

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11

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37

3

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1

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4

29

5

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12

41

16

0

7

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6

22

14

27

11

31

16

17

2

19

3

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1

35

4

39

5

1

3

1

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15

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7

22

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1

3

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4

52

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5 5

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38

10

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20

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0

59

5

5

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4

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30

EQUATOR, in Geography, a great circle of the earth, The circle in the heavens conceived directly over the equa-
equally distant from its two poles, and dividing it into tor, is the equinoctial. See Equinoctial. Thegreatest
two equal parts, or hemispheres, the northern and south- height of the equator above the horizon, is equal to the
ern. The equator is sometimes simply called the line, latitude of the place.

Table for turning Degrees and Minutes into Time, and tlt^Contrary.

D.

H. M.

D.

H. M.

D. | H

M.

D.

H. M.

D.

h

. M.

D.

H

. M.

D.

H. M.

D.

H. M.

D.

H. M.

Mi

M. S.

M.

M. S.

M. 1 M

. s. '

M.

M. S.

M.

M. S.

M.

W

. S.

M.

M. S.

M.

M. S.

M.

M. S,

1

0 4

21

1 24

41

2

44

61

4 4

81

5

24

101

6

44

121

S 4

141

9 24

161

10 44

2

0 8

22

I 28

42

2

48

02

4 8

82

5

28

102

6

48

122

8 8

142

9 28

162

10 48

3

0 12

23

1 32

43

2

52

63

4 12

83

5

32

103

6

52

12.'i

8 12

143

9 32

163

10 52

1

0 16

24

1 36

44

2

56

04

4 16

84

5

36

104

6

56

124

8 16

144

9 36

164

10 56

5

0 20

25

1 40

4.5

3

0

65

4 20

85

5

40

105

7

0

125

8 20

145

9 40

165

11 0

6

0 24

26

1 44

46

3

4

66

4 24

86

5

44

106

7

4

126

8 24

146

9 44

166

11 4

7

0 28

27

1 48

47

3

8

§7

4 28

87

5

48

107

7

8

127

8 28

147

9 4S

167

11 3

8

0 32

28

I 52

48

3

12

68

4 32

88

5

52

108

7

12

12S

8 32

148

9 52

168

11 12

9

0 36

29

1 56

49

3

16

69

4 36

89

0

56

109

7

16

12y

8 36

149

9 56

169

11 16

10

0 40

30

2 0

50

3

20

70

4 40

90

6

0

110

7

20

130

8 40

150

10 0

170

1 1 20

11

0 44

31

2 4

51

3

24

71

4 44

91

6

4

1 11

7

2't

131

8 44

151

10 4

171

11 24

n

0 48

32

2 8

52

3

28

7 2

4 48

92

6

8

112

7

2S

132

8 48

152

10 S

172

11 28

13

0 52

33

2 12

53

3

32

73

4 52

93

6

12

113

7

32

133

8 52

153

10 12

173

11 32

14

0 56

34

2 16

54

3

36

7 1

4 56

94

6

16

114

7

36

134

S 56

154

10 16

174

11 36

15

1 0

35

2 20

55

3

40

7 5

5 0

95

6

20

115

7

40

135

9 0

155

10 20

175

1 1 40

16

1 4

36

2 24

56

3

44

76

5 4

96

6

24

116

7

44

136

9 4

156

10 24

176

11 44

17

1 8

37

2 28

57

3

48

77

5 8

97

6

28

117

7

48

137

9 8

157

10 28

177

11 48

18

1 12

!38

2 32

58

3

52

7 8

5 12

98

6

32

118

7

52

138

9 12

158

10 32

178

11 52

19

1 16

1 30

2 36

59

3

56

79

5 16

99

6

36

119

7

56

139

9 16

159 10 36

179

11 56

20 1 '20

140

2 40

60

4

0

80

5 20

100

6

40

120ls

0

140

9 20

1601 10 40

180

12 0

3Q

2

EQU [ 484 ]

As one whole revolution of the earth, or of the 36"0° of
the equator, is performed in 24 hours, which is at the
rate of 15° per hour ; hence the number of degrees of the
equator, answering to any other given time, or the time
answering to any given number of degrees of the equator,
will be easily found by proportion, viz,

as lh : 15° : : any time : its degrees,

or as 15°: lh: : any degrees: their time.
And thus is computed the foregoing table for turning time
into degrees of the equator, and the contrary.

EQUATORIAL, Universal, or Portable Observa-
tory, is an instrument intended to answer a number of
useful purposes in practical astronomy, independently of
any particular observatory. It may be employed in any
steady room or place, and it performs most of the useful
problems in the science of astronomy. The following is
the description of one lately invented by Mr. Ramsden,
and named the Universal Equatorial, the description of
which was given by the Hon. Stewart Mackenzie.

The principal parts of this instrument (fig. 2, plate viii.)
are, 1st, The azimuth or horizontal circle a, which re-
presents the horizon of the place, and moves on a long
axis b, called the vertical axis. 2d, The equatorial or
hour-circle c, representing the equator, placed at right
angles to the polar axis r>, or the axis of the earth, upon
which it moves. 3d, The semicircle of declination e3 on
which the telescope is placed, and moving on the axis of
declination, or the axis of motion of the line of collima-
tion F ; which circles are measured and divided as in the
following table :

EQU

Measures of the se-
veral circles and

||

Limb di-
vided to

Non. of
30 gives

Divid. on
limb into

Divided by
Non. into

divisions on them

«.=

seconds.

pts. of Inc

pts. of Inc.

Azimuth orl

horizontal \

5-1

15'

30"

45 th

1350th

circle J

Equatorial "1
or hour f

5-1

15',
or 1 m.

30"
2"

45th

1350th

circle J

intime

Vertical se-1

micirclc for !
declination f

5-

15'

r

30"

42d

12(50th

orlatitude. J

4th, The telescope, is an achromatic refractor with a
triple object-glass, whose focal distance is 17 inches, and
its aperture 2*4.5 inc., and it is furnished with 6' different
eye-tubes; so that its magnifying powers extend from 44
to 1 68. The telescope in this equatorial may be brought
parallel to the polar axis, as in the figure, so as to point
to the pole-star in any part of its diurnal revolution; and
thus it has been observed near noon, when the sun has
shone very bright. 5lh, The apparatus for correcting
the error in altitude occasioned by refraction, which is
applied to the end of the telescope next the eye-glass, and
consists of a slide G moving in a groove or dove-tail, and
carrying the several eye-tubes of the telescope, oil which
slide there is an index Corresponding to five small divisions
engraved on the dove-tail; a very small circle, called the
refraction circle n, moveable by a finger-screw at the ex-
tremity of the end of Ihe telescope next the eye-glass ;
which circle is divided into hull' minutes, one whole revo-
lution of it being equal to 3' 18", and by its motion it
raises the centre of the cross hairs on a circle of altitude;
and also a quadrant i of l\ inc. radius, with divisions on

each side, one expressing the degree of altitude of the ob--
ject viewed, and the other expressing the minutes and
seconds of the error occasioned by refraction, correspond-
ing to that degree of altitude. To this quadrant is con-
nected a small round level k, which is adjusted partly by
the pinion that turns the whole of this apparatus, and
partly by the index of the quadrant ; forwhich purpose
the refraction circle is set to the same minute &c, which
the index points to on the limb of the quadrant ; and if
the minute &c, given by the quadrant, exceed the 3' IS"
contained in one entire revolution of the refraction circle,
this must be set to the excess above one or more of its
entire revolutions; then the centre of the cross hairs will
appear to be raised on a circle of altitude to the additional
height which the error of refraction will occasion at that
altitude.

The principal adjustment in this instrument, is that of
making the line of collimation to describe a portion of an
hour-circle in the heavens : in order to which, the azimuth
circle must be truly level ; the line of collimation, or
some corresponding line represented by the small brass
rod m parallel to it, must be perpendicular to the axis
of its own proper motion'; and this last axis must be per-
pendicular to the polar axis. On the brass rod m there
is occasionally placed a hanging level n, the use of which
will appear from the following directions :

The azimuth circle may be made level by turning the
instrument till one of the levels be parallel to an imaginary
line joining two of the feet screws; then adjust that level
with these two feet screws ; turn the circle 180°, or half
round ; and if the bubble be not then right, correct half
the error by the screw belonging to the level, and the
other half error by the two foot screws, repeating this
operation till the bubble come right ; then turn the circle
90° from the two former positions, and set the bubble
right again, if it be wrong, by the foot screw at the end
of the level ; when this is clone, adjust the other level by-
its own screw, and the azimuth circle will be truly level.
The hanging level must then be fixed to the brass rod by
two hooks of equal length, and made truly parallel to it:
for which purpose, the polar axis must be made perpen-
dicular or nearly perpendicular to the horizon; then ad-
just the level by the pinion of the declination semicircle :
reverse the level, and if it be wrong, correct half the er-
ror by a small steel screw that lies under one end of tho
level, and the other half error by the pinion of the decli-
nation-semicircle, repeating the operation lill the bubble
be right in both positions. To make the brass rod, on
which the level is suspended, at right angles to the axis
of motion of the telescope, or line of collimation, make
the polar axis horizontal, or nearly so ; set the declina-
tion semicircle to O1, aiid turn the hour-circle till the
bubble be properly adjusted; then turn the declination-
circle to 00° ; adjust the bubble by raisino; or depressing
the polar axis (first by hand till it be nearly right, after-
wards tighten with an ivory key the socket which runs on
thearch with the polar axis, and then apply the same
ivory key to the adjusting screw at the end of the said
arch till the bubble be properly adjusted) ; then turn the
declination-circle to the opposite .90° ; ami if I he level he
not then right, Correct half the error by the aforesaid ad-
justing screw at the end of the arch, and the other half
error by the two screws that raise or depress the end ot
the brass rod. The polar axis remaining nearly horizontal
as before, and the declination-semicircle at 0°, adjust the

EQU

[ 485 ]

EQ U

bubble by the hour-circle; then turn the declination-
semicircle to 90°, and adjust the bubble by raising or
depressing the polar axis ; then turn the hour-circle 12
hours; and if the bubble be wrong, correct half the error
by the polar axis, and the other half error by the two
pair of capstan screws at the feet of the two supports on
one side of the axis of motion of the telescope; and thus
this axis will be at right angles to the polar axis. The
next adjustment, is to make the centre of the cross hairs
remain on the same object, while the eye-tube is turned
quite round by the pinion of the refraction apparatus : in
order to perform this, set the index on the slide to the
first division on the dove-tail; and set the division marked
IS" on the refraction-circle to its index ; then look through
the telescope, and with the pinion turn the eye-tube quite
round ; then if the centre of the hairs do not remain on
the same spot during that revolution, it must be corrected
by the four small screws, 2 and 2 at a time, which will
be found upon unscrewing the nearest end of the eye-tube
that contains the first eye-glass ; repeating this correction
till the centre of the hairs remain on the spot looked at
during a whole revolution. To make the line of collima-
tion parallel to the brass rod on which the level hangs,
set the polar axis horizontal, and the declination-circle to
§0°, adjust the level by the polar axis; look through the
telescope on some distant horizontal object, covered by
the centre of the cross hairs : then invert the telescope,
which is done by turning the hour-circle half round; and
if the centre of the cross hairs do not cover the same ob-
ject as before, correct half the error by means of the
upper and lower of the 4 small screws at the eye-end of
the large tube of the telescope; this correction will give a
second object now covered by the centre of the hair :
which must be adopted instead of ' the first object ; then
invert the telescope as before; and if the second object
be not covered by the centre of the hairs, correct half
the error by the same two screws as were used before :
this correction will give a third object, now covered by
the centre of the hairs, which must be adopted instead of
the second object; repeat this operation till no error re-
mains ; then set the hour-circle exactly to 12 hours, the
declination-circle remaining at 90° as before ; and if the
centre of the cross hairs do not cover the last object fixed
on, set it to that object by the two remaining small
screws at the eye-end of the large tube, and then the line
of collimation will be parallel to the brass rod. For recti-
fying the nonius of the declination and equatorial circles,
depress the telescope as many degrees &c below 0° or m
on the declination-semicircle, as are equal to the comple-
ment of the latitude; then elevate the polar axis till the
bubble be horizontal ; and thus the equatorial circle will
be elevated to tlic co-latitude of the place: set this circle
to 6 hours; and adjust the level by the pinion of the
declination-circle ; now turn the. equatorial circle exactly
12 hours from the last position ; and if the level be not
right, correct one half of the error by the equatorial
circle, and the other hall by the declination-circle: then
turn the equatorial circle back again exactly 12 hours
from the last position; and if the level be still wrong,
repeat the correction as before, till it be right, when
turned to cither position : which being done, set the nonius
of the equatorial circle exactly to 6 hours, and the nonius
of the declination-circle exactly to 0°.

The principal uses of this equatorial are,

1st, To find the meridian by one observation only ; For

which purpose, elevate the equatorial circle to the co-la-
titude of the place, and set the declination-semicircle to
the sun's declination. for the day and hour of the day re-
quired; then move the azimuth and hour-circles both at
the same time, either in the same or in contrary direction,
till you bring the centre of the cross hairs in the teles-
cope exactly to cover the centre of the sun ; then, the in-
dex of the hour-circle will give the apparent orsolar time
at the instant of observation ; and thus the time is known,
though the sun be at a distance from the meridian ; after
which turn the hour-circle till the index points precisely
to 12 o'clock, and lower the telescope to the horizon, in or-
der to observe some point there in the centre of the glass ;
and that point is the meridian mark, found by one obser-
vation only. The best time for this operation is 3 hours
before, or 3 hours after, 12 at noon.

2d, To point the telescope to a star, though not on the
meridian, in full day-light. — Having elevated the equatorial
circle to the co-latitude of the place, and set the declina-
tion-semicircle to the star's declination, move the index of
the hour-circle till it points to the precise time at which
the star is then distant from the meridian, found in the
tables of "the right ascension of the stars, and the star will,
then appear in the glass.

Besides these uses, peculiar to this instrument, it may
also be applied to all the purposes to which the principal
astronomical instruments are applied ; such as a transit
instrument, a quadrant, and an equal-altitude instru-
ment, &c.

Seethe description and drawing of an equatorial teles-
cope, or portable observatory, invented by Mr. Short, in
the Philos. Trans. No. 493, or vol. 46, pa. 242; and
another by Mr. Nairnc, vol. 6l, pa. 107 ; also the descrip-
tion by Sir Geo. Shuckburgh, in the Philos. Trans, vol. S3,
pa. 67 ; or my Abridg. vol. 17, pa. 299-

EQUIANGULAR Figure, is one that has all its angles
equal among themselves ; as the square, and all regular
polygons. An equilateral figure inscribed in a circle, is
always equiangular. But an equiangular figure inscribed
in a circle, is not always equilateral, except when it has
an odd number of sides: If the sides be of an even num-
ber, then they may either be all equal, orelse half of them
will be equal to each other, and the other half to each;
other, the equals being placed alternately. See the de-
monstration in my Mathematical. Miscellany, pa. 272.

.Equiangular, is also said of any two figures of the
same kind, when each angle of the one is equal to a cor-
responding angle in the other, whether each figure, sepa-
rately considered in itself, be an equiangular figure or not,,
that is, having all its angles equal to each other. Thus,,
two triangles are equiangular to each other, if, ex.gr. one
angle in each be of 30°, a second angle in each of 50°,
and the third angle of each equal to 100 degrees. — Equi-
angular triangles have not ■ their like sides necessarily
equal, but only proportional to each other; but such tri-
angles being always similar to each other.

EQUICRURAL Triangle, is one that has two of its sides-,
equal to each other ; but is more usually called an Isos-
celes triangle.

EQUICULUS,Equuleus, or Eqvvs Minor, aconstel-
lation of the northern hemisphere. See Equuleus.

EQUI DIFFERENT, are such things as have equal dif-
ferences, or arithmetically • proportional. If the terms
have all the same difference, viz, the 1st and 2d, the 2d
and 3d, the 3d and 4th, &c, they are said to be continually,

E Q U

[ 486 ]

E O U

equidifferent ; as the numbers 3, 6, 9, 12, &c, where the
common difference is 3. But if the several different cou-
plets only have the same difference, as the 1st and 2d, the
3d and 4th, the 5th and 6th, &c, they are said to be dis-
cretely equidifferent ; as the terms 3 and 6, 7 and 10, 9
and 12, &c. See Arithmetical Progression and Pro-
portion.

EQUILATERAL Figure, is one that has all its sides
equal to each other. Such as the square, and all regular
polygons, or a triangle that has all its angles equal. See
Equiangular.

Equilateral Hyperbola, is that which has the two
axes equal to each other, and every pair of conjugate dia-
meters also equal to each other. The asymptotes also are
at right angles to each other, and make each half a right
angle with either axis. Also, such an hyperbola is equal
to its conjugate hyperbola, as well as to its opposite hy-
perbola ; so that all the four conjugate hyperbolas are mu-
tually equal to each other.

Moreover, as the 3d proportional to the two axes is the
parameter; therefore, in such a figure, the parameter and
two axes are all three equal to one another. Hence, as
the general equation to hyperbolas is y1 = - (tx+x'2) or =
V (tx + x*), where t is the transverse axis, c the conjugate,
p the parameter, x the absciss, and y the ordinate; then
making t, c, and p all equal, the equation, for the equi-
lateral hyperbola, becomes y2 = tx + x'2 ; differing from
the equation of the circle only in the sign of the term xz,
which in the circle is tx — x2, when t is taken to repre-
sent the diameter of the circle.

EQUILIBRIUM, is an equality between two equal
forces acting in opposite directions; so that they mutually
balance each other ; like the two equal arms, or scales, of
a balance, &c.

" Equilibrium, in solid bodies, forms a considerable
part of the science of Statics. And Equilibrium of fluids,
a considerable part of the doctrine of Hydrostatics.

EQUIMULTIPLES, the products of quantities equally
multiplied.

Thus3« and 3b are equimultiples of a and b;
and mc and md are equimultiples of c and d.

Equimultiples of any quantities, have the same ratio as
the quantities themselves. Thus a : b : : Sa : 3b : : ma : mb.
EQUINOCTIAL, a' great circle in the heavens under
which the equator moves in its diurnal motion. The poles
of this circle are the poles of the world, and it divides the
sphere into two equal parts, the northern and southern.
It cuts the horizon of any place, in the east and west
points; and at the meridian its elevation above the hori-
zon is equal to the co-latitude of the place. It has also
various other properties ; as,

]. When the sun is in this circle, the days and nights
in all parts of the earth are equal; because he then rises
due east, and sets due west. Hence it has the name equi-
noctial. All stars which are under this circle, or have no
declination, also rise due east, and set due west. — 2. All
people living under this circle, or upon the equator, or
line, have their days and nights at all times equal to each
other. — 3. From this circle, on the globe, is counted, upon
the meridian, the declination in the heavens, and the lati-
tude on tiie earth. — 4. Upon the equinoctial, or equator,
is counted the longitude, making in all 360°, that is, 180°
east, and 180° west. — 5. And as the time of one whole

revolution is divided into 24 hours; therefore 1 hour an-
swers to 15°, or the 24th part of 360°. Hence,
1° of longitude answers to 4 min. of time,
15' ----- - - to 1 min. of time,

l'--- ----to 4 sec. of time, &c.

6. The shadows of objects situated under this circle are
cast to the southward one half of the year, and to the
northward during the other half; and twice in a year, viz,
at the time of the equinoxes, the sun at noon casts no
shadow, being exactly in their zenith.

Equinoctial Colure, is the great circle passing through
the poles of the world and the equinoctial points, or first
points of Aries and Libra.

Equinoctial Dial, is one whose plane is parallel to
the equinoctial. Its properties or principles are :

1. The hour-lines are all equally distant from each
other, viz, 15°; and the style is a straight pin, or wire,
set up in the centre of the circle, perpendicular to the
plane of the dial.

2. The sun shines upon the upper part of this dial-plane
from the 21st of March to the 22d of September, and on
the under part the other half of the year.

Some of these dials are made of brass, &c; and set up
in a frame, to be elevated to any given latitude.

Equinoctial Points, ilk the two opposite points where
the ecliptic and equinoctial cross each other; the one
point being in the beginning of Aries, and called the ver-
nal point, or vernal equinox; and the other in the begin-
ning of Libra, and called the autumnal point, or autumnal
equinox. — It is found by observation, that the. equinoctial
points, and all the other points of the ecliptic, are con-
tinually moving backwards, or in antecedentia, i. e. west-
wards. This retrograde motion of the equinoctial points,
arises from the Precession of the equinoxes, and is made
at the rate of 50| seconds every year nearly. Sec Pre-
cession, also the preface to the Nautical Almanac for
the year 1797.

EQUINOXES, the times when the sun enters the
equinoctial points; that is, about the 21st of March and
22d of September: the former being the vernal or spring
equinox, and the latter ihe autumnal equinox. — As the
sun's motion is unequal, being sometimes quicker and
sometimes slower, it hence happens that there are about 8
days more from the vernal to the autumnal equinox, or
while the sun is on the northern side of the equator, than
while he is in moving through the southern signs, from the
autumnal to the vernal equinox, or on the southern side of
the equator. According to the observations of M.Cassini,
the sun is 186'd 14h 53m in the northern signs,

and only 178 14 56' in the southern signs,

so that 7 23 57 is their diff, or nearly S days.

EQUINUS Barbatus, a kind of comet. See Hivpeus.

EQUITANGIyNTIAL Curve, is one that was first con-
structed by Mr. J. Perks, the tangent of which is always
equal to a constant line. It is the involute of the caie-
naria; and is also the curve of traction to a straight line.
— For the various properties of this curve, its mechanical
construction, its relation to the catenaria, and its use in
dividing the meridian in Mercator"s projection, we refer to
the Philos. Trans. No. 8*5 : set' also Brougham's paper
on Porisms, in the Philos. Trans, vol. SS; or my Abridg.
vol. 18, pa. 354, for other curious properties of this curve.

EQUULEUS, Equiculus, and Equus Minor, Equi
lectio, the House's Head, one of the 48 old constellations,
in the northern hemisphere. Its stars in Ptolemy's Cata-

E T H

[ 487 ]

EVA

ogue are 4, in Tycho's 4, in Hevelius's 6, and in Flam- limpid and colourless liquor, of a hot pungent taste,
steed's 10. highly fragrant,, and so volatile that it can hardly be

ERATOSTHENES, a learned Greek philosopher of poured fronvone vessel to another without much loss by

Cyrene, and librarian of Alexandria under Ptolemy Euer
getes, king of Egypt. He was learned in all the sciences,
and first discovered a method of measuring the earth's
magnitude ; whence he was called the Cosmographer and
Surveyor of the Universe. He gave a mechanical solution
to the famous problem of doubling the cube. And he gave
a method of finding prime numbers, in arithmetic, com-
monly called Eratosthenes' Sive, a learned account of
which is given by Dr. Horsley, in the Philos. Trans, of
1772; or in niy Abridgment, vol. 13, pa. 314. Some of his
works remain, which were printed at Oxford, in Greek,
1672, Svo. Eratosthenes died about 194 before the year
Christ, at 81 years of age.

ER ECT Vision. See Vis I ok .

Erect Dials, such as stand perpendicular to the hori-

evaporation, the purest portion of it assuming the state of
elastic fluid.

When exposed to a cold of — 46°, c-ther freezes and
crystallizes. In moderate temperatures it is not affected
by oxygen gas or common air ; but when kindled in a state
of vapour, it burns rapidly with a fine white flame, and
leaves behind a trace of charcoal. In the state of vapour
it detonates with common air and oxygen gas.

EVAPORATION, the act of dissipating the humidity
of a body in fumes or vapour; differing from exhalation,
which is properly a dispersion of dry particles issuing
from a body. — Evaporation is usually produced by heat,
and by the change of air : thus, common salt is formed by
evaporating all the humidity from the brine or salt water;
which evaporation is either performed by the heat of the

zon, and are of various kinds ; as erect direct, when they sun, as in the salt-works on the sea-coast, &c ; or by means

of fire, as at the salt-springs, &c : and it is well known
how useful a brisk wind is in drying wet clothes, or the
surface of the ground ; while in a calm, still atmosphere,
they dry extremely slow.

But, though evaporation be generally considered as an
effect of the heat and motion of the air, yet M. Gauteron,
in the Memoires de l'Acad. des Scienc. an. 1705, shows,
that a quite opposite cause may have the same effect, and
that fluids lose more of their parts in the severest frost than
when the air is moderately warm : thus, in the great frost
of the year 1708, he found that the greater the cold, the
more considerable the evaporation ; and that ice itself lost
full as much as the warmer liquors that did not freeze.

There are indeed few subjects of philosophical investi-
gation that have occasioned a greater variety of opinion
than the theory of evaporation, or of the ascent of water
in such a fluid as air, between 8 and 9 hundred times
lighter than itself, to different heights, according to the
different densities or states of the atmosphere; in which
case it must be specifically lighter than the air through
which it ascends. The Cartesians account for it by sup-
posing, that by the action of the sun upon the water, small
particles of the water are formed into hollow spheres and
tilled with the materia sublilis, which renders them specifi-
cally lighter than the ambient air, so that they are buoyed
up by it.

Dr. Nicuwentyt, in his Religious Philosopher, c'ont. 19,
and several others, have alleged, that the sun emits par-
ticles of fire, which adhere to those of water, and form
molecula;, or small bodies, lighter than an equal bulk of
air, which consequently ascend till they come to a height
where the air is of the same specific gravity with them-
selves; and that these particles being separated from the
fire with which they are incorporated, coalesce and de-
scend in dew or rain.

Dr. Halley has advanced another hypothesis, which has
been more generally received : he imagined, that by the
action of the sun on the surface of the water, the aqueous
particles are formed into hollow spherules, that are filled
with a liner air highly rarefied, so as to become specifi-
cally lighter than the external air. Philos. Trans. No.
192, or my Abridg. vol. 3, pa. 427.

Dr. Desagulicrs, dissatisfied with these hypotheses, pro-
poses another in the Philos. Trans. No. 407, or Abridg,
vol. 7, pa. 323. See also his Course of Experimental
Philosophy, vol. 2, pa. 336. He supposes that heat acts,

face exactly one of the four cardinal points, east, west,
north, south ; and erect declining, when they decline from
the cardinal points. See Dial.

To Erect a Perpendicular, is a popular problem in
practical geometry, and denotes to raise a perpendicular
from a given line, &c, as distinguished from demitting or
letting fall a perpendicular on a line, &c, from some point
out of it. See Perpendicular.

ERIDANUS, the River, a constellation of the southern
hemisphere, and one of the 48 old asterisms. The stars
in this constellation, in Ptolemy's catalogue, are 34, in
Tycho's 19, and in the British catalogue S4.

ERRATIC, an epithet applied to the planets, which are
called erratic or wandering stars, in contradistinction to
the fixed stars.

ESCALADE, or Scalade, a furious attack on a wall
or a rampart; carried on with ladders, to pass the ditch,
or mount the rampart; without proceeding in form,
breaking ground, or carrying on regular works to secure
the men.

ESPAULE, or Epaule. See Epaule.

ESPLANADE, in Fortification, called also Glacis, a
part which serves as a parapet to the counterscarp, or
covert way ; being a declivity or slope of earth, com-
mencing from the top of the counterscarp, and losing it-
self insensibly in the level of the champaign.

Esplanade means also the ground which has been le-
velled from the glacis of the counterscarp, to the first
houses ; or the vacant space between the works and the
houses of the town. The term is also applied, in the
general, to any piece of ground that is made flat or level,
and which before had some eminence that incommoded
the place.

ESTIVAL Occident, Orient, or Solstice. See Occi-
dent, Orient, Solstice.

ESTUARY, an arm of the sea; a frith ; or the mouth
of a lake or river in which the tide reciprocates.

ETHER, an element more fine and subtile than air; or
air refined and sublimed ; or the matter of the highest re-
gions above.

Ether, in Chemistry, a certain very light, volatile,
fragrant, inflammable liquor, procured by distillation, by
heat, from a mixture of alcohol, and sulphuric or some
other acid, in equal quantities.

The specific gravity of ether, is from 0'725 to 0-632,
according to its different degrees of purity. It appears a

E V A

[-'

]

E V A

more powerfully on water than on common air; that the
same degree of heat which rarefies air two-thirds, will
rarefy water nearly 14,000 times ; and thafrfe very small
degree of heat will raise a steam or vapour from, water,
even in winter, while it condenses the air; and thus the
particles of water are converted into vapour by being
made to repel each other strongly, and, deriving electricity
from the particles of air to which they are contiguous, are
repelled by them and by each other, so as to form a fluid
which, being lighter than the air, rises in it, according to
their relative gravities. The particles of this vapour retain
their1 repellent force for a considerable time, till, by some
diminution of the density of the air in which they float,
they are precipitated downwards, and brought within the
sphere of each other's attraction of cohesion, and so join
again into drops of water.

Many objections have been urged against this opinion;
as, by Mr. Clare, in his Treatise of the Motion of Fluids,
pa. 2Q4; and by Mr. Rowning in his System of Philoso-
phy, part 2, diss. 6; to which Dr. Hamilton has added
the two following, viz, That if heat were the only cause of
evaporation, water would evaporate faster in a warm close
room, than when exposed in a colder place, where there
is a constant current of air; which is contrary to experi-
ence; and that the evaporation of water is so far from de-
pending on its being rarefied by heat, that it is carried on
even while water is condensed by the coldness of the air,
till it freezes ; and since it evaporates even when frozen
into hard ice, it must also evaporate in all the lesser de-
grees of cold. And therefore heat does not seem to be
the principal, much less the only cause of evaporation.

Others have more successfully accounted for the phe-
nomena of evaporation on another principle, viz, that of
solution; having shown, from a variety of experiments,
that what we call evaporation, is nothing more than a gra-
dual solution of water in air, produced and supported by
the same means, viz, attraction, heat, and motion, by
which other solutions are effected.

It seems however that the Abbe Nollct first started this
opinion, in his Lecons de Physique Experimentale, pub-
lished in 1743, without pursuing it much further: he
there only conjectures that the air of the atmosphere may
serve as a solvent or sponge, with regard to the bodies that
encompass it, and receive into its pores the vapours and
exhalations that are detached from the masses to which
they belong in a fluid state; and he accounts for their as-
cent on the same principles with the ascent of liquors in ca-
pillary tubes. On his hypothesis, the condensation of the air
contributes, like the squeezing of a sponge, to their descent.
Dr. Franklin, in a paper of Philosophical and Meteoro-
logical Observations, Conjectures, and Suppositions, de-
livered to the Royal Society about the year 1/47, and
read in 1756, suggested a similar hypothesis: he observes,
that air and water mutually attract each other; and
hence he concludes, that water will dissolve in air, as salt
in water; every particle of air assuming one or more particle
of water ; and when too much is added, it precipitates in
rain. But as there is not the same contiguity between the
particles of air as there is between those of water, the so-
lution of water in air is not carried on without a motion
of the air, so as to cause a fresh accession of dry particles.
A small degree of heat so weakens the cohesion of the
particles of water, that those on the surface easily quit it,
and adhere to the particles of air : a greater degree of heat
is necessary to break the cohesion between water and air ;

for its particles being by heat repelled to a greater distance
from each other, they thereby more easily keep the parti-
cles of water, that are annexed to them, from running into
cohesions that would obstruct, refract, or reflect the heat:
and hence it happens that when we breathe in warm air,
though the same quantity of moisture may be taken up
from the lungs as when we breathe in cold air, yet that
moisture is not so visible. On which principles he ac-
counts for the production and different appearances of
fogs, mists, and clouds. And he adds, that if the particles
of water bring with them electrical fire when they attach
themselves to air, the repulsion between the particles of
water electrified, joins with the natural repulsion of the
air to force its particles to a greater distance, so that the
air being more dilated, it rises and carries the water up
with it; which mutual repulsion of the particles of air is
increased by a mixture of common fire in the particles of
water. When air, loaded with surrounding particles of
water, is compressed by adverse winds, or by being driven
against mountains, &c, or condensed by taking away the
fire that assisted it in expanding, then the particles will ap-
proach one another, and the air with its water will descend
as a dew ; or if the water surrounding one particle of air
come in contact with the water surrounding another, they
will coalesce and form a drop, producing rain; and since
it is a well-known fact, that vapour is a good conductor
of electricity, as well as of common fire, it is reasonable to
conclude with Mr. Henley, that evaporation is one great
cause of the clouds becoming at times surcharged with
this fluid. Philos. Trans, vol. 67, pa. 134. See also vol.
55, pa. 182; or Franklin's Letters and Papers on Philo-
sophical Subjects, pa. 42, &c, and pa. 182, ed. 1769.

M.le Roi, of the Academy of Sciences at Paris, has also
advanced the same opinion, and supported it by a variety
of facts and observations in the Memoirs for the year 1751.
He shows, that water undergoes in the air a real dissolution,
forming with it a transparent mixture, and possessing the
same properties with the solutions of most salts in water ;
and that the two principal causes which promote the so-
lution of water in the air, are heat and wind ; that the
hotter the air is, within a certain limit, the more water it
will dissolve; and that at a certain degree of heat the air
will be saturated with water; and by determining at dif-
ferent times the degree of the air's saturation, he estimates
the influence of those causes on which the quantity depends
that is suspended in the air in a state of solution. Ac-
cordingly, the air, heated by evaporating substances to
which it is contiguous, becomes more rare and light, rises
and gives way to a denser air ; and, by being thus removed,
contributes to accelerate the evaporation. The fixed air
contained in the internal parts of evaporating bodies, put
into action by heat, seems also to increase their evapora-
tion. The wind is another cause of the increase of eva-
poration, chief!)' by changing and renewing the air which
immediately encompasses the evaporating substances; and
from the consideration of these two causes combined, it -
appears why the quantity of vapour raised in the night is
less than that raised in the day, since the air is then both
less heated and less agitated. To the objection urged
against this hypothesis, on account of the evaporation of
water in a vacuum, this ingenious writer replies, that the
water itself contains a great quantity of air, which gra-
dually disengages itself, and causes the evaporation ; and
that it is impossible that a space containing water which
evaporates should remain perfectly free from air. To this

EVA

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EVA

objection Dr. Dobson of Liverpool replies, that though air heavier by the cold, fall down in thin flakes of snow
appears, by unquestionable experiments, to be a chemical *'
solvent of water, and, as such, is to be considered as one
cause of its evaporation; heat is another cause, acting
without the intervention of air, and producing a copious
evaporation in an exhausted receiver ; agreeably to an ex-
periment of Dr. Irving, who says, that in an exhausted re-
ceiver water rises in vapour more copiously at" 180° of
Fahrenheit's thermometer, than in the open air at 212°, its
boiling point. Dr. Dobson further adds, that water may
exist in air in three different states; in a state of perfect
solution, when the air will be clear, dry, and heavy, and
its powers of solution still active; in a state of beginning
precipitation, when it becomes moist and foggy, its powers
of solution are diminished, and it becomes lighter in pro-
portion as its water is deposited ; and also, when it is com-
pletely precipitated, which may happen either by a slower
process, when the dissolved water falls in a drizzling rain,
or by a more sudden process, when it descends in brisk
showers. Philos. Trans, vol. 67, pa. 257 ; and Phipps's
Voyage towards the North Pole, pa. 211.

Dr. Hamilton, professor of philosophy in the university
of Dublin, transmitted to the Royal Society in 1765, a
long dissertation on the nature of evaporation, in which
he proposes and' establishes this theory of solution ; and
though other writers had been prior in their conjectures,
and even in their reasoning on this subject, Dr. Hamilton
assures us, that he has not represented any thing as new
which he was conscious had ever been proposed by any
one before him, even as a conjecture. Dr. Hamilton hav-
ing evinced the agreement between solution and evapora-
tion, concludes, that evaporation is nothing more than a
gradual solution of water in air, produced and promoted
by attraction, heat, and motion, just as other solutions are
effected.

In order to account for the ascent of aqueous vapours
into the atmosphere, this ingenious writer observes, that
the lower beds of the air being pressed by the weight of the
upper beds against the surface of the water, and conti-
nually rubbing upon it by its motion, attracts and dissolves
those particles with which it is in contact, and separates
them from the rest of the water. And since the cause of
solution in this case is the stronger attraction of the par-
ticles of water towards the air than towards each other,
those that are already dissolved and taken up, will be still
further raised by the attraction of the dry air that lies over
them, and thus they will diffuse themselves, rising gra-
dually higher and higher, and so leave the lower strata of air
not so much saturated but that it will still be able to dis-
solve and take up fresh particles of water ; which process
is greatly promoted by the motion of the wind. When
the vapours are thus raised and carried by the winds into
the higher and colder parts of the atmosphere, some of
them will coalesce into small particles, which slightly at-
tracting each other, and being intermixed with air, will
form clouds ; and these clouds will float at different
heights, according to the quantity of vapour borne up,
and the degree of heat in the upper parts of the atmo-
sphere: and thus clouds are generally higher in summer
than in' winter. When the clouds are much increased by
a continual addition of vapours, and their particles are
driven close together by the force of the winds, they will
run into drops heavy enough to fall down in rain. If the
clouds be frozen before their particles are gathered into
drops, small pieces of them, being condensed and made
Vol. I.

When the particles are formed into drops before they are
frozen, they become hailstones. When the air is replete
with vapours, and a cold breeze springs up, which checks
the solution of them, clouds are formed in the lower parts
of the atmosphere, and compose a mist or fog, which
usually happens in a cold morning, and is dispersed when
the sun has warmed the air, and made it capable of dis-
solving these watery particles. Southerly winds commonly
bring rain, because, being warm and replete with aqueous
vapours, they are cooled by coming into a colder climate ;
and therefore they part with some of them, which preci-
pitate in rain : whereas northerly winds, being cold, ami
acquiring additional heat by coming into a warmer cli-
mate, are ready to dissolve and receive more vapour than
they before contained ; and therefore are commonly at-
tended with fair weather.

Changes of the air, with respect to its density and rarity,
as well as its heat and cold, will produce contrary effects
in the solution of water, and the consequent ascent or fall
of vapours. Several experiments have been made which
prove that air, when rarefied, cannot keep so much water
dissolved in it as it does in a more condensed state; and
therefore when the atmosphere is saturated with water,
and changes from a denser to a rarer state, the high and
colder parts of it will let go some of the water before it is
dissolved, forming new clouds, and disposing them to fall
down in rain : but a change from a rarer to a denser
state will stop the precipitation of the water, and enable
the air to dissolve, either in whole or in part, some of
those clouds that were formed before, and render their par-
ticles less apt to run into drops and fall down in rain : on
this account, we generally find that the rarefied and con-
densed states of the atmosphere are respectively attended
with rain or fair weather. See more on this subject in
the Philos. Trans, vol. 55, pa. 146; or Hamilton's Philoso-
phical Essays, pa. 33. ,

Dr. Halley, before mentioned, furnished several expe-
riments on the evaporation of water ; the result of which
is contained in the following articles : 1. That water salted
to about the same degree as sea-water, and exposed to a
heat equal to that of a summer's day, did, from a circular
surface of about 8 inches diameter, evaporate at the rate
of 6 ounces in 24 hours : whence by a calculus he finds
that, in such circumstances, the water evaporates l-10th
of an inch deep in 12 hours : which quantity, he observes,
will be found abundantly sufficient to furnish all the rains,
springs, dews, &c. By this experiment, every 10 square
inches of surface of the water, yield in vapour, per diem,
a cubic inch of water : and each square foot half a wine
pint ; every space of 4 feet square, a gallon ; a mile square,
6914 tons; and a square degree, of 69 English miles, will
evaporate 33 millions of tons a day; and thus the whole
Mediterranean, which is computed to contain 160 square
degrees, will yield at least 5 280 millions of tons each
day. Philos. Trans. No. 189 ; or my Abridg. vol. 3, pa.
387- — 2. A surface of S square inches, evaporated purely
by the natural warmth of the weather, without either wind
or sun, in the course of a whole year, 16292 grains of
water, or 64 cubic inches; consequently, the depth of
water thus evaporated in one year, amounts to 8 inches.
But this being too little to answer the experiments of the
French, who found that it rained 19 inches of water in one
year at Paris ; or those of Mr. Townley, who found the
annual quantity of rain in Lancashire above 40 inches ;
3 R

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he therefore concludes, that the sun and wind con-
tribute more to evaporation than any internal heat or agi-
tation of the water. In effect, Dr. Halley fixes the an-
nual evaporation of London at 4S inches ; and Dr. Dob-
son states the same for Liverpool at 3f5| inches. Philos.
Trans, vol. 67, pt 252.

3. The effect of the wind is very considerable, on a
double account ; for the same observations show a very
singular quality in the vapours of water, viz, that of ad-
hering to the surface that exhaled them, which they clothe
as it were with a fleece of vaporous air ; which once in-
vesting the vapour, it afterwards rises in much less quan-
tity. Whence the quantity of water lost in 24 hours,
when the air is very still, was very small, in proportion to
what went off when there was a brisk gale of wind abroad
to dissipate the fleece, and make room for the emission of
vapour; and this, even though the experiment was made
in a place as secure from the wind as could be contrived.
Moreover, this fleece of water, hanging to the surface of
waters in still weather, is the occasion of very singular ap-
pearances, by the refraction of the vapours differing from
and exceeding that of common air; whence every thing
appears more lofly, or raised above its usual level, as
houses, steeples, ships, &c.

4. The same experiments show that the evaporation in
May, June, July, and August, which are nearly equal,
are about three times as great as those in the months of
November, December, January, and February. Philos.
Trans. No. 212, or my Abridg. vol. 3, pa. 6'58.

Dr. Brownrigg, in his Art of making common salt, pa.
189, fixes the evaporation of some parts of England at 738
inches during the months of May, June, July, and August;
and the evaporation of the whole year at more than 140
inches. But the evaporation of the four summer months
at Liverpool, on a medium of 4 years, was found to be
only 18-88 inches. Also Dr. Hales calculates the greatest
annual evaporation from the surface of the earth in Eng-
land- at 6-66 inches; and therefore the annual evaporation
from a surface of water, is to the annual evaporation

account of Pappus and Proclus, was born at Alexandria,
in Egypt, where he flourished and taught mathematics
with great applause, under the, reign of Ptolemy Lagos,
about 280 years before Christ. Some Arabian historians,
however, inform us, that he was born at Tyre, that his
father's name was Naucrates, an inhabitant of Damas.—
The particular place of his nativity appears, therefore, to
be uncertain ; but whether or not Alexandria had the ho-
nour of giving birth to this celebrated mathematician, all
historians agree that he flourished and taught mathema-
tics there, at the time abovemeiitioned ; which city, from
that period to the conquest of it by the Saracens, seems to
have been the residence, if not the birth-place, of all the
most eminent mathematicians of that time. Euclid re-
duced into regularity and order all the fundamental prin-
ciples of pure mathematics, which had been delivered
down by Thales, Pythagoras, Eudoxus, and other mathe-
maticians before him, and added many others of his ovvrt :
on which account it is said he was the first who reduced
arithmetic and geometry into the form of a science. He
likewise applied himself to the study of mixed mathema-
tics, particularly to astronomy and optics.

His works, as we learn from Pappus and Proclus, are
the Elements, Data, Introduction to Harmony, Pheno-
mena, Optics, Catoprics, a Treatise on the Division of
Superficies, Porisms, Loci ad Superficiem, Fallacies, and.
Four books of Conies. The most celebrated of these, is
the first work, the Elements of Geometry ; of which there
have been numberless editions, in all languages ; and a fine
edition of all his works, now extant, was printed in J 703, by.
David Gregory, Savilian professor of astronomy at Oxford.

The Elements, as commonly published, consist of 15
books, of which the last two, it is suspected, are not Euclid's,
but acomment of Hypsicles of Alexandria, who lived 200
years after him. They are divided into three parts, viz,
the Contemplation of Superficies, Numbers, and Solids:
the first 4 books treat of planes only ; the 5th of. the pro-
portions of magnitudes in general ; the 6th of the propor-
tion of plane figures; the 7th, 8th, and 9th, give us the

from the surface of the earth at Liverpool, nearly as 6 to fundamental properties of numbers ; the 10th contains the

1. Philos. Trans, vol, 6'7, ubi supra.

In the Transactions of the American Philosophical So-
ciety, vol. 3, pa. 125, there is an ingenious paper on eva-
poration, by Dr. VVistar : where he shows that evapora-
tion arises when the moist body is warmer than the medium
it is inclosed in: and, on the contrary, it acquires moisture
from the air, the colder the body is. This carrying off,
and acquiring of moisture, it is shown, is by the passage of
heat out of the body, or into it.

EUCLID, of Megara, a celebrated philosopher and
logician; he was a disciple of Socrates, and nourished
about 400 years before Christ. The Athenians having
prohibited the Jvlegarians from entering their city on pain
of death, this philosopher" disguised himself in women's
clothesto attend the lectures of Socrates. After the death of
Socrates, Plato and other philosophers went toEuclid at Me-
gara, to shelter themselves from the tyrants who governed
Athens. Euclid admitted but one chief good ; which he at
different times called God, or the Spirit, or Providence.

This philosopher has, by some, been confounded with
Euclid of Alexandria, the prince of geometricians; but
independently of the different subject of their writings, it
is well known that Euclid of Megara flourished at least a
century before Euclid of Alexandria was born.

Euclid, the celebrated mathematician, according to the

theory of commensurable and incommensurable lines and
spaces ; the 11th, 12th, 13th, 14th, and 15th, treat of the
doctrine of solids.

There is no doubt but, before Euclid, Elements of Geo-
metry were compiled by Hippocrates of Chios, Eudoxus,
Leon, and many others, mentioned by Proclus in the be-
ginning of his second book; for he affirms that Euclid new-
ordered many things in the Elements of Eudoxus, com-
pleted many things in those of Theatetus, and besides
strengthened such propositions as before were' WO slightly,
or but superficially established, with the most firm and
convincing demonstrations. '

History is silent as to the time of Euclid's death, or his
age. But Pappus represents him as a person of a coui-
teotis and agreeable behaviour, and in great esteem with
Ptolemy Lagos, king of Egypt; who one day asking him,
whether there was not any shorter way of coming at geo-
metry than by his Elements, Euclid is said to have an-
swered, " that there was no royal road to geometry."

EUDIOMETEB, an instrument for determining the
purity of the, air, or the quantity of pure and dephlogisti-
cateii or vital air contained in it, chiefly by means of its.
diminution on a mixture with nitrous air.

Instruments of this kind were first made, in consequence
of the experiments and discoveries of Dr. Priestley, for de-

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I 49i ]

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termining the salubrity of different kinds of air. That wri- ing the two airs. — 2. A small glass phial, to serve as a

tor having discovered, that when nitrous air is mixed with measure, and is about one-third the size of the receiver. —

any other air, their original bulk is diminished; and that 3. A small pair of scales that may weigh very exactly.- —

the diminution is nearly, if not exactly, in proportion to 4. Several glass bottles, for containing the nitrous or

its salubrity ; he was hence led to adopt nitrous air as a other air to be used, and which may supply the place of

true test of the purity of respirable air; and nothing more the recipient when broken. The method of using this in-

seemed to be necessary but an easy, expeditious, and ac-
curate method of estimating the, degree of diminution in
different cases ; and for this purpose, the eudiometer was
contrived ; of which several kinds have been invented, the
principal of which are the following.

' I. The eudiometer originally used by Dr. Priestley is a
divided glass tube, into which, after having filled it with
common water, and inverted it into the same, one mea-

strument is as follows: — The receiver is to be- filled with
water, closed exactly with its glass stopper, wiped dry on
the outside, and then weighed very accurately. Being*"
then immerged in a vessel of water, and held with the
mouth downwards, the stopple is removed., and, by means
of a funnel, two measures of common and one of nitrous
air are introduced into it, one after another: these, dimi-
nish as soon as they come into contact; in consequence of

sure or more of common air, and an equal quantity of ni- which the water enters the recipient in proportionable
trous air, are introduced by means of a small phial, which quantity. After being stopped and well shaken, to pro-
is called the measure; and thus the diminution of the vo- mote the diminution, the receiver is to be opened again
lume of the mixture, which is seen at once by means of the under water ; then stopped and shaken again, and so on
graduations of the tube, instantly discovers the purity of for three times successively, after which the bottle is stopped
the air required. ■ for the last time under water, then taken out, wiped very

II. The discovery of Dr. Priestley was announced to clean and dry, and exactly weighed as before. It is plain
the public in the year 1772; and several persons, both at that now, the bottle being filled partly with elastic fluid
home and abroad, presently availed themselves of it, by and partly with water, it must be lighter than when quite
framing other more accurate instruments. The first of full of water; and the difference between those two weights,
these was contrived by M. Landriani; a description of shows nearly what quantity of water would fill the space
which is given in the 6th volume of Rosier's Journal, for occupied by the diminished elastic fluid. But, in making
the year 1775- It consists of a glass tube, fitted, by grind- experiments with airs of different degrees of purity, the
ing, to a cylindrical vessel, to which are joined two glass said difference will be greater when the diminution is less,
cocks and a small bason; the whole being fixed to a wooden or when the air is less pure, and vice versa; by which
frame. In this instrument quicksilver is used instead of means the comparative purity between two different kinds
water; though that is attended with an inconvenience, be- of air is determined.

cause the nitrous air acts upon the metal, and renders the VI. But as this method, notwithstanding the encomi-

«xperiment ambiguous. urns bestowed on it by the inventor, is subject to several

III. In 1777, Mr. Magellan published an account of errors and inconveniences; to remedy all these, another
three eudiometers invented by himself, consisting of glass instrument was invented by M. Cavallo; the description
vessels of rather difficult construction, and troublesome of which, being long, may be seen in his Treatise on the
to use. M. Cavallo observes, that the construction of all Nature and Properties of Air, pa. 344.

the three is founded on a supposition, that the mixture of Other constructions of the eudiometer have also been
nitrous and atmospherical air, having continued for some given by Mr. Cavendish and Mr. Scheele. For further in-
time to diminish, afterwards increases again; which it seems formation, see Magellan's Letter to Dr. Priestley, contain-
is a mistake: nor do they give accurate or uniform results, ing the Description of a Glass Apparatus, &c, and of New
1 in experiments made with nitrous and common air of pre- Eudiometers, &c, 1777, pa. 15, &c; Priestley's Exp. and
cisely the same quality. Obs. on Air, vol. 3, preface and appendix ; the methods of

IV. A preferable and very accurate method of discover- Dr. Ingenhousz in Philos. Trans, vol. 66, art. 15 ; see also
ing the purity of the air by means of a eudiometer, is re- the Philos. Trans, vol. 73 ; and Cavallo's Treatise on Air,
commended by M.Fontana. The instrument is originally pa. 274, 315, 3 16, 317, 328, 340, 344, and 348.
nothing more than a divided glass tube, though the inven- EUDOXUS, of Cnidus, a city of Caria in Asia Minor,
tor afterwards added to it a complicated apparatus, per- flourished about 370 years before Christ. He learned geo-
haps of little or no use. The first simple eudiometer con- metry from Archytas, and afterwards travelled into Egypt
sisted only of a glass tube, uniformly cylindrical, about 18 to learn astronomy and other sciences. There he and
inches long, and 3-4ths of an inch diameter within-side, Plato studied together, as Laertius informs us, for the
the outside being marked with a diamond at such dis- space of 13 years; and afterwards came to Athens, fraught
tances as are exactly filled by equal measures of elastic with all sorts of knowledge, which they had learned from
fluids: and when any parts of these divisions are required, the priests. Here Eudoxus opened a school; which he
the edge of a ruler, divided into inches and smaller parts, supported with so much glory and renown, that even Plato,
is held against the tube, so that the first division of the though his friend, is said to have envied him. Eudoxus
ruler may coincide with one of the marks on the tube, composed Elements of Geometry, from which Euclid li-
The nitrous and atmospherical air are introduced into this berally borrowed, as mentioned by Proclus. Cicero calls
tube, in order to be diminished, and thence the purity of Eudoxus the greatest astronomer that had ever lived : and
the atmospheric air ascertained. Petronius says, he spent the latter part of his life upon the

V. M. Saussure of Geneva also invented a eudiometer, top of a very high mountain, that he might contemplate the
which he thinks is more exact than any of those before stars and the heavens with more convenience and less in-
described; the apparatus of which is as follows: I. A cy- terruption : and we learn from Strabo, that there were
lindrical glass bottle, with a ground stopple, containing some remains of his observatory at Cnidus lo be seen even
about 5{ ounces of air, which serves as a receiver for mix- in his time. He died in the 53d year of his age.

3R 2

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EVECTION, is used by some astronomers for the li-
bration of the moon; being an inequality in her motion,
by which, at or near the quadratures, she is not in a line
drawn through the centre of the earth to the sun, as she is
at the syzygies,or conjunction and opposition, but makes
an angle with that line of about 2° Si'. The motion of
the moon about her axis only is equable, which rotation is
performed exactly in the same time as she revolves about
the earth; for which reason it is that she turns always the
same face towards the earth nearly, and would do so
exactly were it not -that her menstrual motion about the
earth, in an elliptic orbit, is not equable; on which ac-
count the moon, seen from the earth, appears to liberate, a
little upon her axis, sometimes from east to west, and
sometimes from west to east ; or some parts in the eastern
limb of the moon go backwards and forwards a small space,
and some that were conspicuous, are hid, and then appear
again.

The term Evection is used by some astronomers to
denote that equation of the moon's motion, which is pro-
portional to the sine of double the distance of the moon
from the sun, diminished by the moon's anomaly : this
equation is not yet accurately determined ; some state
it at 1° 30', others at 1° 16', &c. It is the greatest of all
the moon's equations, except the equation of the centre.

EVEN Number, is that which can be divided into two
equal whole numbers; such as theseries of alternate num-
bers, 2, 4, 6, 8, 10, &c, and is therefore properly expressed
by the form In.

EVENLY Even Number, is that which can be twice di-
vided by 2, and therefore is expressed by the form 4ra.

Evenly Odd Number, is that which can be only divided
by 2 once, the quotient being an odd number, and is
therefore of the form 4n -+- 2.= 2 {m+ 1), that is, the quo-
tient is of the form 2/J-+-1, which is therefore necessarily
an odd number.

EVERARD's Sliding Rule, a particular sort of rule in-
vented by Mr. Thomas Everard, for the purpose of gaug-
ing. See Sliding Rule.

EULER (Leonard), one of the most celebrated ma-
thematicians of the 18th, or perhaps of any other century.
He was a native of Basil, and was born April 15, 1707.
The years of his infancy were passed at Richen, where his
father was minister. He was afterwards sent to the uni-
versity of Basil; and as his memory was astonishingly re-
tentive, and his application regular, he performed his aca-
demical tasks with great rapidity; and all the time that he
saved by this, was consecrated to the study of mathema-
tics, which soon became his favourite science. The early
progress he made in this branch of study, added fresh ar-
dour to his application ; by which he likewise obtained a
distinguishing mark of the attention and esteem of professor
John Bernoulli, who was then one ofthe most eminent ma-
thematicians in Europe.

In 1723, M. Euler took his degree as master of arts;
and delivered on that occasion a Latin discourse, in which
he drew a comparison between the philosophy of Newton
and the Cartesian system, which was received with the
greatest applause. At his father's desire, he next applied
himself to the study of tluvlogy and the oriental languages:
and though these studies were foreign to his predominant
propensity, his success was considerable even in this re-
spect: however, with his father's consent, he afterwards
returned to mathematics as his principal object. In con-
tinuing to avail himself of the counsels and instructions of

M.Bernoulli, he contracted an intimate friendship with his
two sons Nicholas and Daniel; and it was chiefly in con-
sequence of these connexions that he afterwards became
the principal ornament of the philosophical world.

The project of erecting an academy at Petersburg, which
had been formed by Peter the Great, was executed by
Catharine the 1st ; and the two young Bernoullis being
invited to Petersburg in 1725, promised Euler, who was
desirous of following them, that they would use their en-
deavours to procure for him an advantageous settlement
in that city. In the mean time, by their advice, he made
close application to the study of philosophy, to which he
made happy applications of his mathematical knowledge,
in a dissertation on the nature and propagation of sound,
and an answer to a prize question concerning the masting
of ships; to which the Academy of Sciences adjugded the
accessit, or second rank, in the year 1727- From this lat-
ter discourse, and other circumstances, it appears that
Euler had very early embarked in the curious and useful
study of naval architecture, which he. afterward enriched
with so many valuable discoveries. The study of mathe-
matics and philosophy however did not solely engage his
attention, as he in the mean time attended the medical and
botanical lectures of the professors at Basil.

Euler's merit would have given him an easy admission to
honourable preferment either in the magistracy or univer-
sity of his native city, if both civil ami academical honours
had not been there distributed by lot. This being against
him in a certain promotion, he left his country, set out for
Petersburg, and was made joint professor with his coun-
trymen Hermann and Daniel Bernoulli, in the university
of that city.

At his first setting out in his new career, he enriched the
academical collection with many memoirs, which excited
a noble emulation between him and the Bernoullis ; an
emulation that always continued, without either degene-
rating into a selfish jealousy, or producing the least altera-
tion in their friendship. It was at this time that he car-
ried to new degrees of perfection the integral calculus, in-
vented the calculation by sines, reduced analytical opera-
tions to a greater simplicity, and thus was enabled to
throw new light on all the parts of mathematical science.

In 1730, Euler was promoted to the professorship of
natural philosophy; and in 1733 he succeeded his friend
D. Bernoulli in the mathematical chair. In 1735, a pro-
blem was proposed by the academy, which required ex-
pedition, and for the calculation of which some eminent
mathematicians had demanded the space of some months.
The problem was undertaken by Euler, who completed
the calculation in three days, to the great astonishment of
the academy : but the violent and laborious efforts that he
made to accomplish it, threw him into a fever, that en-
dangered his life, and deprived him of the use ol his right
eye, which afterwards brought on a total blindness.

The Academy of Sciences at Paris, which in 1738 had
adjudged the prize to his memoir Concerning tin- Nature
and Properties of Fire, proposed for the year 1740 the im-
portant subject of the Tides of the Sea; a problem whose
solution comprehended the theory of the solar system, and
required the most arduous calculations. Kuler's solution
of this question was adjudged a master-piece of analysis
and geometry; and it was more honourable for him to
share the academical prize with such illustrious competi-
tors as Maclaurin and Daniel Bernoulli, than to have
carried it away from rivals of less magnitude.' Seldom, if

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ever, did such a brilliant competition adorn the annals of gree, which are furnished by mathematical principles, he

the academy; and perhaps no subject, proposed by that
learned body, was ever treated with such force of reason-
ing and accuracy of investigation, as that which here dis-
played the philosophical powers of this extraordinary tri-
umvirate.

In the year 1741, Euler was invited to Berlin to direct
and assist the academy that was there rising into fame.
On this occasion he enriched the last volume of the Mis-
cellanies (Melanges) of Berlin with five memoirs, which
form an eminent, perhaps the principal, figure in that col-
lection. These were followed, with amazing rapidity, by
a great number of important researches, which are dis-
persed through the memoirs of the Prussian academy ; a
volume of which has been regularly published every year
since its establishment in 1744. The labours of Euler
will appear more particularly astonishing, when it is con-
sidered, that while he was enriching the academy of Berlin
with a profusion of memoirs, on the deepest parts of ma-
thematical science, contained always under .some new
points of view, and often replete with sublime truths, and
sometimes discoveries of great importance ; he still con-
tinued his philosophical contributions to the Petersburg
academy, whose memoirs display the marvellous fecun-
dity of his genius. /

It was with great difficulty that this extraordinary man,
in 176'6, obtained permission from the king of Prussia to
return to Petersburg, where he wished to pass the re-
mainder of his days; but soon after his return, which was
graciously rewarded by the munificence of Catharine the
2d, he was seized with a violent disorder, which ended in
the total loss of his sight: a cataract, formed in his left
eye, which had been essentially damaged by the loss
of the other. eye, and too close an application to study,
deprived him entirely of the use of that organ. It was in
this distressing situation that he dictated to his servant, a
tailor's apprentice, who was absolutely devoid of mathe-
matical knowledge, his Elements of Algebra ; which by
their intrinsic merit in point of perspicuity and method,
and the unhappy circumstances in which they were com-

reduced them to the three ordinates, which determine the
place of the moon: and he divided into classes all the-in-
equalities of that planet, as far as they depend either on
the elongation, of the sun and moon, or on the excentri-
city, or the parallax, or the inclination of the lunar orbit.
All these means of investigation, employed with such art
and dexterity as could only be expected from a genius of
the. first order, were attended with the greatest success ;
and it is impossible to observe without admiration, such
immense calculations on the one hand, and on the other
the ingenious methods employed by this great man to
abridge them, and to facilitate their application to the real
motion of the moon. But this admiration will become
astonishment, when we consider at what period and in
what circumstances all this was effected. It was when our
author was totally blind, and consequently obliged to ar-
range all his computations by the sole powers of his me-
mory and his genius : it was when he was embarrassed in •
his domestic affairs by a dreadful fire, that had consumed
great part of his substance, and forced him to quita ruined
house, every corner of which was known to him by habit,
which in some measure supplied the want of sight. It
was in these circumstances that Euler composed a work
which alone was sufficient to render his name immortal.

Some time after this, the famous oculist Wentzell, by
couching the cataract, restored sight to our author; but
the joy produced by this operation was of short duration.
Some instances of negligence on the part of his surgeons,
and his own impatience to use an organ, whose cure was
not completely finished, deprived him a second time and
tor ever of his sight : a relapse which was also accompanied
with tormenting pain. With the assistance of his sons,
however, and of Messrs. Krafft and Lexell, he continued
his labours : neither the infirmities of old age, the loss
of his sight, nor the acutencss of the pain, could quell the
ardour of his genius. He had engaged to furnish the Aca-
demy of Petersburg with as many memoirs as would be
sufficient to complete its acts for 20 years after his death.
In the space of 7 years he transmitted to the Academy

posed, have equally excited the wonder and applause of above 70 memoirs, and above 200 more, left behind him
the learned. This work, though purely elementary, plainly were revised and completed by a friend. Such of these

exhibits the proofs of an inventive genius; and it is pe
haps here alone that we meet with acomplcte theory of the
Diophantine analysis.

About, this time M. Euler was honoured by the Aca-
demy of Sciences at Paris with the place of one of the
foreign members of that learned body; after which, the
academical prize was adjudged to three of his memoirs,
Concerning the Inequalities in the Motions of the Planets.
The two prize questions proposed by the same Academy
for 1770 and 1772 were designed to obtain from the la-
bours of astronomers a more-perfect Theory of the Moon.
M. Euler, assisted by his eldest son, was a competitor for
these prizes, and obtained them both. In this last me-
moir, he reserved fur further consideration several inequa-
lities of the moon's motion, which he could not determine
in his first theory, on account of the complicated calcula-
tions in which the method he then employed had engaged
him. He afterwards revised his whole theory, with the
assistance of his son and Messrs. Krafft ami Lexell; and pur-
sued his researches till he had constructed the new tables,
which appeared, together with the great work, in 1772.
Instead of confining himself, as before, to the fruitless in-
tegration of three differential equations of the second de-

memoirs as were of ancient date were separated from the
rest, and form a collection that was published in the year
1783, under the title of Analytical Works.

The general knowledge of our author was more exten-
sive than could well be expected, in one who had pursued,
with such unremitting ardour, mathematics and astronomy
as his favourite studies. He had made a very consider-
able progress in medical, botanical, and chemical science.
What was still more extraordinary, he was an excellent
scholar, and possessed in a high degree what is generally
called erudition. He had attentively read the most emi-
nent writers of ancient Rome; the civil and literary hisr
tory of all ages and all nations was familiar to him ; and
foreigners, who were only acquainted with his works, were
•astonished to find in the conversation of a man, whose
long life seemed solely occupied in mathematical and phy-
sical researches and discoveries, such an extensive ac-
quaintance with the most interesting branches of litera-
ture. In this respect, no doubt, he was much indebted to
a very uncommon memory, which seemed to retain every
idea that was conveyed to it, either from reading or from
meditation. He could repeat the jEneid of Virgil, from
the beginning to the end, without hesitation, and indi-

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cate ttie first and last line of every page of the edition he
used. _

Several attacks of a vertigo, in the beginning of bep-
tember 1783, which did not prevent his computing the
motions of the aerostatic globes, were however, the fore-
runners of his mild passage out of this life. While he was
amusing himself at tea with one of his grand-children, he
was struck with an apoplexy, which terminated his illus-
trious career at 70' years of age.

M. Euler's constitution was uncommonly strong and
vigorous. His health was good; and the evening of his
long life was calm and serene, sweetened by the fame that
follows genius, the public esteem and respect that are
never withheld from exemplary virtue, and several domes-
tic comforts, which he was capable of feeling, and therefore
deserved to enjoy.

The catalogue of his works has been printed in 50 pages,
14 of which contain the manuscript works. — The printed
ones consist of works published separately, and works to
be found in the memoirs of several Academies, viz, in 38
volumes of the Petersburg Acts (from 6 to 10 papers in
each volume); — in several volumes of the Paris Acts; —
in 2(5 volumes of the Berlin Acts (about 5 papers to each
volume); — in the Acta Eruditorum, in 2 volumes ;— in
the Miscellanea Taurinensia; — in vol. 9 of the Society of
Ulyssingue; — in the Ephemerides of Berlin ; — in the Me-
moires de la Societe (Economique for 1766; — and in the
Philos. Trans, by seven memoirs, from vol. 44 to vol. 62.

EVOLVENT, in the Higher Geometry, a term used by
some writers for the involute, or curve resulting from the
evolution of a curve, in contradistinction to that evolute,
or curve supposed to be opended or evolved. See Evo-
lute, and Involute.

EVOLUTE, in the Higher Geometry, a curve first pro-
posed by M. Huygens, and since much studied by many
eminent mathematicians. It is any curve supposed to be
evolved or opened, by having a thread wrapped close upon
it, fastened at one end, and beginning to evolve or unwind
the thread from the other end, keeping stretched the part
evolved, or wound off; then this end of the thread will
describe another curve called the Involute. Or the same
involute is described in the contrary way, by wrapping the
thread upon the evolute, keeping it always stretched.

'Thus, if efgh be any curve, and ae either a part of
the curve, or a right line ; then if a thread be wound close
upon the curve from a to h, where it is fixed, and then
be unwound from a ; the curve aefgh, from which it is
evolved, is called the evolute; and the other curve abcd,
described by the end of the thread, as it evolves or un-
winds, is the involute. Or, if the thread hd, fixed at 11,
be wound or wrapped upon the evolute hgfea, keeping
it always stretched, as at the several positions of it hd,
gc, fb, ea, the extremity will describe the involute curve

DCBA.

From this description it appears, 1. That the parts of
the thread at any positions, as ea, fb, gc, hd, &c, are
radii of curvature, or osculatory radii, of the involute
curve, at the points a, b, c, d.

2. The same parts of the thread are also equal to the
corresponding lengths ae, aef, aufg, &c, of the evolute ;
that is,

ae = ae is the rad. of curvature to the point a.

BF = AF - ■ -. - B,

CG = AG - - C,

DH = AH - - D.

3. Any radius of curvature bf, is perpendicular to the
involute at the point b, and is a tangent to the evolute.
curve at the point f, — 4. The evolute is the locus of the
centre of curvature of the involute curve.

The finding the radii of evolutes, is a matter of gTeat
importance in the higher speculations of geometry ; and
is even sometimes useful in practice; as is shown by
Huygens, the inventor of this theory, in applying it to the
pendulum. Horol. Oscil. part 3. For the doctrine of the
oscula of evolutes we are indebted to M. Leibnitz, who
firstshowed the use of evolutes in the measuring of curva
tures.

Tofind the Evolute and Involute Curves, the one from
the other. — For this purpose, put
2 = ab the involute curve,
x = ad the absciss of the involute,
y = db its ordinate,
r = BC its radius of curvature,
0 = ef the absciss of the evolute,
u z=. fc its ordinate, and
a = ae a given line (fig. 2 above).

Then, by the nature of the radius of curvature, it is

r — ... ~ . = bc = ae -+- ec; also by sim. triangles,

yx — xy J °

■ J

JS-xy'

Hence ef = gb — db= . .. ~ . .. — y — v ; and
yx-xy *

fc = ad — ae ■+- gc = x — a -+■ . .; ~ . . = u\ which are
yx — xy

the values of the absciss and ordinate of the evolute curve

ec ; which therefore may be found when the involute is

given.

On the other hand, if v and'w, or the evolute be given :

then, putting the given curve ec = *; since cb = ae -+• ec,

or r = a -+■ s, this gives ?• the radius of curvature. Also,

by similar triangles, there result these proportions, viz,

!■■£> a + s .

sivi'.r: -T- = — — ?) = cb,

•V 5

r'u a + s . , ,

* : u : : r : — = « = gc ; therefore

ad =s ae -+- fc

g c = a ■+• u —

and db = cr, — ef = —. — v — v = y; which arc the

absciss and ordinate of the involute curve, and which may
therefore be found, when the evolute is given. It may
also bc observed that sz = vz -+- B*, and ~- = a1 + y1. And
either of the quantities x, y, may be supposed to flow
equably, in which case the respective second fluxion x or
y will be nothing, and the corresponding term in the de-
nominator )'x — xy will vanish, leaving only the other

E V O

[ 495 ]

E X C

term in it; which will have the effect of rendering the ope-
ration simpler.

For Example. If it be required to find the evolution
EC when the given involute ab is the common parabola,
whose equation hpx=y2, the parameter being jj.

Here y=s/px,j = \*»/f-> and y== ^rv^ making
£=0. ^Then, to find first ae the radius of curvature of
the parabola ab at the vertex, when x = 0, the general
value of the radius of curvature above given becomes

y &0

_■£_ _ O +>' ) 2_ . su(3stituting the value ofi and
-xy -xy

3
(P-

Wv

which is the general value of ror bc, the

radius of curvature, for any value of x or ad; and wherix
or ad is = 0 or nothing, the value of r, or ae, becomes

then a =— ^7- — kV onlv; or half tne parameter of the
axis is the radius of curvature at the vertex of the para-
bola.

Again, in the general values of v and u ahove given, by
substituting the values of y, y, and %, also 0 for x, and ?p
for a; those quantities become

-.?•=?■

y*

and

u = 1 — a -+- -t—rr. = 3x ■+- \p — a = 3r.
—xy ^

Hence then, comparing the values of x*in both of these,
we have x3 = -^ pv ~ = -fa vr, and conseq. Tipv1 == 1 6V ;
which is the equation between the absciss and ordinate of
the evolute curve EC, showing it to be the semicubical
parabola.

In like manner the evolute to any other curve is found.
The evolute to the common cycloid, is an equal cycloid;
a property first demonstrated by Huygens, and which he
used as a contrivance to make a pendulum vibrate in the
curve of a cycloid. See his Horolog. Oscil. See also,
on thesubject of evolute andinvolute curves, the Fluxions
of Newton, Maclaurin, Simpson, DelTIopital, &c ; Wolf.
Elem. Math. torn. 1, &c, &c.

M. Varignon has applied the doctrine of the radius of
the evolute, to that of central forces; so that having the
radius of the evolute of any curve, there may be found
the value of the central force of a body; which, moving
in that curve, is found in the same point where that ra-
dius terminates; or reciprocally, having the central force
given, the radius of the evolute may be determined.
Hist.de l'Acad. an. 1 700'.

The variation of curvature of the line described by the
evolution of a curve, is measured by the ratio of the radius
of curvature of the evolute, to the radius of curvature of
the line described by the evolution. See Maclaurin's
Flux. art. 402, prop. 36.

Imperfect Evoeute, a name given by M. Reaumur to
a new kind of evolute. Mathematicians had hitherto only
considered the perpendiculars let fall from the involute
on the convex side of the evolute: but if other lines not
perpendicular be drawn upon the same points, provided
they be all drawn under the same angle, the effect will
Still be the same; that is, the oblique lines will all inter-
sect in the curve, and by their intersections form (he infi-

nitely small sides of a new curve, to which they would
be so many tangents. — Such a curve is a kind of evolute,
and has its radii ; but it is an imperfect one, since the
radii are not perpendicular to the first curve, or involute.
Hist, de l'Acad. &c, an. 1709.

EVOLUTION, in Arithmetic and Algebra, denotes
the extraction of roots. In which sense it stands opposed
to involution, which is the raising of powers.

Evolution, in Geometry, the opening, or unfolding
of a curve, and making it describe an evolvent.

The equable evolution of the periphery of a circle, or
other curve, is such a gradual approach of the circum-
ference to rectitude, as that its parts do all concur, and
equally evolve or unbend ; so that the same line becomes
successively a less arc of a reciprocally greater circle ;
till at last they change into a straight line.. In the Philos.
Trans. No. 260, a new quadratrix to the circle is found by
this means, being the curve described by the equable evo-
lution of its periphery.

EURYTHMY, in Architecture, Painting, and Sculp-
ture, is a kind of majesty, elegance, and easiness appearing
in the composition of certain members or parts of a body,
building, or painting, and resulting from the fine and exact
proportions of them.

EUSTYLE, is the best manner of placing columns,
with regard to their distance; which, according to Vitru-
vius, should be 4 modules, or 1\ diameters.

EUTOCIUS, a respectable Greek mathematician, of
Ascalon, in Palestine, about the year of Christ 550. He
was one of the most considerable mathematicians that
flourished about the decline of the sciences among the
Greeks, and had for his preceptor Isidorus, the principal
architect of the church of St. Sophia at Constantinople.
He is chiefly known however by his commentaries on the
works of the two ancient authors, Archimedes and Apol-
lonius. Those two commentaries are both excellent com-
positions, to which we owe many useful observations in
the history of the mathematics.

His commentaries on Apollonius are published in
Halley's edition of the works of that author; and those
on Archimedes, first in the Basle edition, in Greek and.
Latin, in 1543, and since in some others, as the late
Oxford edition, in 1792. Of these commentaries, those
rank the highest, which illustrate Archimedes's work on
the Sphere and Cylinder ; in one of which we have a re-
cital of the various methods practised by the ancients in
the solution of the Delian problem, or that of doubling
the cube. The others are of less value ; though it cannot
but be regretted that Eutocius did not pursue his plan of
commenting on all the works of Archimedes, with the
same attention and diligence which he employed in his
remarks on the sphere and cylinder.

EXCENTRIC, is applied to such figures, circles*
spheres, &c, as have not the same centre ; as opposed toi
concentric, which have the same centre.

Excentiuc, or Excentric Circle, in the ancient Ptole-
maic Astronomy, was the very orbit of the planet itself
which it was supposed to describe about the earth, and
which was conceived excentric with it; called also the
deferent. But instead of these excentric circles round
the earth, the planets are now known to describe, elliptic
orbits about the sun ; which accounts for all the irregulari-
tiets of their motions, and their various distances from thfr
eaith, ike, more justly and naturally.

£ X C

C 496 ]

E X C

Excentric, or Excentric Circle, in the ModernAstro-
nomy, is the circle described from the centre of the orbit
of a planet, with halt the greatest axis as a radius ; or it
is the circle that circumscribes the elliptic orbit of the
planet ; as the circle aqb.

Excentric Anomaly, or Ano-
maly oflhe Centre, is an arc aq of
the excentric circle, intercepted
between the aphelion a, and the
right line qh, drawn through the
centre p of the planet perpen-
dicular to the line of the apses

AB.

Excentric Equation, in the
Astronomy of the ancients, is an
angle made by a line drawn from the centre of the earth,
with another line drawn from the centre of the excentric,
to the body or place of any planet. This is the same with
the prosthapheresis ; and is equal to the difference, ac-
counted in an arch of the ecliptic, between the real and
apparent place of the sun or planet. See Equation of
the Centre.

Exc-entric Place of a planet, in its orbit, is the helio-
centric place, or that in which it appears as seen from the
sun.

Excentric Place in the ecliptic, is the point of the
ecliptic to which the planet is referred as viewed from the
sun'; and which coincides with the heliocentric longitude.
EXCENTRICITY, is the distance between the centres
of two circles, or spheres, which have not the same
centre.

Excentricity, in the Old Astronomy, is the distance
between the centre of a planet and the centre of the earth.
— That the planets have such an excentricity, is allowed
on all sides, being evijient from various circumstances ;
and particularly from this, that the planets at some times
appear larger, and at other times less ; which can only
proceed from hence, that their orbits being excentric to
the earth, in some parts of those orbits the planets are
nearer to us, and in others more remote. And as to the
excentricities of the sun and moon, it is thought they are
sufficiently proved, both from eclipses, from the moon's
greater and less parallax at the same distance from the
zenith, and from the sun's continuing longer by 8 days in
the northern hemisphere than in the southern one. ,

Excentricity, in the new Astronomy, is the distance
cs between the sun s and the centre c of a planet's orbit ;
or the distance of the centre from the focus of the elliptic
orbit ; called also the Simple or Single Excentricity.

When the greatest equation of the centre is given, the
excentricity of the earth's orbit may be found by the fol-
lowing proportion ; viz,

As the diameter of a circle in degrees,
Is to the diameter in equal parts;
So is the greatest equation of the centre in degrees,
To the excentricity in equal parts. Thus,
Greatest equat. of the cent. 1° 55' 33"= 1°-925S333 &c.
The diam. of a circ. being 1, its circumf. is 3-1 415.926'.
Then 3-1415926 : 1 : :36(')°: 114°-5yi56'09 diam. in deg.
And 114-59156'Of): 1 :: 1-9258333: 0-016806, the ex.
Hence, by adding this to 1, and subtracting it from 1,
gives roi6806 =: as the aphelion distance, and 0'9S3194
. = bs the perihelion distance.
See Robertson's Elcm of Navig. book 5, pa. 286.

Otherwise, thus : Since it is found that the sun's great-
est apparent semi-diameter is to his least, as 32' 43" to
31' 3S",oras 1963" to I898"; the sun's greatest distance
from the earth will be to his least, or as to SB, as I963
to 1898; of which,

the half dif. is 32* = cs,
and half sum 1930f =cb; therefore,
as 1930i : 32 J : : 1 : :Ol6S35 = cs the excentricity, to the
mean distance or semi-axis 1 ; which is nearly the same a6
before.

The excentricities of the orbits of the several planets,
in parts of their own mean distances 1000, and also in
English miles, are as below, viz, the excentricity of the
orbit of

Parts. Miles.

Mercury - - 210 - - 7,730,000

Venus - - 7 - - 482,000

Earth - - 17 - - 1,618,000

Mars - - 93 - - 13,486,000

Vesta - - 36 - - 8,112,500

Juno - - 254 - - 64,315,000

Pallas - - 246' - - 73,600,000

Ceres - - 78 - - 20,615,000

Jupiter - - 48 - - 23,760,000

Saturn - - 55 - - 49,940,000

Herschel - - 47f - - 86,000,000

Or the excentricities of the orbits in parts of their own

mean distances 1, and the secular variations of the same,

according to Laplace, are as below i

Excentr. Sec. Variat.

Mercury - - 020551494 - - 0-000003S7

Venus - - 0-00685298 - - 0-00006371

Earth - - 0-0l6S351S - - 0-0000416S

Mars - - 009313400 - - 0.0000901 8

Jupiter - - 0-04S17S40 - - 000015935

Saturn - - 0'056l6S30 - - 0-00031240

Herschel 1 _ 0.04gg7030 . . 0-00002507
or Uranus }

Excentricities of the Orbits of the new or Telescopic
Planets, are,

Piazzi, or Ceres (discovered 1801) - - 0078349
Olbers, or Pallas ( - - 1S02) - - 0'2453S4
Harding, or Juno ( - - 1S04) - - 0-254944
Ditto, according to Burckhaidt - - 0-25096
Olbers 2d, or Vesta (discov. 1807) - - 0-093220
The excentricity of the moon's orbit is 0*0548553, the
greater semi-axis being 1.

Double Excentricity, is thedistance between thctwo
foci of the elliptic orbit, and is equal to double the single
excentricity above given.

EXCESS (Spherical) in Trigonometry, is the excess of
the sum of the three angles, of any spherical triangle,
above 2 right angles. Let, a, b, C, be the angles of a sphe-
rical triangle; and a, b, c, the sides opposite to the angles
a, b, c, respectively ; 11= 2 right angles, and r = the ra-
dius of the sphere; then the spherical excess (ce) is ascer-
tained by the following theorem, first given by Simon Lhuil-
lier of Geneva: Tan. |i, or tan. |(a ■+- b + c- n) =
1 o + l+t a + b — c a + c— b b + c — a.

- v'Oan. r±-±-. tan.-^-r-. tan. — — . tan. — — ),

1 s s.— a s — b * — r \ ,

or =-V (tan. -.tan.— .tan. — . tan. a ); where

j_i(h + S -t- c), half the sum of the sides. M. Puissant
gives the following theorems for the excess:

»=(

E X H

-) ars'm. 2c -4- ( — ) c1. sin. 2a, or

[ 497 ]

EXP

i — (—) bk ; where b = the base, and h — the height of

the triangle, also r"= the number of seconds contained
in radius.

In a memoir on this by Burckhardt, in vol. 7 of the
Memoirs of the National Institute of France, it is shown
that the same propertyequally applies to such small trian-
gles on the spheroidal surface of the earth as occur in the
usual geodctical operations ; as no appreciable error will
arise by considering such triangles as spherical ones, nor
in any whose sides are small in comparison with a great
circle of the sphere. It follows, says the author, that
the small triangles traced on the surface of a spheroid
may be calculated on the same principles as such trian-
gles traced on the surface of a sphere ; and both of therri
may be reduced to rectilinear triangles, by diminishing
their angles, each by a quantity equal to one-third of the
area of a similar triangle, described on a sphere whose

liquid bodies; and the term exhalation to the dry ones
emitted from solid bodies; as earth, minerals, &c. In
this sense, exhalations are dry and subtile corpuscles, or
effluvia, loosened from hard terrestrial bodies, either by
the heat of the sun, or the action of the air, or some other
cause; being emitted upwards to a certain height in the
atmosphere, where, mixing with the vapours, they help to
constitute clouds, and return back in dews, mists, rains,
&c. Sir Isaac Newton thinks, that true and permanent
air is formed from the exhalations raised from the hardest
and most compact bodies.

EXHAUSTED Receiver, is a glass, or otiier vessel,
applied on the plate of an air-pump, to have the air
extracted out of it by the action of the pump. — Things
placed in such an exhausted receiver, are said to be in
vacuo.

EXHAUSTIONS, or the Method of Exhaustions,
is a mode of demonstration founded on the principle of
exhausting a quantity by continually taking away certain
parts of it. Themethod of exhaustions was frequently re-

radius is = 1.— The solution of spheroidal triangles, whose soned to by the ancient mathematicians ; as Euclid, Archi

»ides are small with regard to the dimensions ofthe sphe^
Toid, depends therefore immediately on that of rectilinear
triangles ; not only when the spheroid is elliptical and of
revolution, but when it is in any manner irregular, pro-
vided only that it differs but little from a sphere.

EXCH -NGE, in Arithmetic, is the bartering or ex-
changing the money of one place for that of another ; or
finding what quantity of the money of. one place is equal to
agiven sum of another, according to a certain course of ex-
change.—The several operations in this rule are only dif-
ferent applications of the Rule of Three: which may be
seen in most books of arithmetic.

Arbitration of Exchange, is the method of remitting
to, and drawing upon, foreign places, in such a manner
as shall turn out the most profitable.

Arbitration is either simple or compound.

Simple Arbitration respects three places only. Here,
by comparing the par of arbitration between a first and
second place, and between the 1st and a 3d, the rate be-
tween the 2d and 3d is discovered ; whence a person can
judge how to remit or draw to the most advantage, and to
determine what that advantage is.

Compound Arbitration respects the cases in which the
exchanges among three, four, or more places are con-
cerned. A person who knows at what rate \\(t can draw
or remit directly, and also has advice ofthe course of ex-
change in foreign parts, may trace out a path for circu-
lating his money, through more or fewer of such places,
and also in such order, as to make a benefit of his skill
and credit : and in this lies the great ait of such negocia-
tions. See Hutton's Arithmetic, pa. 105; also the arti-
cles Arbitration, and the Chain Rule.

EXCURSION, in Astronomy. See Elongation.

Circles o/' Excursion. See Circles.

EXEGESIS, or Exegetica, in Algebra, is the finding,
either in numbers or lines, the roots of the equation of a
problem, according as the problem is cither numeral or
geometrical.

EXHALATION, a fume or steam, issuing from a body,
and diffusing itself in the atmosphere. The terms exhala-
tion and vapour are often used indifferently ; but the more
accurate writers distinguish them, appropriating the term
vapour to the moist fumes raised from water and other
Vol.. 1.

medes, &c. And it is founded on what I uclid says in the
10th hook of his Elements ; viz, that those .quantities, are
equal, whose difference is less than any assignable quan-
tity. Or it may be otherwise expressed thus : two quan-
tities a and b are equal, when, if to or from om of them
as a, any other quantity as d be subtracted, however
small it may be, the sum or difference thus obtained is re-
spectively greater or less than the other quantity b : d
being an indefinitively small quantity,

if a -+- d be greater than b,

and a — d less than b, then is a equal to b.

This principle is used in the 1st prop, of the 10th book,
which imports, that if from the greater of two quantities
be taken more than its half, and from the remainder more
than its half, and so on ; there will at length remain a
quantity less than either of those proposed. On this
foundation it is demonstrated, that if a regular polygon of
infinite sides be inscribed in a circle, or circumscribed
about it ; that the space, which is the difference bet ween the
circle and the polygon, will by degrees be quite exhausted,
and the circle become ultimately equal to the polygon.
And in this way it is that Archimedes demonstrates, that
a circle is equal to aright-angled triangle, whose two sides
about the right angle are equal, the one to the semidi-
ameter, and the other to the perimeter of the circle. Prop.
1 I)e Dimensione Circuit.

On the method of exhaustions depends themethod of in-
divisibles introduced by Cavalerius, which is but a shorter
way of expressing the method of exhaustions; as also
Wallis's Arithmetic of Infinites, which is a further im-
provement ofthe method of indivisibles; and hence also
the methods of increments, differentials, fluxions, and
infinite series. See some account of the method of ex-
haustions in Wallis's Algebra, chap. 73, and Ronayne's
Algebra, part 3, pa. 395.

EXPANSION, is the dilating, stretching, or spreading
out of a body; whether from any external cause, as the
cause of rarefaction, or from an internal cause, as elas-
ticity. Bodies naturally expand by heat, and are con-
tracted by cold ; and hence it happens that their dimen-
sions and specific gravities are different in different tem-
peratures and seasons of the year. Air compressed or
condeused, as soon as the compressing or condensing force
3 S

EXP

C 498 ]

EXP

is removed, expands itself by its elastic power to its former tract in fixing, after having been melted, considerably

dimensions. more than zinc under similar circumstances.

In some few cases indeed, bodies seem to expand as they But of all known substances, those of the aerial kind

cool, as water in the act of freezing: but this is owing to expand most by an equal degree of heat; and in general

the extrication of a number of air bubbles from the fluid the greater quantity of latent heat that any substance con-

at a certain time; and is not at all a regular and gradual tains, the more easily is it expanded : yet even here no

expansion like that of metals, &c, by means of heat. Mr. general law can be observed. It is indeed certain that the

Boyle, in his History of Cold, says that ice takes up a densest fluids,, such as mercury, oil of vitriol, &c, arc

12th part more space than water; but by MajorWilliams's less expansible than water, spirit of wine, or ether. Which

experiments on the force of freezing water, I have found last is so easily expanded, that were it not for the pres-

it occupies only about the 17th or 18th part more space, sure of the atmosphere it would be in a continual state of

Transact, of the R. Soc. of Edinb. vol.2, pa. 2S. In cer- vapour. And indeed this is the case, in some measure,

tain metals also, an expansion takes place when they pass with perhaps all fluids; as it-has been found, by experi-

from a fluid to a solid state: but this too is not to be ac- ments with the best air-pumps, that water, and other

counted any proper effect of cold, but of the arrangement fluids, ascend in vapours the more as the exhaustion is the

of the parts of the metal in a certain manner ; and is there- more perfect ; from which it would seem that water would

fore to be considered as a kind of crystallization. wholly rise in vapour, in any temperature, if the pressure

The expansion ofditferent bodies by heat is very various; of the atmosphere was entirely taken oft.
and many experiments upon it are to be met with in the After bodies are reduced to a state of vapour, their ex-
volumes of the Philos. Trans, and in other places. In the pansion seems ro go on without any limitation, in pro-
48th vol. in particular, Mr. Smeaton has given a table of portion to the degree of heat applied ; though it may be
the expansion of many different substances, as determined impossible to say what would be the ultimate effect uf that
by experiment, from which the following particulars are principle acting upon them in this manner. 1 he force
extracted. Where it is to be noted, that the quantities of with which these vapours expand on the application of
expansion which answer to 180 degrees of Fahrenheit's high degrees of heat is so great, that no body or force ap-
thermometer, are expressed in ten-thousandth parts of an pears to be sufficient to counteract their effects
English inch, each substance being 1 foot or 12 inches in On this principle depends the construction of steam-en-

length.

White glass barometer tube - 100

Martial regulus of antimony - - 130

Blistered steel - - - - 13S

Hard steel - - - - 147

Iron - - - - - 151

Bismuth - I67

Copper hammered - 201

Copper 8 parts, mixed with I of tin - 218

Cast brass - 225

Brass 16 parts, with tin 1 - . 229

Brass wire --._-. 232

Speculum metal ... 232

Spelter solder, viz, brass 2 parts, zinc 1 - 247

Fine pewter - 274

Grain tin - 298

Soft solder, viz. lead 2, tin 1 - - 301

Zinc 8 parts, tin 1, a little hammered - 323

Lead - 344,

Zinc or Spelter ... 353

373

gines, so much employed in various mechanical opera-
tions; likewise some hydraulic machines ; and the instru-
ments called manometers, which show the variation of
gravity in the external atmosphere, by the expansion or
cendensation of a small quantity of air confined in a proper
vessel. On this principle also, perpetual movements might
be constructed similar to those invented by Mr. Coxe, on
the principle of the barometer. And a variety of other
curious machines may be constructed on the principle
of aerial expansion; an account of some of which is
given under the articles Hydrostatics and Pneu-
matics.

The construction of thermometers depends also on the
principle of the expansion of fluids. See Thermometer.
And for the effects of the different expansions of metals, in
correcting the vibrations of pendulums, see Pendulum.

The expansion of solid bodies is measured by an in-
strument called the pyrometer ; and the force with which
they expand is still greater than that of aerial vapours; the
flame of a small candle produces an expansion in a bar of
iron capable of counteracting a weight of 500 pounds.
The quantity of expansion, however, is so small, that it
has never been applied to the movement of any mechanical

Zinc hammered |. an inch per foot
By other experiments it has likewise been found that
for each thermometrical degree of heat, mercury, water

and air, expand the following parts of their own bulk, viz, engine.

Mercury the 96"00tl,T EXPECTATION, in the Doctrine of Chances, is applied

Water the 6666th > part of its bulk. " to any contingent event, on the happening of which, some

Air the 435th J benefit &c, is expected. This is capable of being reduced

From the foregoing table it appears, that there is no ge- to the rules of computation: for a sum of money in ex-

neral rule for the degree of expansion to which bodies are pectation when a particular event happens, has a deter-

subject by the same degree of heat, either from their spe- minate value before that event happens. Thus, if a person

cine gravity or otherwise. Zinc, which is much lighter is to receive any sum, as 10/. when an event lakes place

than lead, expands more with heat; while glass, which which lias an equal chance or probability of happening and

is lighter than either, expands much less; and copper, failing, the value of the expectation is hall that sum, or

which is heavier than a mixture of brass and tin, expands 5/. : but if there are 3 chances for failing, and only 1 for

less. It seems also that metals observe a proportion of its happening, or 1 chance only in its favour out of all the

expansion in a fluid state, quite different from what they 4 chances; then the probability of its happening is only

do in a solid one : for regulus of antimony seemed to con- 1 out of 4, or J, and the value of tin' expectation is but i

EXP

[ 499

of 10?. which is only 21. 10s. or half the former sum. And
in all cases, the value of the expectation of any sum is
found by multiplying that sum by the fraction expressing
the probability of obtaining it. So the value of the ex-
pectation on 100/. when there are 3 chances out of 5 for
obtaining it, or when the probability of obtaining it is -f-,
is -f of 1001. which is 60/. And if* be any sum expected
on the happening of an event, h the chances for that event
happening, and/ the chances for its failling ; then, there
being h chances out of/-t- h for its happening, the proba-
bility will be

j — =r, and the value of the expectation is- — -. Suppose,

for example, that a person is entitled to 100/. if, in casting
a die, he brings up an ace; and to 50/. if he brings up a
six ; but in all other cases he is to have nothing, to find
the value of his expectation before the die is cast. In
each case the number of chances, both for happening and
failing, are 6, but for the event taking place only 1 ; there-
fore £ of 100/. = 161. 13s. 4id. is the value of his expecta-
tion in the first case, and £ of 50/. = 8/. 6s. 8d. is its worth
in the second case ; consequently, their sum 25/. is the
entire value of his expectation before he undertakes to
throw. See Simpson's or De Moivre's Doctrine of
Chances.

Expectation of Life, in the Doctrine of Life Annui-
ties, is the share, or number of years of life, which a person
of a given age may, on an equality of chance, expect to
enjoy. By the expectation or share of life, says Mr. Simp-
son (Select Exercises, pa. 273), is not here to be under-
stood that particular period which a person hath an equal
chance of surviving; this last being a different, and more
simple consideration. The expectation of a life, to put it
in the most familiar light, may be taken as the number of
years at which the purchase of an annuity, granted upon
it, without discount of money, ought to be valued. Which
number of years will differ more or less from the period
above-mentioned, according to the different degrees of mor-
tality to which the several stages of life are incident. Thus
it is much more than an equal chance, according to the
table of the probability of the duration of life (pa. 254 ut
supra), that an infant, just come into the world, arrives
not to the age of 10 years ; yet the expectation or share
of life due to it, on an average, is near 20 years. The
reason of which wide difference, is the great excess of
the probability of mortality in the first tender years of ljfe,
above that respecting the more mature and stronger ages.
Indeed, if the numbers that die at every age were to be the
same, the two quantities above specified would also be
equal ; but when the said numbers become continually less
and less, the expectation must of consequence be the
greater of the two.

Mr. Simpson has given a table and rule for finding this
expectation, pa. 255 and 273 as below.

For example, if it be required to find the expectation or
share of life, due to a, person of 30 years old. Opposite
the given age in the first column of the table, stands 23'6
in the second col. for the years ill the expectation sought.
See He Moivre's Doctrine of Chances applied to the valu-
ation of annuities, pa. 288; or Dr. Price's Observations on
Reversionary Payments, pa. l68,36'4, 374-, &c ; or Philos.
Trans, vol. 59, pa. 89.

] EXP

A Table of the Expectations of Life in London.

Age.

Expectation

Age.

Expectation.

Age.

Expectation.

1

27-0

28

24-6

55

142

2

32-0

29

24' 1

56

13-8

3

34-0

30

23 6

57

13-4

4

35-6

31

23-1

58

131

5

36-0

32

22-7

59

12-7

6

36-0

33

22'3

60

12-4

7

35-8

34

21-9

61

12.0

8

35-6

35

21-5

62

11-6

9

35'2

36

21-1

63

11-2

10

34-8

37

20-7

64

10-S

11

34-3

38

20'3

65

10-5

12

33-7

39

19'9

66

ioi

13

331

40

19-6

67

9-s

14

32-5

41

19-2

6s

94

15

31-9

42

18-8

69

9-1

1(5

31-3

43

18-5

70

8-8

17

30'7

44

18-1

7\

8-4

18

30-1

45

17-8

72

8-1

19

29-5

46

17-4

73

7-8

20

28-9

47

17-0

74

7-5

2L

28-3

4S

16-7

75

7-2

22

277

49

16-3

76

6-8

23

27'2

50

160

77

.6-4

24

25-6

51

15-6

78

6-0

25

26-1

52

15-2

79

5-5

26

25-6

53

14-9

80

50

27

25-1

54

14-5

EXPERIMENT, in Philosophy, a trial of the effect or
result of certain applications and motions of natural bo-
dies, in order to discover something of their laws and re-
lations, &c. The making of experiments is become a
kind of art; and there are now many collections of them,
mostly under the denomination of Courses of Experimen-
tal Philosophy. Sturmius made a curious collection of
the principal discoveries and experiments of the 17th cen-
tury, under the title of Collegium Experimentale. Other
courses of experiments have been published by Gravesande,
Desaguliers, Helsham, Cotes, &c.

EXPERIMENTAL Philosophy, is that which proceeds
on experiments, or which deduces the laws of nature and
the properties and powers of bodies, and their actions upon
each other, from sensible experiments and observations.
Experiments are of the utmost importance in philosophy;
and the great advantages the modern physics have over
the ancient, is chiefly owing to our making more "experi-
ments, and then founding our theories on the results thus
obtained : whereas the method of the ancients was chiefly
to begin with the assumed causes of tilings, and thence ar-
gue to the phenomena and effects; on the contrary, that of
the modems proceeds from experiments and observations,
whence the properties and laws of natural things are de-
duced, and general theories are formed.

Several of the ancients indeed thought as highly of ex-
periments as the moderns, and practised them also. Plato
omits no occasion of speaking of the advantages of them ■
and Aristotle's history of animals bears ample testimony
for him. Democritus's great employment was to make ex-
periments; and even Epicurus himself owes part of his
glory to the same cause.

Among the moderns, the making of ex-peri men ts was
chiefly begun by Friar Bacon, in the 13th century, who

3 S 2

EXP

[ 500 ]

EXP

spent a grct deal of money and labour in this pursuit. Af- of its condensation when fixed in the gunpowder, and so
ter him, the lord chancellor Bacon is considered as the explodes, and produces all those terrible effects that attend
founder of the present mode of philosophising by experi- the explosion. The elastic fluid generated by fired gun-
ments. And his method has been prosecuted with laudable powder expandsitself with a velocity of almost 10,000 feet
emulation by the Academy del Cimento, the Royal So- per second, and with a force nearly C000 limes greater
ciety, the Royal Academy at Paris; by Mr. Boyle, and, than the pressure of the atmosphere on the same base,
above all, by Sir Isaac Newton, with many other illustri- The electric explosions seem to be still much more
ous names. strong and astonishing; as in the cases of lightning, earth-
Indeed, the making of experiments, within the lastcen- quakes, and volcanoes; and even in the artificial electri-
tury, is so much practised, that nothing will pass in phi- city produced by the ordinary machines. The astonish-
losophy, but what is cither founded on experitnents, or ing strength of electric explosions, which is beyond all
confirmed by them ; so that the new philosophy is almost possible means of measuring it, manifests itself by the many
wholly experimental. tremendous effects we hear of fire-balls and lightning.

Yet there are some, even among the learned, who speak In cases where the electric matter acts like common
of experiments in a different manner, or perhaps rather of fire, the force of the explosions, though very great, is ca-
the abuse of them, and in derision of the pretenders to this pable of measurement, by comparing the distances to
practice. Thus, though Dr. Keil allows that philosophy which bodies are thrown, with their weight. This is most
has received considerable advantages from those who make evident in volcanoes, where the projections of the burning
experiments; yet he complains of their disingenuity, in rocks and lava manifest the immensity of the power, at the
too often wresting and distorting their experiments and same time that they afford a method of measuring it: and
observations, to favour some darling theories they had these explosions are owing to the extrication of aerial va-
espoused. Nay more, M. Hartsoeker, in his Recueil de pours, and their rarefaction by intense heat,
plusieurs Pieces de Physique, undertakes to show, that Next in strength to the aeiial vapours, are those of
such as employ themselves in the making of experiments, aqueous and other liquids. Very remarkable effects of
are not properly philosophers, but as it were the labourers these are observed in steam-engines; and there is one case
or operators of philosophers, that work under them, and from which it has been inferred that aqueous steam is
for them, furnishing them with materials to build their even much stronger than fired gunpowder. This is when
systems and hypotheses on. And the learned M. Dacier, water is thrown upon melted copper : for here the explo-
in the beginning of his discourse on Plato, at the head of sion is so strong as almost to exceed imagination ; and the
his translation of the works of that philosopher, deals still most terrible accidents have happened, even from &o slight
more severely with the makers of experiments. He breaks a cause as one of the workmen spitting in the furnace
out with a kind of indignation at a tribe of idly curious where copper was melting; arising probably from a sud-
people, whose sole employment consists in making expe- den decomposition of the water. Explosions happen also
riments on the gravity of the air, the equilibrium of fluids, from the application of water to other melted metals,
the loadstone, &c, and yet arrogate to themselves the noble though in a lower degree, when the fluid is applied in small
title of philosophers. quantities, and even to common fire itself, as every person's
EXPLOSION, a sudden and violent expansion of an own experience must have informed him; and this seems
elastic fluid, by which it instantly throws off any obstacle to be occasioned by tfte sudden rarefaction of the water into
that happens to be in the way, sometimes with astonishing steam. Examples of this kind often occur when work-
force and rapidity, as the explosion of fired gunpowder, &c. men are fastening cramps of iron into stones ; where there
Explosion differs from expansion, in this, that the latter sometimes happens to be a little water in the hole into
is a gradual and continued power, acting uniformly for which the lead is poured ; in this case the metal will fly
some certain time; whereas the former is alwa}'s sudden, out in such a manner as sometimes to burn them very se,-
and only of momentary or immensurably short duration, verely. Terrible accidents of this kind sometimes occur
The expansions of solid substances do not terminate in in founderies, when large quantities of melted metal is
violent explosions, on account of their slowness of action, poured into wet or damp moulds. In these cases, the
and the small space, through which the expanding suh- sudden expansion of the aqueous steam has thrown out
stance moves; though their strength may be equally great the metal with great violence; and if any decomposition
with that of the most active aerial fluids. Thus we find has taken place at the same time, so as to convert the
that though wedges of wood, when wetted, will cleave aqueous vapour into an aerial one, the explosion must be
solid blocks of stone, they never throw them to any dis- still greater.

ranee, as is the case with gunpowder. On the other To this last kind of explosion must be referred that

hand, it is seldom that the expansion of any elastic fluid which takes place on pouring cold water into boiling or
bursts a solid substance without throwing the fragments of burning oil or tallow, or in pouring the latter upon the
it to a considerable distance, with effects that are often very former; the water however being always used in a small
terrible. quantity. — Another remarkable kind of explosion is that

The greater number of explosive substances are either produced by inflammable and dephlogisticated air, when
aerial, or convertible into such, and raised into an elastic mixed together, and set on fire; a circumstance that often
fluid. Thus gunpowder, whose essence seems to consist in happens in coal-mines, &c. This differs from any of the
common air fixed in the nitre, or at least an air of similar cases before mentioned; for here is an absolute condensa-
elasticity, where it is condensed into a bulk many hundred tion rather than an expansion throughout the whole of the
times less than the naturaj state of the atmosphere ; which operation ; and could the airs be made to take fire through-
air being suddenly disengaged by the firing of the gun- out their whole substance absolutely at the same instant,
powder, and the decomposition of its parts, it rapidly ex- there would be no explosion, but only a sudden produc-
pands itself again with a force proportioned to the degree tion of heat.

EXP

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EXP

Though explosions be sometimes very destructive, they above, and it was Stifelius who first called them expo-
are also of considerable use in life, as in removing obsta- nents.'

cles that could scarcely be overcome by any mechanical Bombelli, whose Algebra was published in 1579, de-
power whatever. The principal of these are the blowing- notes the res, or unknown quantity, by this mark 0, and
up of rocks, the separating of stones in quarries, and other the powers by numeral exponents set over it, thus : _*, £,
purposes of that kind. The destruction occasioned by ^, &c. And

them in times of war, and the machines formed upon the Stevinus, who published his Arithmetic in 1585, and

.principle of explosion for the destruction of the human his Algebra soon afterwards, makes use of a similar me^

race, are well known; arid if we cannot call these useful, thod, but instead of ^ he uses a small circle O, within

they must be allowed at least to be necessary evils. which he places the numeral exponent of the power; thus

The effects of explosions, when violent, are felt at a ©, (j), ©, (§), &c : and in this way he extends his nota-

considerable distance, by reason of the concussions they tion to fractional exponents, and even to radical ones;

give to the atmosphere. Sir VVm. Hamilton relates, that thus (§), <j), ©, ©, &c.

at the explosions of Vesuvius, in 1767, the doors and win- Vieta after this used words again to denote the powers,
dows of the houses at Naples flew open if unbolted, and And Harriot denoted the powers by a repetition of the
one door was burst open that had been locked, though at root ; as a, an, aaa, for the 1st, 2d, and 3d powers. In-
the distance of 6' miles: and the explosion of a powder- stead of which, Descartes again restored the numeral ex-
magazine, or a powder-mill, it is well known, spreads de- ponents, placing them after the root, when the power is
struction for many miles round; and even kills people by high, in order to avoid a too frequent repetition of the
the mere concussion of the air. A curious effect of them same letter ; as (i3 a4, &c, the same as is used at present,
also is, that they electrify the air, and even glass windows, Also Albert Girard, in l62y, used the exponents to roots,
at aconsiderable distance. This is always observable in thus; ^/,y/, I/, &c.

firing the guns at the Tower of London : and some years The notation of powers and roots by the present mode
ago, after an explosion of some powder mills near that of exponents, has introduced a new and general aritbme-
city, many people were alarmed by a rattling and break- tic of exponents or powers; for hence powers are multi-
ing of their china-ware. In this respect, however, the plied by only adding their exponents, divided by sub-
effects of electrical explosions are the most remarkable, trading the exponents, raised to other powers, or roots of
though not in the uncommon way just mentioned ; but it them extracted, by multiplying or dividing the exponent

is certain that the influence of a flash of lightning is dif-
fused for a great way round the place where the explosion
happens, producing very perceptible changes both on the
animal and vegetable creation.

EXPONENT of a Power, in Arithmetic and Algebra,
denotes the number or quantity expressing the degree or
elevation of the power, or which shows how often a given

by the index of the power or root. —
x i.

:o a? x a3 — as, and a"" x a* = a4
1 « I

as -i-a3 = a2, and o4 -J- a1 = a*

the 2d power of a? is ufi,
and the 3d root of a6 is a2.
This algorithm of powers led the way to the invention

the

power is to be divided by its root before it be brought of logarithms, which are only the indices or exponents of
down to unity or 1. Thus, the exponent or index of a powers: and hence the addition and subtraction of loga-
square number, or the 2d power, is 2; of a cube 3; and rithms, answer to the multiplication and division of num-
bers; while the raising of powers, and extracting of roots,
is effected by multiplying the logarithm by ihe index of
the posver, or dividing the logarithm by the index of the
root.

Exponent of a Ratio, is, -by some, understood to be

so on ; the square being a power of the 2d degree
cube, of a 3d, &c. It is otherwise called the Index.

'Exponents, as now used, arc rather of modern inven-
tion. Diophantus, with the Arabian and the. first Euro-
pean authors, denoted the powers of quantities by sub-
joining an abbreviation of the name of the power ; though the quotient arising from the division of the antecedent of
with some variation, and difference from one another, the ratio by the consequent : in which sense, the exponent
The names of the powers, and the marks for denoting of the ratio of 3 to 2 is f; and that of the ratio of 2 to 3

them, according to Diophantus, areas follow: viz,
Names, jj.ovas, api^Mc, Swajxii, xv€o;, 8uvaij.o3vva[JA;,
Marks, //-° , c , $" , v." , h 8", &c ;

which we now denote by I, a, a2, a?, a4, &c.

F. Lucas Paciolus, or De Burgo, for the root, square,
cube, &c, uses the terms cosa, censo, cubo, relato (pri-
mo, secundo, tertio, &c), or the abbreviations co. ce. cu.;
and ]£, for root or radicality.

Cardan used the Latin contractions of the names of the
powers; and other contemporary, and ever, succeeding au-

But others, and those among the best mathemati-
cians, understand logarithmsas the exponents of ratios;
in which sense they coincide with the idea of measures of
ratios, as delivered by Kepler, Mercator, Ilalley,Cotes,&c.

EXPONENTIAL Calculus, the method of differencing,
or finding the fluxions of exponential quantities, and of
summing up those differences, or finding their fluents. See
Calculus, Fluxions, and Fluents.

Exponential Curve, is that whose nature is defined
or expressed by an exponential equation; as the curve de-

thors, especially the Germans, as Stifelius, Scheubelius, noted by nx — y, or by x* =y

Pclitarius, &c, used the like contractions, but somewhat
varied, thus:

*, 3>, 5, #, 55,73, 5tf, &c
or 1, If, 3, «, 55, A 5#, &c.
or !, R, q, &, qq, fs, <7#, &c.
exp. 0, 1, 2, 3, 4, 5, 6, &c.

But besides that method of representation, the same au-
thors also made use of the numbers as in the last line here

Exponential Equation, is one in which is contained
an exponential quantity: as the equation ax = b, or x*
= ab, &c. Exponential equations are commonly best re-
solved by means of logarithms, viz, first taking the log. of
the given equation : thus, by taking the log. of the equa-
tion "x = b, it is x x log. of a = log. of b ; and hence
.r = log. b ■— log. a.

Also, the log. of the equation x* = ab, is x x log. x =

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log. ab; and then x is easily found by trial-and-error, or
the double rule of position.

Exponential Quantity, is that whose power is a vari-
able quantity ; as the expression ax, or xx. Exponential
quantities are of several degrees, and orders, according to
the number of exponents or powers, one above another.
Thus,

ox is an exponential of the 1st order,

ax, is one of the 2d order,

y

ax is one of the 3d order, and so on.
See Bernoulli Oper. torn. 1, pa. 182, &c.

EXPRESSION, in Algebra, is any algebraical quan-
tity, simple or compound : as the expression, 3a, or 2ab,
or \/(a~ i c'2). Sometimes also called a function.

EXTENSION, one of the common and essential pro-
perties of body ; or that by which it possesses or takes up
some part of universal space, called the place of that
body. The extension of a body, is properly speaking in
every direction whatever; but it is usual to consider it as
extended only in length, breadth, and thickness.

EXTERIOR Polygon, or Talus, is the outer or circum-
scribing one. See Polygon, and Talus.

EXTERMINATION, or Exterminating, in Alge-
bra, is to take away certain unknown quantities from de-
pending equations, so as finally to have only one equation
containing one unknown quantity; whence its value may
be determined, and by means of this, and the equations
obtained in the extermination, all the other quantities be-
come known.

The rules for performing this are various, and after all,
much must be left to the ingenuity of the algebraist, as the
application and combination of those particular rules, de-
pend, in a great measure, on the forms and nature of the
equations, out of which the unknown quantities are to be
exterminated. The following however may be considered
as forming principles of the operation.

1. Find the value of one of the unknown letters, in
terms of the other quantities, in each of the given equa-
tions. Next put two of these values equal to each other;
then one of these and a third, fourth, &c ; by which
means one unknown quantity will be exterminated ; and
in the same way others may be taken away, till there re-
main only one equation and one unknown quantity, the
value of which may then be found by the common rules.

2. Find the value of one of the unknown letters in one
of the equations; then substitute this value of it in each
of the other equations into which that letter enters, by
which means it will be exterminated ; and in the same
manner we may take away all but one as before.

3. Multiply or divide the given equations by such
numbers, that the coefficients of one of the unknown
quantities in each of the equations may be equal to each
other. Then by adding or subtracting them as the case
requires, that unknown quantity will be taken away.

The following equations will serve to illustrate all these
rules.

Given ax -+- by =: n, and ex -+- dy = m.

%i r ii n — hi m — (hi
the lirst method, x = -: and x = -:
a c

therefore = - — — , or (ad — be) y = am — en,

which is an equation containing only one unknown
quantity.

Second method. I [e re

x = ^ ; and substituting this value of x in the second

equation, we have

■ — i- dy = m; which gives

(ad — be) y = am — en, as before.

T/iird_method. By multiplying the-first equation by c,
and the second by«, we obtain
acx -+- bey =cn,

and acx -+- ady = am ; then by subtraction we have
(ad — be) y = am — en, still the same result as be-
fore. See Button's Course, Simple Equat. prob. 1, pa.
227, &c.

Many ingenious specimens of the different methods of
extermination are given in Newton's Universal Arithme-
tic, and Maclaurin's Algebra. Sec also an excellent piece
on this subject under the article LI inination in the En-
cyclopedia Methodique.

EXTERNAL Angles, are the angles formed without a
figure, by producing its sides out. In a triangle, any ex-
ternal angle is equal to the sum of both the two internal
opposite angles taken together: amd, in any right-lined
figure, the sum of all the external angles, is equal to 4
right angles, except in the case where one of them tends
inwards. In which case, that angle must be taken from
the sum of the others to leave four right angles. See Hut-
ton's Geom. Theor. 10 and 20.

EXTR A-Constellary Stars, such as are not properly in-
cluded in any constellation.

EXTRA-Mundane Space, is the infinite, empty, void
space, which is by some supposed to be extended beyond
the bounds of the universe, and consequently in which
there is really nothing existing.

EXTRACTION of Roots, is finding the roots of given
numbers, quantities, or equations. The roots of quanti-
ties are denominated from their powers; as the square or
2d root, the cubic or 3d root, the biquadratic or 4th root,
the 5th root, &c ; which are the roots of the 2d, 3d, 4th,
5th, &c powers. The extraction of roots has always made
a part of arithmetical calculation, at least as far back as
the composition of powers has been known : for the com-
position of powers always led to their resolution, or ex-
traction of roots, which is performed by the rules exactly
reverse of the former. Thus, if any root be considered as
consisting of two pans a + r, of which the former a is
known, and the latter x unknown, then the square of this
root is a" -+- lax -+- x*, which is its composition, and this
indicated the resolution, or method of discovering the un-
known part x ; for having subtracted the nearest square a°
from the given quantity, there remains 2n.r ■+- x2 or
(2a ■+- x) x x; therefore divide this remainder by 2«, the
double of the first member of the root, and the quotient
will be nearly equal to x the other member ; thin to 2a
add this quotient x, and multiply the ■sum 2a -+- x by x,
and the product will make up the remaining part '2ax ■+- x*
of the given power.

The composition of the cubic or 3d power next pre-
sented itself, which consists, of these lour terms a3 -»- 3a*.r
h- oux1 -+- xJ; b) means of which the cube roots of num-
bers have been Mi'acU'd; viz, by subtracting the nearest
cube a1 from 'I"' given power, and dividing the remainder
by 3a", which givejs x nearly fir the quotient; then com-
pleting the divisor up to 3a1 ■+■ 3ax -t- x1, multiply it by
x for the other part of the power to be subtracted. And
this was the extent of the extraction of roots in the time

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of Lucas de Burgo, who, from 1470 to 1500, wrote several
pieces on arithmetic and algebra, which were the first
works of this kind printed in Europe.

It was not long however before the nature and compo-
sition of all the higher powers became known, and general
tables of coefficients formed for raising them ; the first of
which is contained in Stifelius's Arithmetic, printed at
Norimberg in 1543, where he fully explains their use in
extracting the roots of all powers whatever, by methods
similar to those for the square and cube roots, as above
described ; and thus completed the. extraction of numeri-
cal roots, at least so far as that method of resolution ex-
tends. Since that time, however, many new modes of
extraction have been devised, and the old method con-
siderably improved.

The extraction of roots of equations followed closely
that of known numbers.. In De Burgo's time they ex-
tracted the roots of quadratic equations, the same way as
at present. Ferreus, Tartalea, and Cardan extracted the
roots of cubic equations, by general rules. Soon after-
wards the roots of higher equations were extracted, at
least in numbers, by approximation. And the late im-
provements in analytics have furnished general rules for
extracting the roots, in infinite series, of all equations
whatever. All which methods may be seen in most
books of arithmetic and algebra ; therefore it nay be
sufficient to give here a short specimen of some of the ea-
siest rules for extracting the roots of quantities and equa-
tions, o

1. To Extract the Square Root.

Separate the given numbers into periods of two figures
each, by setting a point over the place of units, another
over the place of hundreds ; and so on, over every second
figure, both to the left hand in integers, and to the right
hand in decimals.

Find the greatest square in the first period on the left
hand, and set its root on the right hand of the given num-
ber, after the manner of a quotient figure in division.

Subtract the square, thus found, from the said period,
and to the remainder annex the two figures of the next
following period for a dividend. Double the root above
found for a divisor, and find how often it is contained in
the said dividend, exclusive of its right-hand figure, and
set that quotient figure both in the quotient and in the
divisor.

Multiply the whole augmented divisor by the last quo-
tient figure, and subtract the product from the said divi-
dend ; bringing down to it the next period of the given
number for a new dividend.

Repeat the same process over again ; namely, find ano-
ther new divisor, by doubling all the figures now found in
the root; from which, and the last dividend, find the next
figure of the i not as before, and so on through all the
periods to the last.

But the best way of doubling the root, to form the new
divisor, is l.y adding the last figure always to the last
divisor; as appears from the following examples.

When the loot is to be carried into decimals, couplets
of ciphers are to be added, instead of figures, as far as may
be necessary. In which case also a considerable abbrevi-
ation is made, alter the work has been carried on to half
the number of figures, or one more than half, by continuing
it to the other half only by the Contracted method of di-
vision ; as in the following example for the square root of
2 to eight decimals, or nine places of figures.

It may also not be amiss to add fir the information of '
beginners, that when there are an odd number of decimals
in the number whose root is to be extracted, they should
be made even, by annexing a cipher, before the operation
is begun. See the following examples.
. . . root
99856(31(5
9
6l I 98
l 61

6'26
6

3756
3756

2
1

( 1-41421356

24 1
4 1
281
1

100

j 400
281

108

8

. . V root
34-7160(5-892
25

971

86'4 '

2824
4

11900
11296

282S2
2

60400
56564

2S284)

1169

9

10760
10521

11782

2

23900
23564

336

3836 (1356
2828
100S
S48

76o

141

~19
17

2

To extract the square root of a number by continued
fractions, see Continued Fractions.

II. To Extract the Cube Root.

Divide the given number into periods of three figures
each, by setting a point over the place of units, another
over the place of thousands, and so on, over every third
figure, to the left hand in integers, and to the right in de-
cimals. Find the nearest less cube to the first period ; and
set its root in the quotient, subtracting the said cube from
the first period ; and to the remainder bring down the se-
cond period, which forms the resolvend.

To three times the square of the root, thus found, add
three times the root itself, setting this one place more to
the right than the former, and call this sum the divisor.
Then divide the resolvend, wanting the last figure, by the
divisor, for the next figure of the root, which annex to the
former; and let this last figure be called e, and the part
of the root before found a.

Add altogether these three products, namely, three
times a square multiplied by e, thrice a multiplied by e
square, and e cube, setting each of them one place more
to the right than the former, and call this sum the subtra-
hend ; which should not exceed the resolvend ; therefore
if it do, make the last figure e less, and repeal the operation
for finding the subtrahend, till one be found less than the
resolvend.

From the resolvend take the subtrahend, and to the re-
mainder join the next period of the given number for a new
resolvend; to which form a new divisor from the whole root
now found, and from thence another figure of the root as
above directed, and so on, till all the figures are used.
After which the operation may be continued to any lengtk
by bringing always new periods of three ciphers each.

Suppose for ,example it were required to extract the
cube root of 48228-544.

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3 x 32 = 27
3 x 3= 09
Divisor 279

48228-544 ( 36-4 rout

27

21228 resolvend

the roots of numbers in arithmetic, as above taught. Thus,
to extract the square root of 4a1 ■*■ 12ax. ■+- 9X*'

4ar -+- \2ax -t- 9xz( 2a ■+■ 3a: the root

4a2

6 = 162
6'2= 324
63= 216

36a= 3888
36= 108

3S98S

19606" subtrahend
15/2544 resolvend

3 x 36- x 4= 15552 "1

3 x 36 x 4'= 1728 {-add

43= 643

1572544 subtrahend
0000000 remainder

IIL To Extract the Cube Root, or any other Root whatever.
This is easiest done by one general rule, that I have
invented, and published in my new Tracts, vol. 1, pa. 210,
which is to this effect: Let n be any number or power,
whose nth root is to be extracted ; and let r be the nearest
rational root of n, of the same kind, or Rn the nearest ra-
tional power to n, either greater or less than it; then shall
the true root be very nearly equal to
(» + l)s -+- (n — 1)r"

7 ^tt ", ,, „ x r; which rule is general for

(n — 1)n ■+■ (n ■+- 1)r" &

any root whose index is denoted by n. And by expound-
ing n successively by all the numbers 2, 3, 4, 5, &c, this
theorem will give the following particular rules for the
several roots, viz, the

3n ■+■ r1

R ;

2d or squ. root,
3d or cube root,
4th root

4n

3r

+■ 2r3

2n

4rS
3r4

N

5th root

6th root

3n
6n

x r;

7th root
&c

4n

7n -1- 5r'

5n -+- 7r'
8n -i- 6k

5r4

4rs 3n

-^—7 x r, or:rr-
6r5 ' 2k

x r :

R,or

4n -t- 3r7
3n -+- 4R7

r;

6n -1- 8r7
&c.

Or the theorem may be stated in the form of a propor-
tion, thus :

as (» - 1) n + (« + 1) n" : (n + 1) n + (» - 1) r" : : k :
the root sought very nearly.

Suppose for example it was required to find the cube
root of the number 2. Here n = 2, n= 3, and the nearest
power, and root are each 1.

Hence 2n -t- R3 = 4 -+- 1 = 5,
and n -t- 2u;) =2 + 2 = 4;
then 4 : 5 : : 1 : | = 1"25 the first approximation.
Again, taking r = £, and conseq. 113 rz \£J :
Hence 2& -+■ R3 = 4 ■+■ *■& = 3-£J,
and N +rf = 2 + V? = Y* >
then 378 : 381 :: \ : \-l\ — 1-259921, for the cube
root of 2, which is exact in the very last figure.

And again by taking 4^4 '"or tne vului; of R, a great
many more figures may be found.

IV. To Extract the Roots of Algebraic Quantities.— 'Y\\\s
is done by the same rules, and in the same manner as for

4a

12ax
12ai

9*a
9x-

So also the root is carried out in an infinite series, in
imitation of the like extraction of numbers in infinite de-
cimals : thus, the square root of a1 -+- x* is

% 1 1 *

16a5

&C.

_ x!

2a -\

■la
x2

x~

JL.

_ Xs

2a h

a

X*

8a3

-

4a3

X*

X*

A'*

X*

8a3

4a"

Sa*

4a"

. To extract the cube root of a3 — x3 by the general rule
in the 2d article. — Here n = a? — x3, r = a, or r3 = a3 ;
n -+■ 2r3 = 3a3 — xs, and 2n -+- r3 = 3u3 — 2xs ; there-
fore, by the rule,

3ffl3 — x3 : 3a? — 2x3 : : a : a — — ; ^r = l(C,

3a~ 9ar 27a'

which is the cube root of <? — x3 very nearly.

But these kinds of roots are best extracted by the Bi-
nomial Theorem; which see.

V. To Extract the Roots of Equations. — This is the
same thing as to find the value of the unknown quantity
in an equation ; which is effected by various means, de-
pending on the form of the equation, and the dimension
of the highest power of the unknown quantity in it: for
which, see the respective terms, Equation, Root, Qua-
dratic, Cubic, &c.

The most general, as well as the most easy, method of ex-
tracting the roots of all the higher equations, is by double
position, ortrial-and-error; as iteasily applies toallkinds of
equations, however complex the}' may be, even logarithmic
and exponential ones. There are also several other good
methods of approximating to the roots of equations, given
by Newton, Halley, Raphson, De Moivre, Lagrange, &c;
of which the most general is a rule for extracting the root
of the following indefinite equation,

viz, az ■+- bz1 ■+■ cz3 -i- rfz4 ■+■ ezs &c

= gn ■+- hf + *y -+- w + if &c»

given by M. De Moivre in the Philos. Trans, vol. 20, pa.
190, or my Abr. vol. 4, pa. 275.

EXTRADOS, the outside of an arch, or vault, &C.

EXTREME-arad-Mem! Proportion, is when a line, or
any quantity is so divided, as that the whole line is to the
greater part, as that greater part is to the less. Hence,
in any line so divided, the rectangle of the w hole line and
the less segment, is equal to the square of the greater
segment. Euclid shows how to divide a line in exreme-
and-mean ratio, in his Elements, book 2, prop. I], to this
effect: Let 'ab be the given line; to which draw AE'per-
pendicular and equal to half ab; in 1 a produced take
EF = r.11, so shall af be equal to the greater part; conse-
quently, if AG be taken equal to af, the line ab will be
divided in c; as required.

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EYE

The same may be done otherwise thus :

As before, make ae (fig. 2) perpendicular ana = -|.ab;
join f.b, on which take EC = ea, and then take bd = bc,
so shall the line be divided in d as required.

No number can be divided into extreme and mean pro-
portion, so that its two parts shall be rational ; as is well
demonstrated by Clavius, in his Commentary on the gth
book of Euclid's Elements ; and the same thing will also
appear from the following algebraical solution of the same
problem : Let a denote the whole line, and x the greater
part- then shall a-x be the less part, and the rectangle
of the whole and less part being put equal to the square
of the greater part, gives this equation, ,r2= a{a-x) = a2
— ax- hence iz -f- ax = a1 and by completing the square,
and extracting the root, &c, there is at last obtained for
the greater part x - ^~a i conseq. a-x = ~^—a is
the less part. And as the square root of 5, which cannot
be exactly extracted, makes a portion of both these parts,
it is manifest that neither of them can be obtained in ra-
tional numbers.

Euclid makes great use of this problem, in several parts
of the 13th book of the Elements ; and by means of it he
constructs the 10th proposition of the 4th book, which is
to construct an isosceles triangle having each angle at the
base double the angle at the vertex.

EXTREMES Conjunct, and Extremes Disjunct, in
Spherical Trigonometry,' are in the former case the two
circular parts that lie next the assumed middle part, and
in the latter case they are the two that lie remote from the
middle part. These were terms employed by lord Na-
pier, in his universal theorem for resolving all right-angled
and quadrantal spherical triangles, and published in his
Logarithmorum Canonis Descriptio, an. 1614. In this
theorem, Napier condenses into one rule, in two parts, the
rules for all the cases of right-angled spherical triangles,
which had been separately demonstrated by Pitiscus,
Lansbergius, Copernicus, Regiomontanus, and others. In
this theorem, neglecting the right angle, the author calls
the other five parts, circular parts, which are, the two
leos about the right angle, and the complements of the
other three, viz of the hypothenuser and the two oblique
angles. Then, taking any three of these five parts, one of
them will be in the middle between the other two, and
these two are the extremes conjunct when they are imme-
diately adjacent to that middle part, or they are the ex-
tremes disjunct when they are each separated from the
middle one by another part. Thus, the five parts being
ab, ac, and the complements of bc and
of the two angles b and c:' then if
the three parts ab, and the comple-
ments of the angle b and hypothenuse
bc be taken, these three are contiguous
to each other, the angle b lying in the
middle between the other two : there- Sf
fore the complement of b is the middle part, and ab with

Vol. I.

the complement of bc the extremes conjunct. But if the
three sides be taken ; bc is equally separated from the two
legs ab and ac, by the two angles B and c; and there-
fore these two legs ab and ac are the extremes disjunct,
and the complement of bc the middle part. — The author's
rule for resolving each of these cases is in two parts, as
below :

The rectangle contained by radius and the sine of the
middle part, is equal to the rectangle of the tangents of
the extremes conjunct, or equal to the rectangle of the -
sines of the extremes disjunct. Which rule comprehends
all the cases that can happen in right-angled spherical tri-
angles ; in the application of which rule, the equal rect-
angles are divided into a proportion or analogy, so that
the term sought may be the last of the four terms that are
concerned, and consequently its corresponding term in the
same rectangle must be the first of those terms.

EYE, the organ of sight, consisting of several parts, and
of such forms as best to answer the purpose for which it
was designed. Vision or sight being effected by a refrac-
tion of light through the humours of the eye to the bot-
tom or farther internal part of it, where the images of ex-
ternal objects are formed on a fine expansion of the optic
nerve, called the retina, therefore the fore part of the
eye must be of a convex figure, and of such a precise de-
gree of convexity, as the particular refractive power of the
several humours require, for forming the image of an ob-
ject at a given focal distance, viz, the diameter of the eye.
Hence we find,

1 st ; The external part of the eye-ball cr> (Plate 2, fig. S)
is a strong pellucid substance, properly convex, and
which, when dried, has some resemblance to a piece of
transparent horn, for which reason it is called the cornea,
or horny coat of the eye.

2dly ; Immediately behind this coat there is a fine clear
humour which, from its similarity to water, is called the
aqueous or watery humour, and is contained in the space
between cu and gfe.

3dly ; In this space there is a membrane or diaphragm,
called the uvea, with a hole in the middle as at f, called
the pupil, of a muscular contexture for altering the di-
mensions of that hole, in order to adjust or admit a due
quantity of light.

4thly ; Immediately behind this diaphragm is placed a
lenticular-formed substance GB, of a considerable consist-
ence,, called from its transparency the crystalline humour.
This is contained in a fine tunic called the choroides, and
is suspended in the middle of the eye by a ring of muscu-
lar fibres called the ligamentum ciliare, as at g and e ; by
which means it is moved a little nearer to, or farther from,
the bottom of the eye, to alter the focal distance.

5thly ;. All the remaining interior part of the eye, con-
stituting the great body of it, from ghe to imk, is made
up of a large quantity of a jelly-like substance, called the
vitreous or glassy humour ; though it resembles glass in
nothing except its transparency, it being more like the
white of an egg than any other thing.

Gthly ; On one side of the backward part of the eye, as
at k, the optic nerve enters it from the brain, and is ex-
panded over all the interior part of the eye to g and e,
the expansion being named the retina. On this delicate
membrane, the image im of every external object OB, is
formed according to the optic laws of nature, in the fol-
lowing manner.

Let ob be any very distant object. Now a pencil of
3T

EYE

[ 506 ]

EYE

rays proceeding from any point L, will fall on the cornea
CD, anil be refracted by the aqueous humour under it, to
a point in the axis of that pencil continued out. Also the
radius of convexity of the cornea being nearly j of an
inch i and the sine of incidence in air to that of refraction
in the aqueous humour, being nearly as 4 to 3, supposing
the rays parallel, or the object very far distant, the focal
distance after the first refraction, by the proper theorem
(.See Optics) -^— , will be found !§• inch from the cornea:

r being the radius •§- , and m to n as 4 to 3, that is as the
sine of the angle of incidence, to that of refraction.

The rays thus refracted by the cornea, fall converging
on the crystalline humour, and tend to a point 1*228
inch behind it; also the radii of convexity in the said
humour are | and £ respectively; and the sine of inci-
dence to that of refraction of the aqueous into the crystal-
line humour, having been found by experiments to be as

13 to 12; therefore, by this theorem md_;Mi+';„.> tlie focal

distance after refraction in the crystalline, will be 1-02
inch from the fore, part of it: where m= 13, re = 12,
r = \, and d = 1-228,

The rays now pass from the crystalline to the vitreous
humour still in a converging state, and the sines of inci-
dence and refraction being here as 12 to 13, as found by
experiment; and since the surface of the vitreous humour
is concave which receives the rays, and is the same with
the convexity of the hinder surface of the crystalline, the
radius will be the same, viz £ of an inch. Therefore the
focal distance after this third refraction will be found, by
the same1 theorem—^ , to be 6 tenths of an inch

nd — ma+nr

nearly, from the hinder part of the cornea : where m = 12,
n= 13, r = |, and d = -82: the thickness of the lens of
the cornea being nearly f of an inch.

Now experience shows that the distance of the retina
in the back part of the eye, behind the cornea, is nearly
equal to that focal distance; and therefore it follows that
all objects at a great distance have their images formed on
the retina in the bottom or hinder part of the eye, and
thus is distinct vision produced by this wonderful organ.

When the distance of objects is not very great, the focal
distance, alter the last refraction in the vitreous humour,
will be alittle increased; and to do this we can move the
crystalline a little nearer the cornea by means of the liga-
mentuni ciliare, and thus on all occasions it may be ad-
justed for a due focal distance for every distance of ob-
jects, excepting that which is less than 6 or 7 inches, in
persons whose sight is perfect. But many are of opinion,
that this is effected by a power in the eye to alter the
convexity of the crystalline humour as occasion requires;
though this is rather doubtful.

By what has been said it appears, that rays of light
flowing from every part of an object OB, placed at a proper
distance from the eye, will have an image IM thereby
formed on the retina in the bottom of the eye; and since,
the rays OM, Bl, which come from the extreme parts of
the object, cross each other in the middle of the pupil,
the position of the image in will be contrary to that of the
object, or inverted, as in the ease of a lens.

The apparcnl place of any part of an object is in the
axis, and Conjugate focus of that pencil of rays by wjiich
that part or point is formed in the, image. Thus, OM is
the axis, and o the locus proper to the rays by which the

point m in the image is made; therefore the sensation of
the place of that part will be conceived in the mind to be
at o; in like manner the idea of place belonging to the
point i, will be referred in the axis IB, to the proper focus
b ; therefore the apparent place of the whole image im,
will be conceived in the mind to occupy all the space be-
tween o and b, and at the distance al from the eye.

Hence likewise appears the reason, why we see an ob-
ject upright b}' means of an inverted image; for since the
apparent place of every point m will be in the axis mo at
o; and this axis crossing the axis of the eye hl in the
pupil, it follows, that the sensible place o of that point
will lie, without the eye, on the contrary side of its axis, to
that of the point in the eye; and since this is true of all
other parts or points in the image, it is evident that the
position of every part of the object will be on the contrary
side of the axis to every corresponding part in the image,
and therefore the whole object oe will have a contrary po-
sition to that of the image im, or will appear upright.

If the convexity of the cornea CD happens not exactlv
to correspond to the diameter of the eye, considered as
the natural focal distance, then the image will not be
formed on the retina, and consequently no distinct vision
can be effected in such an eye.

If the cornea be too convex, the focal distance in the
eye will be less than its diameter, and the image will be
formed short of the retina. Hence the reason why people
having eyes thus formed are obliged to hold things very
near them, to lengthen the focal distances ; and also why
they use concave glasses to counteract or remedy the ex-
cess of convexity, in order to view distant objects distinctly.
When the eye has less than a just degree of convexity,
or is too flat, as is generally the case with old people, by a
natural deficiency of the aqueous humour, then the rays
tend to a point or focus beyond the retina or bottom of the
eye; and to supply this want of convexity in the cornea,
we use convex lenses in those frames called spectacles, or
visual glasses.

Since the rays of light oa, b a, which constitute the
visual angle oab, will, when they are intercepted by a lens,
be refracted sooner to the axis ; the said angle will thereby
be enlarged, and the object of course become magnified ;
which is the reason why those lenses are called magnifiers,
or reading-glasses.

The dimensions, or magnitude, of an object ob, are
judged of by the quantity of the angle oab which it sub-
tends at the eye. For if the same object be placed at two
different distances h and n, the angles oab, o\b, which in
these two places it subtends at the eye, will be of different
magnitudes, and the lineal dimensions, viz, length and
breadth, will be at N and at l, as the angle o\h is to the
angle oab. But the surfaces of the objects will be as the
squares of those angles, and the solidities as the cubes of
them.

It is found by experience, that two points o, L, in any
object, will not be distinctly seen by the eye, till they are
near enough to subtend an angle OAL of one minute. And
hence when objects, however large they may be, are so re-
mote as not to be seen under an angle of one minute, they
cannot properly be said to have any apparent intensions
or magnitude at all : such is the ease of the large bodies of
the planets, comets, and fixed stars. Hut the science of
optics has supplied means of enlarging this natural small
angle under which most distant objects appear, and
thereby increasing their apparent magnitudes to a very

EYE

[ W 3

EYE

surprising degree, in the instance of that noble instrument applied to the eye in using the instrument. This is usually

the telescope. a lens convex on both sides; but Eustachia Divini long

On the other hand, there are in creation an infinity of since invented a microscope of this kind, the power of
objects, of such small dimensions, that they will not sub- which he places very greatly above that of the common
tend the requisite angle, if brought to the nearest limits of sort; <ind this chiefly depending on the eye-glass, which
distinct vision, viz, 6, 7, or 8 inches from the eye, as found was double, consisting of two plano-convex glasses, so
by experience ; and therefore to render them visible at a posited as to touch one another in the middle of their con-
very near distance, we have a variety of glasses, and in- vex surface. This instrument is much commended by
struments of different constructions, as microscopes, &,c, Fabri in his Optics, as possessing this peculiar excellence,
by which those minute objects appear many thousand that it shows all the objects flat, and not crooked, and
times larger than to the naked eye; and thereby enrich takes in a large area, though it magnifies very much,
the mind with discoveries of the sublimest nature, in regard Bull's EYE, a star of the first magnitude, in the eye
tu-creative power, wisdom, and economy. See Vision. of the constellation Taurus, the bull, and by the Arabs

E.Y E-glass, in Optical Instruments, is that which is called Aldebaran.

F.

F A C

F

l^ACE, or Facade, in Architecture, is sometimes used
for the front or outward part of a building, which im-
mediately presents itself to the eye; or the side where the
chief entrance is, or next the street, &c.

Face, Facia, or Fascia, also denotes aflat member,
havino a considerable breadth, and but a small projec-
ture. Such are the bands of an architrave, larmier, &c.

Faces of a Bastion, in Fortification, are the two fore-
most sides, reaching from the flanks to the outermost point
of the bastion, where they meet, and form the saliant an-
gle of the bastion. These are usually the first parts that
are undermined, or beaten down ; because they reach
the farthest out, are the least flanked, and are therefore
the weakest.

Face of a Place, is the extent between the outermost
points of two adjacent bastions; containing the curtain,
the two flanks, and the two faces of those bastions that
look towards each other. This is otherwise called the
Tcnaille of the place.

Face Prolonged, is that part of a line of defence rasant,
■which is between the angle of the epaule or shoulder of a
bastion and the curtain; or the line of a defence rasant
diminished by the face of the bastion.

FACIA, in Aichitecture, See Face, and Fascia.

FACTORS, in Multiplication, a name given to the two
numbers that are multiplied together, viz, the multipli-
cand and multiplier ; so called, because they are to fa-
cere productum, make or constitute the factum or pro-
duct.

In alocbra, we generally call all those quantities fac-
tors, the product of which constitute any algebraical ex-
pression : thus, (a ■+■ b) and (a — 0) are the factors of the
expression a1 — b ; also a, b, c, d, are the factors of the
quantity abed. But in arithmetic these are commonly
called divisors. See Divisors.

In the Diophantine analysis, and in the theory of num-,
bers, it is frequently very advantageous to be able to re-
solve certain algebraical formula; into their respective
factors, as also in the investigation of the nature of equa-
tions, and in short almost every branch of algebra may
sometimes derive peculiar advantages from such rcsolu-

F A C

tion. Suppose, for example, we had to multiply

— ; 3 by — — — . Now by resolving this into its

x — 'ibx + V J x~ + bx J °

, , ■ , (xs + b!).(x + b).(x-b).(x-li) x° + l'

factors it becomes — —- • — — - = '

(x — 4) . [x — b) . (x + b) . x x

by canceling the common factors; which from the com-
mon mode of operation would have been attended with
considerable labour.

The limits of this work will not however admit of our
entering upon this subject to any extent ; we therefore re-
fer the reader for farther information on this subject to
Euler's Algebra, vol.2, chapter 11; to the additions in
the same vol. by Lagrange; to the Analysis Infinitorum ;
and to Legendre's Theorie des Nombres; where most of
the following theorems are demonstrated in a very mas-
terly manner.

a — b is a factor of am — bm for all values of m.
a -+- b - - - - - am — bm when m is even.
a ■+- b - - - - - am + bm when m is odd.
a — i . . . . _ttm__an for a[| values of m and n,
a -+- 1 - - - - - um — a" when m — n is even.
a ■+■ 1 - - - - - am -t- a" when m — n is odd.
a isanumer. factor ofra — r when a is a prime number.

a r" -*— 1 when « is also prime tor.

a - - - (1 .'2. 3 -- a —•'.!)'+ 1 when a is a prime.

The product of two factors, each of which is the sum
of two squares, is itself the sum of two squares.
That is, (a2 ■+■ bz) . (c2 + dx) =ml + ?f, also
(aa +4'+ c" + d~) .. {'a* + 'ti1 ■+■ 'c2 -t- 'd°) = nr +n- + p-+ a*.

These two last properties were first demonstrated by
Euler, in the Act. Petrop., where there are also many
other ingenious papers on this subject.

Imaginary Factors. See Imaginary.

FACTUM, the product of two quantities multiplied to-
gether. As, the factum of 3 and 4 is 12; and the factum
of 2a and 5b is lOab.

FACUL7E, in Astronomy, a name given by Schemer,
and others after him, to certain bright spots on the sun's
disc, that appear more bright and lucid than the rest of
bis body. — Hevelius assures us that, on July 20, 1634,
he observed a facula whose breadth was equal to a 3d part
3T2

F A L

[ 508 ]

F E L

of the sun's diameter. He says also that the macula?
often change into facula?; but these seldom or never into
macula?. And some authors even contend that all the
macula; degenerate into facula? before they quite disap-
pear. Many authors, after Kircher and Scheiner, have
represented the sun's body full of bright, fiery spots, which
.they conceive to be a kind of volcanos in the body of the
sun: but Huygens, and others of the latest and best ob-
servers, finding that the best telescopes discover nothing
of the matter, they therefore agree in exploding entirely
the phenomena of facula?. All the foundation he could
see for the notion of facula?,- he says, was, that in the
darkish clouds which frequently surround the macula?,
there are sometimes seen little points or sparks brighter
than the rest: the cause of which is attributed by these
authors to the tremulous agitation of the vapours near our
earth; the same as sometimes shows a little unevennessin
the circumference of the sun's disc when viewed through
r telescope. Strictly therefore, the facula? are not eruc-
tations of fire and flame, but refractions of the sun's rays
in the rarer exhalations, which, being condensed near that
shade, seem to exhibit a light greater than that of the
sun.

FAHRENHEIT {Gabriel Daniel), by some writers
said to be a native of Dantzic, and by others to be born at
Hamburgh, in l6'86. He was intended for mercantile af-
fairs. But his inclination led him to the study of natural
philosophy, and he applied particularly to the improve-
ment of barometers and thermometers, constructing great
numbers of them, which were much valued. He im-
proved the thermometer, by substituting mercury instead
of spirits of wine, and formed a new scale for the instru-
ment, founded on accurate experiments, fixing the freez-
ing point of water at 32°, and that of boiling at 212. This
was probably done about the year 1720; as, in his Dis-
sertation on Thermometers, published 1724, he speaks of
having used it some years, to ascertain M. Amonton's ex-
periment, who asserted, that the heat of boiling water was
constant. The English have generally adopted this scale,
though the French preferred that of Reaumur, and very
lately their own centigrade thermometer. Fahrenheit by
some is said to have been living in 1740, while others say
he died in 1736.

FALCATED, one of the phases of the planets, vulgarly
called horned. Astronomers say, the moon, orany planet,
is falcated, when the enlightened part appears in the form
of a crescent, like a sickle, or reapjng-hook, which by the
Latins is called falx. The moon is falcated while she
moves from the 3d quarter to the conjunction, and so on
from hence to the first quarter; the bright part appearing
then like a crescent, viz, during the first and last quarters.
But during the 2d and 3d quarters, the light part appears
gibbous, and the dark part falcated.

FALCON or Faucon, and Falconet or Fauconet,
certain old species of cannon, long since disused.

FALL, the descent or natural motion of bodies towards
the centre of the earth, &c. Galileo first discovered the
law of the acceleration of falling bodies; viz, that the
spaces descended from rest arc as the squares of the times
of descent ; or, which comes to the same thing, that if the
whole time of falling be divided into any number of equal
parts, whatever space it falls through in the first part of
■the time, it will fall 3 times as far in the 2d part of time,
and 5 times as far in the 3d portion of time, and so on,

according to the uneven numbers 1, 3, 5, 7 , Sec. Se«
Acceleration, Descent, Gravity, &c.

FALSE Braye, in Fortification. See Fausse-Bra vr:.

False Position, in Arithmetic. See Position.

False Root, a name given by Cardan, to the negative
roots of equations, and numbers. So the root of y may
be either 3 or — 3, the former he calls the true, and the
latter the false or fictitious root; also of this equation
x'- — x= 6, the two roots are 3 and — 2, the former true,
and the latter false.

FASCIA, in Architecture. See Facia and Face.

FASCIAE, in Astronomy, are certain stripes or rows of
brightparts, observed on the bodies of some of the planets,
like swathes, bands, or belts; especiall}' on the planet Ju-
piter. The fascia?, or belts of Jupiter, are more lucid
than the rest of his disc, and are terminated by parallel
lines. They are sometimes broader and sometimes nar-
rower; and do not always possess the same part of the disc.

M. Huygens also observed a very large kind of fascia in
Mars, in the year lo'ou"; but it was. darker than the rest
of the disc, and occupied its middle part.

FASCINES, in Fortification, are iaggots made of the
twigs and small branches of trees and brush-wood, bound
up in bundles; these, being mixed with earth, serve to
fill up ditches, and to make the parapets of trenches, bat-
teries, &c.

FATHOM, an English measure of the length of 6 feet
or 2 yards.

FATUUS Ignis. 'See Ignis Fatuus.

FAUCON,. and Fauconet, the same as Falcon and
Falconet ; the old names of certain species of ordnance ;
which, as well as many others, are now no longer in use,
as it has been for some time past the practice to denomi-
nate the several sizes of cannon from the weight of their
ball, instead of calling them by those fanciful and unmean-
ing names.

FAUSSE-BRAYE,in Fortification, anelevation of earth,
about three feet above the level ground ; round the foot
of the rampart on the outside, defended by a parapet
about four or five fathoms distant from the upper parapet,
which parts it from the bermc, and the edge of the ditch.
The fausse-braye is the same with what is otherwise called
chemin des rondes, and basse enceinte; its use being for
the defence of the ditch.

FEBRUARY, the 2d month of the year, containing 28
days for three years, and every fourth yea'r 2y days.
These years being called Bissextile, or Leap-years ; and
return every fourth year, except at the commencement of
a new century, which are not bissextile, unless the number
of the century be divisible by4; solSOO was not bissextile,
nor will 1<)00 be so, but 2000 will, because 20 is divisible
by 4. (See Bissextile.) In the first ages of Rome, Febru-
ary was the last month of the year, and preceded January,
till the Decemviri made an order that February should
be the 2d month of the year, and come after January.

FELLOWSHIP, Company, or Partnership, is a
rule in arithmetic, of great use in balancing accounts
among merchants, and partners in trade, teaching how to
assign to every one of them his due share of the gain or
loss, in proportion to the stock he has contributed, and
the time it has been employed, or according to any other
conditions. Or, more generally, it is a method of di-
viding a given number, or quantity, into any number of
parts, that shall have any assigned ratios to one another.

E E R [509

And hence comes this general rule : Having_added into one
sum the several numbers that express the proportions of
the parts, it will be,

As the sum of the proportional numbers :
Is 10 the given quantity to be divided : :
So is each proportional number :
To the corresponding share of the given quantity.
For Example. Suppose it be required to divide the number
120 into three parts that shall be in proportion to each
other as the numbers 1, 2, 3. — Here 120 is the quantity to
be divided, and 0 is the sum of the numbers 1, 2, and 3,
which express the proportions of the parts ; therefore as
f 1 : 20 the 1st part,
6 : 120 : : 1 2 : 40 the 2d part,
(.3 : >6'0 the 3d part.
This rule is usually distinguished into two cases, one in
■which time is concerned, or in which the stocks of part-
ners are continued for different times ; and the other in
which time is not considered ; the latter being called
Single Fellowship, and the former Double Fellowship.

Single Fellowship, or Fellowship without Time, is
the case in which the times of continuance of the shares
of partners arc not considered, being all the same ; and in
this case, the rule will be as above, viz,
As the whole stock of the partners :
Is to the whole gain or loss : :
So is each one's particular stock :
To his share of the gain or loss.

Example. Two partners, a and b, forma joint stock, of
which a contributed 75/, and b 45/; with which they gain
30/: how much of it must each person have ?

a ,nn on f 75 : 18/. 15s. = a's share,

As 1 20 : 30 : : { . , , , , , , , '

£45 : 11/. 5s. = b s share.

Double Fellowship, or Fellow'shitu>2M Time, is the
case in which the times of the stocks continuing are con-
sidered, because they are not all the same. In this case,
the shares of the gain or loss must be proportional, both
to the several shares of the stock, and to the times of their
continuance, and therefore proportional to the products
of the two. Hence this rule: Multiply each particular
share of the stock by the time of its continuance, and add
all the products together into one sum; then say,

As the sum of the products :

Is to the whole gain or loss : :

So is each several product :

To the corresponding share of the gain or loss.

For Example, a had in company 50/. for 4 months,
and B 60/. for 5 months ; and their gain was 24/ : how
must it be divided between them ?
50 CO
4 5

300

200
300

As5O0:24

, f 200 : 5/. 12s. = a's share,
' 1 300 : 14/. 8s. = b's share.
FERGUSON (James), an ingenious experimental phi-
losopher, mechanist, and astronomer, was born in Banff-
shire, in Scotland, 1710, of very poor parents. At the
very earliest age his extraordinary genius began to unfold
itself. He first learned to read, by overhearing his father
teach his elder brother: and he had made this acquisition
before any one suspected it. He soon discovered a pecu-
liar taste for mechanics, which first arose on seeing his

] FER

father use a lever. He pursued his study to a consider-
able length, while he was yet very young; and made a
watch in wood-work, from having once seen one. As he
had at first no instructor, nor any help from books, every
thing he learned had all the merit of an original discovery ;
and such, with inexpressible joy, he believed it to be.

As soon as his age would permit, he went to service ; in
which he met with hardships, that rendered his constitu-
tion feeble through life. While he was servant to a far-
mer (whose goodness he acknowledges in the modest and
humble account of himself which he prefixed to one of
his publications), he contemplated and learned to know
the stars, while he tended the sheep ; and began the study
of astronomy, by laying down, from his own observations
only, a celestial globe. His kind master, observing in
him these marks of ingenuity, procured him the' counte-
nance and assistance of some neighbouring gentlemen,
through whose help and instructions he gained farther
knowledge, having by these means been taught arithmetic,
with some algebra, and practical geometry. He had also
obtained some notion of drawing, and being sent to Edin-
burgh, he there began to take portraits in miniature, -at a
small price ; an employment by which he supported him-
self and family for several years, both in Scotland and
England, while he was pursuing more serious studies. In
London he first published some curious astronomical
tables and calculations; and afterwards gave public
lectures in experimental philosophy, both in London and
most of the country towns in England, with the highest
marks of general approbation. He was elected a fellow
of the Royal Society, and was excused the payment of the
admission fee and the usual annual contributions, on ac-
count of his supposed inability to pay them. He enjoyed
from the king a pension of 50 pounds a year, besides
other occasional presents, which he privately accepted
and received from different quarters, till the time of his
death ; through which, and the fruits of his own labours,
he died worth aboutsix thousand pounds, which astonished
all his friends, who had always entertained an idea of his
great poverty. His death happened in 1776", at 66 years
of age, though he had the appearance of being much
older.

Mr. Ferguson must be allowed to have been a very
uncommon genius, especially in mechanical contrivances
and executions, for he constructed many machines him-
self in a very neat manner. He had also a good taste in
astronomy, as well as in natural and experimental philo-
sophy, and was possessed of a happy manner of explaining
himself in an easy, clear, and familiar way. His general
mathematical knowledge, however, was little or nothing
Of algebra he understood but little more than the nota-
tion; and he has often told me that he could never de-
monstrate one proposition in Euclid's Elements; his con-
stant method being to satisfy himself, as to the truth of,
any problem, with a measurement by scale and com-
passes. He was a man of a very clear judgment in any
thing that he professed, and of unwearied application to
study : benevolent, meek, and innocent in his manners
as a child : humble, courteous, and communicative;
instead of pedantry, philosophy seemed to produce in him
only diffidence and urbanity. The list of Mr. Ferguson's
public works, is as follows :

1. Astronomical Tables and Precepts, for calculating
the true times of New and Full Moons, &c ; 1763.

FER

t 510 ]

F I G

2. Tables and Tracts, Telative to several arts and sciences;
1767. — 3. An Easy Introduction to Astronomy, for Young
Gentlemen and Ladies; 2d edit. 176"£).— 4. Astronomy
explained upon Sir Isaac Newton's Principles ; 5th edit.
1772. — 5. Lectures on Select Subjects in Mechanics,
Hydrostatics, Pneumatics, and Optics ; 4th edit. 1772.
—6. Select Mechanical Exercises ; with a short Account
of the Life of the author, by himself; 1773.— 7- The
Art of Drawing in Perspective made easy ; 1775. — 8. An
Introduction to Electricity ; 1775.— 9- Three Letters to
the Rev. John Kennedy, on the Harvest Moon, 1763 and
1775.

FERMAT (Peter), a celebrated mathematician and
philosopher, and counsellor of the parliament of Tou-
louse, where he was born towards the commencement of
the seventeenth century, or just before ; and notwith-
standing he d"id not make the mathematics the whole, nor
even the principal part of his study, he yet obtained con-
siderable celebrity for his knowledge and discoveries in
this science. He had also a perfect knowledge of the
Greek and Latin languages, beside several of the modern
languages of Europe, as the Italian, Spanish, and Eng-
lish. He also cultivated poetry, some of his works being
still extant. At the same time he fulfilled the duties of
his office as counsellor to the parliament of Toulouse,
with assiduity and honour, enjoying the reputation of an
enlightened judge.

It was the custom of the time in which Fermat flou-
rished, for the mathematicians of different countries to
propose problems to each other for solution, and in this
kind of correspondence we find him deeply engaged, par-
ticularly in theorems' relating to certain properties of
numbers, the demonstrations of which he generally sup-
pressed, if, as is doubtful, he was in possession of them :
however this may be., many of them are even now without
demonstrations, and several others which were left by him
in this state, would perhaps still have remained so had it
not been for the ingenuity and industry of Euler, La-
grange, and some other more modern mathematicians.
• Fermat had a violent dispute with Descartes on the
subject of Maxima and Minima of quantities, of which he
was the author ; as well as on the doctrine of refraction ;
and though a compromise at length took place between
them, they were never good friends after; but in this Des-
cartes was most to blame, as Fermat seems to have much
wished for a complete reconciliation. He died in the
beginning of the year l665. Besides his new edition of
Diophantus, Fermat was author of 1. A Method for the
Quadrature of all sorts of Parabolas.— 2. Another on
Maxima and Minima : which serves not only for the
determination of plane and solid problems; but also for
drawing tangents to f urvc lines, finding the centres of
gravity in solids, and the solution of questions concern-
ing numbers: in short, a method very similar to the
Fluxions of Newton. — 3. An Introduction to Geometric
Loci, plane and solid. — 4. A Treatise on Spherical Tan-
gencies : where he demonstrates in the solids, the same
things as Vieta demonstrated in planes. — 5. A Restoration
of Apollonius's two books on Plane Loci. — 6. A General
Method for the dimension of Curve Lines. Besides a
number of other smaller pieces, and many letters to learned
men ; several of which are to be found in his Opera Varia
Mathematics, printed at Toulouse, in folio, l6'7.9-
FLK MENTATION, an intestine motion, arising spon-

taneously among the small and insensible particles of a
mixed body, thus producing a new disposition, and a
different combination of those parts. Fermentation dif-
fers from dissolution, as the cause from its effect, the
latter being only a result or effect of the former.

FESTOON, in Architecture &c, a decoration in form
of a garland or cluster of flowers.

FtCHANT Flank. See Flank.

Fich ant Line of Defence. See Fixed Line of Defence.

FiELD-.Boo&, in Surveying, a book used for setting
down angles, distances, and other things, remarkable in
taking surveys. The pages of the field-book may be con-
veniently divided into three columns. In the middle co-
lumn are to be entered the angles taken at the several
stations by the theodolite, with the distances measured
from station to station. And the offsets, taken with the
offset-staff, on either side of the station line, are to be
entered in the columns on either side of the middle column,
according to their position, on the right or left, with re-
spect to that line: also on the right or left of these are
to be set down the names and characters of the objects,
with proper remarks, &c. See a specimen in my Trea-
tise on Mensuration," pa. 386, ed. 4th. See also Sur-
veying.

Field-Fort. See Fortine.

FiELD-Pieces, are small cannon, usually carried along
with an army in the field : such as,one-pounders, one-and-
a-half, two-, three-, four-, six-, nine-, and 12-pounders;
which, being light and small, are easily carried.

FiELB-Staff, is a staff carried by the gunners, in which
they screw lighted matches, when they are on service;
which is called arming the field-staffs. See Linstock.

Field of Fiew, or of Vision, is the whole space or ex-
tent within which objects can be seen through an optical
machine, or at one view of the eye without turning it.

The precise limits of this space are not easily ascer-
tained, for the natural view of the eye. In looking at a
small distance, we have an imperfect glimpse of objects
through almost the extent of a hemisphere, or at least for
above 60 degrees each way from the optic axis ; but to-
wards the extremity of this space, objects are very imper-
fectly seen ; and the diameter of the field of distinct vision
does not subtend an angle of more than 5 degrees at most,
so that the diameter of a distinct image on the retina is
less than -j-g-g of an inch ; but it is probably much less.

Field Works, in Fortification, are those that are
thrown up by an army in besieging a fortress, or by the
besieged to defend the place. Such are the fortifications
of camps, highways, &c.

FIFTH, in Music, one of the harmonical intervals or
concords ; called by the ancients diapente.

The fifth is the 3d in order of the concords, and the
ratio of the Chords that produce it, is that of 3 to 2. It
is called fifth, because it contains five terms, or sounds,
between its extremes, and four degrees; so that in the
natural scale of music it comes in the 5th place, or order,
from the fundamental.— The imperfect, or defective fifth,
by the ancients called semidiapente, is less than the fifth
by a mean semitone.

FIGURATK Numhn--, are those numbers which are
formed from the additi.ni of all the leading terms of suc-
cessive series, beginning first with that of the natural
numbers, as in the following table ; where the law of con-
tinuation is manifest.

FIG

1, 2,

3,

4,

5,

6

1, 3,

6,

10,

15,

21

1, 4,

10,

20,

35,

56

1, 5,

15,

35,

70,

126

[ 511 ]

F I G

.(b-m)

1 . 2

n '. (n +

!).(«+ 2)

n..{n+ l) . fri + 2). (ran- 3)

Here, the first series a is th'at of the natural numbers,
of which the general term is n; the second series B, is
that of triangular numbers, of which the general term is

. If now from this general term, which is the nth

1.2 &

term of the series b, there be subtracted the preceding

term of the same series, which is

ill — 1 ) .71

the remainder

will be n, which is the nth term of the series a. Hence
we shall form the nth term of the series b, by adding to
the n — 1 term of the same series, the nth term of the se-
ries A.

The third series c is that of pyramidical numbers, the

-■.0+1). (re + 2) ,

, and it trom this

general term of which

term we subtract the preceding one

(n— 1) .71.(71+1)

the

difference will be

n.{n + l)

which is the nth term of the

series b. Whence we may form the series c by means
of the series b, as this latter is formed from the series a.
And exactly in the same manner the 4th series d, of
(n+l). f«+2). (n + 3)

— , may

which the general term is

1.2.3.4

be formed from the third c, and so on of others.

The general terms given above, as definitions, and from
which the successive law of formation is deduced, contains
the whole theory of figurate numbers, and presents, at
one view, the demonstration of the general proportion men-
tioned by Fermat in his notes on Diaphantus, pa.'l6, and
which he considered as one of his principal discoveries.

So that the figurate numbers of any order may be
found without computing those of the preceding orders ;
which is done by taking the successive products of as
many of the terms of the arithmeticals 1, 2, 3, 4, 5, &c,
in their natural order, as there arc units in the number
which denominates the order of figurates required, and
dividing those products always by the first product : thus,
the triangular numbers are found by dividing the products
1 x 2, 2 x 3, 3 x 4, 4 x 5, &c, each by the 1st pr.
1x2; the first pyramids by dividing the products
1 x 2 x 3, 2 x 3 x 4, 3 x 4 x 5, &c, by the first
1 x 2 x 3. And, in general, the figurate numbers of
any order n, are found by substituting successively 1, 2,
3, 4, 5, &c, instead of x in this general expression
x . x + l . ,r + 2 .x + 3 .&c , . . .

; where the factors in the nume-

1. 2 . 3 .4 . &c

rator and denominator are supposed to be multiplied
together, and to be continued till the number in each be
less by 1 than that which expresses the order of the figu-
rates required. See Maclaurin's Fluxions, art. 351, in
the notes; also Simpson's Algebra, pa. 213; or Malcolm's
Arithmetic, pa. 3.Q-6, where the subject of figurates is
treated in a very extensive and perspicuous manner.

FIGURE, in general, denotes the surface or termi-
nating extremes of a body. — All finite bodies have some
figure, form, or shape ; whence, figurability is reckoned
among the essential properties of body, or matter : abody
without figure, would be an infinite body.

Figures, in Architecture and Sculpture, denote repre-
sentations of things made in solid matter ; as statues, &c.
Figures, in Arithmetic, are the numeral characters,
by which numbers are expressed or written, as the ten
digits, 1, 2, 3, 4, 5, 6, 7, 8, o, 0. These are usually
called the Arabic, and Indian figures, from which people
it is supposed they were derived, being brought into Eu-
rope by the Moors of Spain, and into England about 1130,
as Dr. Wallis conjectures : See his Algebra, pa. 9. How-
ever, from some ancient dates, supposed to consist
wholly or in part of Arabian figures, some have concluded
that these figures, originally Indian, were known and used
in this country at least as earlyas the 10th century. The
oldest date discovered by Dr. Wallis, was on a chimney-
piece, at Helmdon, in Northamptonshire, thus M133,
that is 1133, Other dates discovered since, are 1030,
at Colchester, in Essex; Ml6 or 1016, at Widgel-hall,
near Buntingford, in Hertfordshire ; 10fl on the north
front of the parish church of Rumsey in Hampshire ; and
975 over a gate- way at Worcester.

Dr. Ward, however, has urged several objections against
the antiquity of these dates. As no example occurs of
the use of such figures in any ancient manuscript, earlier
than some copies of Johannes de Sacro Bosco, who died '
in 1256, he thinks it strange that these figures should
have been used by artificers so long before they appear in
the writings of the, learned ; and he also disputes the fact.
The Helmdon date, according to him, should be 1233;
the Colchester date 14i)0 ; that at Widgel-hall has in it no
Arabic figures, the 1 and 6 being i and g, the initial
letters of a name ; and the date at Worcester consists, he
supposes, of Roman numerals, being really mxv. See my
Abridg. Philos. Trans, vol. 8, pa. 32, 31), &c.

Mr. Gibbon observes (in his History of the Decline and
Fall of the Roman Empire, vol. v. pa. 321) that " under
the reign of the caliph Waled, the Greek language and
characters were excluded from the accounts of the public
revenue. If this change was productive of the invention
or familiar use of our present numerals, the Arabic cha-
racters or ciphers, as they are commonly styled, this
regulation has promoted the. most important discoveries
of arithmetic, algebra, and the mathematical sciences."

On- the other hand it may be observed that, " according
to a new, though probable notion, maintained by M. de
Villaison (Anecdota'GrEeca,- torn. ii. pa. 152, 157), our
ciphers are not of Indian or Arabic invention. They
were used by the Greek and Latin arithmeticians long
before the age of Boethius. After the extinction of science
in the West, they were adopted in the Arabic versions
from the original manuscripts, and restored to the Latins
about the 11th century."

FIGURE of the Earth. See Earth.
Figure of an Eclipse, in Astronomy, denotes a re-
presentation on paper &c, of the path or orbit of the sun
or moon, during the time of the eclipse; with the different
phases, the digits eclipsed, and the beginning, middle,
and end of darkness, &c.

Figure, or Delineation, of the full moon, such as,
viewed through a telescope with two convex glasses, is of
considerable use in observations of eclipses, and conjunc-
tions of the moon with other luminaries. In this fioure
are usually represented the maculae or spots of the moon,
marked by numbers; beginning with those that usually
enter first within the shade at the time of the eclipses, and
also emerge the first.

FIG

r sis ]

F 1 G

Figure, in Conic Sections, according to Apollonius,
is the rectangle contained under the latus-rectum and the
transverse axis, in the ellipse and hyperbola.

Figure, in Fortification, is the plan of any fortified
place ; or the interior polygon, &c When the sides,
and the angles, are all equal, it is called a regular figure;
but when unequal, an irregular one.

Figure, in Geometry, denotes a surface or space in-
closed on all sides ; and is either superficial or solid ; su-
perficial when it is inclosed by lines, and solid when it
is inclosed or bounded by surfaces.

Figures are either straight, curved, or mixed, accord-
in" as their bounds are straight, or curved, or both. The
exterior bounds of a figure, are called its sides ; the lowest
side, its base; and the angular point opposite the base,
the vertex of the figure ; also its height, is. the distance
of the vertex from the base, or the perpendicular let fall
upon it from the vertex.

For Figures, equal, equiangular, equilateral, circum-
scribed, inscribed, plane, regular, irregular, similar, &c ;
see the respective adjectives.

Apparent Fibure, in Optics, that figure, or shape,
under which an object appears, when viewed at a dis-
tance. This is often very different from the true figure ;
for a straight line viewed at a distance may appear but as
a point; a surface as a line; a solid as a surface ; and a
crooked figure as a straight one. Also, each of these
may appear of different magnitudes, and some of them of
different shapes, according to their situation with regard
to the eye. Thus, an arch of a circle may appear a
straight line ; a square or parallelogram, a trapezium, or
even a triangle ; a circle, an ellipsis ; angular magnitudes,
round; a sphere, a circle ; &c.

Also any small light, as a candle, seen at a distance in
the dark, will appear magnified, and farther off than it
really is. Add to this, that when several objects are
seen at a distance, under angles that are so small as to
"be insensible, as well as each of the angles subtended by
any one of them, and that next to it ; then all these ob-
jects appear not only as contiguous, but as constituting
and seeming but one continued magnitude.

Figure of the Sines, Cosines, Versed-sines,. Tangents, or
Secants, cfc, are figures made by conceiving the circum-
ference of a circle extended out in a right line, upon
every point of which are erected perpendicular ordinates
equal to the sines, cosines, &cr of the corresponding
arcs ; and then drawing the curve line through the extre-
mity of all these ordinates ; which is called the figure
of the sines, cosines, &c.

It is probable that these figures took their rise from the
circumstance of the extension of the meridian line by Ed-
ward Wright, who computed that line by collecting the
successive sums of the secants, which is the same thing as
the area of the figure of the secants, being made up of all
the ordinates,, or secants, by the construction of the
figure. And in imitation of this, the figures of the other
lines have been invented ; and by means of the figure of
the secants, James Gregory showed how the logarithmic
tangents may be constructed, in his Excrcitationcs Geo-
metrical, 4to, 1668.

Construction^ of the Figures- of Sines, Cosines, i^c

Let ADii &c (fig. I) be the circle, ad an arc, de its

sine, ce its cosine, ae the versed-sine, af the tangent,

gh the cotangent, CI the secant, and cn the cosecant.

Draw a right line aa equal to the whole circumference

adgea of the circle, upon which lay off also the lengths
of several arcs, as the arcs at every 10°, from 0 at u, to
360° at the other end ate; upon these points draw per-
pendicular ordinates, upwards or downwards, according
as the sine, cosine, &c, is affirmative or negative in that
part of the circle, and equal to them ; then drawing a
curve line through the extremities of all these ordinates,
it will form the figure of the sines, cosines, versed-sines,
tangents, cotangents, secants, and cosecants, as in the
annexed figures. Where it may be observed, that the
following curves are the same, viz, those of the sines and
cosines, those of the tangents and cotangents, and those of
the secants and cosecants ; except in the disposition of
some of their parts.

Fig. I .

Sines.

/!

Tangents.

'J

a dS

f

Y

Cotangents.

Secants.

e

)

f

\

r

h

^

a 0

' f

A

a

It maybe known when any of these lines, viz, the sines,,
cosines, &c, are affirmative or negative, i. e. to be set up-
wards or downwards, by observing the following general
rules for those lines in the 1st, 2d, 3d, and 4th quadrants
of the circle.
The sines in the 1st and 2d are affirmative,

in the 3d and 4th negative:

The cosines x in the 1st and 4th are affirmative,

in the 2d and 3d negative :

The tangents in the 1st and 3d are affirmative,

in the 2d and 4th negative;

The cotangents in the 1st and 3d are affirmative,,

in the 2d and 4th negative:
The secants in the 1st and 4th are affirmative,

in the 2d and 3d negative :
The cosecants in the 1st and 2d are affirmative,

in the 3d and 4th negative -
And all the vcrsed-sincs arc affirmative.

F I G

[ 513 ]

F I N

To find the Equation and Area, Sec, to each of these Carves.
Draw any ordinate de ; putting r = AC the radius of
the given circle, x = ad or ad any absciss or arc, and y =
de its ordinate, which will be either the sine de =: s, co-
sine ce=c, versed-sine ae = v, tangent av = t, cotangent
gh =r, secant cf = s, or cosecant ch = cr, according to
the nature of the particular construction. Now, from the
property of the circle are obtained these following gene-
ral equations, expressing the relations between the fluxions
of a circular arc and its sine, or cosine, &c, viz,
rs — re rv r2t

x ~ vV - »-J "" v-V -<?) = </W-~™) ~ 'J + 's

r + T-- s^/lr-r'j «VV - *»)'
And these also express the relation Letween the absciss and
ordinate of the curves in question, each in the order* in
which it stands; where x is the common absciss to all of
them, and the respective ordinates are s, c, v, t, r, s, and a-.
And hence the area &C, of any of these curves may be
found, as follows:

1. In the Figure of Sines. — Here x = ad, and s =
the ordinate de; and the equation of the curve, as above,

is x — —rrr. — . Hence sic = ,, ~° — « is the fluxion of

v{r- - n \/(r - r)

the area; the correct fluent of which is r*:prv/(r'2 — r)
= r°- — re = rv the rectangle of radius and vers. i. e. — or
,-+- as s is increasing or decreasing; which is a general ex-
pression for the area ade in the figure of sines. When
5 = 0, as at a or b, this expression becomes 0 or 2r2; that is
0 at a, and 2r" = the area aeb ; or fl = the area of afg
when ad becomes a quadrant of.

2. In the Figure of Cosines. — Here x = ad and c = de;
and the equation of the curve is x =

— ; =— . Hence ex = „ — is the fluxion of the

v/(r - c1) v'Cf - c)

area ; and the fluent of this is r \/(i~ — c) = rs, the rect-
angle of radius and sine, for the general area adec. When
s = r, or c = 0, this becomes r2 = the area afc, whose ab-
sciss af is equal to a quadrant of the circumference; the
same as in the figure of the sines, upon an equal absciss.

3. In the Figure of Versed-sines. — Here x = ad, and

v ■=. de ; and the equation of the curve is

rv TT . rvv rv*/n

x = — . Hence vx = — = —

y/^iv-vv) </{?rv-vv) y_{lr = v)

is the fluxion of the area ; and the fluent of this is rx — rs
= r x (ad — de) for the area ade in the figure of versed
sines. When ad or ad is a quadrant ss or af, this be-
comes 3-14l6 x i)'2 — r1 = -570Sr2 for the area afg.
And when ad or ad is a semicircle afc, it becomes 3"14.1fjr3
the area abg = in the figure of versed-sines.

4. In the Fig u re of Tangents. — Herex = ad, and t = de;
and the equation of the curve is * =

— . Hence the fluxion of the area is tic c= — ;

7 '■ + i' r -t- t

and the correct fluent of this is f r" x hyp. log. of

—-;- = r* x hyp. log. of — ■ '- = r* x hyp. log. of ^.

And hence the figure of the tangents may be used for con-
structing the logarithmic secants ; a property that was re-
niarked by James Gregory at the end of his Exercit.
Geomet.

When ad becomes a quadrant af, t being then infinite,
this becomes infinite for the area afg. And the same for
the figure of cotangents, beginning at/ instead of a.

Vol. I.

5. For the Figure of the Secants. — Here x = ad, and
s = de; and the equation of the curve is x =

, /, 3 - tt- Hence sx = — — — is the fluxion of the

7-2x hyp. log. of 3"t"v/(s°~'')for

area; thefluent of wl

the general area ade. And when ad becomes the qua-
drant af, this expression becomes infinite for the area afg.
The same process will serve for the figure of cosecants, be-
ginning at/ instead of a.

Hence the meridional parts in Mercator's chart may be
calculated for any latitude a d or ad : For the merid. parts :
are to the arc of latitude a d : : as the sum of the secants :
to the sum of as many radii or : : as the area ade : to ad x
radius ac or ad x ac in the first figure.

FILLET, in Architecture, any little square member or
ornament used in crowning a larger moulding.

F1N..EUS (Orontius), in French, Fine, professor of
mathematics in the. Royal College of Paris, was the son of
a physician, and was born at Briancon in Dauphine in
14-94-. He went young to Paris, where his friends pro-
cured him a place in the college of Navarre. He applied
himself there to the study of philosophy and polite litera-
ture ; but more especially to mathematics, in which he
made considerable progress, particularly in mechanics ;
for, having both a genius to invent instruments, and a skil-
ful hand to execute them, he gained much reputation by
the specimens he gave of his ingenuity.

Finaeus first made himself publicly known by correcting
and publishing Siliceus's Arithmetic, and the Margarita
Philosophica. He afterwards read private lectures in ma-
thematics, and then taught that science publicly in the
college of Gervais ; from the reputation he thus gained,
he was recommended to Francis the 1st, as a proper per-
son to teach mathematics in the new college which that
prince had founded at Paris. And here, though he spared
no pains to improve his pupils, he yet found time lo pub-
lish a great many books on many of the principal parts of
the mathematics. But neither his genius, his labours, his
inventions, nor the esteem which numberless persons
showed him, could secure him from that fate which so
often befalls men of letters. He was obliged to struggle
all his life-time with po\erty ; and when he died, left a
numerous family deeply in debt. However, as merit must
always be esteemed in secret, though it seldom has the
fortune to be rewarded openly; his children found friends
who for their father's sake assisted his family. — He died in
1555, at 6l years of age.

Like all other mathematicians and astronomers of those
times, he was greatly addicted to astrology ; and had the
misfortune to be a long time imprisoned for having pre-
dicted some things, Unit were not acceptable to the court
of France. He was also one of those, who vainly boasted
of having found out the quadrature of the circle, with the
solution of the problem of two mean proportionals, and
that ot the trisection of an angle ; but lie was vigorously
refuted by Nonius, as well as by Butron and Borelli, two
of his own pupils. — Several of his pieces were published
separately; as, his Practical Arithmetic in 1555, and
Practical Geometry in 1544, both in 4to. And an edition
of his whole works, translated into the Italian language,
was published in 4to, at Venice, 1587 ; consisting of Arith-
metic, Practical Geometry, Cosmography, Astronomy, and
Dialling.

FINITE, the property of any thing thut is bounded
3 U

F I R

C 514 ]

F I R

or limited, either in its power, or extent, or duration,
&c ; as distinguished from the property of infinite, or with-
out bounds.

FIRE, is that subtile invisible fluid by which bodies
are made hot to the touch, and expanded or enlarged in
bulk ; by which fluids are rarefied into vapour; or solid
bodies become fluid, and at last are either dissipated and
carried off in vapour, or else melted into glass. It seems
also to be the chief agent in nature on which animal and
vegetable life have an immediate dependence. See Ca-
loric and Heat.

The disputes concerning fire, which long existed among
philosophers, have now in a great measure subsided. Those
celebrated philosophers of the 17th century, Bacon, Boyle,
and Newton, were of opinion, that fire was not a sub-
stance distinct from other bodies, but that it entirely con-
sisted in the violent motion of the parts of any body,
which was produced by the mechanical force of impul-
sion, or of attrition. So Boyle says, when a piece of iron
becomes hot by hammering, " there is nothing to make
it so, except the forcible motion of the hammer impress-
ing a vehement and variously determined agitation on the
small parts of the iron." And Bacon defines heat, which
he makes synonymous with lire, an" expansive undulatory
motion in the minute particles of a body, whereby they
tend with some rapidity from a centre towards a circum-
ference, and at the same time a little upwards." And
according to Newton, fire is a body heated so hot as to
emit light copiously ; for what else, says he, is red-hot
iron, but fire? and what else is a fiery coal than red-hot
wood ? by which he suggests, that bodies which are not
fire, may be changed and converted into it.

On the other hand, the chemists strenuously contend-
ed that fire was a fluid of a certain kind, distinct from all
others, and universally present throughout the whole
globe. Boerhaave particularly maintained this doctrine ;
and in support of it brought this argument, that flint and
stfel would strike fire, and produce the same degree of
heat in Nova Zcmbla as they would do under the equator.
Other arguments were drawn from the increased weight
of metallic calces, which they thought proceeded from
the fixiti" of the element of fire in the substance whose
weight was thus increased. For a long time, however,
the matter was most violently disputed ; but the mechani-
cal philosophers at last prevailed through the deference
paid to the principles of Newton, though he himself had
scarcely taken any active part in the contest.

The experiments of Dr. Black, however, seemed to bring
the dispute to a decision, and that in favour of the che-
mists, concerning what he called latent heat. From these
discoveries it appears, that fire may exist in bodies in such
a manner, as not to discover itself in any other way than
by its action on the minute parts of the body; but that
suddenly this action may be changed so, as no longer to
be directed on the particles of the body itself, but upon
external objects; in which case we then perceive its ac-
tion by our sense of feeling, or discover it by the thermo-
meter, and call it heat, or sensible fire.

From this discovery, and others in electricity, it is now
pretty generally allowed, that fire is a distinct fluid,
capable of being transferred from one body to another.
But when this was discovered, another question ho less
perplexing arose, viz, what kind of a fluid it was; or
whether it bears any analogy to those with which we are
better acquainted. Now there are found two fluids, viz,

the solar light, and the electric matter, both of which oc-
casionally act as fire, and which therefore seem to indi-
cate an original identity ; and popularly the matter has
been long since determined; the solar rays and the elec-
tric fluid having been indifferently accounted elementary
fire. Some indeed have imagined both these fluids to be
mere phlogiston itself, or at least containing a large por-
tion of it; and Mr. Scheele went so far in this way as to
form an hypothesis, which he endeavoured to support by
experiments, that fire is composed of phlogiston and de-
phlogisticated air. But it is now ascertained beyond
doubt, that the result of such a combination is not fire,
but fixed air.

It was long since observed by Newton, that heat was
certainly conveyed by a medium more subtile than the
common air; for two thermometers, one included in the
vacuum of an air-pump, the other placed in the open air,
at an equal distance from the fire, would become equally
hot in nearly the same time. This and other experiments
show, that fire exists and acts where there is no other
matter, and of consequence it is a fluid per se, independ-
ent of every terrestrial substanre, without being Generated
or compounded of any thing we are yet acquainted with.
To determine the nature of the fluid, we have only to con-
sider whether any other can be discovered which will pass
through the perfect vacuum just mentioned, and act there
as fire. Such a fluid is found in the solar light, which is
well known to act even in vacuo as the most violent fire.
The solar light will likewise act in the very ^same manner
in the most intense cold ; for M. de Saussure has found,
that on the cold mountain-top the sun-beams are equally
powerful as on the plain below, if not more so. It ap-
pears therefore, that the solar light will produce heat in-
dependent of any other substance whatever ; that is, where
no other body is present, at least as far as we can judge,
except the light itself, and the body to be acted on. We
cannot therefore avoid concluding, that a certain modifi-
cation of the solar light is the cause of heat, expansion,
vapour, &c, and answers to the rest of the characters given
in the foregoing definition of fire, and that independent of
any other substance whatever.

It is very probable too, that the electric matter is no
other than the solar light absorbed by the earlh, and thus
becoming subject to new laws, and assuming many pro-
perties apparently different from what it has when it acts
as light. Even in this case it manifests its identity with
fire or light, viz, by producing a most intense heat where a
large quantity of it passes through a small space. So that
at any rate, the experiments which have already been
made, and the proofs drawn from the phenomena of nature,
show such a strong affinity between the elements of fire,
light, and electricity, that we may not only assert their
identity on the most probable grounds, but lay i) down as
a position against which at present no argument of any
weight has an existence.

Fiui, Arrow, is a small iron dart, furnished with springs
and bars, and also a match impregnated with sulphur and
powder, which is wound about its shaft. It is elm fly used
by privateers and pirates to fire the sails of the enemy's
ship, and for this purpose it is discharged from a mus-
ketoon, or a swivel gun. The match being kindled by the
explosion, it communicates the flame to the sail against
which it is directed, where the arrow fastens itself by means
of its bars and springs. This weapon is peculiar to hot
climates, particularly the West Indies; the sails being

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very dry, are quickly inflamed, and the fire is soon con-
veyed to the masts, rigging, and finally to the vessel itself.

FiRE-Batlx, in Artillery, are certain balls composed of
combustible matters, such as fine or mealed powder, sul-
phur, saltpetre, rosin, pitch, &c. These are thrown into
the enemy's works in the night-time, to discover where they
are; or to set fire to houses, galleries, or blinds of the be-
siegers, &c. They are sometimes armed with spikes or
hooks oi iron, that they may not roll off, but stick or hang
where they are to take effect.

FiRE-Balls, or Fiery -Meteors, in Meteorology, a kind of
luminous bodies usually appearing at a great height above
the earth, with a splendour surpassing that of the moon,
and sometimes apparently as large. It has not been ob-
served that they follow any regular course or motion, but,
on the contrary, moving in all directions, and with very
different degrees of celerity ; frequently breaking into se-
veral smaller ones ; sometimes making a strong hissing
sound, sometimes bursting or vanishing with a loud report,
and sometimes not.

These luminous appearances doubtless constitute one
part of the ancient prodigies, blazing stars, or comets,
which last they sometimes resemble in being attended with
a train ; but more often they appear round. The first of
these of which we have any accurate account, was ob-
served by Dr. Halley, and some other philosophers at dif-
ferent places, in the year 1719; the height of which above
the surface of the earth was computed at more than 70
miles. Many others have been accurately observed since
that time, and described by different philosophers. See
French Memoirs, and Philos. Trans, vols. 30, 41, 42, 43,
46, 47, 48, 51, 53, 54, 63, 74, &c. The velocities, direc-
tions, appearances, and heights of all these were found to
be very various; though the height of some of them was
supposed above the limits assigned to our atmosphere, or
where it loses its refractive power. The most remarkable
phenomenon of this kind on record, appeared on the 18th
of August 1783, about 9 o'clock in the evening. It was
seen to the northward of Shetland, and took a south-east-
erly direction for an immense space, being observed as far
as the southern provinces of France, and by some it was
said to have been seen at Rome, passing over a space of
1000 miles in about half a minute of time, and at a very
great height. During its course it appeared several times
to change its shape ; sometimes appearing in the form of
one ball, sometimes of two or more; sometimes with a train,
and sometimes without one.

There are divers opinions concerning the nature and
origin of these meteors. The first thing that occurred to
philosophers on this subject was, that they were burning
bodies rising from the surface of the earth, and flying
through the atmosphere with great rapidity. But this hy-
pothesis was soon rejected, on considering that there was
no power known by which such bodies could either be
raised to a sufficient height, or projected with the velocity
which they had by some means acquired. The next hy-
pothesis was, that, instead of one single body, they consist
of a train of sulphureous vapours, extending an immense
way through the atmosphere, and being kindled at one
cjkI, display the luminous appearances in question by the
fire running from one extremity of the train to the other.
Hut it is not easy to conceive how such matters can exist
and be disposed in such lines in so rare a part of the at-
mosphere, and even to burn there, in an almost perfect
vacuum. For which reason this hypothesis was aban-

doned, for another, which was, that those meteors are per-
manent solid bodies, not rising from the earth, but revolv-
ing round it in very ex centric orbits, and thus in their
perigeon moving with vast rapidity. But as the various
appearances of one and the same meteor, to observers at
different places, are not compatible with the idea of a
single body so revolving, this hypothesis has also been given
up in its turn. Again it has been said, that these meteors
are a kind of bodies that take fire as soon as they come
within the atmosphere of the earth. But this cannot be
supposed without implying a previous knowledge of the
nature of these bodies, which it is impossible we can have.
Another hypothesis is, that these fiery meteors are great
bodies of electric matter, moving from one part of the hea-
vens where, to our conception, it is superabundant, to
another where it is deficient: a conclusion attended with
some probability, from the analogy observed between elec-
tricity and the phenomena of these bodies: and hence
they appear to be of the same origin with shooting-stars,
lightning, the aurora-borealis, &c, being all referred to the
same cause, viz, the electricity of the atmosphere. See
Aerolite.

F i re- Engine, is a machine for extinguishing accidental
fires by means of a stream or jet of water. The common
squirting fire-engine consists of a lifting pump placed in a
vessel of water, and wrought by two levers that act always
together. During the stroke, the water raised by the
piston of the pump spouts forcibly through a pipe joined
to the pump-barrel, and made capable of any degree of
elevation by means of a yielding leather pipe, or by a ball
and socket turning every way, screwed on the top of the
pump. The vessel containing the water is covered with a
strainer, to prevent the mud, &c, which is poured into it
with the water, from choking the pump-work. Between
the strokes of this engine the stream is discontinued, for
want of an air-vessel. However, in some cases, engines of
this construction have their use, because the stream,
though interrupted, is much smarter than when the engine
is made to throw water in a continued stream. See these
engines particularly described in Desaguliers's Exper.
Philos. vol. 2, pa. 505 ; or Martin's Philos. Britan. vol. 2,
pa. 69; O. G. Gregory's Mechanics. See also the figure
of them, plate x. fig. 3.

It appears that in the year 1675, Sir Sam. Morland got
a patent for a powerful engine of this kind. This machine,
by the strength of S men, would force water in a conti-
nued stream, from the river Thames, to the top of Wind-
sor castle, and 60 feet higher, at the rate of 6'0 barrels an
hour, which experiment was repeated several times, in the
year 1681, before the king, queen, and court; when his
majesty presented Morland with a medal, with his effigy
set round with diamonds, and appointed him his master of
mechanics, &c.

Fire-Loc^, or Fusil, a small gun or musket, which fires
with a flint and steel; as distinguished from the old
musket, or match-lock, which was fired with a match.
The fire-lock is now in common use with the European
armies, and carried by the foot-soldiers. It is usually
about 3 feet 8 inches in the barrel; and, including the
stock, 4 feet 8 inches, carrying a leaden bullet, of which
29 make 2lb. The diameter of the ball is '55, and that of
the barrel -56 parts of an inch. The time of. the inven-
tion of fire-locks is uncertain ; they were however used
in 1690.

FiRE-PZacw, are contrivances for communicating heat
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to rooms, and also for answering various purposes of art
and manufacture. — The general properties of air and fire,
on which their construction chiefly depends, are the fol-
lowing, viz, that air is rarefied by heat, and condensed by
cold; or, in other words, the same quantity of air takes up
more space when warm than when cold. Air rarefied and
expanded by heat, is specifically lighter than it was before,
and will rise in other air of greater density: so that a fire
being made in any chimney, the air about and over the fire
is rarefied by the heat, thence becomes lighter, and so
rises in the funnel, and goes out at the top of the chimney :
the other air in the room, flowing towards the chimney,
supplies its place, is then rarefied in its turn, and rises
likewise; and the place of the air thus carried out of the
room, is supplied by fresh air coming in through doors and
windows, or, if they be shut, through every crevice with
violence; or if the avenues to the room be so closed up,
that litLle or no fresh supply of air can be obtained, the
current up the funnel must flag; and the smoke, no longer
driven up, float about in the room.

On these principles, various contrivances and kinds of
fire-grates and stoves have been devised, from the old very
open and wide chimney places, down to the present
modish ones, which are much narrowed in the front, open-
ing, by side and back jambs, and a low breast or mantle,
besides the convenience of a flap, called a register, that
covers the top of the fire-stove, but opening to any degree
with a small winch, which lifts the back part sloping up-
wards, and so throws the smoke freely up the funnel, ad-
mitting as little air to pass as you please ; by which simple
means the warm air is kept more in the room, while the very
narrow and sloping orifice promotes the brisk ascent of the
smoke, and yetprevenis its return down again, for the same
reason.

Another ingenious, but more complex, apparatus, called
the Pennsylvania fire-place, was invented by Dr. Franklin,
which keeps a room very warm by a constant supply of
fresh hot air, that passes into it through the stove itself.
See the description in his Letters and Papers on Philoso-
phical Subjects.

Many other forms of fire-grates, often of whimsical fi-
gures, have been recommended by different persons: but,
after all, the common register-grate is far the best.

FiRE-Pot, in the Military Art, is a small earthen pot,
into which is put a grenade, filled with fine powder till the
grenade be covered ; the pot is then closed with a piece
of parchment, and two pieces of match laid across and
lighted. This pot being thrown where it is designed to do
execution, breaks and fires the powder, and this again
fires the powder in the grenade, which ought to have no
fuze, that its operation may be the quicker.

Rasant, or Ruzunt Fire, is a fire from the artillery
and small arms, directed parallel to the horizon, or to
those parts of the works of a place that are defended.

Running FlliE, is when ranks of men fire one after
another ; or when the lines of an army are drawn out to
fire on account of a victory ; in which case each squadron
or battalion takes the fire from that on its right, from the
right of the first line to the left, and from the left to the
right of the second line, &c. This is now commonly
called a.fcit-dc-jaye.

I*'i KV.-Slii.ps, in the navy, are vessels charged with com-
bustible materials or artificial fire-works: which having
the wind of an enemy's ship, grapple her, and set her on

fire. Anderson, in his History of Commerce, vol. I, pit.
432, ascribes the invention to the English, in this in-
stance, viz, some vessels being filled with combustible
matter, and sent among the Spanish ships composing the
Invincible Armada in 1588; and hence arose, it is said,
the terrible invention of fire-ships. But Livy informs us,
that the Rhodians had invented a kind of fire-ships, which
were used injunction with the Roman fleet in their en-
gagement with the Syrians, in the year 190 before Christ :
cauldrons of combustible and burning materials were
hung out at -their prows, so that none of the enemies'
ships durst approach them : for these fell on the enemies'
galleys, struck their beaks into them, and at the same
time set them on fire. Livy, lib. 37, cap. 30.

I¥ild-F ire, is a kind of artificial or factitious fire, that
bums even under water, and that with greater violence than
out of it. It is composed of sulphur, naphtha, pitch, gum,
and bitumen, and it is only extinguishable by vinegar,
mixed with sand and urine, or by covering it with raw
hides. It is said its motion is contrary to that of natural
fire, always following the direction in which it is thrown,
whether it be downwards, sideways, or otherwise.

The French call it Greek fire, or Feu Gregeois, because
first used by the Greeks about the year 660, as is ob-
served by the Jesuit Petavius, on the authority of Nicetas,
Theophanes, Cedrenus, &c. The inventor, according to
the same author, was an engineer of Heliopolis, in Syria,
named Callinicus, who first applied it in the sea-fight com-
manded by Constautine Pogonatcs, against the Saracens,
nearCyzicus, in the Hellespont; and with such effect, that
he burnt the whole fleet, which had on board 30,000
men. But others refer it to a much older date, and as-
cribe the invention to Marcus Gracchus ; an opinion that
is supported by several passages, both in the Greek and
Roman writers ; which show that it was anciently used
by each of these nations in their wars. See Scaliger against
Cardan.

The successors of Constantine used it on several occa-
sions, with great advantage : and it is said that they were
able to keep the secret of the composition to themselves;
so that no other nation knew it in the year 960.

It is recorded by Chorier, in his Hist, de Dauph. that
Hugh, king of Burgundy, demanding ships of the emperor
Leo for the siege of Fresne, desired also the Greek fire.—
F. Daniel gives also a good description of the Greek fire,
in his account of the siege of Damietta. under St. Louis.
Every body, says he, was astonished with the Greek fire,
which the Turks then prepared; and the secret of which
is now lost. They threw it out of a kind of mortar, and
sometimes shot it with an odd sort of cross-bow, which was
strongly bent by means of a handle, or winch, with much
greater force than the bare arm. That which was thrown
from the mortar sometimes appeared in the air of the
size of a tun, with a long tail, and a noise like that of
thunder. The French, by degrees, got the secret of ex-
tinguishing it; in which they succeeded several times.

After all, perhaps the invention of the wild-lire is to be
ascribed to other nations, and to a still older date, and
that it was the same as that used among the Indians in
Alexander's invasion, when it was said the) fought with
thunder and lightning, or shot fire with a. terrible noise.

Ft it E-Works, otherwise called Pyrotechnia, are artificial
fires, or preparations made of gunpowder, sulphur, and
other inflammable and combustible ingredients, which are

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displayed at public rejoicings, and on other occasions.
The principal of these are rockets, serpents, stars, hail,
mines, bombs, garlands, letters, and other devices. The
invention of lire-works is attributed, by M. Mahudel, to
the Florentines and people of Sienna; who also discovered
the method of adding to them decorations of statues, with
fire issuing from their eyes and mouths.

FIRKIN, an English measure of capacity ; being the
4th part of a barrel ; or containing 9 gallons.

F1RLOT, a dry measure used in Scotland. The oat-
firlot contains 21^ pints of that country, or about 85 Eng-
lish pints ; and the barley-firlot, 31 standard pints. The
wheat-firh t contains about 22 11 cubic inches; and there-
fore exceeds the English bushel by 60 cubic inches, or al-
most an English quart.

FIRMAMENT, by some old astronomers, is the orb
of the fixed stars, or the highest of all the heavens. But
in Scripture and common language it is used for the middle
regions, or the space or expanse appearing like an arch
quite around or above us in the heavens. Many ancients
and moderns also accounted the firmament a fluid matter ;
but those who gave it the name of firmament must have
taken it for a solid one.

FIRMNESS, is the consistence of a body; or that state
in which its parts cohere, or are united together, so that
the motion of one part induces a motion of the rest. And "
in this sense firmness stands opposed to fluidity. The firm-
ness of bodies therefore depends on the connexion or co-
hesion of their particles ; and the cause of cohesion the
Newtonians hold to be an attractive force, inherent in
bodies, which binds their small particles together ; exert-
ing itself nnly at the points of contact, or extremely near
them, and vanishing at greater distances.

FISSURES, in the History of the Earth, are certain
interruptions, mostly parallel to each other, that divide or
separate its strata from one another, in nearly horizontal
directions; and the parts of the same stratum in nearly
vertical directions.

FIXED Line of Defence, a line drawn along the face of
the bastion, and terminating in the curtain.

Fixed Signs of the Zodiac, according to some, are the
four signs Taurus, Leo, Scorpio, Aquarius. They arc so
called because the sun passes them respectively in the
middle of each quarter, when that season is more settled
and fixed than under the signs which begin and end it.

Fixed Stars, are such as constantly retain the same
position and distance with respect to each other ; by which
they are contradistinguished from erratic or wandering
stars, which are continually varying their situation and dis-
tance.— The fixed stars only are properly and absolutely
called stars; the other celestial bodies falling under the de-
nomination of planets or comets.

FIXITY, or Fixedness, the quality of a body which
determines it fixed ; or a property which enables it to
endure the fire and other violent agents. A body may be
said to be fixed in two respects: 1st, When on being ex-
posed to the fire, or a corrosive menstruum, its particles
are indeed separated, and the body rendered fluid, but
without being resolved into its first elements. The 2d,
when the body sustains the active force of the fire or men-
struums while its integral parts are not carried off in
fumes. Each kind of fixity is the result of a strong or in-
timate cohesion between the particles.

FLAMIi, the subtlest and brightest part of the fuel,
ascending above it in a pyramidal figure, and heated red-

hot. Sir Isaac Newton defines flame as only red-hot
smoke, or the vapour of any substance raised from it .by
fire, and heated to such a degree as to emit light copiously.
Is not flame, says he, a vapour, fume, or exhalation,
heated red-hot; that is, so hot as to shine? For bodies
do not flame without emitting a copious fume; and this
fume burns in the flame. The ignis fatuus is a vapour
shining without heat ; and is there not the same difference
between this vapour and flame, as between rotten wood
shining without heat, and burning coals of fire ? In dis-
tilling hot spirits, if the head of the still be taken off, the
vapour which ascends will take fire at the flame of a can-
dle, and turns into flame. Some bodies, heated by motion
or fermentation, if the heat grow intense, fume copiously;
and if the heat be great enough, the fumes will shine,
and become flame. Metals in fusion do not flame, for
want of a copious fume. All flaming bodies, as oil, tallow,
wax, wood, fossil coal, pitch, sulphur, &c, by burning,
waste in smoke, which at first is lucid ; but at a little dis-
tance from the body ceases to be so, and only continues
hot. When the flame is put out, the smoke is thick, and
frequently smells strongly: but in the flame i* loses its
smell ; and, according to the nature of the fuel, the flame
is of divers colours. That of sulphur and spirit of wine is
blue; that of copper opened, with sublimate, green; that
of tallow, yellow; of camphire, white; &c. Newton's
Optics, pa. 318.

FLAMSTEED (John), an eminent English astrono-
mer, being indeed the first astronomer-royal, for whose
use the royal observatory was built at Greenwich, thence
called Flamsteed House. He was born at Denby in Der-
byshire the 19th of August 1646; and was educated at
the free-school of Derby, where his father lived. At 14
years of age he was afflicted with a severe illness, which
rendered his constitution tender ever after, and prevented
him then from going to the university, for which he was
intended. He nevertheless prosecuted his school educa-
tion with the best effect; and, in 1662, on quitting the
grammar-school, he pursued the natural bent of his genius,
which led him to the study of astronomy, and closely pur-
sued Sacrobosco's book DeSpha;ra, which fell in his way,
and thus laid the foundation of all that mathematical and?
astronomical knowledge, for which he became afterward
so justly celebrated. He next procured more modern
books on the same subject, and among them Streete's As-
tronomia Carolina, then lately published, from which he
learned to calculate eclipses and the planets' places. Some
of these- being shown to Mr. Halton, a considerable ma-
thematician, he lent him Riccioli's Almagestum Novum,
and Kepler's Tabula; Rudolphina;, from which he gained
much information. In 1669, having calculated some re-
markable eclipses of the moon, he sent them to lord
Brouncker, president of the Royal Society, which were
greatly approved by that learned body, and procured him
a letter of thanks from Mr. Oldenburg their secretary, and
another from Mr. John Collins, with whom, and other
learned men, Mr. Flamsteed for a long time afterwards
kept up a correspondence by letters on literary subjects.
In 167O, his father, observing that he held correspondence
with these ingenious persons, advised him to take a jour-
ney to London, to make himself personally acquainted
with them; an offer which he gladly embraced, and visited
Mr. Oldenburg and Mr. Collins, who introduced him to
Sir Jonas Moore, which proved the means of his greatest
honour and preferment. He here got the knowledge and

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practice of astronomical instruments, as telescopes, mi-
crometers, &c : and on bis return, he called at Cambridge,
and visited Dr. Barrow, Newton, and other learned men
there, and entered himself a student of Jesus-£ollege. In
1672 he made many celestial observations, which, with
calculation* of appulses of the moon and planets to fixed
stars for the year following, he sent to Mr. Oldenburg,
who published them in the Philos. Trans.

In 1673, Mr. Flamsteed wrote a small tract on the true
diameters of all the planets, when at their greatest and
least distances from the earth ; which he lent to Newton
in l6S5, who made some use of it in the 4th book of his
Principia. — In l6~74 he wrote an ephemeris, to show the
falsity of astrology, and the ignorance of those who pre-
tended to it: with calculations of the moon's rising and
setting; also ocoultations and appulses of the moon and
planets to the fixed stars. To which, at Sir Jonas Moore's
request, he added a table of the moon's southings for that
year; from which, and from Philips's theory of the tides,
the high-waters being computed, he found the times come
very near. In l6'74 he also drew up an account of the
tides, for the use of the king. Sir Jonas also showed the
king, and the duke of York, some baijometers and thermo-
meters that Mr. Flamsteed had given him, with the neces-
sary rules forjudging of the weather; and otherwise took
every opportunity of speaking favourably of Flamsteed to
them, till at length he brought him a warrant to be the
king's astronomer, with a salary of 100?. per annum, to be
paid out of the office of Ordnance, because Sir Jonas was
then surveyor general of the Ordnance. This however
did not abate our author's propensity for holy orders, and
he was accordingly ordained at Ely by bishop Gunning.

On the 10th of August 1675, the foundation of the
royal observatory at Greenwich was laid; and during
the building of it, Mr. Flamsteed's temporary observatory
was in the queen's house, where he made his observations
of the appulses of the moon and planets to the fixed stars,
and wrote his Doctrine of the Sphere, which was after-
" wards published by Sir Jonas Moore, in his System of the
Mathematics.

About the year 1684 he was presented to the living of
' Burslow in Surrey, which he held as long as he lived. Mr.
Flamsteed was equally respected by the great men his con-
temporaries, and by those who succeeded him. Dr. Wot-
ton, in his Reflections upon Ancient and Modern Learn-
ing, styles our author one of the most accurate observers
of the heavens, and says he calculated tables of the'eclipses
of the several satellites, which proved very useful to the
astronomers. And Mr. Molyneaux, in his Dioptrica
Nova, speaks very highly of his abilities; and, in the ad-
monition to the reader prefixed to the wort, observes, that
the geometrical method of calculating a ray's progress is
quite new, and never before published; but for the first
hint of it, says he, I must acknowledge myself obliged to
my worthy friend Mr. Flamsteed. He wrote several small
tracts, and had many papers inserted in the Philosophical
Transactions, in almost every volume, from the 4th to
the 2£)th, which are too numerous to be particularly men-
tioned in this place.

But his great work, and that which contained the prin-
cipal operations of his life, was the HistoriaCoulcslis Bri-
fannica, published in 1725, in 3 large folio volumes.
The first of which contains the observations of Mr. Wil-
liam Gascoigne, the original inventor of the method of
measuring angles in a telescope by means of screws, and

the first who applied telescopical sights to astronomical
instruments, in observations made at Middleton, near
Leeds in Yorkshire, between the years 1638 and 16*43 ;
extracted from his letters by Mr. Crabtree ; with some of
Mr. Crabtree's ob-ervaiions about the same time ; and
also those of .Mr. Flamsteed himself, made at Derby be-
tween the years 16*70 and 1675; besides a multitude of
curious observations, and necessary tables to be used
with them, made at the royal Observatory, between the
years 16*75 and l6*S9- — The 2d volume contains his ob-
servations, made with a mural arch of near 7 feet radius,
and 140 degrees on the limb, of the meridional zenith
distances of the fixed sta'rs, sun, moon, and planets, with
their transits over the meridian; also observations of the
diameters of the sun and moon, with their eclipses, and
those of Jupiter's satellites, and variations of the compass,
from 16*89 to 17 1 9 : also tables showing how to render
the calculation of the places of the stars and planets easy
and expeditious. To which are added, the moon's place
at her oppositions, quadratures, &c ; aKo the planets'
places, derived from the observations. — The 3d volume
contains a catalogue of the right-ascensiOus, polar- ■
distances, longitudes, and magnitudes of near 3000 fixed
stars, with the corresponding variations of the same To
this volume is prefixed a preface, at considerable length,
containing an account of all the astronomical < l>si nations
made before his time, with a description of the inst ruments
employed; as also of his own observations and instru-
ments; a new Latin version of Ptolemy's catalogue of
1026 fixed stars; and Ulcgh-beig's places annexed on the
Latin page, with the corrections : a small catalogue of
the Arabs: Tycho Brahe's of about 780 fixed stars: the
Landgrave of Hesse's of 3S6" : Ilcvclius's of 1534 : and a
catalogue of some of the southern fixed stars nol visible
in our hemisphere, calculated from the observn ions made
by Dr. Halley at St. Helena, adapted to the year 1726.

This work he prepared in a great measure fi r the press,
with much care and accuracy : but through a natural
weakness of constitution, and the advance of age, he died
of a strangury before he had completed it, December the
19th, 1719, at 73 years <;f age; leaving the care of finish-
ing and publishing his work to his friend Mr. Hodgson.
A less perfect edition of the Historia Ceelestis had before
been published, but without his consent, viz, in 1712,
in one volume folio, containing his observations to the
year 1705.

FLANK, in Fortification, is that part of the bastion
reaching from the curtain to the face ; and it defends the
curtain, with the opposite face and flank.

Oblique or Second Flank, or Flank of the Curtain,
is that part of the curtain from which the face of the op-
posite bastion can be seen, being contained between the
lines rasant and fichant, or the greater and less lines of
defence; or the part of the curtain between the flank and
the point where the fichant line of defence terminates.

Covered, Low, or Retired Flank, is the platform of
the casemate, which lies hid in the bastion; and is other-
wise called the orillon.

Fichunt Flank, is that from which a cannon playing, _
•fires directly on the face of the opposite bastion.

Rasant or Razant Flank, is the point from which the
line of defence begins, from the conjunction of which
with the curtain, the shot only raseth the face of the
next bastion, which happens when the face cannot be dis-
covered but from the flank alone.

FLO

[ 5'9 ]

FLU

Simple Fl-.ak.es, are lines going from the angle of the
shoulder to the curtain; the chief office of which is for
the defence of the moat and place.

Flank, is also a term of war, used by way of analogy
for the side of a battalion, army, &c, in contradistinction
to the front and rear. So, to attack the enemy in flank,
is to discover and fire upon them on one side.

FLANKED, is used of something that has flanks, or
may be approached on the flank. As

Flanked Angle, or that formed by the two faces of
the bastion, atid forming its point or angle.

Flanked Line of Defence, Flanked Tmaille, &c.

FLANKING, in general, is the discovering and firing
upon the side of a place, body, battalion, &c. To flank
a place, or other work, is to dispose it in such a manner
as that every part of it may be played upon both in front
and rear.

FLAT Bastion, is that which is built on a right line, as
on the middle of the curtain, &c.

FLETCHER (Abraham), an ingenious mathema-
tician, was born at Little Broughton, in Cumberland, in
1714, and brought up in his father's business, husbandry
and tobacco-pipe making. He learned to read and write
by his own endeavours, and afterwards taught himself
arithmetic, and gradually the several branches of mathe-
matics, with botany, touching the medicinal qualities of
herbs. At the agpof 30 he turned schoolmaster, to which
profession was added those of astrologer and country
doctor; in which, by means of industry and oeconomy,
the two handmaids of fortune, he was enabled to leave
his family in possession of 4000/. at his death, which
happened in 1793, in. the 79th year of his age. Mr.
Fletcher published, in 1752, an ingenious treatise, en-
titled The Universal Measurer, in one vol. 8vo ; which was
afterwards (1762) much enlarged and improved, under
the title of The Universal Measurer and Mechanic, being
a general collection of exercises in most of the mathema-
tical sciences.

FLEXIBLE, is the property or quality of a body that
may be bent.

FLEXION, the same as Flexure.

FLEXURE, or Flexion, is the bending or curving
of a line or figure. When a line first bends one way, and
then gradually changes to a bend the contrary way, the
point where the two parts join, or where the hending
changes to the other side., is called the point of inflexion,
or of contrary flexure.

FLIE, or Fly, thatpartof the mariner's compass on
which the 32 points of the wind are drawn, and over
which the needle is placed, and fastened underneath.

FLOAT-.Bo«n/.s, the boards fixed to the outer rim of
undershot water-wheels, serving to receive the impulse of
the stream, by which the wheel is carried round. — There
may be too im:ny of these boards on a wheel. It is
thought to be the best rule, to have their distance asunder
such, that each of them may come out of the water as
soon as possible, after it has received and acted with its
full impulse; or, which comes to the same thing, when
the succeeding one is in a direction perpendicular to the
surface of the water.

Floating Bridge, is a bridge of boats, casks, &c,
covered with planks, firmly bound together for the pas-
sage of men, horses, or carriages, &c.

FLOOD, a deluge or inundation of water.

Flood, is also nsed in speaking of the lide, when it
is rising or flowing up; in contradistinction to the ebb,
which is when it is decreasing or running out.

FLOORING, in Carpentry, is commonly understood
of the boarding of the floors. The measurement of flooring
is estimated in squares, of 100 square feet each, or of 10
feet on each side every way; 10 times 10 being 100.
Hence, the length of the floor being multiplied by its
breadth, in feet, and two figures cut off on the right-
hand, gives the squares, and feet, or decimals cut off.
Thus, a floor being 22 feet long, and 16 wide; then
22 x 16 = 352 square feet, or 3-52 squares, the content-.

FLUENT, or Flowing Quantity, in thte Doctrine of
Fluxions, is the variable quantity which is considered as
increasing or decreasing ; or the fluent of a given fluxion,
is that quantity whose fluxion being taken, according to
the rules of that calculus, shall be the same with the given
fluxion. See Fluxions.

Contemporary Fluents, are such as flow together or
for the same time. And the same is to be understood of
contemporary fluxions. — When contemporary fluents are
always equal, or in any constant ratio ; then also are
their fluxions respectively either equal, or in that same
constant .ratio. That is, if x = y, then is i = y ; or if
x :y : : n : 1, then is x.:j[ : ; n : I ; or if x = ny, then is
x = ny.

It is easy to find the fluxions to all the given forms of
fluents ; but, on the contrary, it is difficult to find the
fluents of many given fluxions ; and indeed there are
numberless cases in which this cannot be effected, except
it be by the quadrature and rectification of curve lines,
or by logarithms, or infinite series.

This doctrine, first invented by Sir Isaac Newton, was
carried by him to a considerable degree of perfection, at
least as to the most frequent, and most useful forms of
fluents; as may be seen in his Fluxions, and in his Qua-
drature-of Curves. Maclaurin, in his Treatise of Fluxions,
has made several inquiries into fluents, reducible to the
rectification of the ellipse and hyperbola : and Dalem-
bert has pursued the same subject, and carried it farther,
in theMemoires del'Acad. de Berlin, torn. 2. p. 200. To
the celebrated M.Euler this doctrine is greatly indebted,
in many parts of his various writings, as well as in the In-
stitutio Calculi Integralis, in 3 vols 4to, Petr. 1768.
The ingenious Mr. Cotes contributed very much to this
doctrine, in his Harmonia Mensurarum, concerning the
measures of ratios and angles, in a large collection of dif-
ferent forms of fluxions, with their corresponding fluents ;
which was farther prosecuted in the same way by Walmes-
ley, in his Analyse des Mesures des Rapportes et des
Angles, a large vol. in 4to, 1749. Besides many other
authors who, by their ingenious labours, have greatly
contributed to facilitate and extend the doctrine of fluents;
as Emerson, Simpson, Landen, Waring, &c, in this
country; Fliopital, and many other learned foreigners.
Lastly, in 1785 was published at Vienna, by M. Paccassi,
a German nobleman, Udhandlung uber enic neue Methode
zu Integriren, being a method of integrating, or finding
the fluents of given fluxions, by the rules of the direct
method, or by taking again the fluxion of the given
fluxion, or the 2d fluxion of the flueni sought; and then
making every flowing quantity, itsfluxion, and 2d fluxion,
in geometrical progression ; a method however, which, it
seems, only holds true in the easiest cases or forms, whoso

FLU

L .020 ]

FLU

fluents are easily had by the most common methods :
and which nearly resembles the method employed by Clai-
raut, in treating what he called Equations of Condition.
As it is only in certain particular forms and cases that
the fluents of given fluxions can be found ; there being no
method of performing this universally a priori, by a direct
investigation, like finding the fluxion of a given fluent
quantity ; we can do little more than lay down a few
rules for such forms of fluxions as are known, from the
direct method, to belong to such and such kinds of fluents
or flowing quantities : and these rules, it is evident, must
chiefly consist in performing such operations as are the
reverse of those by which the fluxions are found to given
flowing quantities. The principal cases of which are as
follow :

I. To find the Fluent of a simple fluxion ; or that in
which there is no variable quantity, and only one fluxional
quantity. This is done by barely substituting the variable
or flowing quantity instead of its fluxion, and is the result
or reverse of the notation only. Thus,

The fluent of ax is ax.

The fluent of ay ■+- 1y is ay -+- 2y.

II. When any power of a flowing quantity is multiplied
by the fluxion of the root. Then, having substituted, as be-
fore, the flowing quantity for its fluxion, divide the result
by the new index of the power. Or, which is the same
thing, take out, or divide by, the fluxion of the root ; add
1 to the index of the power; and divide by the index so
increased.

So if the fluxion proposed be - 3x*x;

Strike out x, then it is - - - 3x5 ;

add 1 to the index, and it is - 3x6;

divide by the index 6, and it is - -|x6 or |x6 ;
which is the fluent of the proposed fluxion 3xix.

In like manner, the fluent
of 4>axx is 2ax2;

1 , |_
of 3x2* is 2x* ;

of axnx is x + ;

n + 1

X . - 1

r -y or 2 *% is — -E or — .

of («3 + z3)Vz is ^ (a3 + a3)s.

III. When the root under a vinculum is a compound quan-
tity; and the index of the part or factor without the vincu-
lum increased by \, is some multiple of that under the vincu-
lum: Put a single variable letter for the compound root ;
and substitute its powers and fluxion instead of those, of
the same value, in the given quantity; so will it be re-
duced to a simpler form, to which the preceding rule, can
then be applied.

2
Thus, if the given fluxion be F = (a* ■+■ x2)Tx x; where

3, the index of the quantity without the vinculum, in-
creased by 1, makes 4-, which is double of 2, the exponent
of x* within the same; therefore putting 2 = oa -1- x2,
thence x2 = 2 — a% and its fluxion is 2xx = z; hence
then x3x = \x*x = |« (z — «"), and the given quantity f
or (a* -t- x2)^x3x is = \£*» (2 - a:) or = &$* —
iaVz; and the fluent of each term gives

1 = fV5* - ■Ati'z* = 3zT (-,^2 - TV); or, by substitu-
ting the value of e instead of it, the same fluent is

9Qr* £}* *y*x2 ~ '8Vi=)' <>r * ("2+ **>* • (*""- *°s>-

J V . H hen there arc several terms involving two or more

variable quantities, having the fluxion of each multiplied by
the other quantity or quantities : Take the fluent of each
term, as if there was only one variable quantity in it,
namely that whose fluxion is contained in it, supposing
all the others to be constant in that term ; then if the
fluents of all the terms so found, be the very same quan-
tity, that quantity will be the fluent of the whole. — Thus,
if- the given fluxion be xy -+- xj. Then, the fluent of xy
is xy, supposing^ constant; and the fluent of xy is also
xy, when x is constant; therefore the common resulting
quantity xy is the required fluent of the given fluxion
xy ■+■ xy.

' And, in like manner, the fluent of

xyz ■+- xjz -+- xyk is xyz.

V. When the given fluxional expression is in this form

, , viz, a fraction including two quantities, being the

fluxion of the former drawn into the latter, minus the fluxion
of the latter drawn into the former, and divided by the square

of the latter

reduces to — = — or xyy -;

y . y y

then the fluent is the fraction -, or of the
. . • V

former quantity divided by the latter. That is,

The fluent of x-^~^ is - ;

- . t y

and the fluent of — ^ — 7 — is — .

y y

Though the examples of this case may be performed
by the foregoing one. Thus the given fluxion

*y-w roJ„„„„ f„ *? __ *y

y'

of which the fluent of - is - when y is constant ; and

y y
the fluent of xjy~2 is -+- xy"1 or - when x is constant;
and therefore, by that case,

- is the fluent of the whole , .

y y

VI. When the fluxion of a quantity is divided by the
quantity itself: Then the fluent is equal to the hyperbolic
logarithm of that quantity; or, which is the same thing,
the fluent is equal to 2-30258509 &c, multiplied by the
common log. of the same quantity.

So, the fluent

or x"1 x is the hyp. log. of x;

is 2 x hyp. log. of x, or = h. 1. of xi;

is a x h. 1. of x, or h. 1. of aa;

is the h. 1. of a -4- x ;

is the h. 1. of a ■+- ,r3.
a + ar

VII. Many fluents may be found by the direct method
of fluxions, thus: Take .the fluxion again of the given
fluxional expression, or the 2d fluxion of the fluent

sought; into which substitute— for x, and - for ;;, &c,

b x y

that is, make x, x, x, as also y, y, y, &t, in continual pro-
portion, or x : x : : x : *, and y : y : :y :'y, &C ; then di-
vide the square of the given fluxional expression by the
2d fluxion, just found, and the quotient will be the fluent
sought in many cases.

Or the same rule may be otherwise delivered thus: In the
given fluxion F, write x for .<-, y for j, &c, and call die
result G, taking also the fluxion of this quantity, G ; then

FLU

[ Ml ]

FLU

make g : f : : g : f, so shall the 4th proportional f be
the fluent as before. And this is the rule of M. Pac-
tassi.

It may be proved whether this be the true fluent, by
taking the fluxion of it again, which, if it agree with the
proposed fluxion, will show that the fluent is right ; other-
wise, it is not.

Thus, if it be proposed to find the fluent of nxn^1x.
Here f = ?ixn lx; write first x for x, and it is nxn~lx .or
nx" = g ; the fluxion of this is g = nzxn~'x; therefore
g : f : : g : f becomes n"xn~I x : nxn~'x : : nxn : x" = f,
the fluent sought.

Again, suppose it be proposed to find the fluent of xy
-+- xj. Here f = xy -+- xy; then, writing x for x} and y
for j, it is xy ■+■ xy or 2xy = g ; the fluxion of which '

ixy

■+■ 2xj = g; then G : F : : g : f becomes 2xy ■+■ 2xy

: xy + xy : : 2xy : xy = f, the fluent sought. See also
Condition, Equations of .

VIII. To find Fluents by means of a table of forms of
Fluxions and Fluents.

In the following table are contained the most usual
forms of fluxions that occur in the practical solution of
problems, with their corresponding fluents set opposite to
them; by means of which, viz, comparing any proposed
fluxion with the corresponding form here, the fluent of it
will be found.

Where it is to be noted, that the logarithms in the said
forms, are the hyperbolic ones, which are found by multi-
plying the common logs, by 2-3025850929940 &c. Also
the arcs whose sine, or tangent, &c, are mentioned, have
the radius 1, and are those in the common tables of sines,
tangents, &c. — And the numbers m, n, &c. are to be some
quantities, as the forms fail when n = 0, or m = 0, &c.

Forms.

Fluxions.

Fluents.

I

n— 1 ,
X X

1

— xn
n

II

i i n\m_1 "-1
[a 4; x ) x x

±—{a±x«r
mn

III

mn — 1 .
X x

1 xmn

(a ± xa)

mna (o± xn)m

IV

m— l
(a ± xn) x

*

- 1 (a ± *»)"■

x

mna xmn

X

VI

VII

VIII

IX

XI

m-l n-l
{mxy ■+■ nxy) x y ,

mx ny\ m

m— l . n— 1 . i— l .

mx xynz'+nxmy yzT + rxmynz x.

, . . .. m— 1 n— 1 r— 1

OT{mxyZ + nxyz-i-rxyz)x y z ,

,mx ny rz
or f H H ) xmv"zT

* y z

— or x x

a ± xn

a ± xn

In- l.

a — xn
Jo— i .

xmya

X™y

log. of X

± — log- of a i xn

-log. of

afj"

—— log. of — ; -r-r-

n</a 6 \/a — Vx
1 . fyAn-V«

log- of — t-z T

n»/a ° t/x" -+-*/a

\n— 1

| a -t- s '
Vol. I.

__ „ arc to tang. J-, or ^ X arc to cosine ^-^

3 X

FLU

[ S22 ]

FLU

Forms.

Fluxions.

Fluents^

XII

in — 1 .
x x

— log. of */x" + \/(+ a -+- x")
re

\/(± a -+-*n)

XIII

in — 1 .
X x

2 . i" 1 2x"
— x arc to sin. v/ — > or — x arc to vers. —
re an a

v/(« - x°)

XIV

— 1,

X X

1 Wof^-^^^

a/(«±0

XV

— 1 .

X X

2 ,xn 2 .a
— ; — x arc to secant */ — , or — — x arc to cosine — ,
n^/a a n</a x"

>i/{— a ■+- x")

1 2a — x11

or — — x arc to cosine

nif/a x"

XVI

X

2

-j- x arc to diam. d, and vers, x

a

^{dx - X1)

XVII

x\/(dx — x2)
x^/(a2 —x-)

— circ. seg. to diam. d and vers, x

2 8

— circ. zone, rad. a, and height from centre x
2 °

XVIII

c^x

CM

n log. c

XIX

X — 1 .

xyx log. y ■+■ xy y

^

m— l.

(a -+- xn) mx x

Putting s = m-t- r

fa ■+- xn-)m+1xln-r ,1 r— l.a r — l.r — 2.os r — ].r-2.r-3.a% .

XX

ns U i — l.xn b— l.i— 2.x-n s — l.s— 2.s— 3xJn
(a -)-xn)m + 1xrn ,1 *+l.xn i+l.! + 2.iIn s+l.s + 2.s + 3.i3"c_N

rraa ' U r + l.« r-H l.r-t-2.a* r+ l.j--t-2.r + 3.a3

The Use of the foregoing Table of Forms of Fluxions and
Fluents. — In the use of this table, it is to be observed, that
the first column serves only to show the number of the
form, as a mark of reference ; in the 2d column are the
several forms of fluxions, which are of different kinds or
classes ; and in the 3d or last column are the correspond-
ing fluents.

The method of using the table is this. Having any
fluxion given, whose fluent it is proposed to find: First,
compare the given fluxion with the several forms of fluxions
in the 2d column of the table, till one of the forms be
found that agrees with it; which is done by comparing
the terms of the given fluxion with the like parts of the
tabular fluxion, viz, the radical quantity of the one, with
that of the other; and the exponents of the variable quan-
tities of each, both within and without the vinculum ; all
which, being found to agree or correspond, will give the
particular values of the general quantities in the tabular
form. Then substitute these particular values, for the
same quantities in the general or tabular form of the fluent,
and the result will be the particular fluent sought; after
it is multiplied by any coefficient the proposed fluxion may
have.

For Example. To find thefluentof Ihefluxional expression
3 x 'i. This agrees with the first form ; and by compa-
ring the fluxions, it appears that x = x, and re — 1 = j, or

n = !>; which being substituted in the tabular fluent, or

i . 8

-xn, gives, after multiplying by 3 the coefficient, 3 x |-xT

or §x 3 for the fluent sought.

Again, To find the fluent of 5xV.y/c3 — x3, or 5x^

(c3 — x3)5. This belongs to the 2d form ; for a = c3, and
— xn — — x3, re = 3 under the vinculum, and m — 1 = i,
or m r= |, also the exponent re — 1 of xn_I without the
vinculum, by using 3 for re, is re— 1 =2, which agrees
with x2 in the fluxion given ; and therefore all the parts of
the form are found to answer. Then, substituting these

values into the general fluent, (a — x")m, gives

--x'fc5- x3)* = - V (c3 - x3)*.

Thirdly, To find the fluent of 'r * ,. This agrees with

the Sth form ; where ± x" = -+■ x3 in the denominator,
or re = 3 ; and the numerator xn~ * then becomes x'\ which
agrees with the numerator in the given fluxion, also a — 1.
Hence then, by substituting in the general form of the

fluent - logarithm of a ■+■ ,in, it becomes \ logarithm of

1 ■+■ x3.

IX. To find Fluents by mentis of Infinite Series. — When a
finite form cannot be found to agree with a proposed

FLU

t

fluxion, it is then usual to throw it into an infinite series,
either by division, or extraction of roots, or by the bino-
mial theorem, &c ; after which, the fluents of all the terms
are taken separately.

hence by subtraction

FLU

For Example. To find the fluent of

Here,

by dividing the numerator by the denominator, this be-
comes * — 2xx + 3x'2x — Sx^ic -+- 8x*x &c ; and, thefluents
of all the terms being taken, givex — xa -t- xJ — ^ x4 -t- -|xs
&c, for the fluent sought.

To Correct a Fluent. — The fluent of a given fluxion,
found as above, sometimes wants a correction, to make it
contemporary with that required by the problem under
consideration, &c : for the fluent of any given fluxion, as
x, may be either x (which is found by the rule) or it may
be x + c, that is x plus or minus some constant quantity
c; because both x andx 4; c have the same fluxion x : and
the finding of. this constant quantity c, is called correcting
the fluent. " Now this correction is to be determined from
the nature of the problem in hand, by which means to
know the relation which the fluent quantities have to each
other at some certain point or time. Reduce therefore
the general flucntial equation, found by the rules above,
to that point or time ; then if the equation be true at that
point, it is correct; but if not, it wants a correction, and
the quantity of that correction is the difference between
the two general sides of the equation when reduced to that,
particular state. Hence the general rule for the correc-
tion is this :

Connect the constant, but indeterminate, quantity c,
with one side of the fluential equation, as determined by
the foregoing rules; and, in this equation, substitute for
the variable quantities such values as they are known to
have at any particular state, place, or time ; and then
from that particular state of the equation find the value
of c, the constant quantity of the correction.

Example. To find the correct fluent of z = «x3x. First
the general fluent of this is z = £ax4, orz = -|<m4 -+- c,
taking in the correction c. — Now if it be known that z and
x begin together, or that z = 0, when x = 0; then writing
0 both for x and z, the general equation becomes 0 = 0 + c,
or c = 0 ; so that, the value of c being 0, the correct
fluents are z =^ax4.

But if when z = 0, x = b, any known quantity ; then
substituting 0 for z, and b for x, in the general equation,
it becomes 0 = $ub* + c, from which is found c = — ^abA;
and this being written for it in the general equation, this
becomes z = -|a(x'' — b4), for the correct, or contemporary
fluents.

Or lastly, if it be known that when z = d,x = b; then
substituting d for z, and b for x, in the general fluential
equation z = |-ax4 ■+- c, it becomes d = ±ab' ■+- c ; and
hence is deduced the value of the correction, viz,
c = d — ^abA ; consequently, writing this value for c in
the general equation, it becomes z = \a(x* — 64) -+- d, for
the correct equation of the fluents in this case.

And hence arises another easy and general way of cor-
recting the fluents, which is this: In the general equation
ot the fluents, write the particular values of the quantities
which they are known to have at any certain time; then
subtract the sides of the resulting particular equation,
from the corresponding sides of the general one, and the
remainders will give the correct equation of the fluents
sought. So, as above,

the general equation being - - z = ax4;
write d for z, and b for x, then - d=abi;

z — d = ax4 — ab',
or - - - - - - z = ax4 — aft4 ■+- d,

the correctfluents as before. — See other methods of finding
o rtransforming fluents, in my Course of Mathematics.

FLUID, or Fluid Body, according to Newton, is that
whose parts yield to the smallest force impressed, and by
yielding are easily moved among each other; in whicii
sense it stands opposed toft solid. This is the definition
of a perfect fluid : if the fluid require some sensible force
to move its parts, it is imperfect in proportion to that
force ; and such perhaps are all the fluids we know of in
nature. See Fluidity.

That fluids have vacuities in their substance is evident,
because certain bodies may be dissolved in them without
increasing their hulk. Thus, water will dissolve a certain
quantity of salt; after which it will receive a little sugar,
and after that a little alum ; and all this without increas-
ing its first dimensions. Which shows that the particles
of these solids are so far separated as to become smaller
than those of the fluid, and to be received and contained
in the interstices between them.

Fluids are either elastic, such as air ; or non-elastic, as
water, mercury, &c. These latter occupy the same space,
or are of the same bulk, under all pressures or forces ; but
the former dilate and expand themselves continually by '
taking off the external pressure from them ; for which
reason it is that the density and elasticity of such fluids,
are proportional to the force or weight that compresses
them. The doctrine and laws of fluids are of the greatest
extent and importance in philosophy : the properties of
elastic fluids constituting the doctrine of Pneumatics;
those of the non-elastic fluids, that of Hydrostatics ; and
the laws of their motions, Hydraulics. For which see these
respective articles. — For the laws of the pressure and gra-
vitation in fluids specifically heavier or lighter than the bo-
dies immerged, see Specific Gravity. — For the laws of
the resistance of fluids, or the retardation of solid bodies
moving in fluids, see Resistance. — For the ascent of
fluids in capillary tubes, or between glass planes, &c, see
Ascent, and Capillary Tubes.

FLUTES, or Flutings, are certain channels or cavi-
ties cut along the shaft of a column, or pilaster.

FLUIDITY, that state or affection of bodies, which do-
nominates or renders them fluid; or that property by
which they yield to the smallest force impressed: in con-
tradistinction to solidity or firmness. Fluidity is to be care-
fully distinguished from liquidity or humidity, which lat-
ter implies wetting or adhering. Thus, air, ether, mer-
cury, and other melted metals, and even smoke and flame
itself, are fluid bodies, but not liquid ones; while water,
beer, milk, urine, &c, are both fluids and liquids at the
same time.

The nature and causes of fluidity have been variously
assigned. Tlje Gasscndists, and ancient corpuscularians,
require only three conditions as necessary to it; viz, a
smallness and smoothness of the particles of the body,
vacuities between these particles, each of which being a
spherical figure. The Cartesians, and after them Dr.
Hooke, Mr. Boyle, &c, besides these circumstances, re-
quire also a certain internal or intestine motion of the
particles, as chiefly contributing to fluidity. Thus, Mr.
Boyle, in his History of Fluidity, argues from various ex-
periments : for example, a little dry powder of alabaster,
or plaster of Paris, finely sifted, being put into a vessel
over the fire, soon begins to boil like water; exhibiting all

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the motions ami phenomena of a boiling liquor: it tumbles
variously in great waves; will bear stirring with a stick or
ladle, without resisting; and if strongly stirred near the
-side of the, vessel, its waves will apparently dash against it,
thus resembling the boiling of water : yet it is all the while
a dry parched powder.

The same is observed in sand ; a dish of which being set
on a drum-head, briskly beaten by the sticks, or on the
upper stone of a mill, it in all respects emulates the pro-
perties of a fluid body. A heavy body, ex. gr. will imme-
diately sink in it to the bottom, and a light one emerge to
the top : each grain of sand has a constant vibratory and
dancing motion; and if a hole be made in the side of the
dish, the sand will spin out like water.

The Cartesians bring divers considerations to prove that
the parts of fluids are in continual motion: as 1st, The
change of solids into fluids, ex. gr. ice into water, and vice
versa; the chief difference between the body in those two
states consisting in this, that the parts, being fixed and at
rest in the one, resist the touch ; whereas in the other,
being already in motion, they give way to the slightest im-
pulse. 2dly, The effects of fluids, which commonly pro-
ceed from motion: such are the insinuation of fluids among
the pores of bodies ; the softening and dissolving hard bo-
dies; the actions of corrosive menstruums; &c: to which
they add, that no solid can be brought to a state of flu-
idity, without the intervention of some moving or move-
able body, as fire, air, or water. Air, these philosophers
hold to be the first spring of the causes of fluidity, it being
this that gives motion to fire and water, though itself re-
ceives its motion and action from the ether, or subtile me-
dium.

But Boerhaave pleads strenuously that fire is the first
mover, and the great cause of all fluidity in other bodies,
as air, water, &c : without this, he says, that the atmo-
sphere itself would fix into one solid mass. And in like
manner, Dr. Black, of Edinburgh, mentions fluidity as an
effect of heat. The different degrees of heat which are re-
quired to bring different bodies into a state of fluidity, he
supposes may depend on some particulars in the mixture
and composition of the bodies themselves : which is ren-
dered farther probable from considering that the natural
state of bodies in this respect is changed by certain mix-
tures; thus, when two metals are compounded, the mixture
is commonly more fusible than either of them separately.

Newton's idea of the cause of fluidity is different: he
considers it to be the great principle of attraction. The
various intestine motion and agitation among the particles
of fluid bodies, he thinks is naturally accounted for, by
supposing it a primary law of nature, that as all the par-
ticles of matter attract each other when within a certain
distance ; so at all greater distances, they avoid and fly
from one another. For then, though their common
gravity, together with the pressure of other bodies upon
them, may keep them together in a mass, yet their con-
tinual endeavour to avoid one another singly, and the ad-
ventitious impulses of heat and light, or other external
causes, may make the particles of fluids continually move
round each other, and so produce this quality. As there-
fore the cause of cohesion of the parts of solid bodies ap-
pears to be their mutual attraction; so, on this principle,
the chief causeofiluidity seems to be a contrary motion, Im-
pressed on the particles of fluids; by which they avoid and
fly from each other, as soon as they come at, and as long as
they keep at, a certain distance. It is observed also in

all fluids, that the direction of their pressure against the
vessels which contain them, is in lines perpendicular to
the sides of such vessels; which property, being the ne-
cessary result of the spherical figure of the particles of any
fluid, shows that the parts of all fluids are so, or of a figure
very nearly approaching to it.

FLUX, in Hydrography, a regular and periodical motion
of the sea, happening twice in 24 hours and 4S minutes,
nearly. The flux, or flow, is one of the motions of the
tide : the other, by which the water sinks and retires, being
called the reflux, or ebb. See Tide.

Between the flux and reflux there is a kind of rest or
cessation, of about half an hour; during which time the
water is at its greatest height, called high-water.

The fluxof the sea follows chiefly the course of the moon -r
and is always highest and greatest at new and full moons,
particularly near the time of the equinoxes. In some parts,
as at Mount St. Michael, the water rises 18 or 20 feet,
though in the open sea it never rises above a foot or two ;
and in some places, as about the Morea, there is no flux at
all. It runs up some rivers above 120 miles : though up
the river Thames it goes only about SO, viz, near to Kings-
ton in Surrey. Above London-bridge, the water flows 4
hours, and ebbs 8; and below the bridge, it flows 5 hours,
and ebbs 7.

FLUXION, in the Newtonian analysis, denotes the rate
or proportion at which a flowing or varying quantity in-
creases its magnitude or quantity; and it is proportional
to the magnitude by which the flowing quantity would be
uniformly increased, in a given time, by the generating
quantity continuing of the invariable magnitude it has at
the moment of time for which the fluxion is taken :
by which it stands contradistinguished from fluent, or the
flowing quantity, which is gradually, and indefinitely in-
creasing, after the manner of a space which a body in mo-
tion describes.

Mr. Simpson observes, that there is an advantage in
thus considering fluxions, not as mere velocities of increase
at a certain point, but as the magnitudes which would be
uniformly generated in a given finite time : the imagina-
tion is not here confined to a single point, and the higher
orders of fluxions are rendered much more easy and intel-
ligible. And though Sir Isaac Newton defines fluxions
to be the velocities of motions, yet he has recourse to the
momeifls or increments, generated in equal particles of
time, to determine those velocities, which he afterwards
directs to expound by finite magnitudes of other kinds. As
to the illustration of this definition, and the rules for find-
ing the fluxions of all sorts of flowing quantities, see the
following article, or the Method of Fluxions.

Mtthod of Fluxions, is the algorithm and analysis of
fluxions and fluents, or flowing quantities. Most foreign-
ers define this as the method of differences or differentials,
being the analysis of indefinitely small quantities. But
Newton, and other English authors call these infinitely
small quantities, moments; considering them as the mo-
mentary increments of variable quantities; as of a line con-
sidered as generated by the flux or motion of ;i point, or of
a surface generated by the flux of a line. Accordingly,
the variable quantities are called fluents, or flowing quan-
tities; and the method' eff finding either the fluxion, or the
fluent, the method of fluxions. But INI. L< ibnitz considers
the same infinite!} small quantities as the differences, or
differentials, of quantities ; and the method of finding those
differences, he calls the Differential Calculus. Besides

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this difference in the name, there is another in the nota-
tion. Newton expresses, the fluxion of a quantity, as of .r,
by a point placed over it, thus x; while Leibnitz expresses
his differential of the same r, by prefixing the initial letter
d, as clx. And except in these circumstances, the two
methods may be considered as the same.

The Method of Fluxions is probably one of the great-
est, most subtle, and sublime discoveries of any age : it
opens a new world to our view, and extends our know-
ledge, as it were, to infinity; carrying us beyond the
bounds that seemed to have been prescribed to the human
mind, at least infinitely beyond those to which the ancient
geometry was1 confined.

The history of this important discovery, recent as it is,
is a little dark and embroiled. Two of the greatest men
of the last age have both of them claimed the invention,
Sir I. Newton, and M. Leibnitz; and nothing can be more
glorious for the method itself, than the zeal with which
the partisans of either side have asserted their title.

To exhibit ajust view of this dispute; and of the pre-
tensions of each party, we may here advert to the origin of
the discovery, and mark where each claim commenced,
and how it was supported.

The principles on which the Method of Fluxions is
founded, or which conducted to it, had been laying, and
gradually developing, from the beginning of the 17th cen-
tury, by Fermat, Napier, Barrow, Wallis, Slusius, &c, who
had methods of drawing tangents, of maxima and minima,
of quadratures, &c, in certain particular cases, as of ra-
tional quantities, on nearly the same principles. And it
was not wonderful that such a genius as Newton should
soon after raise those faint beginnings into a regular and
general system of science, which he did about the year
1665, or sooner.

The first time however that the method appeared in
print, was in 1(584, when M. Leibnitz gave the rules of it
in the Leipsic Acts of that year: but without the demon-
strations. The two brothers however, John and James
Bernoulli, being struck with this new method, applied
themselves diligently to it, found out the demonstrations,
and applied the calculus with great success.

But before this, M. Leibnitz had proposed his Differ-
ential Method, viz, in a letter, dated Jan. 21, 1677, in
which he exactly pursues Dr. Barrow's method of tan-
gents, that had been published in 1670; and Newton com-
municated his method of drawing tangents to Mr. Collins,
in a letter dated Dec. 10, 1672; which letter, together
with another dated June 13, 1676, were sent to M. Leib-
nitz by Mr. Oldenburgh, in 1676. So that .there is a
strong presumption that he might avail himself of the in-
formation contained in these letters, and other papers
transmitted with them, and also in ]6*75, before the pub-
lication of his own letter, containing the first hint of his
differential method. Indeed it sufficiently appears that
Newton had invented his method before the year l66'9,and
that he actually made use of it in his Compendium of
Analysis and Quadrature of Curves before that time. His
attention seems to have been directed tliis way, even be-
fore the time of the plague which happened in London in
l6(5.5 and 1666, when he was about 28 years of age.

This is all that is heard of the method, till the year
1687, when Newton's great work the Principia appeared,
which is almost wholly founded on the same calculus.
The common opinion then was, that Newton and Leibnitz
had each invented it about the same time: and what

seemed lo confirm it was, that neither of them made any
mention of the other; and that, though they agreed in the"
substance of the thing, yet they differed in their ways -of
conceiving it, calling it by different names, and using dif-
ferent characters. However, foreigners having first learned
the method through the medium of Leibnitz's publication,
which spread the method through Europe, they were in-
sensibly accustomed to consider him as the sole, or prin-
cipal inventor, and became ever after strongly prejudiced
in favour of his notation and mode of conceiving it.

The two great authors, themselves, without any seeming
concern, or dispute, as to the property of the invention,
enjoyed the glorious prospect of the progresses continually
making under their auspices, till the year 1699, when
the peace began to be disturbed.

M.Facio, in a treatise on the Line of Swiftest Descent,
declared, that he was obliged to own Newton as the first
inventor of the differential calculus, and the first by many
years; and that he left the world to judge, whether Leib-
nitz, the second inventor, had taken any thing from him.
This precise distinction between first and second inventor,
with the suspicion it insinuated, raised a controversy be-
tween M.Leibnitz, supported by the editors of the Leipsic
Acts, and the English mathematicians, who declared for
Newton. Sir Isaac himself never appeared on the scene ;
his glory was become that of the nation; and his adhe-
rents, warm in the cause of their country, needed not his
assistance to animate them.

Writings succeeded each other but slowly, on either
side ; probably on account of the distance of places ; but
the controversy grew still stronger and stronger: till at
length M. Leibnitz, in the year 1711, complained to the
Royal Society, that Dr. Keil had accused him of pub-
lishing the Method of Fluxions invented by Sir I. Newton,
under other names and characters. He insisted that no-
body knew better than Sir Isaac himself, that he had
taken nothing from him ; and required that Dr. Keil
should disavow the ill construction which might be put
upon his words.

The Society, thus appealed to as a judge, appointed a
committee to examine all the old letters, papers, and do-
cuments, that had passed among the several mathemati-
cians, relating to this subject ; who, after a strict exami-
nation of all the evidence that could be procured, gave in
their report as follows : " That M.Leibnitz was in Lon-
don in 1673, and kept a correspondence with Mr.
Collins by means of Mr. Oldenburgh, till Sept. 1676,
when he returned from Paris to Hanover, by way of
London and Amsterdam : that it did not appear that
M. Leibnitz knew any thing of the differential calculus
before his letter of the 21st of June, l677, which was
a year after a copy of a letter, written by Newton in
tire year 1672, had been sent to Paris to be communi-
cated to him, and above 4 years after Mr. Collins began
to communicate that letter to his correspondents ; in
which the Method of Fluxions was sufficiently explain-
ed, to let. a man of his sagacity into the whole matter:
and that Sir I. Newton had even invented his method
before the year 1669, and consequently 15 years be-
fore M. Leibnitz had given any thing on the subject in
the Leipsic Acts." From which they concluded that
Dr. Keil had not at all injured M. Leibnitz in what he had
said.

The Society printed this their determination, together
with all the pieces and materials relating to it, under the

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title of Commercium Epistolicum de Analysi Promota,
Svo, Lond. 1 7 1 '2 . This book was carefully distributed
through Europe, to vindicate the title of the English na-
tion to the discovery ; for Newton himself, as already
hinted, never appeared in the affair : whether it was that
he trusted his honour with his compatriots, who were
zealous enough in the cause ; or whether he felt himself
even superior to the glory of it.

M. Leibnitz and his friends however could not show the
same indifference; he was accused of a plagiarism ; and
the whole Commercium Epistolicum either expresses it in
terms, or insinuates it. Soon after the publication there-
fore, a loose sheet was printed at Paris, in behalf of M.
Leibnitz, then at Vienna. It is written with great zeal
and spirit ; and it boldly maintains that the Method of
Fluxions had not preceded the Method of Differences; and
even insinuates that it might have arisen from it. The
detail of the proofs however, on each side, would be too
long, and could not be understood without a large com-
ment, which must enter into the deepest geometry; M.
Leibnitz had begun to work upon a Commercium Episto-
licum, in opposition to that of the Royal Society; but
he died before it was completed.

A second edition of the Commercium Epistolicum was
printed at London in 1722 ; when Newton, in the pre-
face, account, and annotations, which wereadded to that
edition, particularly answered all the objections which
Leibnitz and Bernoulli were able to make since the Com-
mercium first appeared in 1712 ; and from the last edition
of the Commercium, with the various original papers
contained in it, it evidently appears that Newton had dis-
covered his Method of Fluxions many years before the
pretensions of Leibnitz. See also Raphson's History of
Fluxions.

There are however, according to the opinion of some,
strong presumptions in favour of Leibnitz; i.e. that he
was no plugiary : for that Newton was at least the first
inventor, is past all dispute; his glory is secure; the
reasonable part, even among the foreigners, allow it : and
the question is only, whether Leibnitz took it from him,
or fell upon the same thing with him ; for, in his theory
of abstract notions, which he dedicated to the Royal Aca-
demy in 1671, before he had seen any thing of Newton's,
he already supposed infinitely small quantities, some
greater than others ; which is one of the great principles
of his system. But, before prosecuting farther the history
and improvements of this science, it will be proper to
premise somewhat of the principles and practice of it, ac-
cording to the ideas of the inventor.

Principles of the Method of Fluxions.

1. In the doctrine of fluxions, magnitudes or quanti-
ties, of all kinds, are considered, not as made up of a
number of small parts, but as generated by continued
motion, by means of which they increase or decrease : as
a line by the motion of a point; a surface by the motion
of a line ; and a solid by the motion of a surface : which
is no new principle in geometry ; having been used by
Archimedes, &c. So likewise, time may be considered
as represented by a line, increasing uniformly by the mo-
tion of a point. And quantities of all kinds whatever,
which are capable of increase and decrease, may thus
be represented by lines, surfaces, or solids, considered
as generated by motion.

2. Any quantity, in this manner generated, and va-
riable, is called a Fluent, or a flowing quantity. And the

rate or proportion according to which any flowing quantity
increases, at any position or instant, is the fluxion of the .
said quantity, at that position or instant: and it is pro-
portional to the magnitude by which the flowing quantity
would be uniformly increased, in a given time, with the
generating celerity uniformly continued during that time.

3. The small quantities thus generated or described,
in airy given small portion of time, and by any continued
motion, either uniform or variable, are called Increments.

4. Hence, if the motion of increase be uniform, by
which increments are generated, the increments will in
that case be proportional, orequal, to the measures of the
fluxions : but if the motion of increase be accelerated,
the increments so generated, in a given finite time, will
exceed the fluxion ; and if it be a decreasing motion, the
increment so generated, will be less than the fluxion.
But if the time be indefinitely small, so that the motion
be considered as uniform for that instant ; then these
nascent increments will always be proportional or equal
to the fluxions, and may be substituted for them) in any
calculation.

5. To illustrate these definitions : Suppose a point m be
conceived to move from the position a, and to generate a

line ap, with a motion any-how regulated ; and suppose
the celerity of the point m, at any position p, to be such,
as would, if from thence it should become, or continue,
uniform, be sufficient to describe, or pass uniformly over,
the distance pp, in the given time allowed for the fluxion :
then will the said line vp represent the fluxion of the said
fluent or flowing line ap, at that position.

6. Again, suppose the right line mn to move, from the
position ab, continually parallel to itself, with any con-
's Q- <? C "

-4. E P D

tinued motion, so as to generate the fluent, or flowing
rectangle abqp, whilst the point m describes the line ap ;
also let the distance vp be taken, as above, to express the
fluxion of the line or base ap ; and complete the rect-
angle PQap. Then, in the same manner as pp is the
fluxion of the line ap, so the small parallelogram p^is
the fluxion of the flowing parallelogram, aq ; both these
fluxions or increments being uniformly described in the
same time.

7. In like, manner, if the solid aerp be conceived as
generated by the plane pqr moving, from the position
DE R /-

A P P 3

ABE, always parallel to itself, along the line ad; and if
Pp denote the fluxion of the line ap. Then, as the paral-
lelogram p</, or pp x vq, expresses the fluxion of the
flowing rectangle aq, so likewise shall the fluxion of the
variable solid or prism An be expressed by the prism vr,
or pp x the plane pr. And in both these last two cases,
it appears that the fluxion of the generated rectangle, or
prism, is equal to the product of the gencrant, whether

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line or plane, drawn into the fluxion of the line along points p and q, the contemporaneous positions being p,q,

which it moves

S. Hitherto the generant, or generating line or plane,
has been considered as of a constant or invariable magni-
tude; in which case the fluent, or quantity generated, is
a. parallelogram, or a prism, the former being described
fay the motion of a line, and the latter-by the motion of a
plane. In like manner are other figures, whether plane
or solid, conceived to be described, by the motion of a
variable magnitude, whether it be a line or a plane.
Thus, let a variable line pq be carried with a parallel
motion alung ap, or while a point pis carried along, and
describes, the line ap, suppose another point q to be car-

ried by a motion perpendicular to the former, and to de-
scribe the line pq: let pq be another position of pq, inde-
finitely near the former ; and draw Qr parallel to ap. Now
in this case there are several fluents or flowing quantities,
with their respective fluxions : viz, the line or fluent ap,
the fluxion of which is vp, or Qr ; the line or fluent pq,
the fluxion of which is qr; the curve, or oblique line aq,
described by the oblique motion of the point, the fluxion
of which is qc/ ; and lastly the surface apq, described by
the variable line pq, and the fluxion of which is the rect-
angle PQr/>, or pq x pjb. And in the same manner may
any solid be conceived to be described, by the motion of
a variable plane parallel to itself, substituting the variable
plane for the variable line ; in which case, the fluxion of
the solid, at any position, is represented by the variable
plane, at that position, drawn into the fluxion of the line
along which it is carried.

9. Hence then it follows generally, that the fluxion of
any figure, whether plane or solid, at any position, is equal
to the section of it, at that position, drawn into the fluxion
of the axis, or line along which the variable section is sup-
posed to be perpendicularly carried ; i. e. the fluxion of
the figure asp, is equal the plane pq, x pp when that
figure is a solid, or to the ordinate pq x pjj when the
figure is a surface.

10. It also follows, from the same premises, that, in any
curve, or oblique line, aq, whose absciss is ap, and ordi-
nate is pq, the fluxions of these three form a small right-
angled plane triangle Qqr ; for Qr = vp is the fluxion of
the absciss ap, qr the fluxion of the ordinate pq, and aq
the fluxion of the curve or right line aq. And conse-
quently that, in any curve, the square of the fluxion of the
curve, is equal to the sum of the squares of the fluxions of
the absciss and ordinate, when these two lines are at right
angles to each other.

11. From the premises it also appears, that contempo-
raneous fluents, or quantities that flow or increase together,
which are always in a constant ratio to each other, have
their fluxions also in the same constant ratio at every
position. For, let ap and bq be two contemporaneous

ap T=n x bq,
Ap = n x By;
vp = n x Q q;
Q q : : n : 1,
BQ : : n : 1 ; or

A P p B Q q

fluents, described in the same time by the motion of the

and p, q; and let ap be to bq, or Ap to By, in the constant

ratio of n to 1.

Then is ....

and ....

therefore by subtraction,

that is, the fluxion pjd : fluxion

the same as fluent ap : fluent

the fluxions and fluents are in the same constant ratio.

But if the ratio of the fluents be variable, so will that of
the fluxion^ be also, chough not in the same variable ratio
with the former, at every position.

The Notation, fyc, in Fluxions.

12. To apply the foregoing principles to the determina-
tion of the fluxions of algebraic quantities, by means of
which those of all other kinds are determined, it will be
necessary first to premise the notation used in this science,
with some observations. As, first, that the final letters of
the alphabet z,y, x, iv, &c, are used to denote variable or
flowing quantities; and the initial letters a, b, c, d, &cr
constant or invariable ones : thus, the variable base ap of
the flowing rectangular figure ABQP,at art. 6, may be re-
presented by x: and the invariable altitude pq, by o : also
the variable base or absciss ap, of the figures in art. 8,
may be represented by x; the variable ordinate pq, by y;
and the variable curve or line aq, by z.

Secondly, that the fluxion of a quantity denoted by a
single letter, is represented by the same letter with a point
over it : Thus, the fluxion of x is expressed by *, that of y
hy y, and that of a by «. As to the fluxions of constant or
invariable quantities, as of a, b, c, &c, they are equal to
0 or nothing, because they do not flow, or change their
magnitude.

Thirdly, that the increments of variable or flowing quan-
tities, are also denoted by the same letters with a small (')
over them : So the increments of x, y, z, are x, y, z.

13. From these notations, and the foregoing principles,,
the quantities and their fluxions, there considered, will be
denoted as below.

In all the foregoing figures, put
the variable or flowing line - - - ap = xr

in art. 6, the constant line - - - pq = a,

in ait. 8, the variable ordinate - - fQ.z=yT

the variable curve or right line - - aq = z;

Then shall the several fluxions be thus represented,, viz,
x = vp the fluxion of the line ap,
ax = PQqp the fluxion of abqp in art. 6,
yx. ■= VQrp the fluxion of apq in art. 8,
z = Q q = \/( x~~h y9) tne fluxion of aq,
and ax = pr the flux, of the solid in art. 7, if

a denote the constant generating plane pan.
Also;u:= BQ in the figure to art. 1 1,
and nx =o.q the fluxion of the same.

14. The principles and notation being thus laid down,
we may proceed to the practice and rules of this doctrine,
which consists of two principal parts, called the direct and
inverse method of fluxions; viz, the direct method, which
consists in finding the fluxion of any proposed fluent or
flowing quantity ; and the inverse method, which consists
in finding the fluent of any proposed fluxion. As to the
former of these two problems, it can always be determined,
and that in finite algebraic terms ; but the latter, or find-
ing of fluents, only in some certain cases, except by means
of infinite series. — First then, of

FLU

[ 5& ]

FLU

V

:f

^

y

X

y

x 1

4.

P 2

77(c' Direct Method of Fluxions.

15. To find the fluxion of the product or rectan-
gle of two variable quantities ; let
AR<iP = xy be the flowing or varia-
ble rectangle, generated by two
lines rq and pq. moving always per-
pendicular to each other, from the
positions av and ar; denoting the
one by x, and the other by y; and
suppose x and y to be so related, that the curve aq may
always pass through their intersection q, or the opposite
angle of the rectangle.

Now this rectangle consists of the two trilineal spaces
APa, and arq, of which the fluxion of the former is pa x
pr/ or xy, and that of the latter is Ra x rc or xj, by art. 8 ;
therefore the sum of the two, xy -+- xy, is the fluxion of the
whole rectangle xy or arqp.

Tlte same otherwise. — Let the sides of the rectangle, x and
y, by flowing, become x + x' and y +y: then the product
of the two, or xy •+- xy -t- yy -+- xy will be the new or con-
temporaneous value of the flowing rectangle PR or xy;
subtract the one value from the other, and the remainder
xy -+- x'y ■+- xy , will be the increment generated in the same
time as x or y; of which the last term xy is nothing, or in-
definitely small in respect of the other two terms, because
x' and y are indefinitely small in respect of x and y ; which
term being therefore omitted, there remains xy ■+■ xy for
the value of that increment: and hence, by'substituting x
andi for x and y', to which they are proportional, there
arises xy ■+■ xy for the value of the fluxion of xy ; the same
as before.

17. Hence may be derived the fluxions of all powers
and products, and of all other forms of algebraic quanti-
ties whatever. And first for the continual products of any
number of quantities, as xyz, or wxyz, or vwxyz, &c. For
xyz put q or pz, so that p = xy, and xyz — pz=.q. Now,
taking the fluxion of q = pz, by the last article, it is

q = pz + pz; but p = xy, and so p = xy -+- xj by the
same article; substituting therefore these values of p and
p instead of them, in the value of q, this becomes
q = xyz -f- xjz -+- xyz, the fluxion of xyz required ; which
is therefore equal to the sum of the products arising from
the fluxion of each letter or quantity multiplied by the
product of the other two.

Again, to determine the fluxion of wxyz, the continual
product of four variable quantities; put this product, viz,
wxyx or qw =r, where q = xyz as above; then, taking the
fluxion by the last article, r = qw -h qw; and this, by sub-
stituting for q and q their values as above, becomes
r = wxyz -+- wxyz ■+■ wxyz ■+■ wxyz, the fluxion of wxyz as
required ; consisting of the fluxion of each quantity drawn
into the products of the other three.

In the very same manner it is discovered that the fluxion
of vwxyz is

vwxyz ■+■ vwxyz + vwxyz ■+■ vwryz -+- vwxyz ; and so on, for
any number of quantities whatever; in which it is always
fqund that there are as many terms as variable quantities in
the proposed fluent, and that these terms consist of the
fluxion of each variable quantity multiplied by the product
Ol all the rest of the quantities.

18. From hence is easily derived the fluxion of any
power ol a variable quantity, as of x\ or x*, or x\ &c.

For, in the rectangle or product xy, if x = y, then is the
product xy = xx or x2, and also its fluxion
xy ■+■ xy = sex -t-" x.r or 2xx, me fluxion x1.

Again, if all the three x, y, z be equal ; then is the pro-
duct of the three xyz = xxx or xs; and its fluNion
xyz ■+■ xjz + xyz = xxx ■+- xxx •+- xxx or 3x*x, the fluxion
of x3.

And in the same manner it will appear that the
fluxion of x4 is = 4x3x,

that of x5 is = 5x4x,

that of xn is = ?u"n_'x;
where n is any positive whole number. That is, the. fluxion
of any positive integral power, is equal to the exponent of
the power (n), multiplied by the next less power of the
same quantity (xn_I), and by the fluxion of the root (x).

19. Next, for the fluxion of any fractiou, as -, of one
variable quantity divided by another ; put the proposed
fraction -= q ; then, multiplying by the denominator,
x = yy; and taking the fluxions, x=qy-i- qy, or qy=x — qy-
and, by division, q = — = (by substituting the value

. 3TS X Xy xy — xy ■ ■ : X

of a, or - ) -4 = — — s-^r the fluxion or -, as requi-

- y y - y~ . y . y

red. That is, the fluxion of any fraction, is equal to the
fluxion of the numerator drawn into the denominator,
minus the fluxion of the denominator drawn into the nu-
merator, and the remainder divided by the square of the
denominator.

20. Hence too is easily derived the fluxion of any ne-
gative integer power of a variable quantity, as of x-n

or — which is the same thing. For here the numerator

.x,u
of the fraction is 1, whose fluxion is nothing; and there-
fore, by the last article, the fluxion of such a fraction, or
negative power, is barely equal to minus the fluxion of the
denominator, divided by the square of the said denomi-
nator. That is, the fluxion of x , or — — ,

_n_ 1

nx x nx -("+))

x; which is the

same rule as before for integral powers.

Or, the same thing is otherwise derived immediately
from the fluxion of a rectangle or product, thus: put the

proposed fraction, or quotient, — — q; then is i^x" = 1 ;

and, taking the fluxions, jxn -+- qnxn~* x = 0; hence

qxn = — rpx"-1*, and then dividing by x" we have

qnx ,, ... 1 „ . — nx
q = = (by substituting — for <7), or

— nx~n~' x ; the same as before.

21. Much in the same manner is obtained the fluxion
of any surd, or fractional power of a fluent quantity, as of

Xn or J/xm. For, putting the proposed quantity 1 « = q,
then, raising each to the n power, - - xm = qn;
take the fluxions, »ixm" "* — nqn~' q •

,. • , , ._, • mxm-'* _ mxm-'x __ m ~-i.
divide by nqn , q = — = — ,„_ „, — ~x ":

which is still the same rule as before, for finding the fluxion
of any power of a fluent quantity, and which is therefore
general, whether the exponent be positive or negative, in-
tegral or fractional.

FLU

22. For the Fluxions of Loga-
rithms: Lot a be the principal
vertex of an hyperbola, having its
asymptotes CD, cp, with the ordi-
nates da, ea, pq, &c, parallel to
them. Then, from the nature of r: S 'S'F
the hyperbola, and of logarithms, it is known that any
space abpq is the log. of the ratio of cb to cp, to the
modulus abcd. Now put cb = ba = 1 the side of the
square or rhombus db; m = the modulus, or area of db,
or sine of the angle c to the radius 1 ; also the absciss
cp = x, and the ordinate pq = y. Then, by the nature
of the hyperbola, cp x pq is always equal to db, that is,

xy = m; hence y = -, and the fluxion of the space, or

xy is — = PQgp the fluxion of the log. of x, to the mo-
dulus m. And in the ordinary hyp. logarithms the- mo-
dulus m being 1, therefore - is the fluxion of the hyp. log.

of x; which is therefore equal to the fluxion of the quan-
tity, divided by the quantity itself. And the same might
be shown in several other ways, independent of the figure
of the hyperbola.

23. By means of the fluxions of logarithms, are deter-
mined those of exponential quantities, i. e. quantities
which have their exponent also a flowing or variable quan-
tity. These exponentials are of two kinds, viz, when the
root is a constant quantity, as ex; and when the root is
variable, as yx.

In the former case, put the proposed exponential ex = z,
a single variable quantity; then take the logarithm of
each, so shall log. of z = x '. log. of e; take the fluxions

of these, so shall - = x . log. of e; hence 2 = 2*. log.

of e = ex x • log. of e, the fluxion of the proposed expo-
nential ex; and which therefore is equal to the said pro-
posed quantity, drawn into the fluxion of the exponent,
and also into the log. of the root.

24. And in the 2d case, put the exponential yx =2 z;
then the logarithms give log. z = x x log. y, and the

fluxions give - = x . los. v -+- x . -'> hence ^ = zx. log.

* z ° J y

y ■+■ — = (by substituting yx for z) yxx • log. y ■+■ xyx~'j,

is the fluxion of the proposed exponential yx; which
therefore consists of two terms, of which the one is the
fluxion of the proposed quantity considering the exponent
only as constant, and the other is the fluxion of the same
quantity considering the root as constant.

Of Second, Third, £,c Fluxions. — Having explained
the manner of considering and determining the firstfluxions
of flowing or variable quantities ; it remains now to consi-
der those of the higher orders, as 2d, 3d, 4th, &c, fluxions.

25. If the rate or celerity with which any flowing quan-
tity changes its magnitude, be constant, or the same, at
every position ; then is the fluxion of it also constantly
the same. But if the variation of magnitude be continually
changing, either increasing or decreasing; then will there
be a certain degree of fluxion peculiar to every point or
position ; and the rate of variation or change in the flux-
ion, is called the fluxion of the fluxion, or the second
fluxion of the given fluent quantity. In like manner, the
variation or fluxion of this 2d fluxion is called the third
fluxion of the hist proposed fluent quantity ; and so on.
And these orders effluxions arc denoted by the fluent let-

Vol. I.

[ 529 ]

FLU

ter or quantity, with the corresponding number of points
over it; viz, 2 points for the 2d fluxion, 3 for the 3d
fluxion, 4 for the 4th fluxion, and so on. So the dif-
ferent orders of the fluxions of x, are.*, x, &, &c ; where
each is the fluxion of the one next before it.

26. This description of the higher orders of fluxions
may be illustrated by the three figures at the Sth article ;
where, if x denote the absciss ap, and y the ordinate pq ;
and if the ordinate pq or y flow along the absciss ap orr,
with an uniform motion; then the fluxion of x,'viz,i- = p/>
or Qr is a constant quantity, or i= 0, in all the figures.
Also, in fig. 1, in which aq is a right line, 3/ isz=rq, or
the fluxion of pq, is a constant quantity, or y = 0 : for,
the angle q, = the angle a, being constant, Qr is to rq, or
x toy, in a constant ratio. But in the 2d figure, rq, or
the fluxion of pq, continually increases more and more;
and in fig. 3 it continually decreases more and more ; and
therefore in both these cases y has a 2d fluxion, being
positive in fig. 2, but negative in fig. 3 : and so on for the
other orders of fluxions.

27. Thus if, for instance, the nature of the curve be
such, that xais everywhere equal to a^y ; then, taking the
fluxions, it is (fj = 3x"x; and, considering x as a constant
quantity, and taking the fluxions again, the equations of
the several orders of fluxions will be as below ; viz,

the 1 st fluxions a^y = 3.rz,r,

the 2d fluxions ay =z 6xi2,

the 3d fluxions o*y = 6x3,

the 4th fluxions a2y = 0,
and all the higher fluxions = 0 or nothing.

Also the higher orders of fluxions are found in the same
manner as the lower ones. Thus,
3fyi

■3J/V+ 6yy z

ISyyy' -t- 6y\

The 1st flux, ofy is - -
its 2d flux, or flux, of 3y- ' ,
considered as the rect-
angle of 3y2 and y, is - J
and theflux.ofthisagain.or 1 „ i:,
the 3d fluxion of y3, is j ^ y
28. In the foregoing articles, it has been supposed that
the fluents increase ; or that their fluxions are positive ;
but it often happens that some fluents decrease, and that
therefore their fluxions are negative : and whenever this
is the case, the sign of the fluxion must be changed, or
made contrary to that of the fluent. So, of the rectangle
xy, when both x and y increase together, the fluxion is
xy ■+• xj: but if one of them, as y, decrease, while the
other, x, increases; then the fluxion of y being —y, the
fluxion of xy will in that case be xy — xy. This may
be illustrated by the annexed rect-
angle apqii = xy, supposed to be
generated by the motion of the line
pq from a towards c, and by the mo-
tion of the line rq from b towards a :
for, by the motion of PQ, from a to-
wards c, the rectangle is increased,
and its fluxion is ■+■ xy ; but by the
motion of hq, from b towards a, the
rectangle is decreased, and the fluxion of the decrease is
xj; therefore, taking the fluxion of the decrease from that
of the increase, the fluxion of the rectangle ary, when x
increases and y decreases, is xy — xy.

For the Inverse Method, or the finding of fluents, see
Fluent. And for the several applications of this science
to Maxima and Minima, the drawing of Tangents,
&c, see the respective articles.
3 Y

X

0.

y »

X

J

L 3

i» ti

FLY

L 530 }

F O C

Since tlic first invention of this calculus by Sir I. New-
ton, it has received various improvements from many emi-
nent authors, particularly John Bernoulli, who treated
of the fluents belonging to the fluxions of exponential ex-
pressions; James Bernoulli, Craig, Cheyne, Cotes, Man-
i'redi, Iliccati, Taylor, Fagnanus, Clairaut, Dalembert,
Euler, Maclaurin, Condorcet, Leseur and Jacquier, Bou-
ganville, Walmesley, Lagrange, Emerson, Simpson, Lan-
den, Waring, &c. There are several other treaties also
on the principles of fluxions, by Hayes, Newyentyt, L'Ho-
pital, Hudson, Rowc, Lyons, Dealtry, &c, &c, delivering
the elements of this science in an easy and familiar manner.

FLY, in Mechanics, a heavy weight applied to certain
machines, to regulate their motions, as in a jack; or to
increase their effect, as in the coining engine, &c; by
means of which the force ofthepower is not only preserved,
but equally distributed throughout the revolution.

The fly is either like a cross with heavy weights at the
ends of its arms ; or like a heavy wheel at right-angles to
the axis of motion. It may be applied to various kinds of
engines, whether they be moved by men, horses, wind, or
water; and it is of great use in those parts of an engine
which have a quick circular motion, and where the power
or resistance acts unequally in the different parts of a
revolution. In this case the fly becomes a moderator,
making the motion of revolution almost everywhere equal.

FLYERS, in Architecture, such stairs as go straight,
■and do not wind round, nor have the steps made tapering,
but equally broad at both ends. Hence, if one flight do
not rise to the top of the story, &c, there is a broad half
pace, and then commonly another set of flyers.

FLYING, the progressive motion of a bird, or other
winged animal, through the air. The parts of birds chiefly
concerned in flying, are the wings and tail : by the former,
the bird sustains and wafts himself along; and by the
latter he is assisted in ascending and descending, to keep
his body poised upright and steady. The wings are ex-
tended or stretched quite out, and then struck forcibly
downwards against the air, which by its resistance raises
the bird upwards; then, to make another stroke, the wing,
by means of its joints, readily closes in some degree, pre-
senting the sharp edge of the pinion foremost to cut the
air, and drawing the collapsed feathers after it like a flag,
to diminish the resistance to the ascent as much as pos-
sible; the wing and feathers are then stretched out hori-
zontally again, and another downward stroke is given,
which raises the bird still higher; and so on as far as he
pleases, or at least as far as the density of the air will sus-
tain him ; performing those motions of the wings very ra-
pidly, that the flight may be the quicker.

Artificial Flying, is that attempted by men, &c, by
the assistance of mechanics. The art of flying has been
attempted by several persons in all ages. Friar Bacon,
about .000 years ago, not only asserts the possibility of
flying, but affirms that he himself knew how to make a
machine with which a man might be able to convey him-
self through the air like a bird; and further adds, that it
had been tried with success. Though the fact is to be
doubted, if, as it was said, it consisted in the. following
method ; viz, in a couple of large thin hollow copper
globes, exhausted of air; which, being much lighter than
air, would sustain a chair on which a person might sit.
Father Francisco Lnna, in his Pr-odromo, proposes the
same thing, us his own idea. lie computes, that a round
vessel of plate-brass, \\ feet in diameter, weighing bounces

the square foot, will only weigh 184-S ounces ; whereas a
quantity of air of the same bulk will weigh near 2156
ounces ; so that the globe will not only be sustained in the
air, but will carry with it a weight ot 304 ounces; and by
increasing the size of the g'obe, the thickness of the metal
remaining the same, he adds, a vessel might be made to
carry a much greater weight. But the fallacy is obvious?
a globe of the dimensions he describes, as Dr. Hooke ob-
serves, would not sustain the piessure of the air, but be
crushed inwards. Indeed it is not even probable that a
globe can be made sufficiently thin to float in the atmo-
sphere after it is exhausted of air, and yet be strong enough
to sustain the compressing force of the atmosphere. But
for this purpose it seems that the globe should be rilled
with an air as elastic or strong as the atmosphere, and yet
be very much lighter; such as is used in the mongoltiers
and balloons; the former of which is filled with common
air heated, so as to be more elastic, and less heavy; and
the latter with inflammable air, which is as elastic as the
common air, with only about one-tenth of its weight. And
thus the idea of flying, or rather floating, in the air, has
been realized by the moderns, using however a different
kind of air. See Aerostation.

The same author describes a machine for flying, invented
by the Sieur Besnier, a smith of Sable, in the county of
Maim See Philos. Collec. No. 1.

By the foregoing method however, at best, only a me-
thod of floating can be obtained, like a log floating in a
current ; but not of flying, which consists in moving
through the air, independent of any current ; and which
must be effected by something in the nature of wings.
Attempts of this latter kind also have indeed been made
by several persons ; but it does not appear that any of
them have been attended with such success as to induce
the authors of those attempts to make them public. The
philosophers in the reign of King Charles the 2d were
much employed on this subject; and thecelebrated Bishop
Wilkins was so confident of success in it, that he says,
he does not question but in future ages it will be as usual
to hear a man call for his wings, when he is going a jour-
ney, as it is now to call for his boots.

The story of the flight of Dasdalus is well known.

Flying Pinion, is part of a cli ck having a fly or fan
with which to gather the air, and so check the rapidity of
the clock's motion, when the weight descends in the strik-
ing part.

FOCAL Distance, the distance of the focus, which is
sometimes understood as its distance from the vertex, as
in the parabola ; and sometimes its distance from the cen-
tre, as in the ellipse or hyperbola.

FOCUS, in Geometry and the Conic Sections, is ap-
plied to certain points in the ellipse, hyperbola, and para-
bola, where the rays reflected from all parts of tin si' curves
concur or meet ; that is, rays issuing from a luminous
point in the one focus, and falling on all p ints of the
curves, are reflected into [he other focus, or into the line
directed to the other focus, viz, into the other focus in
the ellipse and parabola, and directly from it in the hyper-
bola: which is the reason of the name focus, or Burn-
ing-point. Hence, as the one focus of the parabola is at
an infinite distance; and consequently all rays drawn
from it, to any finite part of the curve about the \ertax.
are parallel to one another; therefore, if rays from the
sun, or any other object so distant that the rays may be
considered as parallel, fall upon the curve of a parabola

F O C

[ 531 ]

F O C

or concave surface of aparaboloidul figure, those rays will
all be reflected into its focus.

Thus, the rays of vf, from the focus f, are reflected in
the direction pf, into the other focus f, in the ellipse and
parabola, and form the focus / into pq, in the hyperbola.
— In all the three curves, the double ordinate cd drawn
through the focus F, is the parameter of the axis, or a 3d
proportional to ab and ab, the transverse and conjugate
axes. — In t lie ellipse and parabola, the transverse axis is
equal to the sum of the two lines pf ■+- vf, drawn from the
foci to any point P in the curve ; but in the hyperbola, the
transverse is equal to the difference of those two lines.
That is,

ab = pf ■+- P/in the ellipse and parabola,

AB = PF — P/in the hyperbola.

In the ellipse and parabola, the square of the distance
between the foci, is equal to the difference of the squares
of the two axes ; and in the hyperbola, it is equal to the
sum of their squares: that is

ifx = ab1- abz in the ellipse and parabola.

f/'! = ab2 -+- ab3 in the hyperbola.

Therefore the two semi-axes, with the distance of the
focus from the centre, form always a. right-angled triangle
f«e, or A(ZE.

In all the curves, the conjugate semi-axis is a mean pro-
portional between the distances of either focus from either
end of the transverse axis : that is,
af : e« : : e« : fb,
or eo1 = af . FB.

If t litre be any tangent to these curves, and two lines
drawn from the foci to, the point of contact ; these two
lines will make equal angles with that tangent.
So, if gpg touch the curve at p ;
then is the angle fpg = /.fro.

If a line be drawn from either focus, perpendicularly
upon a tangent ; the distance of their intersection from the
centre will be equal to the semi-transverse axis. So,' if
Fit or fa be perpendicular to the tangent ph ; then is
eh = ea or eb. Consequently, the circle described on
the diameter ab, will pass through all the points h, h.

The foregoing are the principal properties that are com-
mon to the foci of all the three conic sections. To which
Tiiay be added the following properties which are peculiar
«o the parabola : viz, In the parabola, the distance from

t e focus to the vertex, is equal to | of the parameter, or
half the ordinate at the focus: viz, af = £fc. Also, a
line drawn from the focus to any point in the curve, is
equal to the sum of the focal distance from the vertex and
the absciss of the ordinate to that point : i. e. fp = af -+-
ai = 2af ± Ft = cf + fi. If from any point of a pa-
rabola there be drawn a tangent, and a perpendicular to
it pk, both to meet the axis produced ; then the focus will
be equally distant from the two intersections and the point
of contact: i. e. fg = fp = fk. Also the ^.kpf= kp/.
Hence also the subnormal ik is = 2af = f'c the semi-
parameter. — The line drawn from the focus to any
point of the curve, is equal to £ the parameter of the dia-
meter to that point : i.e. fp = i the parameter of the dia-
meter Tf. — If an ordinate to any diameter pass through
the focus, it will be equal to half its parameter; and its
absciss equal to .§ of the same parameter; or the absciss
equal to half the ordinate: i.e. pl =|mn =|lm or£i,N.
Focus, in Optics, is a point in which several rays meet,
and are collected, after being either reflected or refracted.
It is so called, because the rays being here brought toge-
ther and united, their force and effect are increased, inso-
much as to be able to burn ; and therefore it is that bodies
are placed in this point to be burnt, or to show the effect
of burning-glasses, or mirrors. — It is to be observed, how-
ever, that in practice, the focus is not an absolute point,
but a space of some small breadth, over which the rays
are scattered ; owing to the different nature and refrangi-
bility of the rays of light, and to the imperfections in the
figure of the lens, &c. However, the smaller this space is,
the better, or the nearer to perfection the instrument ap-
proaches. Huygens shows that the focus of a lens convex
on both sides, has its breadth equal to £ of the thickness
of the lens.

Virtual Focus, or Point of Divergence, so called by Mr.
Molyneux, is the point from which rays tend, after refrac-
tion or reflection; being in this respect opposed to the or-
dinary focus, or point of concurrence, where rays are made
to meet after refraction or reflection. Thus, the foci of an
hyperbola are mutually virtual foci to each other; but, in
an ellipse, they are common foci to each other; for the
rays are refleeted from the other focus in the hyperbola,
but towards it in the ellipse; as appears by the figures at
the beginning of this article.

In Dioptrics, let
abc be the conca-
vity of a glass, whose
centre is n, and axis
de: Let fg be a ray
of light falling on the
glass, parallel to the axis de ; this ray fg, after it has
passed through the glass, at its emersion at g will not pro-
ceed directly to h, but be refracted from the perpendicular
dg, and will become the ray gk, which being produced to
meet the axis in e, this point e is the virtual focus, as the
ray is refracted directly from this point.

Rules for the Foci of Lenses and Mirrors.

I. la Catoptrics, or Lenses.

1. The focus of a convex glass, i.e. the point where

parallel rays unite when transmitted through a convex

glass whose surface is the segment of a sphere, is distant

from the pole or vertex of the glass, nearly a diameter

and half of the convexity. — 2. In a plano-convex glass,

the focus of parallel rays is distant from the pole of the

glass a diametcj of the convexky, if the segment do not

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exceed 30 degrees. Or the rule in plano-convex glasses
is, As 107 : 193 : : so is the radius of convexity : to the
refracted ray taken to its concourse with the axis; which
in glasses of large spheres is nearly equal to the distance of
the focus taken in the axis. — 3. In double-convex glasses
of the same sphere, the focus is distant from the pole of
the glass about the radius of the convexity, if the segment
be but 30 degrees. But when the two convexities are un-
equal, or segments of different spheres, then the rule is, As
the sum of the radii of both convexities : is to the radius
of either convexity alone : : so is double the radius of the
other convexity : to the distance of the focus. But here
it must be observed, that the rays which fall nearer the
axis of any glass, are not united with it so near the pole of
the glass as those farther off : nor will the focal distance be
so great in a plano-convex glass, when the convex side is
towards the object, as when the plane side is towards it.
And hence it is truly concluded, that, in viewing any ob-
ject by a plano-convex glass, the convex side should always
be turned outwards ; as also in burning by such a glass.
II. For the Virtual Focus.

1. In concave glasses, when a ray passes through air
parallel to the. axis, the virtual focus, by its first refraction,
becomes at the distance of a diameter and a half of the
concavity. — 2. In plano-concave glasses,- when the rays
fall parallel to the axis, the virtual focus is distant from
the glass, the diameter of the concavity. — 3. In plano-
concave glasses, as 107 : 193 : : so is the radius of
the concavity : to the distance of the virtual focus. —
4. In double concaves of the same sphere, the virtual
focus of parallel rays is at the distance of the radius of
the concavity. But, whether the concavities be equal
or unequal, the virtual focus, or point of divergency
of the parallel rays, is determined by this rule; As the
sum of the radii of both concavities : is to the radius of
cither concavity : : so is double the radius of the other
concavity : to the distance of the virtual focus. — 5. In
concave glasses, exposed to converging rays, if the point
to which the incident ray converges, be farther distant
from the glass than the virtual focus of parallel rays, the
rule for finding the virtual focus of this ray, is this ; As
the difference between the distance of this point from the
glass, and the distance of the virtual focus from the glass:
is to the distance of the virtual focus : : so is the dis-
tance of this point of convergence from the glass : to the
distance of the virtual focus of this converging ray. — 6. In
concave glasses, if the point to which the incident ray
converges, be nearer to the glass than the virtual focus of
parallel rays, the rule to find where it crosses the axis, is
this ; As the excess of the virtual focus, above this point of
convergency : is to the virtual focus : : so is the distance
of this point of convergency from the glass : to the distance
of the point where this ray crosses the axis.

III. Practical Rules for finding the Foci of Glasses.

1. To find, by experiment, the focus of a convex sphe-
rical glass, being of a small sphere. Apply it to the end
of a scale of inches and decimal parts, and expose it to
the sun; then on the scale may be seen the bright inter-
section of the rays measured out : or, fix it in the hole of
a dark chamber ; and where a white paper receives (lie
distinct representation of distant objects, there is the locus
of the glass. — 2. For a glass of a pretty long focus, ob-
serve some distant object through it, and recede from the
glass till the eye perceives all in confusion, or the object
begins to appear inverted; then the eye is in the focus. —

3. For a plano-convex glass: make it reflect the sun
against the wall ; then on the wall will be seen two sorts
of light, a brighter within another more obscure: with-
draw the glass from the wall, till the bright image be in its
least dimensions ; then is the glass distant from the wall
about a fourth part of its focal length. — <1. For a double
convex : expose each side to the sun in like manner ; and -,
observe both the distances of the glass from the wall : then
is the first distance about half the radius of the convexity
turned from the sun ; and the second is about half the
radius of the other convexity. The radii of the two con-
vexities being thus known, the focus is then found by this
rule; As the sum of the radii of both convexities : is to
the radius of either convexity : : so is double the radius of
the other convexity : to the distance of the focus.
IV. To find the Foci of all Glasses Geometrically.
Dr. Halley has given a general method for finding the
foci of spherical glasses of all kinds, both concave and
convex ; exposed to any kind of rays, either parallel, con-
verging, or diverging; as follows: To find the focus of
any parcel of rays diverging from, or converging to, a
given point in the axis of a spherical lens, and making the
same angle with it; the ratio of the sines of refraction
being given.

Suppose gl a lens; p a point in its surface; v its pole;
c the centre of the spherical segment ; o the object, or
point in the axis, to or from which the rays proceed ; and
op a given ray : and suppose the ratio of refraction to be
as r to s. Then making cr to co, as s to r for the im-
mersion of a ray, or as r to s for the emersion (i.e. as the
sines of the angles in the medium which the ray enters, to
the corresponding sines in the medium out of which it
comes) ; and laying en from c towards o, the point u will
be the same for all the rays of the point o. Lastly, draw-
ing the radius PC, continued if necessary ; with the centre
r, and distance op, describe an arc intersecting pc in Q.
Then the line qr, being drawn, it will be parallel to the
reflected ray; and pf, being drawn parallel to it, will in-
tersect the axis in the point r, the focus sought. — Or, say
as cq : cp : : cr : CF, which will be the distance of the
focus from the centre of the sphere. — And from this ge-
neral construction, he adverts to a number of particular
simple, cases. — Dr. Halley gave also a universal algebrai-
cal theorem to find the focus of all sorts of optic glasses,
or lenses. Sec also the Philos. Trans. No. 205, or my Abr.
vol. 3, pa. 593.

V. In Culoptrics, or Foci hy Reflection.

These are easily found for any known curve, from this
principle, that the angle of incidence is always equal to the
angle of reflection. The same are also easily found by ex-
periment, being exposed to any object.

The increase of heat from collecting the sun's rays into
a focus, has been found in many cases of burning-glasses,
to be astonishingly great ; the effect being increased as the
square of the diameter of the glass exceeds that of the
focus. If, for instance, there be a burning-glass of 12

F O N

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FOR

inches diameter; this will collect or crowd together all the for the Vortices of Descartes; on which Voltaire says,

rays of the sun which fall upon the glass into the compass " We must excuse his comedies, on account of his great

of about | part of an inch: then, the areas of the two age; and hjs Cartesian opinions, as they were those of

spaces being as the square of 12 to the square of £, or as his youth, when they were universally received all over

the square of 96 to the square of 1, that is, as 92l6 to 1 ; Europe."

it follows, that the heat in the focus will be 92l6 times' In his poetical performances and Dialogues of the Dead,

greater than the sun's common heat. And this will have the spirit of Voiture was discerned, though more extended

an effect as great as the direct rays of the sun would have and more philosophical. His Plurality of Worlds is a work

on a body placed at the 96th part of the earth's distance singular in its kind: his design in this was, to present that

from the sun; or the same as on a planet that should part of philosophy to view in a gay and pleasing dress,

move round the sun at but a very little more than a dia- Fontenelle applied himself also to mathematics and na-

meter of the sun's distance from him, or that would never tural philosophy ; in which he proved not less successful

appear farther from him than about 36 minutes. than he had been in polite literature. Having been ap-

Besides Dr. Halley's method for finding the foci, several pointed perpetual secretary to the Academy of Sciences,

other authors have written upon this subject ; as Mr. he discharged that trust above 40 years with universal ap-

Ditton, in his Fluxions ; Dr. Gregory, in his Elements of plause : his History of the Academy often throws great

Dioptrics; M.Carre, and Guisnee, in the Memoires de light on their Memoirs, which are sometimes obscure; and

l'Acad. ; Dr. Barrow and Sir I. Newton have also neat and it has been said, he was the first who introduced elegance

elegant ways of finding geometrically the foci of spherical into the sciences. The Eloges, which he pronounced on

glasses; which may be seen in Barrow's Optical Lectures, the deceased members of the Academy, have this peculiar

FOLIATE, a name given by some to a curve of the 2d merit, that they excite a respect for the sciences as well as

order, expressed by the equation 3? -+- y3 = axy, being one for the authors.

species of defective hyperbolas, with one asymptote, and On the whole, Fontenelle must be considered as the great

consisting of two infinite legs crossing each other, forming master of the new art of treating abstract sciences, in a

a kind of leaf. It is the 42d species of Newton's Lines of manner that make their study at once easy and agreeable:

the 3d order. nor are any of his works of other kinds void of merit. All

FOLKLiS (Martin), an English mathematician, phi- those talents which he possessed from nature, were assisted

losopher, and antiquary, was born at Westminster, about by a good knowledge of history and languages : and he

I69O; and was greatly distinguished as a member of the
Royal Society in London, and of the Academy of Sciences
at Paris. He was admitted into the former at 24- years of
age; made one of their council two years after; named
vice-president by Sir Isaac Newton himself; and, after Sir
Hans Sloane, became president. Coins, ancient and mo-
dern, were a great object with him ; and his last produc-
tion was a book upon the English Silver Coin, from the
Conquest to his own times. He died at London in 1754.
Dr. Birch had drawn up materials for a life of Mr. Folkes,

perhaps surpasses all men of learning who have not had
the gift of invention.

Besides his poetical and theatrical works, with those of
belles-lettres, &c, he published Elemens de Geometrie de
l'lnfini, in 4to, 1727; also the Theorie des Tourbillons
Cartesiens; and Discours Moraux et Philosophiques. All
his different works were collected in eleven volumes 12mo,
under the title of CEuvres Diverges.

FOOT, a measure of length, divided into 12 inches, and
each inch supposed to contain 3 barley-corns in length.

which are preserved at large in the Anecdotes of Bowyer, Geometricians divide the foot into 10 digits, and the digit
pa. 562. There are many memoirs of Mr. Folkes's in tire into 10 Lines, &c.

Philos. Trans, from vol.30 to vol. 46, both inclusive; viz,
1. Account of an Aurora Borealis, vol.30. — 2. Of Lieu-
wenhoek's curious Microscope, vol. 32. — 3. On the Stand-
ard Measures in the Capitol at Rome, vol. 39. — 4. Obser-
vations of three Mock-suns, vol. 40. — 5. On the Fresh-

It seems this measure has been taken from the length of
the human foot; but it is of different lengths in different
countries. The Paris royal foot is to the English foot, as
4263 to 4000, or nearly 16 to 15, and exceeds the English
by 9f lines. The. ancient Roman foot of the Capitol con-

water Polypus, vol. 42.-6. On human bones petrified, sisted of 4 palms; equal toll inches and T7^ English; the

vol. 43. — 7- On a passage in Pliny's Natural History,
44. — 8. On an Earthquake at London, vol. 46. — 9. Ditto
at Kensington, vol. 46. — 10. Ditto at Newton, vol. 46.

FOMAHAUT, or FoMALHAUT^astar of the first mag-
nitude in the water of the constellation Aquarius, or in the
mouth of the southern fish. Its latitude is 21° 6' 28"
south, and mean longitude to the beginning of 1/60,
1150° 28' 55".

FONTENELLE (Bernard de), a celebrated French
author, was born at Rouen in 1657, and died in 1756,
having nearly attained his 100th year. He was a univer-

Rhinland, or Ley den foot, used by the northern nations, is
to the Roman foot, as 19 to 20. For the proportions of
the foot of several nations, compared with the English, see
the article Measure.

SjKareFooT,isasquarewhosesideis 1 foot, or 12 inches,
and consequently its area is 144 square inches.

Cubic Foot, is a cube, whose side is one foot, or 1 2 inches,
and consequently it contains 123 or 1728 cubic inches.

FooT-bank, or Foor-stcp, in Fortifi. See Banquette.

FORCE, Vis, or Power, in Mechanics, Philosophy, &c,
denotes the cause of the change in the state of a body,

sal genius : at a very early age he wrote several comedies with respect to motion, rest, pressure, &c ; as well as its

and tragedies of considerable merit; and he did the same endeavour to oppose or resist any such change. Thus,

at a very advanced age. Voltaire declares him the most whenever a body, which was at rest, begins to move; or

universal genius the age of Lewis the 14th produced ; and when its motion is either not uniform, or not direct ; the

compares him to lands situated in so happy a climate, as cause of this change in the state of the body, is what is

to produce all sorts of fruits. His last comedies, though called Force, and is an external cause. Or, while a body

they showed the elegance of Fontenelle, were, however, remains in the same state, either of rest, or of uniform and

little fitted for the stage ; he then also produced an Apology rectilinear motion, tke cause of its remaining in such state,

FOR

[ 534 ]

FOR

is in the nature of the body, being an innate internal force, centre of oscillation, and by his dissertations, in answer
and is called its inertia. t0 tr>e objections of the abbot Catalan, one of which whs

Mechanical forces may be reduced to two sorts; one of published in 1684. This eminent mathematician had oe

a body at rest, the other of a body in motion

The force, &c, of a body at rest, is that which we con-
ceive to be in it when lying on a plane, or hanging by a
rtrpe, or supported by a spring, &c ; and this is called
Pressure, Tension, Force, or Vis Mortua, Solicitatio, Co-
natus Movendi, Conamen, &c ; which kind of force may
be always measured by a weight, viz, the weight that sus-
tains it. To this class of forces may also be referred cen-
tripetal and centrifugal forces, though they reside in a body
in motion; because these forces are homogeneous to
weights, pressures, or tensions of any kind. The pressure,
or force of gravity in any body, is proportional to the
quantity of matter contained in it.

The force of a body in motion, is a power residing in
that body, so long as it-continues its motion ; by means of
which, it is able to remove those obstacles which are op-
posed to it ; or to lessen, destroy, or overcome the force
of any other moving body, which meets it in an opposite
direction ; or to surmount any the largest dead pressure
or resistance, as tension, gravity, friction, &c, for some
time; but which will be lessened or destroyed by such re-
sistance as lessens or destroys the motion of the body.
This is called Vis Motrix, Moving Force, or Motive Force,
and by some late writers Vis Viva, to distinguish it from
the Vis Mortua before mentioned ; and by these appella-
tions, however different, the same, thing is understood by
all mathematicians; namely, that power which is inherent

monstrated, that in the collision of two bodies that are
perfectly elastic, the sum of the products of each body
multiplied by the square of its velocity, was the same after
the shock as before (though the same thing is true of
the sums of the products of the bodies multiplied simply
by their velocities). Now that proposition is so far ge-
neral as to obtain in all collisions of bodies that are per-
fectly elastic : and it is also true', when bodies of a perfect,
elasticity strike any immoveable obstacle, as well as when
they strike one another ; or when they are constrained by
any power or resistanceto move in directions different from
those in which they impel one another. These considera-
tions might have induced Huygens to lay it down as a ge-
neral rule, that bodies constantly preserve their ascen-
sional force, i.e. the product of their mass by the height
to which their centre of gravity can ascend, which is as
thesquare of the velocity ; and therefore, ina given system
of bodies, the sum of the squares of their velocities will
remain the same, and not be altered by the action of the
bodies among themselves, nor against immoveable obsta-
cles. Leibnitz's metaphysical system led him to think
that thesame quantity of action or force subsisted in the
universe; and finding this impossible, if force were esti-
mated by the quantity of motion, he adopted Huygens's
principle of the preservation of the ascensional force,
and made it the measure of moving forces. But it is to
be observed that Huygens's principle, above-mentioned, is

in a moving body, and which accompanies it in whole, or general only wheji bodies are perfectly elastic : and in
in part, as long as the body is in motion ; so as to displace,
or to withstand opposite moving forces, or by which it
overcomes any dead resistance, &c, which force is other-
wise called Percussion, or Percussive Force, or Momentum.
But concerning the measure of this kind of force, ma-
thematicians have been divided into two parties. It is al-
lowed on both sides, that the measure of this force de-
pends partly on the mass of matter in the body, or its
weight, and partly on the velocity of its motion ; so that
on any increase of either weight or velocity, the moving
force becomes greater. It is also agreed, that the velo-
city being given, or being the same in two moving bodies,
their forces are in proportion to their masses or weights.
But, when two bodies are equal, and the velocities with
which they move are different, they no longer agree about
the measure of the moving force.

The Cartesians and Newtonians maintain, that, in this
case, the moving force is in proportion simply as the ve-
locity with which a body moves; so that with a double
velocity it has a double force, &c : But the Leibnitians
assert, that the moving force is proportional to the square
of the velocity ; so that, with a double velocity a qua-
druple force will be produced, &c. Or, when the bodies
are different, the former hold, that the momentum or
moving force of bodies, is in the compound ratio of their
weights and velocities: while the latter maintain, that it
is in the compound ratio of the. weights ami squares of the
velocities.

Though Leibnitz was the first who expressly asserted,
that the force of a body in motion is as the square of its
velocity, which was in a paper inserted in the Leipsic Acts
for the year lo'So", yet it is thought that Huygens led him
into that notion, by some demonstrations in the 4th part
eff his book Uc Ilorologio Oacillatorio, relating to the

some other cases, which Maclaurin has endeavoured to
distinguish: showing at the same time that no useful con-
clusion in mechanics is affected by the disputes concern-
ing the measure of the force of bodies in motion, which
have been objected to mathematicians. Analyst, Query Q.
See Maclaurin's Fluxions, vol. 2, art. 533; Huygens, Oper.
torn. 1, pa. 248; &c.

Leibnitz's principle was adopted by several persons ; as
Wolfius, the Bernoullis, &c. Mr. Dan. Bernoulli, in his
Treatise, has assumed the preservation of thevisascen-
dens of Huygens, or, as others express it, the conservatio
virium vivarum ; and, in Bernoulli's own expression,
a?qualitas inter descensum actualem ascensumque poten-
tialem, as an hypothesis of wonderful use in mechanics.
But Dr. Jurin contends, that the conclusions drawn from
this principle are oftener false than true. See De Con-
servat. Virium Vivarum Dissert. Lond. 1744.

Catalan and Papin answered Leibnitz's paper published
in 1086 ; and from that time the controversy became
more general, and was carried on for several years by
Leibnitz, John and Daniel Bernoulli, Poleni, Wolfius,
Gravesandc, Camus, Muschenbroek, &c, on one side; and
Pemberton, Eames, Desaguliers, Dr. S. Clark, M. de Mai-
ran, Jurin, Maclaurin, Robins, c\:e, on the other. See
Act. Erud. 10\S6', 16.90, 1 6" t> i , lt>\95; Nouv. de la Rep.
des Let. Sept. \686, l6"S7, art. 2; Comm. Epist. inter
Leibn. et Bern. Ep.24, p. 143 ; Discourses sur les Loix de
la Comm. du Mouvement, Oper. torn. 3, & Diss, de vera
Notionc Virium Vivarum, ib.; Act. Petropol. torn. 1,
pa. 131, &CJ Ilydronaniica, sect. 1 ; Herman, in Act. Pe-
trop. torn. 1, p. 2, &c ; Polcn. deCastcllis ; Wolf, in Act.
Petrop. torn. 1, p. 217, &c, and in Cosmol. Gener. ;
Graves, in Journ. Lit. et Phys. Elem. Math. 1742, lib. 2,
cap. 2 and 3; Memoir, de l'Acad. des Sciences, 17'18;

FOR

[ 535 ]

FOR

Muschenbr. Int. ad Phil. Nat. 1762, vol.1, p. 83 &c ;
Pemb. &c, in Phil. Trans. No. 371, 375, 376, 396,
400, 401, or my Abridg. vol. 6, p.570 &c. Mairan in
Mem. de l'Acad. des Sc. 1.7.28, Phil. Trans. No. 459,
or Abridg. vol. 7, p. 637, Philos. Trans, vol.43, p.423
&c; Maclaurin's Ace. of Newton's Discoveries, p. 117
&c, Flux, ubi supr. and Recueil des Pieces qui ontemporte
le Prix &c. torn. 1 ; Desagul. Course Exp. Philos. vol.1,
p. 393 &c, vol.2, p. 49 &c; and Robins's Tracts, vol. 2,
p. 135.

The nature and limits of this work will not admit of a
full accounfof the arguments and experiments that have
been urged on both sides of the question ; but they may
be found chiefly in the preceding references. A few of
them however may be considered, as follows.

The defenders of Leibnitz's principle, besides the argu-
ments above-mentioned, refer to the spaces that bodies
ascend to, when thrown upwards, or the penetrations of
bodies let fall into soft wax, tallow, clay, snow, and other
soft substances, which spaces are always as the squares of
the velocities of the bodies. On the other hand, their op-
ponents retort, that such spaces are not the measures of
the force in question, which is rather percussive and mo-
mentary, as those above are passed over in unequal times,
and are indeed the joint effect of the forces and times.

Desaguliers brings an argument from the familiar ex-
periment of the balance, and the other simple mechanic
powers, showing that the effect is in proportion to the ve-
locity multiplied by the weight; for example, 4 pounds
being placed at the distance of 6 inches from the centre
of motion of a balance, and 2 pounds at the distance of 12
inches; these will have a vis viva if the balance be put
into a swinging motion. Now it appears that these forces
are equal, because, with contrary directions, they soon
destroy each other ; and they are to each other in the
simple ratio of the velocity multiplied by the mass, viz
4 x 6 = 24, and 2 x 12 — 24 also.

Mr. Robins, in his remarks on J. Bernoulli's treatise,
entitled, Discours sur les Loix de la Communication du
Mouvemrnt, informs us, that Leibnitz adopted this opi-
nion through mistake; for though he maintained that the
quantity of force is always the same in the universe, he
endeavours to expose the error of Descartes, who also as-
serted, that the quantity of motion is always the same ;
and in his discourse on this subject in the Acta Erudito-
rum for 1 6*86, he says that it is agreed on by the Carte-
sians, and all other philosophers and mathematicians, that
there is the same force requisite to raise a body of 1 pound
to the height of 4 yards, as to raise a body of 4 pounds
to the height of 1 yard; but being shown how much he
was mistaken in taking that for the common opinion,
which would, if allowed, prove the force of the body to be
as the square of the velocity it moved with, he afterwards,
rather than own himself capable of such a mistake, en-
deavoured to defend it as true ; since* he found it was the
necessary consequence of what he had once asserted; and
he maintained, that the force of a body in motion was
proportional to the height from which it must fall, to ac-
quire that velocity ; and the heights being as the square of
the velocities, the forces would be as the masses multi-
plied by them ; whereas, when a body descends by its gra-
vity, or is projected perpendicularly upwards, its motion
may be considered as the sum of the uniform and conti-
nual impulses of the power of gravity, during its falling
in the former case, and till they extinguish it in the latter.

Thus when a body is projected upwards with a double ve-
locity, these uniform impulses must be continued for a
double time, in order to destroy the motion of the body ;
and hence it follows, that the body, by setting out with a
double velocity, and ascending for a double time, must
arise to a quadruple height, before its motion is exhausted.
But this proves that a body with a double velocity moves
with a double force, because it is produced or destroyed
by the same uniform power continued for a double time,
and not with a quadruple force, though it rises to a qua-
druple height; so that the error of Leibnitz consisted in
his not considering the time, since the velocities alone are
not the causes of the spaces described, but the times and
the velocities together ; yet this is the fallacious argument
on which he first built his new doctrine ; and those which
have been since much insisted on, and derived from the
indentings or hollows produced in soft bodies by others
falling into them, are much of the same kind. Robins's
Tracts, vol.2, p. 178.

But many of the experiments and reasonings, that have
been urged on both sides of this controversy, have been
founded in the different senses applied to the term Force.
The English and French philosophers, by the word Force,
mean the same thing as they do by momentum, motion,
quantity of motion, percussion, or instantaneous pressure,
which is measured by the mass drawn into the velocity,
and may be known by its effect ; and when they consider
bodies as moving through a certain space, they allow for
the time in which that space is described : whereas the
Dutch, Italian, and German philosophers, who have
espoused the new opinion, mean by the word Force, or
Force inherent in a body in motion, that which it is able
to produce ; or, in other words, the force is always mea-
sured by the whole effect produced by the body in motion,
until its wholeforce be entirely communicated or destroy-
ed, without any regard to the timeemployed in producing
this total effect. Thus, say they, if a point runs through
a. determinate space, and presses with a certain given
force, or intensity of pressure, it will perform the same
action whether it move fast or slow, and therefore the
time of the action in this case ought not to be regarded.
Gravesande, Phys. El. Math. § 723 — 728.

Euler observes, with regard to this dispute concerning
the measure of vivid force, or living force, as it is some-
times called, that we cannot absolutely ascribe any force
to a body in motion, whether we suppose this force pro-
portional to the velocity, or to the square of the velocity:
for the force exerted by a body, striking another at rest,
is different from that which it exerts in striking the same
body in motion; so that this force cannot be ascribed to
any body considered in itself, but only relatively to the
other bodies it meets with. There is no force in a body
absolutely considered, but its inertia, which is always the
same, whether the body be in motion or at rest. But if
tli is body be forced by others to change its state, its inertia
then exerts itself as a force, properly so called, which is
not absolutely determinable; because it depends on the
changes that happen in thestate of the body.

A second observation which has been made by some
eminent writers, is, that the effect of a shock of two or
more bodies, is not produced in an instant, but requires a
certain interval of time. Admitting this to be the case,
the heterogeneity between the vis viva and niortua, or
living and dead force, willvanish; since a pressure may
always be assigned, which in, the same time,, however little,

FOR

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shall produce the same effect. If then the vis viva be ho-
mogeneous to the vis mortua, and having a perfect mea-
sure and knowledge of the latter, we need require no
other measure of the former than that which is derived
from the vis mortua equivalent to it.

Now that the change in the state of two bodies, by their
shock, does not happen in an instant, appears evidently
from the experiments made on soft bodies : in these, per-
cussion forms a small cavity, visible after the shock, if the
bodies have ho elasticity. Such a cavity cannot certainly
be made in an instant. And if the. shock of soft bodies
require a determinate time, we must certainly say as much
of the hardest, though this time may be so small as to be
beyond all our ideas, Neither can an instantaneousshock
agree with that constant law of nature, by virtue of which
nothing is performed per saltum. But it is needless to in- ,
sist farther upon this, since the duration of any shock may
be determined from the most, certain principles.

The force of percussion, resulting from the pressures
that bodies exert on each other, while the collision lasts,
may be perfectly known, if these pressures be determined
for every instant of the shock. The mutual action of the
bodies begins the first moment of their' contact; and is
then least; after which this action increases, and becomes
greatest when the reciprocal impressions are strongest. If
the bodies have no elasticity, and the impressions they
have received remain, the forces will then cease. But if
the bodies be elastic, and the parts compressed restore
themselves to their former state, then will the bodies con-
tinue to press each other till they separate. To compre-
hend therefore perfectly the force of percussion, it is requi-
site first to define the time the shock lasts, and then to as-
sign the pressure corresponding to each instant of this time ;
and as the effect of pressures in changing the state of any
body may be known, we may thence come at the true
cause of the change of motion arising from collision. The
force of percussion therefore is no more than the opera-
tion of a variable pressure during a given time; and to
measure this force, we must have regard to the time, and
to the variations according to which the pressure increases
and decreases.

Euler has given some calculations relative to these par-
ticulars ; and he illustrates their tendency by this instance:
Suppose that the hardness of the two bodies, a and b, is
equal; and such, that being pressed together with the
force of 1001b, the impression made on each is of the
depth of -f-g^th .part of a foot. Suppose also that b is
fixed, and that a strikes it with the velocity of 100 feet in
a second; then, according to this author, the greatest force
of compression will be equivalent to400lb, and this force
will produce in each of these bodies an impression equal
to Jy of a foot; and the duration of the collision, that is,
till the bodies arrive at their greatest compression, will be
about V3B of a second. M. Euler, in his calculations,
supposes the hardness of a body to be proportional to the
force of pressure requisite to make a given impression on
it ; so that the force by which a given impression is made
on a body, is in a compound ratio of the hardness of the
body and of the. quantity of the impression. But he ob-
serves, that regard must be had to the magnitude of the
boilic s, as the same impression cannot be made on the least
bodies as on the greatest, from the defect of space through
which their component particles must be driven: he con-
siders therefore only the least impressions, and supposes
the bodies of such magnitudes, that the impressions with

respect to them may be considered as nothing. What he
supposes concerning the hardness of bodies, neither implies
elasticity nor the want of it, as elasticity only produces a
restitution of figure and impression when the pressing
force ceases; but this restitution need not be here consi-
dered. It is also supposed, that the bodies which strike
each other, have plane and equal bases, by which they
touch each other in the collision; so that the impression
hereby made diminishes the length of each body. It is
farther to be observed, that in M. Euler's calculations,
bodies are supposed so constituted, that they may not only
receive impressions from the forces pressing them, but that
a greater force is requisite to make a greater impression.
This excludes all bodies, fluid or solid, in which the same
force may penetrate farther and farther, provided it has
time, without ever being in equilibrio with the resistance :
thus a body may continually penetrate farther into soft
wax, though the force impelling it be not increased : in
these, and the like cases, nothing is required but to sur-
mount the first obstacles; which being once done, and
the connexion of parts broken, the penetrating body always
advances, meeting with the same obstacles as before, and
destroying them by an equal force. But this celebrated
genius only considers the first obstacles which exist before
any separation of parts, and which are doubtless such,
that a greater impression requires a greater force. Indeed
this chiefly takes place in elastic bodies ; but it seems like-
wise to obtain in all bodies when the impressions made on
them are small, and the contexture .of their parts is not al-
tered.

These, things being premised, let the mass or weight
of the body a be expressed in general by a, and let its
velocity before the shock be that which it might ac-
quire by falling from the height a. Further, let the hard-
ness of a be expressed by m, and that of b by n, and
let the area of the base, on which the impression is
made, be cc ; then will the greatest compression be made

with the force */ '(- — — x ao). Therefore, if the hard-

ness of the two bodies, and the plane of their contact
during the whole time of their collision be the same, this
force will be as ^/a«, that is, as the square root of the
vis viva of the striking body a. And as ^/ a is proportional
to the velocity of the body a, the force of percussion will
be in a compound ratio of the velocity and of the subdu-
plicate ratio of the mass of the body striking; so that in
this case neither the Leibnitian nor the Cartesian proposi-
tions take place. But this force of percussion depends
chiefly on the hardness of the bodies; the greater this is,
the greater will the force of percussion be. If m = N, this
force will be as ^/(mcc x ah), that is, in a compound sub-
duplicate ratio of the vis viva of the striking body, of the
hardness, and of the plane of contact. But if M, the hard-
ness of one of the bodies, be infinite, the force of percus-
sion will be as ^/(kcc x m); at the same time, if m=n,
this force will be as x/(|ncc x ao). Therefor.', all other
things being equal, the force of percussion, if the striking
body be infinitely hard, will be to the force of percussion
when both the bodies are equally hard, as x/ '2 to 1.

Euler further deduces from his calculation, that, the im-
pression received by the bodies a and b will be as follows;

N X Aa M X Ao .

v:„ ac ./ and </■ r respectively.

viz, as y (M + Nj x Mrr v ^M + Nj x N„ i J

If therefore the hardness of A, that is M, be infinite, it will

sullcr no impression ; whereas that on a will extend to the

FOR [537

But if the hardness of the two bodies

FOlt

depth ofy'—

be the same, or m = n, they will each receive equal im-
pressions of the depth v^TT- • So that the impression
received by the body b in this case, will be to theimpres-
sion it receives in the former, as 1 to ^/2.

Euler has likewise considered and computed the case
where the striking body has its anterior surface convex,
with which it strikes an immoveable body whose surface
is plane. He has also examined the case when both bodies
are supposed immoveable; and from his formulse he de-
duces the known laws of the collision of elastic and non-
elastic bodies. He has also determined the greatest pres-
sures the bodies receive in these cases ; and likewise the
impressions made on them. In particular he shows, that
the impressions received by the body struck, or b, if move-
able, is to the impression received by the same body when
fixed, as ^b to ^/(a -+- b).

There are several curious, as well as useful observations,
in Desaguliers's Experimental Philosophy, concerning the
comparative forces of men and horses, and the best way of
applying them. A horse draws with the greatest advan-
tage when the line of direction is level with his breast; in
such a situation, he is able to draw 2001b for S hours a
day, walking about 2f miles an hour. But if the same
horse be made to draw 2401b, he can work only 6 hours a
day, and cannot go quite so fast. On a carriage indeed,
where friction alone is to be overcome, a middling horse
will draw 10001b. But the best way to try the force of a
horse, is to make him draw up out of a well, over a single
pulley or roller; and in that case, an ordinary horse will
draw about 200lb, as before observed.

It is found that 5 men are of equal force with one horse,
and can with equal ease push round the horizontal beam
of a mill, in a walk 40 feet wide; whereas 3 men will do
it in a walk only 19 feet wide.

The worst way of applying the force of a horse is to make
him carry or draw up hill : for if the hill be steep, 3 men
will do more than a horse, each man climbing up faster
with a burden of 1001b weight, than a horse that is loaded
with 3001b : a difference which is owing to the position of
the parts of the human body being better adapted to climb,
than those of a horse.

On the other hand, the best way of applying the force of
a horse, is the horizontal direction, in which a man can
exert the least force : thus, a man that weighs 1401b, when
drawing a boat along by means of a rope coming over his
shoulders, cannot draw above 27lb, or exert above l-7th
part of the force of a horse employed to the same purpose;
so that in this way the force of a horse is equal to that of has been in motion by the action of the variable force, v

Force is distinguished into motive and accelerative or
retardive.

Motive Force, otherwise called Momentum, or Force of
Percussion, is the absolute force of a body in motion, &c ; •
and is expressed by the product of the weight or mass of
matter in the body multiplied by the velocity with which
it moves. But

Accelerative Force, or Retardive Force, is that which
respects the velocity of the motion only, accelerating or re-
tarding it ; and it is denoted by the quotient of the motive
force divided by the mass or weight of the body. So, if m
denote the motive force, and b the body, or its weight, and

/the accelerating or retarding force, then is/ as -.

Again, forces are either constant or variable.
Constant Forces are such as remain and act continually
the same for some determinate time. Such, for example,
is the force of gravity, which acts constantly the same upon
a body while it continues at the same distance from the
centre of the earth, or from the centre of force, wherever
that may be. In the case of a constant force f acting upon
a body b, for any time/, we have these following theorems;
putting

/= the constant accelerating force = f 4- b,
v = the velocity at the end of the time t,
s = the space passed over in that time, by the con-
stant action of that force on the body:
andg = 16-jSj- feet, the space generated by gravity in 1
second, and calling the accelerating force of gravity 1 ;

then is s = \tv = gftr = -— - ;
v=2gft = y = ,/tgfs;

Variable Forces are such as are continually changing
in their effect and intensity ; such as the force of gravity
at different distances from the centre of the earth, which
decreases in proportion as the square of the distance in-
creases. In variable forces, theorems similar to those above
may be exhibited by using the fluxions of quantities, and
afterwards taking the fluents of the given fluxipnal equa-
tions. And herein consists one of the great excellencies
of the Newtonian or modern analysis, by which we are
enabled to manage, and compute the effects of all kinds of
variable forces, whether accelerating or retarding. Thus,
using the same notation as above for constant forces, viz.
/the accelerating force at any instant, t the time a body

7 men.

The best and most effectual posture in a man, is that of
rowing ; when he not only acts with more muscles at once
for overcoming the resistance, than in any other position;
but also as he pulls backwards, the weight of his body as-
sists by way of lever. See Desaguliers's Exp.Philos. vol. 1,
pa. 241, where several other observations are made relative
to force acquired by certain positions of the body; from
which that author accounts for most feats of strength and
activity. See also a memoir on this subject by M. Lahire,
intheMem. Roy. Acad. 1729; or in Desaguliers's Exp. &c.
pa. 2b'7 &c, who has published a translation of part of it
with remarks.

Vol. I.

the velocity generated in that time, a the
in that time, and g = 1GTV feet; then is

space

*gf

Bg/i

**igfl;

S. <V _ r *■ u u

t ~ T ~ 2^7' J ~ *gi ~ %«' '

In these four theorems, the force/, though variable, is
supposed to be constant for the indefinitely small time } ;
and they arc to be used in all cases of variable forces, ;is
the former ones in constant forces ; viz, from the circum-
stances of the problem under consideration, deduce a ge-
neral expression for the value of the force/, at any inde-
finite time t ; then substitute il in one of these theorems,
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which shall be proper to the case in hand ; and the equa_
tion thence resulting will determine the corresponding va
lues of the other quantities in the problem. It is also to
be observed, that the foregoing theorems equally hold good
for the destruction of motion and velocity, by means of re-
tarding or resisting forces, as for the generation of the same
by means of accelerating forces. Many applicationsof these
theorems may be seen in my Mathematics, vol. 2.

There are many otherdenominations and kinds of forces;
such as attractive, central, centrifugal, &c, &c ; for which
see the respective words.

FORCES, the Equilibrium of, or the Composition and
Resolution of, is the conspiring and opposing of two or
more forces, so as to balance one another, or to keep
-in equilibrio the body or point against which they all
mutually act. — This equilibrium may be of two, or of
three, or of any other number of forces. In the case of
two forces only, it is manifest that they must be both
equal in quantity, and in directions exactly opposite. For,
if they be not directly opposite, the body or point of
meeting must move towards one side; and if they be not
equal, the greater will prevail over the less, and carry this
along with it, with the -difference of the two forces. Hence,
in all (he cases of equilibrium and compositions of three
or more forces, the practice must be to compose and re-
duce all the given forces into one, of a given quantity, and
having a given direction ; for then taking another, of an
equal quantity, and giving it the opposite direction, it will
be in equilibrio with all the former given forces, so com-
pounded together. — This will divide into two or three or
any other number offerees, as below.

Parallelogram or Triangle of Forces, is when two ob-
lique forces are given, to compound them, in order to
find a third force to balance them. So, if e '••.
ab, AC be the two given forces, acting at
the common point a : draw bd parallel
and equal to AC, and join ad ; then is
ad a force compounded of, and equiva-
lent to, the two given forces ab, ac.
Hence if, in ad produced, there be
taken ae = ad ; then aE will be the equivalent balancing
force to ad, and consequently to the two given forces ab,
ac — From the circumstance of the two given forces ab
and ac, or a B and bd, forming, with their equivalent com-
pound ad, a triangle abd, arises the phrase Triangle of
Forces. And from the circumstance of a parallelogram
abdc being formed by joining CD, that the expression
Parallelpgram of Forces has arisen. — From the foregoing
method of compounding two forces into one, will easily
follow the composition of any number whatever of forces
into one; viz, first compound two together, then that and
a 3d, next that and a 4th, and so on to the last, what-
ever their number may be; which may be symmetrically
called the Polygon of Forces, as below.

Polygon of Forces, is the compounding together any
number of forces into one. As, -r

the lour forces ab, ac, ad, ae,
all acting at the point a. From
the extremity B of the first, draw
BF parallel and equal to the 2d
ac ; then re parallel and equal to
the 3d ai> ; lastly on parallel and
equal to the 4th ae ; then ah, be-
ing jftined, i- the quantity and di-
rection of a single force, which is

compounded of, and equivalent to, the four first given
forces; and consequently, taking ai opposite and equal
to ah, then ai, balancing ah, will also balance all the
four given forces. For, by the principle of the Triangle of
forces, ab and ac, or bf, are compounded into .«; then
at and ad, or fg, arc compounded into ag ; lastly ag
and ae, or gh, are compounded into aii : and so on, if
there be ever so man}' forces.

Force of Running Water. This must be as the square
of its velocity, or as the height from which a heavy body-
must fall, to acquire that velocity; because the effect is
first as the velocity of each particle, and also as the num-
ber of them, which is as the velocity again.

FORCER, in Mechanics, is properly a piston without
a valve. For, by drawing up such a piston, the air is
drawn up, and the water follows; then pushing the piston
down again, the water being prevented from descending by
the lower valve, is forced up to any height above, by means
of a side branch between the two. — See the ways of making
these in Desagulier's's Exper. Philos. vol. 2, pa. l6l &c.
See also Clare's Motion of Fluids, pa. 60.

Forcing Pump, one that acts, or raises water, by a
forcing piston. See above.

FORELAND, or Foreness, in Navigation, a point of
land jutting out into the sea.

Foreland, in Fortification, is a small piece of ground
between the wall of a place and the moat; called also
Berme and Liziere.

FORE-STAFF, an instrument formerly used at sea, for
taking the altitudes of the heavenly bodies. It is so called,
because the observer, in using it, turns his face forward
or towards the object, in contradistinction to the back-staff,
with which he turns his back to the object. It is also called
the cross-staff, because it consists of several pieces set
across a staff. See Cn.oss-Staff.

A-^HZi

The fore-staff is formed of a straight square staff ae, of
about 3 feet long, having each of its lour sides graduated
like aline of tangents, and four crosses, or vanes, ff, ee,
dd, cc, sliding upon it, of unequal lengths, the halves of
which represent the radii to the lines of tangents on the
different sides of the staff. The first or shortest of these,
vanes, ff, is called the ten cross, or ten vane, and belongs
to the 10 scale, or that side of the instrument on which
the divisions begin at 3 degrees, and end at 10. The next
longer cross, ee, is called the 30 cross, belonging to that
side of the staff where the divisions begin at 10 degrees, and
end at 30, called the 30 scale. The third vane dd, is
tinned the 6'0 eros^', and belongs to that side where the
divisions begin at 20 degrees, and end at 6*0. The last or
longest vane cc, called the .':() cross, belongs to the side
when the divisions begin at 30 degrees, and end at 90.

The chief use of this instrument, is to take the height
of the sun, and stars, or the distance, between two stars:
and the 10, 30, 6*0, or flO cross is to be used, according as
the altitude is more or less; that is, if the altitude be less
than 10 degrees, the 10 cross is to be used; if above 10,
but less than 30, the 30 cross is to he used ; and so on.

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To observe an Altitude with a Fore-stuff. Apply tlie flat
end of the staff to the eye, and slide one of the crosses
backwards and forwards upon it, till over the upper end
of the cross be just seen the centre of the sun or star, and
over the under end the extreme horizon; then the degrees
and minutes cut by the cross on the side of the staff pro-
per to the vane in use, gives the altitude above the
horizon.

In like manner, for the Distance between two luminaries ;
the staff being set to the eye, bring the cross just to sub-
tend or cover that distance, by having the one luminary
just at the one end of it, and the other luminary at the
other end of it; and the degrees and minutes, in the dis-
tance, will be cut on the proper side of the staff, as before.

FORMULA, a theorem or general rule, or expression,
for resolving certain particular cases of some problem, &c.
So *s ■+■ ^d is a general formula for the greater of two quan-
tities whose sum- is s and difference d; and £s — -|rf is
the formula, or general value, for the less quantity. Also
^/ (dx — x2) is the formula, or general value, of the or-
dinate to a circle, whose diameter is d, and absciss x.

FORT, a small castle or fortress; or a place of incon-
siderable extent, fortified cifher by nature or art. The
fort is usually encompassed with a moat, rampart, and
parapet, to secure some high ground, or passage of a river ;
to make good or strengthen an advantageous post; or to
fortify the lines and quarters of a siege.

Field Fort, otherwise called Fortin, or Fortlet,and some-
times Sconce, is a small fort, built in haste, for the defence
of a pass or post; but particularly constructed for the de-
fence of a camp in the time of a siege, where the principal
quarters are usually joined, or made to communicate with
each other, by lines defended by fortins-and redoubts. Their
figure and size are various, according to the nature of the
situation, and the importance of the service for which
they are intended ; but they are most commonly made
square, each side about 100 toises, the perpendicular 10,
and the faces 25; the ditch about this fort may be 10 or 12
toises wide ; the parapet is made of turf, and fraised, and
the ditch pallisadoed when dry. There may be made a
covert-way about this fort, or else a row of pallisadcs
might be placed on the outside of the ditch. Some of these
are fortified with bastions, and some with demi-bastions.
— A fort differs from a citadel, as this last is erected to
command and guard some town ; and from a redoubt, as
it is closed on all sides, while the redoubt is open on oneside.

Royal Fort, is one whose line of defence is at least 26
fathoms long.

Star Fort, is a sconce or redoubt, constituted by re-
entering and saliant angles, having commonly from five to
eight points, and the sides flanking each other.

Forts are sometimes made triangular, only with half
bastions; or of various other figures, regular or irregular,
and sometimes in the form of a semicircle, especially when
they are situated near a river, or the sea, as at the entrance
of a harbour, for the convenience of firing at ships quite
around them on that side. In the construction of all forts,
it should be remembered, that the figure of fewest sides and
bastions, that can possibly answer the proposed defence,
is always to be preferred; as works on such apian are
sooner executed, and with less expense ; besides, fewer
troops will serve, and they are more readily brought to-
gether in case of necessity.

FORTIFICATION, callcdalso Military Architecture^
the art of fortifying or strengthening a town or other place,

by making certain works around it, to secure and defend it
from the attacks of enemies. Fortification has doubtless
been practised by all nations, and in all ages ; being at
first very rude and simple, and varying in its nature and
manner, according to the mode of attack, and the weapons
made use of. Thus, when villages and towns were first
formed, it was found necessary, for .the common safety, to
encompass them with walls and ditches, to prevent any
violence or sudden surprises from their neighbours. When
offensive and missive weapons came to be used, walls were
raised as a defence against the assailants, and look-holes or
loop-holes made through the same to annoy the enemy, by-
shooting arrows, &c, through them. But finding that as
soon as the enemy got close to the walls, they could no-
longer be seen or discomfited by the besieged, these added
square towers along the wall, at proper distances from
each other, so that all the intervening parts of the wall
might be seen and defended from the adjacent sides of the
towers. However, this manner of inclosing towns was-
found to be rather imperfect, because there remained still
the outer face of the towers which fronted the field, that
could not be seen and defended from any other part. To
remedy this imperfection, they next made the towers
round instead of square, as seeming better adapted both
for strength to resist the battering engines, and for being
defended from the other parts of the walk. Nevertheless, a
small part of these towers still remained unseen, and inca-
pable of being defended, "for which reason they were again
changed for square ones, as before, but with this difference,
that now they presented an angle of the square outwards
to the field, instead of a face or side ; and thus such a dis-
position of the works was obtained, as that no part could
be approached by the enemy without being seen and at-
tacked.

Since the use of gunpowder, it has been found neces-
sary to add thick ramparts of earth to the walls, and the
towers have been enlarged into bastions, as well as many
other things added, that have given a new appearance to
the whole art of defence, and the name of fortification, on
account of the strength afforded by it, which was about the
year 1500, when the round towers were changed into
bastions.

But notwithstanding all the improvements made in this
art since the invention of gunpowder, that of attacking is
still superior to it : the superiority of the besiegers' fire, to-
gether with the greater number of men,sooneror later com-
pels the besieged to submit. A special advantage was
added to the art of attacking by M. Vauban, at the siege
of Ath, in the year l6"97, viz, in the use of ricochet firing,
or at a low elevation of the gun, by which the shot was
made to run and roll a great way along the inside of the
works, to the great annoyance of the besieged.

The chief authors who have treated of fortification,
since it has been considered as a particular art, are the
following, and mostly in the order of time: viz, LaTreille,
Alghisi, Marchi, Pasino, Ramelli, Cataneo, aud Speckle,
who, as Mr. Robins says, was one of the greatest geniuses
that has applied to this art: he was architect to the city
of Strasburgh, and died in the year 1589 : he published a
treatise on fortification in the German language, which
was reprinted at Leipsic in 1736. Afterwards, Errard,
who was engineer to Henry the Great of France ; Ste-
vinus, engineer to the prince of Orange ; with Marolois,
the chevalier de Ville, Lorini, Coehorn, the count de
Pagan, and the marshal de Vauban ; which last two noble
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authors Lave contributed greatly to the perfection of the
art; besides Scheiter, Mallet, Belidor, BJondel, Muller,
Montalambert, &c. Also a list of several works on the
art of fortification may be added, as follows: viz. Melder's
Praxis Fortificatoria: Les Fortifications du Comte de
Pagan: L'Ingenieur Parfait du Sieur de Ville: Sturmy's
Architectura Militaris Hypothetical.: Blondel's Nouvelle
Manic-re de Fortifier les Places : the Abbe de Fay's Veri-
table Maniere de Bien Fortifier: Vauba'n's lngenieur
Francois : Cochorn's Nouvelle Fortification tant pour un
Terrain bas et humide, que sec et eleve : Alexander de
Grotte's Fortification: Donatus Roselli's Fortification:
Medrano's lngenieur Francois: the Chevalier de St. J u-
lien's Architecture Militaire: Lwnsberg's Nouvelle Ma-
niere de Fortifier les Places: an anonymous treatise in
French, called Nouvelle Maniere de Fortifier les Places,
tiree des Methodes du Chevalier de Ville, &c: Ozanam's
Traite de. Fortification: Memoires de l'Artillerie de Su-
rirey de St. Remy: Muller's treatises of Elementary and
Practical Fortification: Riou's Elem. of Fortification,
printed anonymously : and Montalambert's Fortification
Perpendiculairc.

Maxims in Fortification. From the nature and cir-
cumstances of this art, certain general rules, or maxims,
have been drawn, and laid down. These may indeed be
multiplied to any extent, but the principal of them are
the following: viz,

1. That the manner of fortifying should be accommo-
dated to that of attacking. So that no one manner can be
assured always to hold, unless it be assured that the man-
ner of besieging is incapable of being altered. Also, to
judge of the perfection of a fortification, the method of be-
sieging at the time when it was built must be considered.

2. All the parts of a fortification should be equally strong
on all sides, where there is equal danger; and they should
be able to resist the most powerful machines used in be-
sieging.

3. A fortification should be so contrived, as to be de-
fended with the fewest men possible: which consideration,
when well attended to, saves a great deal of expense.

4. That the defendants may be in the better condition,
they must not be exposed to the enemies' artillery ; but the
aggressors must be exposed to theirs.

5. All the parts of a fortification should be so disposed,
as that they may defend each other. In order to this,
every part ought to be flanked, i.e. seen sideways, capable
of being seen and defended from some other part ; so that
there be no place where an enemy can lodge himself, either
unseen, or under shelter.

6. All the champaign around must lie open to the de-
fendants; so that no hill or eminence be allowed, behind
which the enemy might shelter himself from the guns of the
furtification ; or from which he might annoy them, with his
own. Hence, the fortress is to command entirely the
place surrounding it ; and consequently the outworks must
all be lower than the body of the place.

7. No line of defence must exceed the point-blank
musket-shot, which is from 120 to 150 fathoms.

8. The more acute the angle at the centre, the stronger
is the place; as consisting of the more sides, and conse-
quently more defensible.

9. All the defences should be as nearly direct as possible.

10. The works that are most remote from the centre of
ihe place, ought always to be open to those that are nearer.

These arc the general laws and views of fortification. As

to the particular ones, or such as respect the several mem-
bers or parts of the work, they are given under those arti-
cles respectively.

Fortification is either theoretical or practical.
T/ieore/zcf^FoRTiFiCATioN, consists in tracing the plans
and profiles of a work on paper, with scales and compasses ;
and in examining the systems proposed by different authors,
to discover their advantages and defects.

Practical Fortification, consists in forming a pro-
ject of a work according to the nature of the ground, and
other necessary circumstances, tracing it on the ground, and
executing the project, with all the military buildings; as
magazines, storehouses, bridges, &c.

Again, fortification is either defensive or offensive.
Defensive Fortification, is the art of defending a town
that is besieged, with all the advantages the fortification of
it will admit. And

Offensive Fortification, is the same with theattack of
a place, being the art of making and conducting all the
different works in a siege, in order to gain possession of the
place.

Fortification is also used for the place fortified; or
the several works raised to defend and flank it, and keep
off the enemy. All fortifications consist of lines and
angles, which have names according to their various of-
fices. The principal lines are those of circumvallation, of
contravallation, of the capital, &c. The principal angles
are those of the centre, the flanking angle, flanked angle,
angle of the epaule, &c.

Fortifications are either durable or temporary.
Durable Fortification, is that which is built and in-
tended to remain a long time. Such are the usual forti-
fications of cities, frontier places, &c. And a

Temporary Fortification, is that which is erected on
some emergent occasion, and only for a short time. Such
are field-works, thrown up for the seizing and maintaining
a post, or passage; those about camps, or in sieges; as
circumvallations, contravallations, redoubts, trenches, bat-
teries, &c.

Again, fortifications are either regular or irregular.
Regular Fortification, is that in which the bastions
are all equal; or which is built in a regular polygon, the
sides and angles of which are usually about a musket-shot
from each other. A regular fortification, having the parts
all equal, has the advantage of being equally defensible ;
so that there are no weak places.

Irregular Fortification, is that in which the bastions
are unequal, and unlike; or the sides and angles not all
equal, and equidistant. In an irregular fortification, the
defence and strength being unequal, it is necessary to re-
duce the irregular shape of the ground, as near as may be,
to a regular figure: i. e. by inscribing it in an oval, instead
of a circle; so that one half may be similar and equal to
the other half.

Marine Fortification, is sometimes used for the art
of raising works on the sea-coast, &c, to defend harbours
against the attacks of shipping. — See a neat treatise on Ma-
rine Fortification, at the end of Robertson's Elements of
Navigation.

There are many modes of fortification, &c, that have
been much esteemed and used ; a small specimen of a com-
parative view of the principal of these, is represented in
plate xii, viz, those of Count Pagan, and Mess. Vauban,
Coehom, Belidor, and Blondel ; the explanation of which
is as follows :

FOR

[ »» J

FOS

I. Pagan's System.
a Half bastions.
B Ravelin and counterguard.
c Counterguards before the bastions.
d The ditch.
E The glacis.
g The place of arms.
■ h Retired flanks.
a Line of defence.

2. Vauban's System.
b Angle of the bastion, or flanked aifgle.
c Angle of the shoulder.
d Angle of the flank.
e Saliant angle.
f Face of the bastion.
g The flank.
h The curtain.
i Tenailles.
k Traverses in the covert way.

3. Coehom's System.

1 Concave flanks.

2 The curtains.

3 Redoubts in the re-entering angles

4 Traverses.

5 Stone lodgments.

6 Round flanks.

7 Redoubt.

8 Coffers planked on the sides, and above covered

overhead with a foot of earth.

4. Belidor's System.
i Cavaliers.

k Rams-horns, or Tenailles.

l Retrenchments within the detached bastions.

m Circular curtain.

N The ravelin.

p Lunettes with retired batteries.

q Redoubt.

r Detached redoubt.

s An arrow.

t Small traverses.

5. Blondel's System.
I Retired battery.

m Lunettes.

n Ravelin, with retired bastion.

o Orillons.

Another, or new method of fortification has lately been
proposed by M. Montalambert, called Fortification Pcr-
pendiculaire, because the faces of the works are made by a
series of lines running zigzag perpendicular to each other.

Profile of a Fortification, is a representation of a
vertical section of a work ; serving to show those dimen-
sions which cannot be represented in plans, and are ne-
cessary in the building of a fortification. The names and
dimensions of the principal parts are as follow (see fig. 8,
p. ix), where the numbers or dimensions are all expressed
in feet.

ab The level of the ground plane;

ac = 27;

cd = 18, and cw = l6§ ; also dn is parallel to ab;

de = 30;

ef = 2, l'G = 3, gh = 3, hi = 4£, il = 1£;

LK = 18, KM = 2£, NP = 36, NO = 5, PR = 7 ;

rs = 1, st = 12 or 18, ov = 9, vn = 120;

mz = 3, nu = 3, mc =z 30, cd = 2 ;

de = 3, ef = Z,fl = 4£, rg = 120, Ih = 1;

aw The interior talus or slope of the rampart ;

•we, or de, The terre-plein of ditto;

fg The talus, gh the upper part, of the banquette; '

hl The interior side of the parapet ;

lm The upper part of ditto ;

N The cordon of 1 foot radius ;

np The depth, and pre the breadth of the ditch ;

oq Interior side of revetement;

nr The scarp or exterior side of ditto ;

st The depth of the foundation ;

yz Revetement of the parapet;

nu The counterscarp;

mc The covert-way;

ce Talus of the banquette ;

ef The upper part of ditto;

fh Parapet of the covert-way ;

hg The glacis.

Other sections are at fig. 2, pi. 14.

Fortified Place, a fortress, or fortification, i.e. a
place well flanked, and sheltered with works.

Fortin, or Fortlet, a diminutive of the word Fort,
meaning a small fort, or sconce, called also Field Fort.

Star Fort in, is that whose sides flank each other, &c.

FORTRESS, the same as Fort, or a Fortification.

FOSTER (Samuel), an English mathematician, was
born in Northamptonshire, and admitted a sizer at Ema-
nuel-college Cambridge in I6'l6. He took the degree of
bachelor of arts in lfjiy, and of master in 1623. He ap-
plied early to the mathematics, and attained a great pro-
ficiency in them, of which he gave the first specimen in
1624, in a treatise on the use of the quadrant.

On the death of Mr. Gellibrand, our author was chosen
to succeed him, in 1636", as astronomy-professor in Gre-
sham-college, London. He quitted it again however the
same year, though for what reason does not appear, and
was succeeded by Mr.Mungo Murray, professor of philo-
sophy at St. Andrews in Scotland. But this gentleman
marrying, the professorship again became vacant, and Mr.
Foster was re-elected in 1641.

Mr. Foster was one of those gentlemen who held private
meetings for cultivating philosophy and useful knowledge,
which afterwards gave rise to the Royal Society. In 1646,
Dr. Wallis, who was one of those associating gentlemen,
received from Mr. Foster a theorem De triangulo spherico,
which he published in his Mechanics. Neither was it only
in this branch of science that he excelled, but he was also
well versed in the ancient languages; as appears from his
revising and correcting the Lemmata of Archimedes, which
had been translated into Latin from an Arabic manuscript
by Mr. Jphn Greaves. He made also several curious obser-
vations on eclipses of the sun and moon, in various places;
and was particularly noted for inventing, as well as improv-
ing, astronomical and other mathematical instruments.
After a long declining state of health, he died at Gresham-
college in 16*52.

His printed works are as follow ; of which the first two
articles were published before his death, and the rest of
them after it. >

1. The Description and Use of a small Portable Qua-
drant, for the easy finding the Hour of Azimuth ; 4to, 1624.
Originally published at the end of Gunter's Description of
the Cross-Staffe, as an appendix to it.

2. The Art of Dialling ; 4to, 1638. Reprinted, with
additions, in 1675.

3. Posthuma Fosteri ; byWingate; 4to, 1652.

4. Four Treatises of Dialling; 4to, 1654.

5. Miscellanies, or Mathematical Lucubrations. Pub-

FOU

[ 542 ]

FOU

liilied by John Twysden, with additions of his own ; and
an appendix by Ley bourne ; folio, 16"59.

6. The Sector altered, and other Scales added, &c.
Published by Leybourne in l66l, in 4to.

There have been two other persons of the same name,
who have published some mathematical pieces. The first
was,

FOSTER (William), a disciple of Mr. Oughtred,
and afterwards a teacher of the mathematics in London.
He distinguished himself by a book, which he dedicated
to Sir Kenelm Digby, entitled., The Circles of Proportion,
and the Horizontal Instrument, &c; 4to, 1633. — The
other was,

FOSTER (Mark), who lived later in point of time than
either of the other two, and published a treatise entitled,
Arithmetical Trigonometry: being the Solution of all the
usiral Cases in Plain Trigonometry by Common Arithme-
tic, without any tables whatsoever. 12mo, l6y0.

FOUNDATION, that part of a building which is un-
derground : or the mass which supports a building, and
on which it stands : or it is the coffer or bed dug below
the level of the ground, to raise a building on.

FOUNTAIN, in Philosophy, a spring or source of
water rising out of the ground. See Spring.

Fountain, or Artificial Fountain, in Hydraulics, a
machineor contrivance by which water is violently spouted
or thrown up ; called also a jet d'eau.

There are various kinds of artificial fountains, but all
formed by a pressure of one kind or other, on the water
&c, viz, either the pressure, or weight of a head of water,
or the pressure arising from the spring and elasticity of the
air, &c. When these are formed by the pressure of a
head of water, or any other fluid of the same kind with the
fountain, or jet, &c, then will this spout ufk nearly to the
same height as that head, abating only a little for the re-
sistance of the air, with that of the adjutage &c, in the
fluid's rushing through ; but, when the fountain is pro-
duced by any other force than the pressure of a column of
the same fluid with itself, it will rise to such a height as
may be nearly equal to the altitude of a column of the
same fluid whose pressure is equal to the given force that
produces the fountain.

To Construct an Artificial Fountain, playing by the pres-
sure of the water. This is to be effected by making a close
connexion between a head or elevated piece of water, and
the lower place, where the fountain is to play ; which
may be done in this manner: Having a head of water,
naturally, or, for want of such, make an artificial one,
raising the water by pumps, or other machinery : from
this head convey the water in close pipes, in any direction,
down to the place where the fountain is to play ; and
there let it issue through an adjutage or small hole, turned
upwards, by which means it will spout up nearly as high as
the head of the water it comes from, as above mentioned.
To Construct an Artificial Fountain, playing by the spring
or elasticity of the air. A vessel proper for a reservoir, as
aij, fig. 4, plate x, is provided either of metal, or glass,
or the like, ending in a small neck c, at the top : through
this neck is put a tube cd, till the lower end come near
the bottom of the vessel, this being about half full of
water. The neck is so contrived, that a syringe, or con-
densing pipe, may be screwed on the tube; by means of
which a large quantity of air may be intruded through the
tube into the water, out of which it will disengage itself,
and emerge into the vacant part of the vessel, and lie over
the surface of the water cd.

Now, the water in the vessel being thus pressed by the
air, which is, for ex. double the density of the external
air; and the elastic force of air being proportional to its
density, or to its gravitating force, the effect will be the
same, as if the weight of the column of air, over the sur-
face of the water, were double that of the column pressing
in the tube; so that the water must be forced upwards
through the tube, when the syringe is removed, with a
force equal to the excess of pressure of the included air,
above that of the external, that is, in this case, with a force
equal to the pressure of an entire column of the atmo-
sphere ; which being equal to the pressure of a column of
33 or 34 feet of water, it follows that the fountain will
play to nearly 33 or 34 feet high.

These aerial or aquatic fountains may be applied in
different ways, so as to exhibit various appearances ; and
from these alone arises the greatest part of artificial water-
works: even the engine for extinguishing fire, is a foun-
tain playing by the force of confined air.

A Fountain spouting the water in yarious directions.
Suppose ab the vertical tube, or spout, in which the wa-
ter rises, (fig.5, pi. x) ; into this let several other tubes
be fitted ; some horizontal, others oblique, or inclining,
or reclining, &c, as at e, ii, l, n, p, &c. Then, as all
water retains the direction of the aperture through which
it comes, that issuing through a will rise perpendicularly;
and the rest will tend different ways, describing arches of
different magnitudes.

Or thus : Suppose the vertical tube ab (fig. 6) through
which the water rises, to be stopped at the top, as in a ;
and, instead of pipes or cocks, let it be only perforated
with small holes all around, or only half round its sur-
face : then will" the water spout out, in all directions,
through the little holes, to different distances.

And hence, if the tube ab be about the height of a
man, and having a turn-cock at c; on opening this cock,
the spectators will be sprinkled unexpectedly with a
shower.

Fouktain playing by drawing the breath. Suppose ab
(fig. S, plate viii), a globe of glass, or metal, in which is
fitted a tube cd, having a small orifice in c, and reaching
almost to d, the bottom of the globe. Then if the air be
sucked, or drawn with the mouth, out of the tube CD, and
the orifice c be immediately immerged under cold water,
the water will ascend through the tube into the sphere.
Thus proceeding, by repeated exsuctions, till the vessel be
above half full of water ; then applying the mouth to c,
and blowing air into the tube, on removing the mouth,
the water will spout forth. — Or, if the globe be put into
hot water, the air being thus rarefied, will make the water
spout as before.

And this kind of fountain is called Pila Hcronis, or
Hero's Ball, from the name of its inventor.

Fountain, whose stream ra/sts and plays a brass ball.
Provide a hollow brass ball a (fig.<), pi. viii), made very
thin, that its weight may not be too great for the force of
the water; and let the tube bc, through which the water
rises, be exactly perpendicular to the horizon. Then the
ball, being laid in the bottom of the cup or bason b, will
be carried up by the stream, and sustained at a considera-
ble height, playing a little up and down.

Fountain which spouts water inform of a shower. To
the tube in which the. water is to rise, lit a spherical or
lenticular head as (fig. 1, pi. alii) made of a plate of metal,
and perforated at the top with a great number of small
holes. The water rising violently towards ab, will be

F O U

[ 543 ]

F O U

there divided into innumerable little threads, and after-
wards broken, and dispersed into the finest drops.

Fountain which spreads the water infoim of a table'
cloth. To the tube ab (fig. 2, pi. xiii) solder two spherical
segments, c and d, almost touching each other, with a
screw e, to contract or amplify the interstice or chink, at
pleasure. Some choose to make a smooth and even cleft,
in a spherical or lenticular head, fitted upon the tube.
The water spouting through this chink or cleft will expand
itself like a cloth. — And thus, the fountain may be made
to spout out in the figure of men, or other animals.

Fountain, which, when it has done spouting, may be
turned like an hour-glass. Provide two glasses, a and b,
(fig. 3, pi. xiii) to be so much the larger as the fountain is to
play the longer, and placed at so much the greater distance
from each other as the water is desired to spout the higher.
Let cde bi a crooked tube, furnished in e with a jet ; and
ghi another bent tube, furnished with a jet in i : of and
kl are to be other lesser tubes, open at both ends, and
reaching near the bottom of the vessels a and b, to which
the tubes CD and gh are likewise to reach.

Now if the vessel a be filled with water, it will descend
through the tube CD ; and it will spout up through the
jet e, by the pressure of the column of water CD. But
unless the pipe G f were <-pen at G, to let the air run up to
F, and press at the top of the surface of the water in the
cavity a, the water could not run down and spout at e.
After its fall again, it will sink through the little tube kl,
into the vessel b, and expel the air through the tube gi.
At length, when all the water is emptied out of the vessel
a, by turning the, machine upside-down, the vessel b will
be the reservoir, and make the water spout up through the
jet i, the pipe kl supplying b with air to let the water
descend in the direction ghi.

Hence, if the vessels a and b contain just as much wa-
ter as will be spouted up in an hour's time, we shall have
a spouting clepsydra, or water-clock ; which may be gra-
duated or divided into quarters, minutes, &c.

Fountain of Command. This depends on the same
principles with those of the former : cae (fig. 4, pi. xiii) is
a vessel of water secured against the entrance of the air,
except through the pipeGF, when the cock c, by which
it is filled, is shut. There is another pipe edhb, going
from the bottom of the water to the jet b in the bason db ;
but this is stopped by the cock h. At the lowest part
of the bason db, there is a small hole at i, to let tlie wa-
ter of the bason db run into the bason gh under it ; there
is also a small triangular hole or notch, in the bottom of
the pipe fg, at g. Turn the cock h, and the fountain
will play for some time, then stop, then play again alter-
nately for several times together. When those times of
playing and stopping are known before-hand, you may
command the fountain to play ors'.op; whence its name.
The cause of this phenomenon is as follows : the water
coining down the pipe ediib, would not come out at b, if
the air ss, above the water, were not supplied as it di-
lated: now it is supplied by the pipe gf, which takes it
in at the notch G, and delivers it out at r ; but after some
time the water, which was spouted out at B, falling down
into the bason db, rises high enough to come above the
notch g, which stops the passage of the air ; so that the
air W, above the water in the vessel cae, wanting a sup-
ply, cannot sufficiently press, and the fountain ceases
playing. — Rut when the water of nn has run down into
the lower bason gh, through the hole i, till it falls below

the top of the notch g, the air runs up into the upper rer
ceptacle, and supplies that at ss, and the fountain plays
again. This is seen a little before-hand, by a skin of wa-
ter on the notch g, before the air finds a passage, and then
yen may command the fountain to play. It is evident
that the hole i must be less than the hole of the jet, or
else all the water would run out into the lower bason,
without rising high enough to stop the notch G.

Fountain that begins to play upon the lighting of candles,
and ceases as they go out. Provide two cylindrical vessels,
ab and cd, (fig. 7, pi. xiii,) connect them by tube;, open
at both ends, kl, be,&c, so that the air may descend out
of the higher into the lower: to the tubes solder candle-
slicks, h, &c, and to the hollow cover of the lower vessel
ce, fit a little tube or jet, fg, furnished with a cock o,
and reaching almost to the bottom of the vessel-. In o let
there be an aperture, furnished with a screw, by which
water may be poured into CD. Then, on lighting the
candles H, &c, the air in the contiguous tubes becoming
rarefied by the heat, the water begins to spout through gf.

By the same contrivance may a statue be made to shed
tears on the shining of the sun, or on lighting a candle,
&c : all that is here required, being only to lay tubes from
the cavity where the air is rarefied, to some other cavities
placed near the eyes, full of water.

A Fountain by the Rarefaction of the Air, may be made
in the following manner: Let ab and cd, fig. 5, pi. xiii, be
two pipes fixed to a brass head c, made to screw into a glass
vessel e, which having a little water in it, is inverted till the
pipe's are screwed on; then reverting it suddenly, so as to put a,
the lower end of thespouting pipe ab, into a jar of water a,
and the lower end of the descending pipe CD, into a receiv-
ing vessel d, the water will spout up from the jar a into the
tall glass vessel e, from which it will go down at the mouth,
c, through cd, into the vessel D, till the water is wholly
emptied out of a, making a fountain in e, into d. The rea-
son of the play of the fountain is this i the pipe cd, being
2 feet 9 inches long, lets down a column of water, which
rarefies the air 1-1 2th part in the vessel e, where it presses
against the water spouting at b with 1- 1 2th of the force by
which the water is pushed up at the hole a, by the pressure
of the common air en the water in the vessel a ; so that the
water spouts up into e, when the air is rarefied 1- 12th,
with the difference of the pressure of the atmosphere, and
the forementioned rarefied air; i.e. of 33 to 2|-, or of 12 to
1. This would raise tire water 2 feet 9 inches ; but the length
of the pipe a, ofp inches, being deducted, the jet will only
rise 2 feet. This, says Desaguliers, may be called a syphon
fountain, where ab is the driving leg, and CD the issuing
leg.

Fountain of Hero of Alexandria, so called; because it
was contrived by him. In the second fountain above de-
scribed, the air is compressed by asyringe; in this, (see fig.
6, pl.xiii) the air, beingonly compressed by theconcealcd
fall of water, makes ajet,which, after some continuance, is
considered by the ignorant as a perpetual motion; because
they imagine that the same rater which fell from the jet
rises again. The boxes cf. and dyx, being close, we see
only the bason abW, with a. hole at w, intowhich the water
spouting at b falls; but that water does not come up
again; for it runs down through the pipc-wx into the box
dyx, whence it drives out the air, through the ascending
pipe yz, into the cavity of the box ce, where, pressing up-
on the water that is in it, it forces it nut through the spoilt-
ing-pipe ob, as long as there is any water in ce; so that this

F R A

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hav-

]

F R A

whole play is only while the water contained in ce, hav-
ing spouted out, tails down through the pipe \vx, orof the
boxes c e and r> Y above one another ; the height of the wa-
ter, measured from the bason abw to the surface of the
water in the lower box dyx, is always equal to the height
measured from the top of the jet to the surface of the water
in the middle cavity at ce. Now, since the surface ce is
always falling, and the water in dy always rising, the height
of the jet must continually decrease, till it is shorter by
the depth of the cavity ce, which is emptying, added to
the depth of the cavity dy, which is always filling; and
when the jet is fallen so low, it immediately ceases. The
air is represented by the points in this figure.

To prepare this fountain for playing, which should be
done unobserved, pour in water at w, till the cavity dxy
is filled ; then invert the fountain, and the water will run
from the cavity dxy into the cavity CE, which may be
known to be full, when the water runs out at b held down.
Set the fountain up again, and, to make it play, pour in
about a pint of water into the bason abw ; and as soon as
it has filled the pipe wx it w ill begin to play, and continue
as long as there is any water in ce. You may then pour
back the water left in the bason abw, into any vessel, and
invert the fountain, which being set upright again, will be
made to play, by putting back the water poured out into
abw; and so on as often as you please.

SpoutingFovtiT a in, or Jet d'Eau, is any fountain whose
water is darted forth impetuously through jets, or ajutages,
and returns in form of rains, nets, folds, or the like.

Fount ain-Fct, is a pen contrived to contain a quan-
tity of ink, and let it flow very gently, so as to supply the
writer a long time without the necessity of taking fresh

ink.

The fountain-pen, represented fig. S, pi. mi, consists ot gether : as If, or 12f.
divers pieces of metal, f, g, h, the middle piece f carrying The artihmetic of fractions consists in the reduction, ad

ger is divided; and the upper number 3, is called the Nu-
merator, and shows how many of those equal parts are con-
tained in the fraction. Hence it follows that as the nume-
rator is to the denominator, so is the fraction itself, to the
whole of which it is a fraction ; or as the denominator is to
the numerator, so is the whole or integer, to the fraction :
thus, the integer being denoted by 1, as 4 : 3 : : 1 : J the
fraction. — And hence there may be innumerable fractions
all of the same value, as there may be innumerable quan-
tities all in the same ratio, viz, of 4 to 3 ; such as 8 to 6", or
12 to 9, &c. So that if the two terms of any fraction, i.e. the
numerator and denominator, be either both multiplied or
both divided by any number, the resulting fraction will
still be of the same value : thus, |- or -| or -^ or 4-f &c, are
all of the same value with each other.

Fractional expressions are usually distinguished into
proper and improper, simple and compound, and mixt
numbers.

A Proper Fraction, is thatwhose numerator is less than
the denominator; and consequently the fraction is less than
the whole or integer; as ^.

An Improper Fraction, is when the numerator is either
equal to, or greater than, the denominator; and conse-
quently the fraction either equal to, or greater than, the
whole integer, as ±, which is equal to the whole; or^,
which is greater than the whole.

Simple Fractions, or Single Fractions, are such as
consist of only one numerator, and one denominator; as
§, or i, or 4-f .

Compound Fractions are fractions of fractions, and
consist of several fractions, connected together by the word
of: as f of J, or 4 of f , of |-.

A Mixt Number consists of an integer and a fraction to-

the pen, which is screwed into the inside of a small pipe;
and this again is soldered into another pipe of the same size
as the lid°Cr ; in which lid is soldered a male screw, for
screwing on the cover ; as also for stopping a little hole at
the place, and hindering the ink from passing through it :
at the other end of the piece f isa small pipe, on the out-
side of which may be screwed the top cover H. A portc-
craion goes in the cover, to be screwed into the last-men-
tioned pipe, to stop the end of the pipe into which the ink
is to be poured by a funnei.

To use the pen, the cover G must be taken off, and the
pen a little shaken, to make the ink run more freely.

FOURTH, in Music, one of the harmonic intervals, or
concords. It consists in the mixture of two sounds, which
are in the ratio of 4 to 3; i.e. of two sounds produced by
chords, whose lengths are to each other as 4 to 3.

FRACTION, or Broken Numher, in Arithmetic and Al-
gebra, is a part, or some parts, of another number or quan-
tity considered as a whole, but divided into a certain num-
ber of parts; as 3-4ths, which denotes 3 parts out of 4, of
any quantity. Fractions are usually divided into vulgar,
decimal, duodecimal, and sexagesimal. For the last three
sorts, see the respective words.

Vulgar Fractions, called also simple Fractions, are
usually denoted by two numbers, the one set under the
other, with a small line between them : thus -J denotes the
fraction three-fourths, or 3 parts out of 4, of some whole
Considered as divided into 4 equal parts.

The lower number 4, is called the Denominator of the
fraction, showing into how many parts the whole or intc-

dition, subtraction, multiplication, and division of them.

Reduction o/Fractions is of several sorts; as 1. To re-
duce a given whole number into a fraction of any given deno-
minator. Multiply the given integer by the proposed de-
nominator, and the product will be the numerator. Tims,
it is found that 3 = £, and 5 = z~.

If no denominator be given, or it be only proposed to ex-
press the integer fraction-wise, or like a fraction ; set 1 be-
neath it, for its denominator. So3=-f-,and 5 = -f, and 7=t-

2. To reduce a given fraction to another fraction equal to
it, thai shall have a given denominator. Multiply the nu-
merator by the proposed denominator, and divide the pro-
duct by the former denominator, then the quotient set over
the proposed denominator will form the fraction required.
Thus, if it be proposed to reduce -|- to an equal fraction
whose denominator shall be S ; then 3 x 8 = 24, and
24 -^ + = 6 the numerator, so that f is the fraction sought,
being = f , and having S for its denominator.

3. To abbreviate, or reduce fractions to lotvcr terms.
Divide their terms, i.e. the numerator and denominator, by
any number that will divide them without a rei lainder, so
shall the quotients be the corresponding terms of a new
fraction, equal to the former, but in smaller numbers.
In like manner abbreviate these new term! again and so
on till there be no number greater than ; that will ivide
them without a remainder, and then the fraction is said to
be in its least terms. Thus, to abbreviate-^; first divide
both terms by 5, and the fraction becomes-^ ; next divide
these by 3, and it becomes £ : so that ££= -/-i =4, '" its
least terms.

F R A

[ 545 ]

FRA

To reduce a fraction at once to its lowest terms; divide
both numerator and denominator by their greatest com-
mon measure : to find which, see the article Common
Measure.

It may further be observed, that a proper fraction is" in
its lowest terms, when the numerator and denominator
are not both divisible by their difference, or by some sub-
multiple of their difference, without a remainder.

4. Toreduce fractions toollierequivalentonesoftliesame de-
nominator. Multiply each numerator, separately taken,
by all the denominators except its own, and the products
will be the new numerators; then multiply all the dencr-
minators continually together, for the common denomi-
nator, to these numerators. Thus, f and f reduce to4f
and -if; and f, |, and ■£ reduce to £§, f-§> and £§•

5. To Jind the value of a fraction, in the
known parts of its integer. Multiply always
thenumerator by the numberof parts of'the
next inferior denomination, and divide the '") 1!^0 (11*

9

20

16

20
l6_

4
12
) 4iT( 3d
48

products by the denominator. Do the same
with the remainders through all the de-
nominations. So to find the value of —^ of
a pound sterling ; multiply 9 by 20 for shil-
lings, and dividing by 16", gives 11 for the
shillings ; then multiply the remainder 4 by c
12 pence, and dividing by 16 gives 3 for
pence : so that 1 Is. 3d. is the value of -^l.

6. To reduce a mixt number to an equivalent improper
fraction. Multiply the integer by the denominator, and
to the product add the numerator, for the new numerator,
to be set over- the same denominator as before. Thus 3-|
becomes 2£.

7 . To reduce an improper fraction to its equivalent whole
or mixed number. Divide the numerator by the denomina-
tor ; so shall the quotient be the integral part, and the re-
mainder set over the denominator will form the fractional
part of the equivalent mixed number. Thus *■£ reduces
to 3$, and *£ = 8.

8. To reduce a compound fraction to a simple one. Mul-
tiply all the numerators together for the numerator, and
all the denominators together for the denominator, of the
simple fraction sought. Thus, ■£ of | = $, and § of £ of -J

= TTS'

To reduce a vulgar fraction to a decimal. See Deci-
mals. And for several other particulars concerning Re-
duction, as well as the other operations in fractions; see
my Arithmetic.

Addition of Fractions. First reduce the fractions to
their simplest form, and reduce them also to a common de-
nominator, if their denominators are different; then add
all the numerators together, and set the sum over the com-
mon denominator, for the sum of all the fractions required.
Thus, |. -+- -|- = ■§•= 1 ;
And T -h x = -j-g. ■+■ |-|. = l-j. = 1-j-g.. i

Subtraction of Fractions. Reduce the fractions the

Thus f X * = _8T;

And I x X x j , _ ioi ii j ,

T n- -J t*7 — 48° — ^ — ^

lo Divide Fractions. Divide the numerator by the
numerator, and the denominator by the denominator, if
they will exactly divide. Thus, ^ -=- f = f .

But if they will not divide without a remainder, then
multiply the dividend by the reciprocal, of the divisor,
that is, by the fraction obtained by inverting or changing
its terms. Thus, £. -j- $ = A * f = ¥;•

Algebraic Fractions, or Fractions in Species, are
exactly similar to vulgar fractions, in numbers, and all
the operations are performed exactly in the same way;
therefore the rules need not be repeated, and it may be
sufficient here to set down a few examples to the foregoing
rules. Thus, 5

1. The fraction-— abbreviates to—.

6a3 — gaar

3.

! -+- Sax ia -

' — a*x + ax' — x3

See Common Measure.

mon denominator.

, a, c ad + be

5. - + - =-

h d

-, by dividing by 3a.

, by dividing by a — x.

6.

X

7. % x

Id
s — bx

Continued Fraction, is used for a fraction whose de-
nominator is an integer with a fraction, which latter frac-
tion has for its denominator an integer and a fraction,
and the same for this last fraction again, and so on, to
any extent, whether supposed to be infinitely continued,
or broken off after any number of terms. Euler, Analys;
Inf. vol. 1, p. 295.

orf + ,

'As I 7 -)- i

T -t- &C

Or, using letters instead of numbers,

d + « + &c. d + &c.

When these series are not far extended, it is not diffi-
cult to collect them by common arithmetic.

Lord Brounker, it appears, was the first who consi-
dered continued fractions, or at least, who applied them

same as for addition; then subtract the one numerator t0 tnt3 quadrature of curves, in Wallis's Arith. Infin. prop

J91, vol. 1, pa. 46"9 &c, where this author explains the
manner of forming them, giving several numeral examples,
in approximating ratios, as well as the general series

from the other, and set the difference over the common Aq-
nominator.

To Multiply Fractions together. Reduce them all to
the form of simple fractions, if they be not so; then mul-
tiply all the numerators together for the numerator, and
all the denominators together for the denominator of the
product sought.

Vol. I.

— b
a — c

y -j &c, as he denotes it.

Iluygens also used it for the like purpose, viz, to approxi-
mate the ratios of large numbers, in his Descrip. Autom.
4 A

FRA

[ 546 ]

FRA

Planet.in Opor. Reliq. pa. 173 &c, edit. Amst. 1728. And
a special treatise on Continued Fractions was given by
Euler, in his Analys. Infin. vol. 1, pa. 295 &c.

This subject is perhaps capable of much improvement,
though it has been rather neglected, as very little use has
been^made of it, except, by those authors, in approxima-
ting to the value of fractions, and ratios, that are expressed
in large numbers; besides a method of Goniometry by ,
De Lagny, explained in the Introduction to my Loga-
rithms, pa. 78; as also some use I have made of it in
summing very slowly converging series, in my Tracts, vol.
1, pa. 203.

As to the reducing of common fractions, and ratios,
that are expressed in large numbers, to continued frac-
tions, it is no more than the common method of finding
the greatest common measure of those two numbers, Ly
dividing the greater by the less, and the last divisor always
by the last remainder ; for then the several quotients are
the denominators of the fractions, the numerators being
always 1 or unity. Thus, to find approximating values
of the fraction ^oooooooo'01 to the ratio of 3 141 5926535
to 10000000000, being the ratio of the circumference of
a circle to its diameter, by means of a continued fraction ;
or to change the said common fraction to a continued
fraction : Dividing the greater term always by the less,
"the same as to find the greatest common measure of the
said numbers or terms, the several quotients will be 3, 7,
15, 1, 292, 1, 1, &c, which, after the first, will be the
denominators, to the common numerator 1 ; and there-
fore the said fraction will be changed into this continued
fraction,

3 + t + 1

" + t+ ,

T5~Z 4- I

+ T + &C.

Hence, stopping at any part of these single fractions, one
after another, will give several values of the proposed ra-
tio, all successively nearer and nearer the truth, but al-
ternately too great and too little. So, stopping at j-, it is
3| = y- = 3-1428571, too great, or 22 to 7, the ratio of
the circumference to the diameter as given by Archime-
des. Again, leaving off at r\, it is

3-i- = 3-^sV = Ui = 3-141509 &c, too little.
But stopping at -J.,

it is 3 — ^—7 = 3— = 3tjtt = |J| (the ratio of Me-
tius) = 3-1415929 &c, which is rather too great. And
so on, always nearer and nearer, but alternately too great
and too little.

And, in like manner is any algebraic fraction thrown
into a continued fraction. As the fraction

ojSj/J + af>d 4- art ,.,,.'.,., ..

which being in like manner di-

/»

b' c' d'

On the other hand, any continued fraction being «iven
its equivalent common fraction will be found, by beoin-
ning at the last denominator, or lowest end of the given
continued fraction, and gradually collecting the fractions
backwards, till we arrive at the first, when the whole will
thus be brought to one common fraction; as was done
above in collecting the fractions

Y5T -t- &C.

And in like manner the continued fraction

cals of the former, and so will become -,

0'

and

hence the proposed common fraction is equal to this ter-
minate continued,

collects into the fraction

a/SyJ + afid 4- art

*&y$ 4- afid + art + lyi + bd'

When the given continued fraction is an infinite one
collect it successively, first one term, then two together,
three together, &c, till the sum is sufficiently exact! C%
if these collected sums be found to converge too slowly to
the true value, having collected a fewTof the terms into
successive sums, these being alternately too crlcat and too
little, the true value will be found as near as you please by
the method of arithmetical means, explained inmy Tracts
vol. 1, Tract 8, pa. 176*.

When the denominators of a continued fraction recur
in any certain order, as in the following

«/SyJ + a/id + art + lyi + W'

a 16 y $

vided, the quotients are -, -, -, -; which single frac-

' ' a b c d &

tions being considered as denominators to other fractions
whose common numerator is 1, these will be the recipro-

v/

y/~

&C. ■*■ Va 4- &c

its value may be computed by making it first equal to x,
and then solving the equation : thus, suppose the value
of the first was required ; then put the whole = x, and

we get — = x, or — = .r : hence r -+■ 1

' qr 4- qx + 1

1 +■ ~r~

1 r + x

= qrx + qx" 4- x, and $? + w =0, from which x,

and consequently the value of the fraction, may be deter-
mined, by solving the quadratic.

In the same manner may the values of other quanti-
ties, which run on in infinitum, be found if the factors
recur.

Example. Required the value of

J a + ^h

</>

a 4- Vb 4- &c. in infinitum.

Let

V

!~V\

by squaring both sides a

J

- V a 4- </ I,
J /> +

&c = x j then

v;

Vb 4- &c

ss x1; and

v/"

Vb •+■ &c = x" — a; there-

fore b + Sra h- Vb '■*■ &P = -r' ~ 2«x2 4- a
I) -\- x = x'1 — Qaxz ■+■ «*» hence x* — 2ax'i — x 4-
b = 0, whence the value of X may be found.
If the letters be all equal, as

+J a 4- a 4- V a

Va &

equation, so shall xx = a

C. Put it = x, then square

F R A

■V a -+-

[ 547 ]

F R A

a + V a !+. •%/

a &c;

hence Xs — a =

«y a ■+- V a +Vo &c) = a- the first value, because
it reverts to the same quantity; and from this quadratic
equation we find x = i -^^/(a + |).
Again, to find the value of

/ v/ 7/===

squaring, we have x

a x \/6 &c : putting it = x, and
= a ^/ b%/ aV b ike, or =

J

ywbs

b a*/ b &c ; then this squared, and divided by b,

gives ~ = (VaWav'i^) = x; or x4 = a'24x,
or x3 = cfb, and x = \/ orb. — If b = a, or the radicals
be always the same letter a, then x = J/a3 = a.

In like manner, „j V v

a 6 ev aV/ iv'f &c is =
v/a4i1c, or = J/a7 = a, when b and c are taken == a.

Vanishing Fractions. Such fractions as have both
their numerator and denominator vanish, or equal to 0,
at the same time, may be called Vanishing Fractions. We
are not to conclude that such fractions are equal to no-
thing, or have no value ; for that they have a certain de-
terminate value, has been shown by the best mathemati-
cians. The idea of such fractions as these, first origi-
nated in a very severe contest among some French mathe-
maticians, in which Varignon and Rolle were the two
chief opposite combatants, concerning the then new or
differential calculus, of which the latter gentleman was a
strenuous opponent. Among other arguments against it,
he proposed an example of drawing a tangent to certain
curves at the point where the two parts cross each other;
and as the fractional expression for the subtangent, by
that method, had both its numerator and denominator
equal to 0 at the point proposed, Rolle considered it as an
absurd expression, and as an argument against the method
of solution itself. The. seeming mystery however was soon
explained, and first of all by John Bernoulli. See an ac-
count of this controversy in Montucla, Hist. Math. vol. 2,
pa. 366", 1st ed.

Since that time, such kind of fractions have often been
contemplated by mathematicians. As, by Maclaurin, in
his fluxions, vol. 2, pa. 6QS : Saunderson, in his Algebra,
vol. 2, art. 4()'9 : De Moivre, in Miscel. Anal. pa. 105 :
Emerson, in his Algebra, pa. 212: and by many others.
The same fractions have also proved a stumbling-block to
more mathematicians than one, and the cause of more vio-
lent controversies: witness that between Powell and Wa-
ring, at the time when Mr. Maseres was a competitor of
the latter for the professorship at Cambridge. In the
specimen of a work published on occasion of that com-

petion, by Waring, was the fraction-— — , whichhesaid

became 4 when p was = 1. This was struck at by
Powell, as absurd, because he said whenjj=l, then the

P — p" 1 — i o ,., ,, ,,,

traction — = = -. Waring replied that

i — p 1-1 o a '

■-"^- is =p •+■ pr +p3 + p' (by common division)

— 1 ■+■ 1 -i- 1 -+- I = 4, when p is = 1. See the contro-

versial pamphlets that passed between those two gentlemen
at that time.

There are several methods of finding the value of such
fractions, that have been given by the gentlemen above
quoted. The one is by considering the terms of the frac-
tion as two variable quantities, continually decreasing, till
they both vanish together; or finding the ultimate value
ot the ratio denoted by the fraction. In this way of con-
sidering the matter, it appears that, as the terms of the
fraetion are supposed to decrease till they vanish, or be-
come only equal to their fluxions or their increments, the
value of the fraction at that state, will be equal to the
fluxion or increment of the numerator divided by that of
the denominator. Hence then, taking the example x ~ *" "
whenx=l; the fluxion of the numerator is x — dx'x,
and of the denominator — x ; therefore

X — 5x*X 1 — 5a ' 5.T4 — 1

= 5x4 — 1 = 5 - 1 = 4,

the value of the fraction -

i
-V

i — j?"

— of

when x = ].
i — x*

-Or, thus,

because x = 1 therefore — = - — — ■ then the

l — x l — x

fluxion of the numerator, — 4x3*, divided by the fluxion
of the denominator, or — *, gives 4x3 or 4, the same as
before.

Another method is by reducing the given expression to
another, or simpler form, and then substituting the values
of the letters. ' So in the above example ~ '. or ~J ,

, r 1 — X 1 — x'

when x = 1 ; divide the numerator by the denominator,
and it becomes 1 ■+■ x ■+- x2 ■+- x3, which when x= 1, be-
comes 4, for the given fraction, the same as before. —

Again, to find the value of ■

-^— when x is = a, in
a— >/ax

which case both the numerator and denominator become

= 0. Divide the numerator by the denominator, and the

quotient is ^ ax •+- x ■+■ x^/-; which when x = a, be-
comes a -+- a h- a = 3a, for the value of the fraction in
that state of it.

Mr. Woodhouse, in his Principles of Analytical Calcu-
lation, has endeavoured to prove, that the method of ex-
panding these fractions for estimating their values, is an
arbitrary operation; that there is nothing contained in the
significancy of such expressions which renders any such
step necessary. He admits, that in the investigations of
extension and of motion, we are necessitated, from the
nature of the case, to compute a vanishing fraction by a
process exactly similar to that just adverted to ; but con-
tends the necessity of such operation when it is abstract-
edly proposed to find its value. For, says he; ' Propose

the question separately, what does : — become, when

x = a? and the obvious answer to any mind undebauched

with mathematical sophistry is = -. A different

1 J a" — n" o

result can be obtained by a different process, but why
ought that process to be followed ? No satisfactory an-
swer can be given to this question, when it is abstractedly
proposed to find its value.'

In order to show somewhat the importance of this
question, let us suppose that in the investigation of a
problem, we obtained for the. fluent corrected

5 x x' — a'

s = xT — (x2 + u'X1 x , what would he the value

" ■ ' x — a

4 A 2

FRA [ 548 ]

of z when x = a ? It is evident that a case is here pre-
sented, in which the computation of a vanishing fraction
becomes an abstract proposition : but does it hence fol-
low, that to expand the expression for effecting such com-
putation is an arbitrary process? Surely not. The bare
assertion that there is>nothing contained in the significancy
of the fraction which renders the expansion of the same
necessary for the estimation of its value, is no proof what-
ever against the truth of the method ; or that it is not a
necessary step ; since if it were true, that the fraction did
not contain a value in this case, such truth would evi-
dently appear in the result of that operation ; that is
neither more nor less than nothing would be expressed by

the quotient. Now it is obvious that if hasavalue,

a — a

we have no other means of determining it than by ex-
panding the expression : for in the form , we can

derive nothing as a true and precise value of it. Let
— — — be put = -.; it may be so, and we see even in this

case that the fraction may be equal to some finite quan-
tity; that is to say, this expression does not preclude the
possibility of its having some definite value, though it af-
fords no clue for computation in this particular state.
Hence, since it must be granted, that a fraction and its
equivalent whole number are in all cases equal to each
other; it follows that if in any case a value be exhibited
in one form, it is likewise contained in the other, and con-
sequently must be, given to that other; therefore, the

equivalent whole number of = -, having some de-

a — a o s

finite value, " ~° itself, has1 also the same value. And

a — a

hence the true value of z, in the case where x = a is

a* - (a* + b*)i x 2a, or a x [a*/a - V O2 + b-)].

It is not therefore true that to expand the fractions to
obtain its true value, is an arbitrary operation, but a pro-
cess of necessity, rendered unavoidable by something con-
tained in the significancy of the expression. To admit
the necessity of expanding the fraction when its quantity
is to be computed, in any investigation of extension or of
motion ; and to deny that necessity when it is abstractedly-
proposed to find its value, is confessing, that a fraction and
its equivalent whole number are to be considered equal
only in particular cases, which is absurd.

FRAISE, in Fortification, a kind of defence, consisting
of pointed stakes, driven almost parallel to the horizon
into the retrenchments of a camp, &c, to ward off and
prevent any approach orscalade.

FRANKLIN (Dr. Ben jamin), one of the most cele-
brated philosophers and politicians of the ISth century,
was born at Boston in North America in the year 1706',
being the youngest of 13 f children. His father was a tal-
low-chandler in Boston, and young Franklin was taken
bom school at 10 years of age, to assist him in his busi-
ness. In this situation he continued two years ; but dis-
liking that occupation, he was bound apprentice to an
elder brother, who was then a printer in Boston, but had
learned that business in London, and who in the year 1721
began to print a newspaper, being the second ever pub-
lished in America; the copies of which our author was
sent to distribute, after having assisted in composing and
printing it. On this occasion, our young philosopher en-
joyed the secret and singular pleasure of being the much-
admired author of many essays in this paper; a circum-

F R A

stance which he had the address to keep a secret even from
his brother himself; and this when he was only 15 years
of age. The frequent ill usage from his brother, induced
our author to quit his service, which he did, at the age
of 17, and went to New York. But not meeting employ-
ment here, he went forward to Philadelphia, where he
worked with a printer a short time; after which, at the
instance of Sir Willliam Keith, governor of the province,
he returned to Boston to solicit pecuniary assistance from
his father, to set up a printing-house for himself at Phila-
delphia, on the promise of great encouragement from Sir
William, &c. His father, however, thought fit to refuse
such aid, alleging that he was yet too young (18 years
old) to be intrusted with such a concern ; so our author
again returned' to Philadelphia without it. Upon this,
Sir William said he would advance the sum that might be
necessary, and our young philosopher should go to Eng-
land, and purchase all the types and materials, for which
purpose he would give him letters of credit. He could
never however get these letters ; yet bj' dint of fair pro-
mises of their being sent on board the ship after him, he
sailed for England, expecting these letters of credit were
in the governor's packet, which he was to receive on its
being opened. In this however he was cruelly deceived,
and thus he was sent to London without either monev,
friends, or credit, at IS years of age.,.

Under these circumstances his only resource was that
of employment, which he shortly obtained, as a journey-
man printer, first at a Mr. Palmer's, and afterwards with
Mr. \Vatts, with whom he worked a considerable time, and
by whom he was greatly esteemed, being also treated with
such kindness, that it was ever after most gratefully re-
membered by our philosopher.

After a stay of IS months in London, he returned to
Philadelphia, in 1726, with a merchant of that town, as
his clerk, on a salary of 50 pounds a year. But his master
dying the year after, he again engaged to direct the print-
ing business of the same person with whom he had worked
before'. After continuing with him the best part of a year,
Franklin, in partnership with another young man, at
length set up a printing-house himself.

A short time previous to this, our author had gradually
associated a number of persons, like himself, of a rational
and philosophic turn of mind, and formed them into a
club or society, to hold meetings, to converse together for
their mutual improvement in all kinds of useful know-
ledge, which was in high repute for many years after.
Among many other useful regulations, they agreed to bring
such books as they had into one place, lo form a common
library. This resource being found defective, at Frank-
lin's persuasion they resolved to contribute a small sum
monthly towards the purchase of books for their use from
London. Thus their stock began to increase rapidly; and
the inhabitants of Philadelphia, being desirous of having
a share in their literary knowledge, proposed that the
books should be lent out on paying a small sum for the
indulgence. Thus in a few years the society became
rich, and possessed more books than were perhaps to be
found in all the other colonies : the collection was ad-
vanced into a public library ; and the other colonics, sen-
sible of its advantages, began to form similar plans;
whence, originated the libraries at Boston, Now York,
Charlestown, &c ; that of Philadelphia being now not in-
ferior to any in Europe.

About 1728 or 172;), our author set up a newspaper,
the second in Philadelphia, which proved very profitable,

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and otherwise useful, as affording an opportunity of ma-
king himself known as a political writer, by inserting seve-
ral of his writings of that kind in it. In addition to his
printing-house, he set up a shop to sell books and sta-
tionary ; and in 1730, entering into a matrimonial en-
gagement, his wife proved very useful in assisting to ma-
nage the shop, &c. He afterwards began to have some
leisure, both for reading books, and writing them, of which
he gave many specimens from time to time. In 1732 he
began to publish Poor Richard's Almanac, which was
continued for many years. It was always remarkable for
the numerous and valuable concise maxims which it con-
tained, for thececonomy of human life, all tending to ex-
hort to industry and frugality : and in the almanac for the
last year, all the maxims were collected in an address to
the reader, entitled, The Way to Weal th.. This has been
translated into various languages, and inserted in different
publications. It has also been printed on a large sheet,
proper to be framed, ,and hung up in conspicuous places
in all houses, as it justly deserves to be. Mr. Franklin
became gradually more known for his political talents,
and in the year 1736, he was appointed clerk to the Ge-
neral Assembly of Pennsylvania; and was re-elected by
succeeding assemblies for several years, ti!l he was chosen
a representative for the city of Philadelphia ; and in 1737
he was appointed post-master of that city. In 173S, he
formed the first fire-company there, to extinguish and
prevent fires and the burning of houses : an example
which was soon followed by other persons, and in other
places. And soon after, he suggested the plan of an as-
sociation for insuring houses and ships from losses by fire,
which was adopted; and the association continues to this
day. In the year 1744, during a war between France
and Great Britain, some French and Indians made inroads
on the frontier inhabitants of the province, who were un-
provided for such an attack : the situation of the province
was at this time truly alarming, being destitute of every
means of defence. At this crisis Franklin stepped forth,
and proposed to a meeting of the citizens of Philadelphia,
a plan of a voluntary association for the defence of the
province. This was approved of, and signed by 1200 per-
sons immediately. Copies of the same were circulated
through the province; and in a short time the number of
signers amounted to 10,000.' Franklin was chosen co-
lonel of the Philadelphia regiment; but he did nof think
proper to accept of the honour.

Pursuits of a different nature now occupied the greatest
part of his attention for some years. Being always much
addicted to the study of natural philosophy ; and the dis-
covery of the Leyden experiment in electricity having
rendered that science an object of general curiosity ; Mr.
Franklin applied himself to it with great diligence, and
soon began to distinguish himself very eminently in that
way. He engaged in a course of electrical experiments
with all the ardour and thirst for discovery which cha-
racterized the philosophers of that.day. By these, he was
enabled to make a number of important discoveries, and
to propose theories to account for various phenomena ;
which have been generally adopted, and which will pro-
bably endure for ages. His observations he communicated,
in a series of letters, to his friend Mr. Collinson ; the first
of which is dated March 28, 1747- In these, he makes
known the power of points in drawing and throwingoff the
electric matter, which had hitherto escaped the notice
!l tricians. He also made the discovery of a plus and
minus, o.- of a positive and negative state of electricity;

from which he explained in a satisfactory manner the
phenomena of the Leyden phial, first observed by Cuneus
or Muscbenbroeck, which had much perplexed philoso-
phers. He showed that the bottle, when charged, con-
tained no more electricity than before, but that as much
was taken from one side as was thrown upon the other ;
and that, to discharge the same, it was only necessary to
make, a communication between the two sides, by which
the equilibrium might be restored, and that then no signs
of electricity would remain. He afterwards demonstrated
by experiments, that the electricity did not reside in the
coating, as had been supposed, but in the pores of the
glass itself. After a phial was charged, he removed the
coating, and found that upon applying a new coating the
shock might still be received. In the year 1749", he first
suggested his idea of explaining the phenomena of thun-
der-gusts, and of the aurora borealis, upon electrical
principles. He points out many particulars in which
lightning and electricity agree ; and likewise adduces
many facts, and reasoning from facts, in support of his '
positions. In the same year he conceived the bold and
grand idea of ascertaining the truth of his doctrine, bv
actually drawing down the forked lightning, by means of
sharp-pointed iron rods raised into the region of the
clouds ; whence he derived his'method of securing build-
ings and ships from being damaged by lightning. It was
not until the summer of 1752 that he was enabled to com-
plete his grand discovery — the experiment of the electrical
kite, which being raised up into the clouds, brought
thence the electricity or lightning down to the earth; and
M. D'Alibard made a like experiment about the same
time in France, by following the track which Franklin
had before pointed out. The letters which he sent to Mr.
Collinson, it is said, were refused a place among the papers
of the Royal Society of London ; and Mr. Collinson pub-
lished them in a separate volume, under the title of New
Experiments and Observations on Electricity, made at
Philadelphia, in America; which were read with avidity,
and soon translated into different languages. His theories
were at first opposed by several philosophers, and by the
members of the Royal Society of London; but in 175.5,
when he returned to that city, they voted him the gold
medal which is annually given to the person who presents
the best paper on some interesting subject.- He was also
admitted a member of the society, and had the degree of
doctor of laws conferred on him by different universities:
butat this time, by reason of the war which broke out be-
tween Britain and France, he returned to America, and
interested himself in the public affairs of that country.
Indeed he had done this long before ; for although philo-
sophy was a principal object of Franklin's pursuit for se-
veral years, he did not confine himself to it alone. In
the year 1747 he became a member of the General As-
sembly of Pennsylvania, as a burgess for thecity of Phila-
delphia. Being a friend to the rights of man from his in-
fancy, he soon distinguished himselfas a steady opponent
ot the unjust schemes of the proprietaries. He was looked
up to as the head of the opposition ; and to him have been
attributed many of the spirited replies of the assembly, to
the messages of the governors. His influence in the body
was very great. This arose not from any superior powers
of eloquence ; he spoke but seldom, and he never was
known to make any thing like an elaborate harangue.
His speeches often consisted of a single sentence, or of a
well-told story, the moral of which was always obviously
to the point. He never attempted the flowery fields of

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oratory. His manner was plain and mild. His style in
speaking was, like that of his writings, simple, unadorned,
and remarkably concise. With this plain manner, and
his penetrating and solid judgment, he was able to con-
found the most eloquent and subtle of his adversaries, to
confirm the opinions of his friends, and to make converts
of the unprejudiced who had opposed him. With a single
observation, he has rendered of no avail a long and ele-
gant discourse, and determined the fate of a question of
importance.

In the year 1749 he proposed the plan of an academy,
to be erected in the city of Philadelphia, as a foundation
for posterity to erect a seminary of learning, more exten-
sive and suitable to future circumstances; and in the be-
ginning of 1750, three of the schools were opened, namely,
the Latin and Greek school, the Mathematical, and the
English schools. This foundation soon after gave rise to
another more extensive college, incorporated by charter
May 27, 1755, which still subsists, and in a very flou-
rishing condition. In 1752, he was instrumental in the
establishment of the Pennsylvania Hospital, for the cure
and relief of indigent invalids, which has proved of the
greatest use to that class of persons. , Having conducted
himself so well as postmaster of Philadelphia, he was, in
1753, appointed deputy postmaster-general -for the whole
British colonies.

These being much exposed to depredations on theirfron-
tier by the Indians and the French, at a meeting of com-
missioners from several of the provinces, Mr. Franklin
proposed a plan for the general defence, to establish in the
colonies a general government, to be administered by a
president-general, appointed by the crown, and by a grand
council, consisting of members chosen by the representa-
tives of the different colonies ; a plan which was unani-
mously agreed to by the commissioners present. The
plan however had a singular fate : it was disapproved of by
the ministry of Great Britain, because it gave too much
power to the representatives of the people ; and it was re-
jected by every assembly, as giving to the president-general,
who was to be the representative of the crown, an influ-
ence greater than appeared to them proper, in a plan of
government intended for freemen. Perhaps this rejection,
on both sides, is the strongest proof that could be adduced
of the excellence of the system, as suited to the situation
of Great Britain and America at that time. It appears to
have steered exactly in the middle, between the opposite
interests of both. Whether the adoption of this plan would
have prevented the separation of America from Great
Britain, is a question which might afford much room for
speculation.

In the year 1755, General Braddock, with some regi-
ments of regular troops, and provincial levies, was sent, to
dispossess the French of the posts on which they had seized
in the back settlements. After the men were all ready, a
difficulty occurred, which had nearly prevented the expe-
dition. This was the want of waggons. Franklin now
stepped forward, and with the assistance of his son, in a
very short time procured 150. After the defeat of Brad-
dock, our author introduced into the assembly a bill for
organizing a militia, and had the dexterity to gel it passed.
In consequence of this act a very respectable militia was
formed; and Franklin was appointed colonel of a regiment
in Philadelphia, which consisted of 1200 men; in which
capaoity he acquitted himself with much propriety, and was
of singular service; though this militia was soon after dis-
banded by order of the English ministry.

In 1757, he was sent to England, with a petition tothe
king and council, against the proprietaries, who refused to
bear any share in the public expenses and assessments;
which he got settled to the satisfaction of the state. After
the completion of this business, Franklin remained at the
court of Great Britain for some time, as agent for the pro-
vince of Pennsylvania; and also for those of Massachu-
setts, Maryland, and Georgia. Soon after this, he pub-
lished his Canada pamphlet, in which he pointed out, in a
very forcible manner, the advantages that would result
from the conquest of this province from the Frepch. An
expedition was accordingly planned and fitted out, and the
command given to General Wolfe ; the success of which
is well known. He now divided his time indeed between
philosophy and politics, rendering many services to both.
While here, he invented the elegant musical instrument
called the Harmonica, formed of glasses played on by the
fingers.. In the summer of 1762 he returned to America;
on his passage to wdiich he observed the singular effect
produced by the agitation of a vessel, containing oil floating
on water: the upper surface of the oil remaining smooth and
undisturbed, while the water was agitated with the utmost
commotion. On his return he received the thanks of the
Assembly of Pennsylvania, which having annually elected
him a member in his,absence, he again took his seat in this
body, and continued a steady defender of the liberties of
the people.

In IJbl, by the intrigues of the proprietaries, Franklin
lost his seat in the assembly, which he had possessed for
" 14 years; but was immediately appointed provincial agent
to England, for which country he presently set out. In
1766 he was examined before the parliament relative to
the stamp-act; which was soon after repealed. The same
year he made a journey into Holland and Germany; and
another into France; being everywhere received with the
greatest respect by the literati of all nations. In 1773 he
attracted the public attention by a letter on the duel be-
tween Mr. Whately and Mr. Temple, concerning the pub-
lication of Governor Flutchinson's letters, declaring that
he was the person who had discovered those letters. On
the 29th °' January next year, he was examined before the
privy-council, on a petition he had presented long before,
as agent for Massachusetts Bay, against Mr. Hutchinson:
but this petition being disagreeable to ministry, it was pre-
cipitately rejected, and Dr. Franklin was soon after re-
moved from his office of postmaster-general for America.
Finding now all efforts to restore harmony between Great
Britain and her colonies useless, he returned to America in
1773; just after the commencement of hostilities. Being
named one of the delegates to the Continental Congress, he
had a principal share in bringing about the revolution and
declaration of independency on the part of the colonics.
In 1776 he was deputed by Congress to Canada, to nego-
ciatc with the people of that country, and to persuade
them to throw off the British yoke; but the Canadians had
been so much disgusted with the hot-headed zeal of the
New Englanders, who had burnt some of their chapels,
1h.1t ihey refused to listen to the proposals, though enforced
by all the arguments Dr. Franklin could make use of. On
his return to Philadelphia, Congress, sensible how much
he was esteemed in France, sent him thither to put a
finishing hand to the private negoci.ations of Mr. Silas
Deane; and this important commission was readily ac-
cepted by the doctor, though then in the 7 1st year of his
age. The event is well known; a treaty of alliance and
commerce was signed between France and America; and

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M. Le Roi assorts, that the doctor had a great share in the
transaction, by strongly advising M. Maurepas not to lose
a single moment, if he wished to secure the friendship of
America, and to detach it from the mother-country. In
1777 he was regularly appointed plenipotentiary from
Congress to the French court; but obtained leave of dis-
mission in 1780. Having at length seen the full accom-
plishment of his wishes by the conclusion of the peace in
1783, which gave independency to America, he became
desirous of revisiting his native country. He therefore re-
quested to be recalled; and, after repeated solicitations,
Mr. Jefferson was appointed in his stead. On the arrival
of his successor, he repaired to Havre de Grace, and
crossing the channel, landed at Newport in the Isle of
Wight ; whence, after a favourable passage, he arrived
safely at Philadelphia in September 1785. He was re-
ceived amidst the acclamations of a vast multitude, who
flocked from all parts to see him, and who conducted him
in triumph to his own house; where in a few days he was
visited by the members of the Congress and the principal
inhabitants of Philadelphia. He was afterward twice
chosen president of the Assembly of Philadelphia; but his
increasing infirmities obliged him to ask permission to re-
tire, and to spend the remainder of his life in tranquillity;
which was granted, in 1788. After this, the infirmities of
age increased fast upon him ; he became more and more
afflicted with the gout and the stone, till the time of his
death, which happened the 17th of April 1790, about 11
at night, at 84 years of age ; leaving one son, governor Wil-
liam Franklin, a zealous loyalist, who resided in London,
on a pension from the English government, till his death
in the year 1813;. and a daughter, married to Mr. William
Bache, merchant in Philadelphia.

Doctor Franklin was author of many .tracts on electri-
city, and other branches of natural philosophy, as well as
on politics, and miscellaneous subjects. He had also
many papers inserted in the Philosophical Transactions,
from vol. 47 to vol. 6-1.

FREEZli, or Frize, in Architecture, a large flat mem-
ber, being that'part of the entablature of columns that se-
parates the architrave from the cornice.

FREEZING, or Congelation, the fixing of a fluid body
into a firm or solid mass by the abstraction of its caloric :
in which sense the term is applied to water when it freezes
into ice ; to metals when they resume their solid state, af-
ter being melted by heat; or to glass, wax, pitch, tallow,
&c, when they harden again after having been rendered
fluid by heat. But it differs from crystallization, which is
rather a separation of the particles of a solid from a fluid,
in which it had been dissolved more by the moisture than
the action of heat. ,

The process of congelation is always attended with the
emission of heat or caloric, as is found by experiments on
the freezing of water, wax, spermaceti, ike ; for in such
cases it is always found, that a thermometer dipt into the
fluid mass, keeps continually descending as this cools, till
it arrive at a certain point, being the point of freezing,
which is peculiar to each fluid, where it is rather station-
ary, and then rises for a little, while the congelation pro-
ceeds. But by what means it is that fluid bodies should
thus be rendered solid by cold, or loss of caloric ; or fluid
by the absorption of ibis substance; or what is introduced
into the bodies by either of these principles, are matters
which philosophers have never' yet been able to discover,
or to satisfy themselves upon. The following phenomena
however ate usually observed. Water, and some other

fluids, suddenly dilate and expand in the act of freezing,
so as to occupy a greater space in the form of ice than be-
fore, in consequence of which it is that'ice is specifically
lighter than the same fluid, and floats in it. And the de-
gree of expansion of water, in the state of ice, is by some
authors computed at about TJW of its volume. Oil how-
ever is an exception to this property, and quicksilver too,
which shrinks and contracts still more after freezing.
Boyle relates several experiments of vessels made of metal,
very thick and strong; in which, when filled with waler,
close stopped, and exposed to the cold, the w'ater being
expanded in freezing, and not finding either room or vent,
burst the vessels. A strong barrel of a gun, filled with
water, and close stopped and frozen, was rent the whole
length. Huygens, to try the force with which water ex-
pands when freezing, filled a cannon with it, whose sides
were an inch thick, and then closed up the mouth and
vent, so that none could escape; the whole being exposed
to a strong freezing air, the water froze in about 12 hours,.,
and burst the piece in two places. Mathematicians have ■
computed the force of the ice upon this occasion; and they
say, that such a force would raise a weight of 27720
pounds. Lastly, Major Edward Williams, of the Royal
Artillery, made many experiments on the force of freezing
water at Quebec, in the years 1784 and 1785. He filled
all sizes of iron bomb-shells with this fluid, then plugged,
the fuze hole close up, and exposed them to the strong
freezing air of the winter in that climate; som'etimes driv-
ing in the iron plugs as hard as possible with a sledge ham-
mer; and yet they were always thrown out by the sudden
expansion of the water in the act of freezing, like a ball
shot by gunpowder, sometimes to the distance of 4 or
500 feet, though they weighed nearly 3 pounds; and when
the plugs were screwed in, or furnished with hooks or
barbs, to lav hold of the inside of the shell by, so that they
could not possibly be forced out, in this case the shell was
always split in two, though the thickness of the metal of
the shell was an inch and three quarters. It is further re-
markable, that through the circular crack, round about
the shells, where they burst, there stood- out a thin film or
sheet of ice, like a fin ; and in the cases when the plugs
were projected by freezing water, there suddenly issued
out from the fuze-hole, a bolt of ice, of the same diame-
ter, and stood over it to the height sometimes of S inches
and a half. And hence we need not be surprised at the
effects of ice in destroying the substance of vegetables and
trees, and even splitting rocks, when the frost is carried to
excess. See other similar instances in the article Cold.

It is also observed that water loses of its weight by
freezing, being found lighter after thawing again, than be-
fore it was frozen. And indeed it evaporates almost as
fast when. frozen, as when it is fluid.

We likewise find it asserted, that water 'does not freeze
in vacuo; requiring for that^purpose the presence and
contiguity of the air. But this circumstance is liable to
some doubt, and it may be suspected that the degree of
cold has not been carried far enough in these particular
instances; as it is found that mercury in thermometers has
even been frozen, though it requires a vastly greater de-
gree of cold to freeze this fluid than water. Besides, the
assertion has been lately disproved by the experiments of
Professor Leslie and Dr.Marcet, &c, who have frozen not
water only, but even mercury, in the exhausted receiver,
and that only by simple evaporation.

That water which has been boiled freezes more readily
than that which has not been boiled ; and that .'. slight

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commotion of the fluid disposes it to freeze more speedily ;
having sometimes been cooled several degrees below the
freezing point, without congealing when kept quite still,
but suddenly freezing into ice on the slightest agitation.

That the water, being covered over with a surface of oil
of olives, does not freeze so readily as without it; and
that nut oil absolutely preserves it under a strong frost,
when olive oil would not.

That rectified spirit of wine, nut oil, and oil of turpen-
tine, seldom freeze.

That the surface of the water, in freezing, appears all
wrinkled; the wrinkles being sometimes in parallel lines,
and sometimes like rays, proceeding from a centre to the
circumference. See the articles Congelation", Cold,
and Ice.

Freezing Mixture, a preparation for the artificial con-
gelation of water, and other fluids. According to Mr.
Boyle, all kinds of salts, whether alkaline or acid ; and
even all spirits, as spirit of wine, &c ; as also sugar, and
saccharum saturni, mixed with snow, are capable of freez-
' ing most fluids; and the same effect is produced, in a very
high degree, by a mixture of oil of vitriol, or spirit of nitre,
with snow.

The same is also remarked by M. Homberg of ecpjal
quantities of corrosive sublimate, and sal ammoniac, with
four times the quantity of distilled vinegar.

Boerhaave gives a method of producing artificial frost
without either snow or ice : we must have for this purpose,
at any season of the year, the coldest water that can be
procured'; this is to be mixed with a proper quantity of
any salt (sal ammoniac will answer the intention best), at
the rate of about 3 ounces to a quart of water. Another
quart of water must be prepared in the same manner with
the first; the salt, by being dissolved in each, will make-
file water much colder than it was before. The two
quarts are then to be mixed together, and this will make
them colder still. Two quarts more of water prepared and
mixed in the same manner are to be mixed with these,
which will increase the cold to a much higher degree in all.
The whole of this operation is to be carried on in a cold
cellar ; and a glass of common water is then to be placed
in the vessel of the fluid thus artificially cooled, and it will
be turned into ice in the space of 12 hours.

There is also a method of making artificial ice by means
of snow, without any kind of salt. For this purpose fill a
small pewter di*h with water, and upon that set a common
pewter plate filled, but not heaped, with snow. Bring this
simple apparatus near the fire, and stir the snow in the
plate : the snow will dissolve, and the ice will be formed
on the back of the plate, which was set in the dish of water.
M. Reaumur tried the effect of several salts, and exa-
mined the various degrees of cold by an ice thermometer,
which being placed in the fluid to be frozen, showed very
exactly the degree of cold by the descent of the spirit.

Nitre, or saltpetre, usually passes for a salt that may be
very serviceable in these artificial congelations; but the
experiments of this gentleman prove that this opinion is
erroneous. The most perfectly refined saltpetre employed
in the operation sunk the spirit in the thermometer only
',i\ degrees below the fixed point. Less refinqd nitre de-
pressed the thermometer lower, and gave a greater degree
of cold ; owing to the common or sea-salt that it contains
when less pure, which has a greater effect than the pure
saltpetre itself.

Two parts of common salt being mixed with three parts
of powdered ice in very hot weather, the spirit in the ther-

mometer immediately descended 15 degrees, which is half
a degree lower than it would have descended in the severest
cold of our winters. M. Reaumur then tried the salts all
round, determining with great regularity and exactness,what
was the degree of cold occasioned by each in a °iven dose.
Among the neutral salts, none produced a greater decree
of cold than the common sea salt. Among the alkalies,
sal ammoniac sunk the thermometer only to 13 decrees.
Pot-ashes sunk it just as low as well-refined saltpetre.

For the common uses of the table, the ice is not re-
quired to be very hard, or such as is produced by lone
continuance of violent cold: it is rather desired to be like
snow. Saltpetre, which is no very powerful agent in freez-
ing, is therefore more fit for the, purpose than a more po-
tent salt. It is not necessary that the congelation should
be very suddenly made ; but that it may retain its form as
long as may be, when made, is of great importance. If it
be desired to have ices very hard and firm, and very sud-
denly prepared, then sea salt is of all others most to be
chosen for the operation. The ices thus made will be very
hard, but they will soon run. Pot-ashes afford an ice of
about the hardness that is usually required. This forms
indeed very slowly, but then it will preserve a long time.
And common wood-ashes will perform the business very
nearly in the same manner as the pot-ashes; but for this
purpose, the wood which is burnt, ought to be fresh.

The strong acid spirits of the neutral salts act much
more powerfully in these congelations than the salts them-
selves, or indeed than any simple salt can do. Thus, spi-
rit of nitre mixed with twice its quantity of powdered ice,
immediately sinks the spirit in the thermometer to 19 de-
grees, or 4 degrees more than that obtained by means of
sea salt, the most powerful of all the salts in making arti-
ficial cold. A much greater degree of cold may be given
to this mixture, by piling it round with more ice mixed
with sea salt. This gives a redoubled cold, and sinks the
thermometer to 24 degrees. If this whole matter be co-
vered with a fresh mixture of spirit of nitre and ice, a still
greater degree of cold is produced, and so on; the cold
being by this method of fresh additions to be increased
almost without bounds: but it is to observed, that every
addition gives a smaller increase than the former.

It is very remarkable in the acid spirits, that though
sea salt is so much more powerful than nitre in substance
in producing cold, yet the spirit of nitre is much stronger
than that of sea salt; and another not less wonderful phe-
nomenon is, that spirit of wine, which is little else than
liquid fire, has as powerful an effect in congelations, or very
nearly so, as the spirit of nitre itself.

The several liquid substances which produce cold, in
the same manner as the dry salts on being mixed with ice,
are much more speedy in their action than the salts : be-
cause they immediately and much more intimately come
into contact with the particles of the ice, than the salts
can. Of this nature are spirit of nitre, spirit of wine, &c.
To produce the expected degree of cold, it is always ne-
cessary-that the ice and the additive matter, whatever it
may be, should both rush together, and intimately uniting,
form one clear fluid. It is hence that no new cold is pro-
duced with oil, which, though it melts the ice, yet can-
not mix itself into a homogeneous liquid with it, but must
always remain floating on the surface of the water that is
produced by the melting of the ice. Mem. Acad. Scienc.
Par. 1 734.

It has been discovered, that fluids standing in a current
of air, grow by this means much colder than before. Fall-

FRE

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PRE

renheit had long since observed, that a pond, which stands
quite calm, often acquires a degree of cold much beyond
what is sufficient for freezing, and yet no congelation en-
sued : but if a slight breath of air happens in such a case
to glide over the surface of the water, it stiffens the whole
in an instant. It has also been discovered, that all sub-
stances become colder by the evaporation of the fluids
they contain, or with which they are mixed. If both
these methods therefore be practised on the same body at
the same time, they will increase the cold to almost any de-
gree of intenseness we please.

Most extraordinary instances of artificial freezing have
since been made in Russia and at Hudson's Bay, and other
parts, by which quicksilver was frozen into a solid massof me-
tal. This indeed had before been effected by the natural cold
of the atmosphere alone, in Siberia. In the winter of 1733,
Professor Gmelin, with two other gentlemen of theRussian
academy, were sent by Anne Ivanouna, the new empress,
to explore and describe the different parts of her Asiatic
dominions, with the communication of Asia and America.
In the same season of the year 1734-5, M. Gmelin being
at Yeneseisk in 5S° 30' north lat. and 92° long, east from
Greenwich, first observed such a descent of the mercury,
as must have been attended with congelation, being far be-
low its freezing point, now fixed at —40 of Fahrenheit's
thermometer. " Here," says he, " we first experienced the
truth of what various travellers have related, with respect

found it at —102 ; but ascending with such rapidity, that
in the space of half an hoar it had risen to — 19°. This
phenomenon, which appeared so surprising, doubtless de-
pended on the expansion of the mercury frozen in the
bulb of the thermometer, and which, now melting, forced
upwards the small thread in the stem. Similar appea'r-
ances were observed afterwards, when the thread of quick-
silver in the thermometer was separated about 6 degrees.

A second instance, where a natural congelation of mer-
cury has certainly been observed, is recorded in the Trans-
actions of the Royal Academy of Sciences at Stockholm,
as made by Mr. Andrew Hellant. The weather, in Ja-
nuary 1760, was remarkably cold in Lapland ; so that on
the 5th of that month, the thermometers fell to —75,
— 12S, or lower; on the 23d and following days they fell
to— 58, —79)— 92, and below -238 entirely into the ball.
This was observed at four different places in Lapland, si-
tuated between the 65th and 7Sth degrees of north lat.
and the 21st and 28th degrees of east longitude.

But the congelation of quicksilver, by an artificial freez-
ing mixture, was first observed, and put beyond doubt,
by Mr. Joseph Adam Braun, professor of philosophy at
Petersburg. This gentleman, wishing to try how many
degrees of cold he could produce, availed himself of a good
opportunity, which offered for that purpose, on the 14th of
December 1759, when the mercury in the thermometer
stood in the natural cold at —34, which it is now known

to tKe extreme cold of Siberia; for, about the middle of is only 5 or 6 degrees above its point of congelation. As-

December, such severe weather set in, as we were sure had
never been known in our time at Petersburg. The air
seemed as if it were frozen, with the appearance of a fog,
which did not suffer the smoke to ascend as it issued from
the chimneys. Birds fell from the air as dead, and froze
immediately, unless they were brought into a warm room.
Whenever the door was opened, a fog suddenly formed
round it. During the day, short as it was, parhelia and
haloes about the sun were frequently seen; and in the
night mock moons, and haloes about the moon. Finally,
our thermometer, not subject to the same deception as the
senses, left us no doubt of the excessive cold ; for the quick-
silver in it was reduced, on the 5th of January, old style,
to —120° of Fahrenheit's scale, lower than it had ever
been observed in nature."

The next instance of congelation occurred at Yakutsk, in
(52° north lat. and 150° east longitude. The weather here
was unusually mild for the climate, yet the thermometer
fell to —72°; and one person informed the professor by
a note, thai the mercury in his barometer was frozen. He
hastened immediately to his house to behold such a sur-
prising phenomenon ; but though he was witness to the
fact, observing that the mercury did not continue in one
column, but was divided in different places as into little
cylinders, which appeared frozen, yet the. prejudice he had
entertained against the possibility of the congelation, would
not allow him to believe it.

Another set of observations, in the course of which the
mercury must frequently have been congealed, were made
by Professor Gmelin at Kirenga-fort, in .57|- north hit.,
and 108 east long. ; his thermometer, at different times,
standing at — 108, —86, — 100, — 113, and many other
intermediate degrees ; in the course of the winter of 1 737-8.
On the 27th of November, after the thermometer had been
standing for two days at —46°, he observed it sunk at
noon to — 10S. Suspecting some mistake, after he had
noted down the observation, he instantly ran back, and

Vol. I.

sisting this natural cold therefore, with a mixture prepared
of aquafortis and pounded ice, his thermometer was sunk
to —6<). Part of the quicksilver must now have been
really congealed, but unexpected by him, and he only
thought of pursuing his object of producing still greater
degrees of cold ; and having expended all his pounded ice,
he was obliged to use snow instead of it. With this fresh
mixture the mercury sunk to —100, —240, and —350°.
Taking the thermom&ter out, he found it whole, but the
quicksilver fixed, and it continued so for 12 minutes. On
repeating the experiment, with another thermometer which
had been graduated no lower than —220, all the mercury
sunk into the ball, and became solid as before, and did not
re-ascend till after a still longer interval of time.' Mr.
Braun now suspected that the quicksilver was really frozen,
and prepared for making a decisive experiment. This was
accomplished on the 25th of the same month, and the bulb
of the thermometer broken as soon as the metal was con-
gealed ; when it appeared that the mercury was changed
into a solid and shining metallic mass, which flatted and
extended under the. strokes •, a pestle, being rather less
hard than lead, and yielding a dull sound like that metal.
Mr. yEpinus made similar experiments at the same lime.
employing as well thermometers as tubes of a larger bore;
in which last he remarked, that the quicksilver fell sensibly
on being frozen, assuming a concave surface, and also,
that the congealed pieces sunk in fluid mercury: also, in
their further experiments, they invariably found that the
mercury sunk lower when the whole of it was congealed,
than if any part of it remained fluid: all showing that,
contrary to water, mercury contracted in freezing. It was
further observed, that the mercury when congealed looked
like the most polished .silver, and when beaten flat, it was
easily cut with a penknife, like soft thin sheet lead.

The fact being thus established, and fluidity no longer
to be considered as an essential property of quicksilver,
Mr. Braun communicated an account of his experiments
4 B

FRE

[ 554 ]

FRE

to the Petersburg academy, on the 6th of September menon we speak of. On the 6th and 7th of December

] 760 ; of which a large extract was inserted in the Philos.
Trans, vol. 52, pa. 156. He afterwards declared that he
never suffered a winter to pass without repeating the ex-
periment of freezing quicksilver, and never failed of suc-
cess when the natural cold was of a sufficient strength for
the purpose ; and this degree of natural cold he supposes
at —10 of Fahrenheit; though some commencement of
the congelation might be perceived when the temperature
of the air was as high as -1- 2.

The results of all his experiments were, that with the
abovementioned frigorific mixtures, and once with rectified
spirits and snow, when the natural cold was at —28°, he
congealed the quicksilver, and discovered that it is a real
metal, that melts with a very small degree of heat. How-
ever, not perceiving the necessary consequence of its great
contraction in freezing, he always confounded its point of
congelation, with that of its greatest contraction in freez-
ing, and thus marked the former a great deal too low.

In the process of his observations, Mr. Braun found
that double aquafortis was more effectual than spirit of
nitre; but with this simple spirit, which seldom brings
the mercury Ipwei than — 148, this metal may be frozen
in the following manner : Six glasses being filled wilh'snow
as usual, and the thermometer placed in one of them, the
spirit of nitre was poured upon it; when the mercury
would fall no lower in this, the thermometer was removed
to the second, and so on to the third and fourth, in which
fourth immersion the mercury was congealed.

Mr. iEpinus gives the following direction for using the
fuming spirit of nitre : Take some of this spirit, cooled as
much as possible, and put it into a wine-glass till it be
about half full, filling it up with snow, and stirring them
till the mixture become of the consistence of pap ; by
which means you obtain, almost -in an instant, the ne-
cessary degree of cold for the freezing of quicksilver.

It is remarked by Mr. Braun, that by the mixture of
snow and spirit of nitre, which congealed the mercury, he
never was able to bring thermometers, filled with the
most highly rectified spirit of wine, lower than —148:
so that the cold which will freeze mercury, will not freeze
spirit of wine; and therefore spirit thermometers are the
most fit to determine the degree of coldness in frigorific
mixtures, till we can construct solid metallic thermome-
ters with sufficient accuracj'. This author tried the ef-
fects of different fluids in his frigorific compositions : he
always found that Glauber's spirit of nitre and double
aquafortis were the most powerful; and from a number
of experiments made when the temperature of the air was
between 21 and 28 of Fahrenheit, he concludes, that
spirit of salt poured upon snow increased the natural cold
36°; spirit of sal ammoniac, 12; oil of vitriol, 42;
Glauber's spirit of nitre, 70; aquafortis, 48; simple spirit
of nitre, 36; dulcified spirit of vitriol, 2f; Hoffman's
anodyne liquor, 38 ; spirit of hartshorn, 12; spirit of sul-
phur, 12; spirit of wine rectified, 24; camphorated
spirit, IS; French brandy, 14; and several kinds of wine
increased the natural cold to 7, 8, or 'J degrees.

The most remarkable congelation 0/ mercury, by na-
tural cold, that has ever been observed, was that related
by Dr. Peter Simon Pallas, who had been sent by the
empress of Russia, with some other gentlemen, on an ex-
pedition similar to that of M.Gmclin. Being at Kras-
noyarsk in the year 1772, in north lat. 56° 3<J, and east
long. £)3°, he had an opportunity of observing the pheno-

that year, says he, there happened the greatest cold I have
ever experienced in Siberia : the air was calm at the time,
and seemingly thickened ; so that, though the sky was in
other respects clear, the sun appeared as through a fog.
I had only one small thermometer left, in which the scale
went no lower than —7°; and on the 6th in the morning,
I remarked that the quicksilver in the tube sunk into the
.ball, except some short columns which stuck fast to its
surface. When the ball of the thermometer, as it hung
in the open air, was touched with the finge.r, the quick-
silver rose ; and it could plainly be seen that the solid co-
lumns stuck and resisted a good while, and were at length
pushed upward with a seeming violence. He also placed
upon the gallery, on the north side of his house, some
quicksilver in an open bowl. Within an hour he observed
the edges and surface of it frozen solid ; and some minutes
afterward the whole was condensed by the natural cold
into a soft mass very much like tin. While the inner part
was still fluid, the frozen surface exhibited a great variety
of branched wrinkles ; but in general it remained pretty
smooth in freezing. The congealed mercury was more
flexible than lead; but on being bent short, it was found
more brittle than tin; and when hammered out thin, it
appeared somewhat granulated. When the hammer was
not perfectly cooled, the quicksilver melted away under
it in drops ; and the same thing-happened when the metal
was touched with the finger, by which also the finger was
immediately benumbed. When the frozen masswas broken
to pieces in the cold, the fragments adhered to each other •
and to the bowl in which they were placed. In the warm
room it thawed on its surface gradually, by drops, like
wax on the fire. Though the frost seemed to abate a little
towards night, yet the congealed quicksilver remained un-
altered, and the experiment with the thermometer could
still be repeated. On the 7th of December he had an op-
portunity of making the same observations all day; but
some hours after sunset, a north-west wind sprung up,
which raised the thermometer to —4-6°, when the mass
of quicksilver began to melt.

The experiments of Mr. Braun were successfully re-
peated at Gottingcn, in 1774, by Mr. John Frederick
Blumeubach ; being encouraged to this attempt by the
excessive cold of the winter that year, especially the night
of January the 11th, when he made the experiment, the
thermometer standing at —10 in the open air. Mr. Blu-
menbach at 5 in the evening, put 3 drachms of quicksilver
into a small sugar-glass, and covered it with a mixture of
snow and Egyptian sal ammoniac, setting the glass out in
the air upon a mixture also of sal ammoniac. At one 'he
next morning, the mercury was found frozen quite solid,
and hard to the glass; and did not melt again tjll 7 or
8 the next morning. The colour of the frozen mercury
was a dull pah' while with a blueish cast, like zinc, very
different from the natural appearance of quicksilver.

By similar means, quicksilver was twice frozen by Mr.
Hutchins, governor ot Albany-fort, in UuiUon's-bay, in
the months of January and February of 177.'. And the
same was done on the 28th of January 1776, by Dr. Lam-
bert Bicker, secretary of Rotterdam. The temperature
of the atmosphere was then at +2°; and the lowest it
could reduce the thermometer bj artificial cold, was — <)4;
when, on breaking the glass, the mercury was found con-
gealed.

In the beginning ol the year 17S0, M. Von Elterlcin of

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FRE

Vytegra, a town of Russia, in lat. 6l° north, and long.
36° east, froze quicksilver by natural cold. On the 4th
of January 17SO, the cold being increased to —34 that
evening at Vytegra, he exposed to the open air 3 ounces
of very pure quicksilver in a china teacup, covered with
paper pierced full of holes. Next day, at 8 in the morn-
ing, he found it solid, and looking like a piece of cast
lead, with a considerable depression in the middle. On
attempting to loosen it in the cup, his knife raised shavings
from it as if it had been lead, which remained sticking up;
and at length the metal separated from the bottom of the
cup in one mass. He then took it in his hand to try if it'
would bend : it was stiff like glue, and broke into two
pieces; but his fingers immediately lost all feeling, and
could scarcely be restored in an hour and a half by
rubbing them with snow. At 8 o'clock the thermometer
stood at — 57; but half after f) it was risen to —40 ; 'and
then the two pieces of mercury which lay in the cup had
lost so much of their hardness, that they could no longer
be broken, or cut into shavings, but resembled a thick
amalgam, which, though it became fluid when pressed
by the fingers, immediately afterwards resumed the con-
sistence of pap. With the thermometer at —39, the
quicksilver became fluid. The cold was never less on the
5th than —28, and by 9 in the evening it had increased
again to — 33. This experiment seems to fix the freezing
point of mercury at — 40 of Fahrenheit's thermometer, or
40 below 0; which is 72° below the freezing point of water.

During the winter of 1781 and -82,' Mr. Hutchins re-
sumed the subject of freezing quicksilver by artificial cold,
with such success, that from his experiments and those
of M. Von Elterlein, last mentioned, the freezing point of
mercury is now almost as well settled, viz, at —40, as that
of water is at 4- 32.

An instance of the natural congelation of quicksilver
also occurred in Jemptland, one of the provinces of
Sweden, on the 1st of January 1782; and lastly, on the
26th of (he same month, Mr. Hutchins observed the sarae_
effect of the cold at Hudson's-bay ; when be found that at
the point of its freezing a mercurial thermometer stood at
— 40, and a spirit thermometer at —30.

On this subject, see the article Cold ; see also the Phi-
los. Trans, vol.51, pa. 6*72 ; vol. 52, pa. 156; vol.66, pa.
174; vol.73, pa. 303 and 325 ; vol.76, pa. 241 ; vol. 77,
pa. 285 ; vol.78, pa. 43 ; and several others, particularly
vol. 79< Pa- 199> &c> beingexperimentson the congelation
of quicksilver in England, by Mr. Richard Walker, where
he proves that mercury may be frozen not only' in Eng-
land in summer, but even in the hottest climate, at any
season of the year, and without the application of ice or
snow.

About the year 1810, Professor Leslie, of Edinburgh,
simply by evaporation, succeeded in freezing water in-
closed in an exhausted receiver, to promote a quick eva-
poration. The water is placed over an open vessel, con-
taining sulphuric acid, which absorbs the vapour as it ex-
hales from the water ; the whole being inclosed within the
receiver of an air-pump, the water cools as the exhaustion
proceeds ; and is ultimately changed into ice. Mr. Leslie
afterwards succeeded in freezing mercury by a similar pro-
cess ; viz, by investing the bulb of a mercurial thermo-
meter with a thin coat of ice, and exposing this to the
joint effect of exhaustion and of sulphuric acid.

Dr. Ma"rcet too has since effected the congelation of
mercury with great facility and quickness, wmply by sub-

stituting the evaporation of ether, instead of that of water,
in the process. See the particulars in Nicholson's Journal,
vol.34, p. 119.

Dr. W. H. Wollaston has even succeeded in freezing
water at a distance from the mixture. His method con-
sists of a tube, with a ball at each end, bent at right
angles : one of the balls contains a little water ; and the
remainder of the space is to be as perfect a vacuum as can
be obtained. When this instrument has been successfully
exhausted, if the empty ball be immersed in a freezing
mixture of salt and snow, the water in the other ball,
though at the distance of 2 or 3 feet, will be frozen solid
in the course of a very few minutes. The vapour contained
in the empty bait is condensed by the common operation
of cold, and the vacuum produced by this condensation
gives opportunity for a fresh quantity to arise from the
opposite ball, with proportional reduction of its tempera-
ture. See Philos. Trans. 1S13, pt. 1.

Freezing Point, denotes the point or degree of cold,
shown by a mercurial thermometer, at which certain
fluids begin to freeze, or, when frozen, at which they
begin to thaw again. On Fahrenheit's thermometer, this
point is at -f-32 for water, and at —40 for quicksilver;
these fluids congealing at those two points respectively.
It would also be well if the freezing points for other fluids
were ascertained, and the whole arranged in a table.

It may here be proper to remark, that though the freez-
ing point of water be 32°, yet it may be cooled down in
favourable circumstances considerably below that tempera-
ture, before it begins to shoot into crystals. Experiments
were made on this subject by Mairan and Fahrenheit ;
but it is to Sir Charles Blagden that we are indebted for
the fullest investigation of it. He succeeded in cooling
water down to 21° before it froze, by exposing it slowly
to the action of freezing mixtures. The experiment suc-
ceeds best when the water tried is well purged of its air.
It ought also to be transparent ; for opaque bodies floating
therein cause it to shoot into crystals when only a few de-
grees below the freezing point. When a piece of ice is
thrown into water thus cooled, it causes it instantly to
shoot out into crystals. The same effect is produced by
throwing the liquid into a tremulous motion ; but not by
stirring it. It freezes also when cooled down too suddenly.
Freezing Rain, or Raming Ice, a very uncommon
sort of shower which sometimes falls during winter. A
remarkable shower of this kind happened in December
1672, in the west of England : of which some accounts
arc given in the Philos. Transact. No. 90. This rain, as
soon as it touched any thing above ground, as a bough,
or the like, immediately settled into ice ; and by enlarging
and multiplying the icicles, it broke all down with its
weight. The rain that fell on the snow, immediately con-
gealed, without sinking in the snow at all. It made also
an amazing destruction of trees, beyond any thing in all
history. " Had it concluded with some gust of wind,"
said a gentleman on the spot, " it might have been of ter-
rible consequence. Having weighed the sprig of an ash-
tree, the wood of which was just three quarters of a
pound, the ice upon it amounted to 16 pounds. Some
were frighted with the noise in the air; till they discerned
it was the clatter of icy boughs dashing against each
other." Dr. Bcale observes, that there was no considera-
ble frost perceived on the ground during the whole of the
shower; from which he concludes, that a frost maybe
very fierce and dangerous on the tops of some hills, while
4B 2-

F RI

[ .w6 ]

F R t

in other places it keeps at some feet above the ground ;
and may wander about very furiously in some places, and
be remiss in others not far off. This rain was followed by
glowing heats, and a great forwardness of vegetation.

FRLNICLE (Bernard), a celebrated French mathe-
matician of the 17th century. He was the contemporary

angle, or on an horizontal support and between two upright
pins; or best of all between little wheels, called friction
wheels, moveable on their respective axes: for by this con-
trivance, the friction is transferred from the circumfe-
rence of those wheels to their pivots, and is so much less
as these are smaller. In like manner the attrition or re-

and companion of Descartes, Fermat, and the other learned sistance may be still farther diminished, by makin^ the ax-
mathematicians of their time. He was admitted Geome- is of those wheels rest upon other friction wheels that tur-,
trician of the French Academy in }666; and died in 1675. round with them. This was long since recommended by
Frenicle had many papers inserted in the Ancient Memoirs P. Casabus ; and experience confirms the truth of it. Hence

of the Academy, of l6'6"6", particularly in vol. 5 of that
collection, viz, l.A method of resolving problems by Ex-
clusions.— 2. Treatise of right-angled Triangles in Num-
bers.— 3. Short tract on Combinations. — 4. Tables of
JMagic Squares — 5. General method of making Tables of
Magic Squares

also it is, that a pulley moveable on its axis resists less
than if it were fixed, and the cord sliding over the circum-
ference. And the same may be observed of the wheels of
coaches, and other carriages. Indeed about 40 years ago
friction balls or rollers were -placed within the naves of
carriage-wheels by some persons, particularly a Mr. Varlo;

FRESCO, is a sort of painting which is made upon the and lately Mr. Garnett had a patent for an improved man-

plastering of walls before it is dry

FRIABILITY, the quality of a body that is friable.

FRIABLE, a quality of bodies by which they are ren-
dered tender and brittle, easily crumbled or reduced to
powder between the fingers; their force of cohesion being
such as easily exposes them to such solution. Such are
pumice, and all calcined stones, burnt alum, &c.

It is supposed that friability arises from hence, that the

ner of applying friction wheels to any axis, as of carriages,
blocks or pulleys, scale beams, &c, in which the inclosed
wheels or rollers are kept always at the same distance by-
connecting rods or bars.

From these principles, with the assistance of the higher
geometry, Olaus Roemer determined the figure of the
teeth of wheels, that should make the least resistance pos-
sible, which he found should be epicycloids. And the same

body consists wholly of dry parts irregularly combined, was afterwards demonstrated by Lahire, and Camus
and which are readily separated, as having nothing unc- M. Amontons by experiment attempted, to settle a

tuous or glutinous to bind them together. foundation for the precise calculation of the quantity of

FRICTION, the act of rubbing or grating the surfaces friction; which M. Parent, endeavoured to confirm from

of bodies against or over each other, called also attrition, reasoning and geometry. Amontons' principle is, that the

—The phenomena arising from the friction of divers bo- friction of two bodies depends only on the weight or force

dies, under different -circumstances, are very numerous with which they press each other, being always more or

and considerable. Mr. Hawksbee gives a number of ex- less in proportion to that pressure; esteeming it a vulgar

periments of this kind ; particularly of the attrition or fric- error, that the quantity of friction has any dependence on

tion of glass, under various circumstances; the result of the extent of the surface that is rubbed, or that the fric-

which was, that it yielded light, and became electrical, tion increases with the surface; arguing that it will re-

Indeed all bodies by friction are brought to conceive heat; quire the same weight to draw along upon aplane, apiece

many of them to emit light ; particularly a cat's back, su- of wood on its narrow edge, as on its broad and flat side;

gar, beaten sulphur, mercury, sea- water, gold, copper, &c, because, though on the broad side there be 4 times thenum-

but above all diamonds, which when briskly rubbed against ber of touching particles, yet each particle is pressed with

jlass, gold, or the like, yield a light equal to that of a
live coal when blowed by the bellows.

Friction, in Mechanics, denotes the resistance amov-
ing body meets with from the surface on which it moves.

Friction arises from the roughness or asperity of the sur-

but -J- of the weight bearing on those of the narrow side;
and since 4 times the number multiplied by i- of the weight
is equal to i of the number multiplied by 4 times the
weight, it is plain that the effect, that is, the resistance,
is equal in both cases, and therefore requires the same

face of the body moved on, and that of the body moving : force to overcome it

for such surfaces consisting alternately of eminences and On the first proposal of this paradox, M. Lahire very
cavities, either the eminences of the one must be raised properly had recourse to experiments, as the best test, had
over those of the other, or they must be both broken and they been judiciously performed: such as they were, how-
worn off : but neither can happen without motion, nor can ever, they succeeded in favour of this system. He laid
motion be produced without a force impressed. Hence the several pieces of unpolished wood on a rough table, whose
force applied to move the body is either wholly or partly sizes were unequal ; and afterwards placed weights on them,
spent on this effect; and consequently there arises a re- so as to render them all equally heavy : and he found that
sistance, or friction, which will be greater as the eminen- the same precise force, or weight, applied to them by a
ces are greater, and the substance the harder,- and as the small pulley, was required to put each in motion, not-
body, by continual rubbing, becomes more and more po- withstanding the inequality of the surfaces. The experi-
lished, the friction in consequence diminishes. meiit succeeded in tliesame manner with pieces of marble,
As the friction is less in a body that rolls, than when on a marble table. After this, by reasoning, M. Lahire
it slides, hence in machines, lest the friction should em- gave a physical solution of the effect. And M. Amontons
ploy a great part of the power, care is to be taken that no settled a calculus of the value of friction, with the loss sus-
part ofthe machine slide along another, if it can be avoided; tained by it in machines, on the foundation of this new
but rather that they roll, or turn upon each other, or at principle. In wood, iron, lead, and brass, which are the
leastslideas little as possible. With this view it may be chief materials used in machines, he makes the resistance
proper to lay the axes of cylinders, not in a groove or con- caused by friction to be nearly the same in all, when those
cave matrix, as usual, but between two cheeks forming an materials are anointed with oil or fat : and the quantity of

F ft I

MM* ]

F H I

this resistance^ independent of the magnitude of the sur-
face, he finds nearly equal to a third part of the weight of
the body moved, or of the force with which the two bodies
are pressed together. Others have asserted that if the sur-
faces be hard and well polished, the friction will be less
than a third part of the weight; bu-t if the parts be soft or
rugged, it will be much greater. It was farther observed,
that in the cylinder moved on two small gudgeons, or on
a small axis, the friction would be diminished in the same
proportion as the diameter of these gudgeons is less than the
diameter of the cylinder; because in this case, the parts
on which the cylinder moves and rubs, will have less velo-
city than the power which moves it in the^ame propor-
tion, which is in effect making the friction to be propor-
tional to the velocity. So that, from the whole of then-
observations, this general proposition is deduced, viz, That
the resistances arising from friction, are to each other in
a ratio compounded of the pressures of the rubbing parts,
and the velocities of their motions. Principles which, it
is now known from better experiments, are both erroneous;
notwithstanding the hypothesis of M. Amontons has been
adopted, and attempted to be confirmed by Camus, Desa-
guliers, and others.

M. Muschenbroek and the abbe Nollet, however, on the
other hand, have concluded from experiments, that the
friction of bodies depends on the magnitude of their sur-
face, as well as on their weight. Though the former ob-
serves, that in small velocities the friction varies very
nearly as thevelocity, but that ingreatvelocities the propor-
tion increases faster : he has also attempted to prove, that
by augmenting the weight of a body, the friction does not
always increase exactly in the same ratio. Introd. ad Phil.
Nat. vol. l,c. .9; and Lect. Phys.E.xp. torn. l,p. 241. Hel-
sham and Ferguson, from the same kind of experiments,
have endeavoured to prove, that the friction does not vary
by changing the quantity of surface on which the body
moves; and the latter of these asserts, that the friction in-
creases very nearly as the velocity ; and that by increas-
ing the weight, the friction is also increased in the same
ratio. Indeed there is scarce any subject of experiment,
with regard to which, different persons have formed such
various conclusions. Of those who have written on the
theory, no one has established it altogether on true prin-
ciples, till the experiments lately made by Mr. Professor
Vince of Cambridge: Euler, whose theory' is extremely
elegant, and would have been quite satisfactory had his
principles been founded on good experiments, supposes the
friction to vary in proportion to the velocity of tiie body,
and its pressure upon the plane; neither of which is true:
and others, though they have justly imagined that friction
is a uniformly retarding force, have yet retained the other
supposition, and so rendered their solutions not at all ap-
plicable to the cases for which they were intended.

For these reasons a new and ingenious set of experiments
was successfully instituted by the Rev. Mr.Vince, Plumian
Professor of Astronomy and Experimental Philosophy in
the university of Cambridge, which are published in the
75th vol. of the Philos. Trans. p.-l65. The object of these
experiments was to determine, 1st, Whether friction be a
uniformly retarding force. — 2d, The quantity of friction.
— 3d, Whether friction varies in proportion to the pressure
or weight. — 4th, Whether the friction be the same, on
whichever of its surfaces a body moves.

The professor says, " The experiments were made with
the utmost care and attention, and the several results

agreed so very -exactly with each other, that I do not
scruple to pronounce them to be conclusive."— "A plane
was adjusted parallel to the horizon, at the extremity of
which was placed a pulley, which could be elevated or
depressed in order to render the string which connected
the body and the moving force parallel to the plane or
horizon. A scale accurately divided was placed by the
side of the pulley perpendicular to the horizon, by the
side of which the moving force descended; on the scale
also was placed a moveable stage, which could be ad-
justed to the space through which the moving force de-
scended in any given time, which time was measured by a
well regulated pendulum clock vibrating seconds. Every
thing being thus prepared, the following experiments were
made to ascertain the law of friction. But let me first
observe, that if friction be a uniform force, the difference
between it and the given force of the moving power must be
also uniform, and therefore the moving body must descend
with a uniformly accelerated velocity, and consequently the
• spaces described from the beginning" of the motion must be
as the squares of the times, just as when there was no fric-
tion, only they will be diminished on account of the fric-
tion." Accordingly the experiments are then related, which
are performed agreeably to these ideas, and from them are
deduced these general, conclusions, which may be consi-
dered as established and certain facts or maxims, viz,

1st, That friction is a uniformly retarding force in hard
bodies; and is not subject to alteration by the velocity;
except when the body is covered with cloth, woollen, &c,
in which case the friction increases a little with thevelocity.
2dly,- Friction increases in a less ratio than the quantity
of matter, or weight of the body. This increase however
is different for the different bodies, more or less; nor is it
yet sufficiently known, for any one body, what proportion
the increase of friction bears to the increase of weight.

3dly, The smallest surface has the least friction; the
weight being the same. But the ratio of the friction to the
surface is not yet accurately known.

It is observed, that the professor's experiments con-
sisted in determining how far the sliding bodies would be
drawn, in given times, by a weight hanging freely over a
pulley. This method would both show him if the friction
were a constant retarding force, and the other conclusions
above stated. For, as the spaces described by any con-
stant force, in given times, are as the squares of the times;
and as the weight drawing the body is a constant force, if
the friction, which acts in opposition to the weight, should
also be a constant force, then their difference, or the force
by which the body is urged, will also be constant, in which
case the spaces described ought to be as the squares of the
times; which happened accordingly in the experiments.

Mr. Vince adds some remarks likewise on the nature
of the experiments which have been made by others."
These, he observes, the authors " have instituted, to find
what moving force would just put a body at rest, in mo-
tion : and they concluded from thence, that the acceleru-
tive force was then equal to the friction ; but it is mani-
fest, that,any force which will put a body in motion must
be greater than the force which opposes its motion, other-
wise it could not overcome it; and hence, if there were no
other objection than this, it is evident that the friction
could not be very accurately obtained ; but there is an-
other objection, which totally destroys the experiment, so
far as it tends to show the quantity of friction, which is,
the strong cohesion of the body to the plane when it lies

F R I

[ 558 ]

FRO

at rest." This he confirms by several experiments, and
then adds, " From these experiments therefore it appears,
how very considerable the cohesion was, in proportion to
the friction when the. body was in motion: it being, in
one case almost -f, and in another it was found to be very
nearly equal to the whole friction. All the conclusions
therefore deduced from the experiments, which have been
instituted to determine the friction from the force necessary
to put a body in motion (and I have never seen any de-
scribed but upon such a principle) have manifestly been
totally false; as such experiments only show the resistance
which arises from the cohesion and friction conjointly/'
Philos. Trans, vol. 75, pa. 165.

• Emerson, in his Principles of Mechanics, deduces from
experiments the following remarks relating to the quantity
of friction : When a cubic piece of soft wood, of 8 pounds
weight, moves upon a smooth plane of soft wood, at the
. rate of 3 feet per second, its friction is about ■§■ of the
weight; but if the plane be rough, the friction is little
less than half the weight: on the same supposition, when
both the pieces of wood are very smooth, the friction is
about A of the weight : the friction of soft wood on hard,
or of hard wood upon soft, is f-or-|. of the weight ; of hard
wood upon hard wood, y or A ; of polished steel moving
on steel or pewter, ?; moving on copper or lead, J of the
weight. He observes in general, that metals of the same
kind have more friction than those of different kinds ; that
lead makes much resistance ; that iron or steel running in
brass makes the least friction of any; and that metals
oiled' make the friction less than when polished, and twice
as little, as when unpolished. Desaguliers also observes
that, in M. Camus's experiments on small models of
sledges in actual motion, there are more cases in which
the friction is less than where it is more than -J of the
weight. See a table, exhibiting the friction between va-
rious substances, formed from his experiments in Desag.
Exp. Philos. vol. 1, pa. 193 &c; also pa. 133 to 138, and
pa. 1S2 to 254, and pa. 45S to 4,60. On the subject of
friction, sec several vols, of the Philos. Trans, as vol. 1,
pa. 206; vol. 34, pa. 77; vol. 37, pa. 394; vol. 53,
pa. 139, &c.

FRIDAY, the Gth day of the week, so called from Friga,
or Friya, a goddess worshipped by the Saxons on this day.
Jt is a fast-day in the church of England, in memory of our
Saviour's crucifixion, unless Christmas-day happen to fall
on Friday, which is always a festival.

Good Friday, the Friday next before Easter, repre-
senting the clay of our Saviour's crucifixion.

FRIGID Zone, the space about either pole of the earth
to which the sun never rises for one whole day at least in
their winter. These two zones extend to about 23f de-
grees everyway from the pole considered as their centre.

FRIGORIF1C, in Physics, something belonging to, or
that occasions cold. — Some philosophers, as Gassendus,
and other corpuscularians, denying cold to be a mere pri-
vation, or absence of heat, contend that there are actual
frigorific corpuscles or particles, as well as fiery ones:
whence proceed cold and heat. But later philosophers
allow of no other frigorific particles beside those nitrous
salts, &c, which float in the air in cold weather, and oc-
casion freezing.

FRISI, or I'RISIUS (Gumma), appears to have been a
native of the Flemish Netherlands, and flourished the
beginning of the l6"th century. As Werner had proposed
to find the longitude by observations on the moon; so

Gemma Frisi, in a tract entitled, De Principiis Astrono-
mice et Cosmographiaj, printed at Antwerp 1530, advised
keeping the time by means of small clocks or watches, for
the purpose, then, as he says, lately invented-. He also
contrived a new kind of cross-staff, described in his tua-
tise De Radio Astronomico et Geometrico, printed at the
same place 1545, and in his additions to Peter Apiau's
Cosmography, gives the figure of an instrument, he calls a
nautical quadrant, as very useful in navigation, promising
to write largely on the subject : accordingly, in <m edition
he made in the year 1553, of his above-mentioned book,
De Principiis Astronomic, &c, he. delivers several nauti-
cal axioms, as tie calls them, which with some alterations
were repeated by his son Cornelius Gemma, in the lather's
posthumous piece on the Universal Astrolabe, published in
1556". — Gemma Fiisi died in 1555, at 4.5 years of age.

FR1ZE, FuiezE; or Freeze, in Architecture, a part
of the entablature of columns, between the architrave and
cornice. ,

FRONT, in Architecture, denotes the principal face or
side of a building; or that presented to their chief aspect
or view.

Front, in Perspective, a projection or representation
of the face, or forepart of an object, or of that part di-
rectly opposite to the eye, called also more usually Or-
thography.

FRONTISPIECE, in Architecture, the portale.or prin-
cipal face of a fine building,

FRONTON, in Architecture, an ornament among us
more usually called Pediment.

FROST, such a state of the atmosphere as causes the
congelation or freezing of water, or other fluids into ice.

The nature and effects of frost in different countries, are.
mentionetl under the articlesCoNGELATiON,andFREEziNG.
In the more northern parts of the world, even solid bodies
are affected by frost, though this is only or chiefly in con-
sequence of the moisture they contain, which being frozen
into ice, and so expanding as water is known to do when
congealed, it bursts and lacerates any thing in which it is
contained, as plants, trees, stones, and large rocks. Some
fluids expand by frost, as water, which expands about
■j%th part of its natural bulk, for which reason ice floats in
water ; but others again contract, as quicksilver, and hence
frozen quicksilver sinks in the fluid metal.

Frost, being derived from the atmosphere, naturally
proceeds from the upper parts of bodies downwards, as the
water and the earth : so, the longer a frost is continued,
the thicker the ice becomes upon the water in ponds, and
the deeper into the earth the ground is frozen. In about
16 or 17 days frost, Mr. Boyle found it had penetrated 14
inches into the ground. At Moscow, in a hard season,
the frost will penetrate 2 feet deep in the ground; and
Capt. James found it penetrated 10 feet deep in Charlton-
island, and the water in the same island was frozen to the
depth of 0' feet. Scheffer assures us, that in Sweden the
frost pierces 2 cubits, or Swedish ells, into the earth, and
turns what moisture is found there into a whitish substance,
like ice ; and standing waters to 3 ells, or more. The same
author also mentions sudden cracks or rifts in the ice of
the lakes of Sweden, 9 or 10 feet (hep, and many leagues
long: the rupture, being accompanied with a noise not less
loud than if many guns were discharged together. By
such means, however, the fishes are furnished with air; so
that they are rarely found dead.

The several natural histories of frosts funnsh very ex-

FRO

[ 559 J

F R U

traordinary effects of them. The trees are often scorched,
and burnt up, as with the most excessive heat; and split
or shattered. In the great frost in 1683, the trunks of oak,
ash, walnut, &c, were miserably split and cleft, so that they
might be seenx through, and the cracks often attended with
dreadful noises like the explosion of fire-arms. Philos.
Trans. No. 105.

The close of the year 170S, and the beginning of 1709,
were -remarkable, through the greatest part of Europe, for
a severe frost. Dr. Derham says, it was the greatest in
degree, if not the most universal, in the memory of man;
extending through most parts of Europe, though scarcely
felt in Scotland or Ireland.

In very cold countries, meat may be preserved by the
frost 6 or 7 months, and prove tolerably good eating. See
Capt.Middleton's observations made in Hudson's-bay, in
the Philos. Trans. No. 46"5, sect. 2. In that climate the
frost seems never out of the ground, it having beer, found
hard frozen in the two summer months. Brandy and
spirit of \vine, placed in the open air, congeal to solid ice
in 3 or 4 hours. Lakes and standing waters, not above 10
or 12 feet deep, are frozen to the ground in winter, and all
their fish perish. But in rivers, where the current of the
tide is strong, the ice does not reach so deep, and the fish
are preserved. Id. ib. Some remarkable instances of frost
in Europe, and chiefly in England, are recorded as below:
In the year
.220, Frost in Britain that lasted 5 months.
260, The Thames frozen .9 weeks.
291, Most rivers in Britain frozen 6 weeks.
359, Severe frost in Scotland for 14 weeks.
508, The rivers in Britain frozen for 2 months.
558, The Danube quite frozen over.
695, Thames frozen 6' weeks; booths built on it.
759, Frost from Oct. 1. till Feb. 26", 760.
827, Frost in England for 9 weeks.
859, Carriages used on the Adriatic sea.
90S, Most rivers in England frozen 2 months.
923, The Thames frozen 13 weeks.
987, Frost lasted 120 days: began Dec. 22.
99%i The Thames frozen 5 weeks.
1035, Severe frost on June 24: the corn and fruits de-

1 stroyed.

1063, The Thames frozen 14 weeks.
1076, Frost in England from Nov. till April.
1114, Several wooden bridges carried away by ice.
1205, Frost from Jan. 14 till March 22.
1407, Frost that lasted 15 weeks.
1434, From Nov. 24 till Feb. 10. Thames frozen down

to Gravesend.
1683, Frost for 13 weeks.
170|, Severe frost for many weeks.
1715, The same for many weeks.
1739, One for 9 weeks. Began Dec. 24. -
1742, Severe frost for many weeks.
1747, Severe frost in Russia. -
1754, Severe one in England.
1760, The same in Germany.
1776", The same in England.
1788, Thames frozen below bridge; booths on it.
1 79+5 Hard frost of many weeks.

1813, Severe frost from the middle of December to the
middle of March 1814. The Thames from
Blackfriars to London bridge, completely frozen
over, booths and thousands of people upon it.

H oar Frost, is the dew frozen or congealed, early in cold
mornings; chiefly in autumn: though many Cartesians
will have it formed of a cloud ; and either congealed in the
cloud, and so let fall ; or ready to be congealed as soon as
it arrives at the earth.

This kind of frost, M. Regis observes, consists of an as-
semblage of small parcels of ice crystals ; which are of
various figures, according to the different disposition of the
vapours, when met and condensed by the cold.

FRUSTUM, in Geometry, is the part of a solid next the
base, left by cutting off the top, or segment, hy a plane pa-
rallel to the base : as the frustum of a pyramid, of a cone,
of a conoid, of a spheroid, or of a sphere, which is any part
comprised between two parallel circular sections ; and the
middle frustum of a sphere, is that whose ends are equal
circles, having the centre of the sphere in the middle of it,
and equally distant from both ends.

For the Solid Content of the Frustum of a cone, or of any
pyramid, whatever figure the base may have. Add into one
sum, the areas of the two ends and the mean proportional
between them' ; then -j of that sum will be a mean area, or
the area of an equal prism, of the same altitude with the
frustum ; and consequently that mean area being multi-
plied by the height of the frustum, the product will be the
solid content of it.

That is, if a denote the area of the greater end,
a that of the less, and h the height ;
then (a -t- a -+- ^/.\a) x y/i is the solidity.

Other rules for pyramidal or conic frustums may be seen
in my Mensuration, part 3.

The curve Surface of the Zone or Frustum of a sphere, is
had by multiplying the circumference of the sphere by the
height of the frustum. Mensur. pa. 146.

And the Solidity of the same Frustum is found, by adding
together the squares of the radii of the two ends, and -J of
the square of the height of the frustum, then multiplying
the sum by the said height and by the number l-5708.
That is (r.- +rJ + |7i'2) x ±ph is the solid content of the
spheric frustum, whose height is h, and the radii of its ends
B. and r, p being = 3,1416". Mensur. pa. 152.

For the frustums of spheroids, and conoids,, either para-
bolic or hyperbolic, sec Mensur. part 3. sect. 5, 6, 7- And
in pa. 352, &c, are general theorems concerning the frus-
tum of a sphere, cone, spheroid, or conoid, terminated by
parallel planes, when compared with a cylinder of the
same altitude, on a base equal to the middle section of the
frustum made by a parallel plane. The difference be-
tween the frustum and the cylinder is always the same
quantity, in different parts of the. same, or of similar so-
lids, or whatever the, magnitude of the two parallel ends
may be ; the inclination of those ends to the axis, and the
altitude of the frustum being given; and the said constant
difference is |- part of a cone of the same altitude with the
frustum, and the radius of its base is to that altitude, as
the fixed axis is to the revolving axis of the frustum. Thus,
if bec be any conic section, or a
right line, or a circle, whose axis,
or a part of it, is a-d ; ab and cd
the extreme ordinatcs, fe the mid-
dle ordinate, ap being = rn; then
taking, as ad to dk, so is the whole
fixed axis, of which ad is a part, to its conjugate axis;
and completing the parallelogram aghd: then if the
whole figure revolve about the axis ad, the line bec will
generate the frustum of the cone or conoid, according as it

FUL

[ 560 ]

FUN

is a right line or a conic section, or it will generate the
whole solid when ab vanishes, or a and B meet in the
same point; likewise aghd will generate a cylinder, and
adk a cone: then is the 4th part of this cone always equal
to the difference between the said cylinder generated by
aghd, and the solid or frustum generated by abecd ; hav-
ing all the same altitude or axis ad.

In the parabolic conoid, this difference and the cone va-
nish, and the frustum, or whole conoid abecd, is always
equal to the cylinder aghd, of the same altitude. — In the
sphere, or spheroid, the frustum abecd is less than the cy-
linder aghd, by J of the cone AKD.-r-And, in the cone or
hyperboloid, that frustum is greater than the cylinder, by
i of the said cone akd, which is similar to the other
cone ibcd.

- These relations are likewise true, whether the ends of the
frustum are perpendicular or oblique to the axis. And
the same will obtain for the frustum of any pyramid, whe-
ther right or oblique; and such a frustum of a pyramid
will exceed the prism, of the same altitude, and on the
middle section of the frustum, by £ of the same cone.

It has been observed, that the difference, or -J of the cone
akd, is. the same, or constant, when the altitude and in-
clination of the ends of the frustum remain the same. But
when the inclination of the ends varies, the altitude being
constant ; then the said difference varies so as to be always
reciprocally as the cube of the conjugate to the diameter
ad. And when both the altitude and inclination of the
ends vary, the differential cone is as the cube of the alti-
tude directly, and the cube of the said conjugate diameter
reciprocally : but if they vary so, as that the altitude is al-
ways reciprocally as that diameter, then the difference is a
constant quantity.

Another general theorem for Frustums, is this. In the
frustum of any solid, generated by the revolution of any
conic section about its axis, if to the sum of the two ends
be added 4 times the middle section, -g- of the last sum will
be a mean area, which being drawn into the altitude of the
solid, will produce the content. That is (ab -+■ dc ■+■
4fe) x ^ ad is. the content of abcd.— And this theorem
is general for all frustums, as well as the complete solids,
whether right or oblique to the axis, and not only of the
solids generated from the circle or conic sections, but also
of all pyramids, cones, and in short of any solid whose pa-
rallel sections are similar figures. — The same theorem also
obtains for any parabolic area abecd, and is very nearly
true for the area of any other curve whatever, or for the
content of any other solid than those above mentioned.

FUGUL, in music, is when the different parts of a mu-
sical composition follow each other, each repeating in order
what the first had performed.

FULCRUM, or Prop, in Mechanics, is the fixed point
about which a lever &c moves.

FULGURATING Phosphorus, a term used" by some
English writers, to express a substance of the phosphorus
kind. It was prepared both in a dry and liquid state, but'
the preparation it appears was not well known to any but
the inventor of it. This matter not only shone in the dark
in both states, but communicated its light to any thing it
was rubbed on. When inclosed in a glass vessel well
stopped, it would frequently fulgurate, or throw out little
flashes of light, and sometimes fill the whole phial with
waves of flume. It does not require having its light re-
cruited at the fire, or in the sunshine, like, the phosphorus
of the Bolognian stone, but of itself continues in a state of

shining for several years together, and is seen as soon as
exposedin the dark ; the solid or dry matter always resem-
bling a burning coal of fire, though not consuming itself.
Philos. Trans/No. 134.

FULIGINOUS, an epithet applied to fhick smoke or
vapour replete with soot or other crass matter. — In the
first fusion of lead, there exhales a great deal of fuliginous
vapour, which being retained and collected, makes what
is called Litharge. And lampblack is what is gathered
from the fuliginous vapours of pines, and other resinous
wood, when burnt.

FULMINANT, Fulminans, or Fulminating, a
term applied to any thing thundering, or that makes a
noise like thunder.

Aram Fuiminaks. See Aurum.

P'ulvis Fulmin ans, is a composition of 3 parts of nitre,
2 parts of salt of tartar, and 1 of sulphur. — Both the au-
rum and pulvis fulminans produce their effect chiefly down-
wards; in which they differ from gun-powder, which acts
in orbem, or all around, but principally upwards. When
the composition is laid in brass ladles, and so set on fire,
after fulmination, the -ladles are often found perforated.
It differs also from gunpowder in this, that it does not re-
quire to be confined, in order to fulminate, and it must be
slowly and gradually heated. Some instants before explo-
sion, a light blue flame appears on its surface, proceeding
from the vapours beginning to kindle. No more fire or
flame is perceived during the fulmination, being suffocated
and extinguished by the quickness and violence of the com-
motion. Nor does the fulminating powder generally kin-
dle the combustible bodies in contact with it, because the
time of its inflammation is too short.

Fulminating Damp. See Damp.

FULMINATION, or Fulguration, a vehement noise
or shock resembling thunder, caused by the sudden explo-
sion and inflammation of divers preparations; as aurum
fulminans, &c, when set on fire.

FUNCTION, a term used in analytics, for an algebrai-
cal expression any how compounded of a certain letter or
quantity with other quantities or numbers: and the ex-
pression is said to be a function of that letter or quantity.
Thus a — 4x, or ax -+- 3x!, or 2x — ay' (a1 — a'2), or xe,
or cx, is each of them a function of the quantity .v.

Functions of Calculus, is that branch ofanalysis which
treats of the doctrine of functions.

Landen, we believe, was the first who proposed to treat
the method of (fluxions as a branch of pure algebra, inde-
pendently of any metaphysical considerations, or of such
as are drawn from the principles of motion. Soon after
him, M. Lagrange endeavoured to establish the analytical
principles in a memoir published in the Berlin Acts for
the year 1/72: and a farther developement of his ideas
was presented in 17^6, under the title of Theorie des Fonc-
tions Analytiqucs, a work in which it is the object of the
author to fix the principles of the differential calculus, in-
dependently of the consideration of infinitely small, or va-
nishing quantities, of limits, or of flowing quantities. An-
other work also by M. Lagrange, entitled Lefons sur le
Calcul des Fonctions, has since been published, which is
meant to serve as a commentary and supplement to the first
part of the Theorie des Fonctions Analytiqu.es, and offers a
course, ofanalysis on that branch of modern calculus which
is commonly called Infinitesimal or Transcendental, and
which, in theestimation of this very distinguished mathema-
tician, is alone properly the calculus of functions.

FUN

[ 561 ]

FUN

The following extract will suffice to convey an idea,
both of the nature of analytic functions, and of part of the
notation employed by this author.

" The earlier analysts (says Lagrange) have only employ-
ed the word functions to design the different powers of the
same quantity ; it was afterwards extended in signification
to every quantity formed in any manner whatever from
another quantity ; and it is now generally adopted to de-
note that the value of a quantity depends, according to a
given law, on one or many other given quantities.

" Under, this point of view, algebra must be regarded as
the science of functions; and it is easy to perceive that the
resolution of equations consists, generally, in finding the
values of the unknown quantities in determinate functions
ofkuown quantities. These functions represent, then, the
different operations which must be performed on known
quantities, to obtain the values of those which are sought,
and they are, properly, only the ultimate result of the
calculation.

" But, in algebra, we only consider ihe functions in as
much as they result from arithmetical operations, genera-

provided that the other quantities with which it is blended
remain constant: hence is introduced naturally, with re-
spect to functions, the distinction of quantities into variable
and constant quantities.

" In ordinary algebra, quantities are simply distinguished
into known and unknown, and it is customary to denote
the former by the first letters of the alphabet, and the other
by the latter. The application of algebra to the theory of
curves first caused the distinction of the quantities which
enter the equation of a curve into given ones, such as the
axes, the parameters, &c, and indeterminates, such as the
co-ordinates. Afterwards these quantities became viewed
under the more natural aspect of constant and variable
quantities; and the consideration of functions leads us na-
turally to regard the quantities which compose them under
the same point of view.

" We call therefore, simply, a function of one or of seve-
ral quantities, every expression in which these quantities
enter in any manner whatever, connected or not with other
quantities considered as having given and invariable values,
while the quantities of the functions are deemed capable

lized and transferred to the letters ; while in the calculus of of receiving all possible values.

functions, properly speaking, we consider the functions
which result from the algebraic operation of the de\ elope-
ment in series, when we attribute to one or to several
quantities of the function indeterminate augmentations.

" The developenaent of functions, viewed in a general
manner, gives rise to derivative functions of different orders ;
and the algorithm of these functions once discovered, they
may be considered in themselves and independent of the
series whence they result. Thus, a given function being
regarded as primitive, we may deduce, by simple and uni-
form rules, other functions which I call derivative; and when
we have an equation between several variables, we may
pass successively to the derivative equations, and return
again from these to the primitive equations. These trans-
formations correspond to differentiations and integrations;
but in the theory of functions they depend alone on ope-
rations purely algebraical, founded on the simple principles
of the calculus.

" The derivative functions present themselves naturally in
geometry, when we consider areas, tangents, radii of curva-
ture, &c; and in mechanics, when we consider velocities
and forces. If we regard, for example, the area of a curve
as a function of the abscissa, the ordinate is then the first
derivative function, or prime function; the relation of the
ordinate to the subtangent is expressed by the prime func-
tion of the ordinate, and of consequence by the second
derivative function or second function of 'the area; the ra-
dius of the osculatory circle depends on two prime deri-
vative functions of the ordinate, and so of others. In like
manner, regarding the space run over as a function of the
time, the velocity is then the prime function, and the acce-
lerating force is the second function. It is not perhaps
one of the least advantages of the calculus of functions, that
it furnishes for these elements of the geometry of curves,
and of mechanics, expressions as simple and intelligible
as are the algebraic expressions of powers and roots.

" When we consider a function with respect to one of the
quantises which compose it, we make our abstraction of
the value of that quantity, and consider it only as to the
manner in which it enters the function ; that is to say, in
which it is combined with itself, and with the other quan-
tities. Thus, the function is considered as remaining the
same, while this quantity varies in any manner whatever,
Vol. I.

" We ordinarily design the variable parts of functions
by the final letters of the alphabet x, y, &c ; and the con-
stant parts by the initial letters a, b, c, &c. And to denote
a function of a single variable x, we merely place before
that quantity the characteristic letter/or f ; but when we
would design the function of a quantity already composed
of that variable, as x-, or a -+- bx, &c, we include this
quantity in a parenthesis. Thus/x indicates a function of
.r, f(x-), f(a -f- bx), &c, denote functions of x1 and of
a ■+■ bx.

" To express a function of two independent variables as
x,y, we write/(x, y), and so of others.

" If two functions of two different variables, x, y, that
is to say, the one of x, the other of y, are composed in the
same manner, and with the same constant quantities ; these
functions will be similar, and may be designed in the same
investigation by the same characteristic, thus fx, and fy,
will be two similar functions which become identical when
x =y. But if, while the two functions are composed in
the same manner, the constant quantities which they con-
tain are different, then we can no longer, generally speak-
ing, represent them by the same characteristic in the course
of the same calculation. Still, however, if the two func-
tions differ only, for example, by the value of a constant
quantity, which will be a in the one, and b in the other,
we may ytt design them by the same characteristic/repre-
senting them by f(x, a), and/(^, b), as similar functions
of x, a, and of y, b. Thus, in this case, the quantities a
and b enter likewise into the expression of the function;
since, though they are constant quantities in each function,
they may be regarded as variables from one function to the
other."

For other particulars of the notation of the Calculus of
Functions,'and the doctrine itself, we must refer the reader to
the works of Lagrange already named, where he will find
his curiosity for such researches amply satisfied. On the
subject of functions, their divisions, transformations, ex-
plication by infinites, &c, see also Euler's " Analys. 1 11—
finitorum," c. 1, where this branch of analysis is likewise
fully treated. See also preface to Woodhouse's Princi-
ples of Analytical Calculation, for many very proper re-
marks on the Theorie des Fonctions Analytiqucs of M.
Lagrange.

4C

F U S

[ 562 1

F U Z

FUNDAMENTAL, something that serves as a base or
foundation for another.

Fundamental, in 3Iusic. A fundamental sound is
that which forms the lowest note of the chord, and from
which are deduced the harmonical relations of the rest ;
or it is that which serves for a key to the tone.

Fundamental Chord, is that which consists of the
three fundamental consonances, the 3d, the 5th, and the
Sth, or their inversions.

■ Fundamental Bass, is that formed by the funda-
mental notes of every perfect chord that constitutes the
harmony of the piece; so that under each chord it causes
to be heard, or understood, the fundamental sound of that
particular chord ; that is to say, the sound from which it
is derived by the rules of harmony. Whence we may see,
that the fundamental bass can have no other contexture
than that of a regular and fundamental succession, with-
out which the procedure of the upper parts would be ille-
gitimate.

FUNICULAR Curve, the same as the Catenary;
which see.

Funicular Polygon, is that formed by a chain
hanging freely by its extremities.

FURLONG, an English long-measure, containing 660
feet, or 220 yards, or 40 poles or perches, or the 8th part
of a mile.

FURNITURE, in Dialling, certain additional points
and lines drawn on a dial, by way of ornament: such as
the signs of the zodiac, length of days, parallels of declina-
tion, azimuths, points of the compass, meridians of chief
cities, Babylonic, Jewish, or Italian hours, &c.

FUSAROLE, in Architecture, a small round member
cut in form of a collar, with oval beads, under the echinus,
or quarter round, in the Doric, Ionic, and Composite ca-
pitals.

FUSEE, 'or Fusy, in Watch-work, is that part re-
sembling a low cone with its sides a little sunk or concave,
which is drawn by the spring, and about which the chain
or string is wound.

The spring of a watch is the first mover. It is rolled
up in a cylindrical box, against which it acts, and causes
to turn round in unbending itself. The chain, which at
one end is wound about the fusee, and at the other fast-
ened to the spring-box, disengages itself from the fusee in
proportion as the box is turned. And hence the motion
of all the other parts of the spring-watch. The effort or
action of the spring is continually diminishing from first
to last; and unless that inequality were rectified, it would
draw the chain with more force, and wind a greater quan-
tity of it on the box, at one time than another ; so that
the movement would never keep equal time. To correct
this irregularity of the spring, it was very happily con-
trived to have the spring applied to the arms of levers,
which are continually longer as the force of the spring is
weaker: this foreign assistance, always increasing as it is
most needed, maintains the action and effect of the spring
in an equality. It is for this reason then that the fusee

A P

Jjm

»

_— -^^ B

% ^ .

c

is made tapering somewhat conically, its radius at every
point of the axis answering to the corresponding strength
of the spring.

Now if the action of the spring diminished equally, as
the parallels to the base of a triangle do ; the cone, which
is generated of a triangle, would be the precise figure re-
quired for the fusee ; but it is certain that the weakening
of the spring is not in that proportion ; and therefore the
fusee should not be exactly conical ; and in fact expe-
rience shows that it should be a little' hollowed about the
middle, because the action of the spring is not there suffi-
ciently diminished of itself.

M. Varignon has investigated the figure of the fusee,,
or the nature of the curve by whose revolution about its
axis, shall be produced the solid whose figure the fusee is
to have. This curve, it may easily be shown, is an hyper-
bola, whose asymptote is the axis of the fusee. Thus,
let dfe be the curve of
the fusee, its axis being
abc : let ad express
the greatest strength of
the spring when the
watch is quite wound
up, or when the spring K
acts at d, and bg the least strength when the watch is
down, or when the spring acts at e ; so as that be : ad : :
ad : bg, orBE x eg = ad1; join dg, producing it to
meet" the axis produced in c; then shall hi denote the
strength of the spring acting at the corresponding point f
of the fusee; and the nature of it must be such that the
rectangle hi x he be equal to a constant quantity, oc
he must be reciprocally as hi, or hi : bg : : be : iif ;
but because ... at>, hi, eg,

are directly proportional to - ca, cii, cb,

therefore these are reciprocally propor. to ad, iif, be;
and consequently the curve dfe is an hyperbola, whose
centre is c, and asymptotes AC and kl : so that the figure
of the fusee is the solid generated by an equilateral hyper-
bola about its asymptote. See also Martin's Mathem.
Instit. vol. 2, pa. 364.

Fusee, Fuse, or Fuze, in Artillery, is a wooden tap
or tube used to set fire to the powder in a bomb-shell-
The bore of this tube is filled with a composition, of sul-
phur one part, saltpetre 3 parts, and mealed powder 3, 4,
or 5 parts. The tube is driven hard into the hole in the
shell, having first cut it to the exact length, answering to
the time of the intended flight of the shell, so that, the
composition in the fuse catching fire by the discharge of
the shell from the mortar, it just burns down to its lower
end, and so sets fire to the powder in the shell, thereby
bursting it at the moment when it arrives at the end of its
range or flight.

FUST, in Architecture, the shaft of a column, or that
part comprehended between the base and the capital,,
called also the Naked.

FUZE, or Fuzee. See Fusee.

[ 563 j

G.

GAG

GAG

/^JlABIONS, in Fortification, arc large cylindrical bas-
^"^ kets, open at both ends, made of ozier twigs, of 3 or
4 feet in diameter, and from 3 to 6 feet high. These, being
filled with earth, are sometimes used as merlons for the
batteries, and sometimes as a parapet for the lines of ap-
proach, when the attacks are carried on through a stony
or rocky ground, and to advance them with extraordinary
vigour. They serve also to make lodgments in some
posts, and to secure other places from the shot of the
enemy; who, on their part, endeavour to burn and destroy
the gabions, by throwinglighted pitch-faggots among them.
GABLE, or GA~BLE-end, of a house, is the upright
triangular end, from the cornice or eaves to the top of its
roof.

GAGE, in Hydrostatics, Pneumatics, &c, is an instru-
ment for ascertaining measures of various kinds.

Gage of the Air-pump, is adapted for showing the de-
gree to which the air is rarefied, or the receiver is exhaust-
ed, at any time by the air-pump. This is either the com-
mon barometer-gage, both long and short, or the pear-
gage, which at first was thought a great improvement, but
afterwards it was discovered that its seeming accuracy
was founded on a fallacy, which gave an erroneous indica-
tion of exhaustion. See AiR-pump.

Gage of the Barometer, is a contrivance for estimating
the exact degree of the rise and fall of the mercury in the
tube of that instrument. It is well known that while the
mercury rises in the tube, it sinks in the cistern, and vice
versa; and consequently, the divisions on the scale, fixed
near the top of the tube, had their distance from the sur-
face of' the mercury in the cistern always various; from
which there must often happen errors in determining the
height of the mercury in the tube. To remedy this incon-
venience, a line is cut upon a round piece of ivory, which
is fixed near the cistern : this line is accurately placed at
a given distance from the scale; for example, at 27 inches;
and a small float of cork, with a cylindrical piece of ivory
fixed to its upper surface, on which a line is cut at the
exact distance of 2 inches from the under side of the cork,
is left to play freely on the quicksilver, and the cylinder
works in a groove made in the other piece. From this
construction it appears, that if these marks are made to
coincide, by raising or lowering the screw which acts on
the quicksilver, then the divisions on the scale will ex-
press the true measure of the distance from the surface.

Gage of the Condenser, is a glass tube of a particular
construction, adapted to the condensing engine, and de-
signed to show the exact density and quantity of the air
contained at any time in the condenser. See Desaguliers's
Kxper. Philos. vol. 2, pa.394.

Sea Gage, an instrument for sounding the depth of the
sea. Several kinds of these have been invented, by Dr.
Hales, Dr. Desaguliers, and others. Formerly, the ma-
chines for this purpose consisted of two bodies, the one
specifically lighter, and the other specifically heavier,
than the water, so joined together, that as soon as the
licavy one came to the bottom, the lighter should get loose
from it, and emerge ; and the depth was to be estimated
by the time the compound was in falling from the top to

the bottom of the water, together with the time the lighter
body was in rising, reckoned from the disappearing of the
machine, till the emergent body was seen again. But no
certain conclusion could be drawn from so precarious and
incomplete an experiment.

But that invented by Drs. Hales and Desaguliers was
of a more exact nature, depending on the pressure of the
*fluid only. For as the pressure of fluids, in all directions,
is the same at the same depth, a gage which discovers
what the pressure is at the bottom of the sea, will show
what the true depth of the sea is in that place, whether
the^ time of the machine's descent be longer or shorter.

Dr. Hales, in his Vegetable Statics, describes his gage
for estimating the pressures made in opaque vessels ; where
hcThey being poured over the surface of mercury in an
open vessel, rises upon the surface of the mercury as it is
pressed up into a tube whose lower orifice is immersed into
the honey and mercury, and whose top is hermetically
sealed. Now, as by the pressure, the air in the tube is
condensed, and the mercury rises, so the mercury comes
down again when the pressure is taken off, and would leave
no mark of the height to which it had risen ; but the
honey (or treacle, which does better) which is upon the
mercury, sticking to the inside of the tube, leaves a mark,
which shows the height to which it had risen, and conse-
quently gives the quantity of pressure, and the height of
the surface of the fluid.

Desaguliers's addition to this machine, consisted in a
contrivance to carry it, down to the bottom of the sea by
means of a heavy weight, which was immediately disen-
gaged by striking the bottom, and the gage, made very
light for the purpose, re-ascended to the top.

Dr. Hales afterwards made more experiments of this
kind, and proposed another sea gage for vast depths,
which is described in the Philos. Trans. No. 405, and is to
this effect. Suppose a pretty long tube of copper or iron,
close at the upper end, to be let down into the sea, to any
depth, the water will rise in the tube to a height bearing
a certain proportion to the depth of the sea, to which the
machine is sunk. And this proportion is as follows : 33
feet of sea water being nearly equal to the mean pressure
of the atmosphere, therefore at 33 feet deep, the air in
the tube will be compressed into half the length of the
tube, or the water will rise and fill one half of the tube ;
in like manner, at 66 feet deep, the water will occupy §.
of the tube; at 99 feet deep it will fill | of the tube; at
132 feet deep it will fill ■£ of the tube ; and so on. Hence
therefore, by knowing the height to which the water rises
in the tube, there will be known the consequent depth of
the sea.

But, in very great depths, the scale near the top of the
tube would be so small, and the divisions so close, that
there would be no accuracy in the experiment, unless the
tube were of a very great length, and this again would
render it both liable to be broken, and quite impractica-
ble. To remedy this inconvenience, the Doctor made
the following contrivance: To the bottom of the tube he
screwed a large hollow globe of copper, with a small ori-
fice, or a short pipe at bottom of the globe, to lot in the
4 C 2

GAL

[ 564 ]

GAL

water; by which means he had a very great quantity. of
air, and the scale enlarged. See also Desagul. Exp. Phil.
vol.2, pa. 224 and 241.

Bucket Sea Gage, is an instrument contrived also by
Dr. Hales, to find the different degrees of coolness and
saltness of ihe sea, at different depths. This gage consists
of a common pail or bucket, with two beads ; each of
which has a round hole in the middle, about 4 inches in
diameter, covered with square valves opening upward.
That these heads may both open and shut together, there
is a small iron rod, having one end fixed to the upper side
of the lower valve, and the other end to the lower side of
the upper valve: so that as the bucket descends with its
sinking weight into the sea, both the valves may open by
the force of the water, which by that means has a free
passage through the bucket. But when the bucket is
drawn up, then both the valves shut by the force of the
water at the upper end of the bucket ; and consequently
the bucket will be filled with the lowest sea water to which
it has descended, and immediately the mercurial thermo-
meter, fixed within it, is examined, to see the degree of
temperature ; and the degree of saltness is afterwards as-
certained at leisure. Philos. Trans. No. 9, pa. 149, and
No. 24, pa. 447-

Lord Charles Cavendish adapted a thermometer for the
temperature of the sea water, at different depths. See
Philos. Trans, vol. 50, pa. 300, and Phipps's Voyage to-
wards the North Pole, pa. 142, &c.

Agueo-mercurial Gage, is the hame of an apparatus
contrived by Dr. Hales, and applied, in various forms, to
the branches of trees, to determine the force with which
they imbibe moisture. Vegetable Statics, vol. 1 , ch. 2,
pa. 84.

Sliding Gage, a tool used by mathematical-instrument-
makers, for measuring and setting off distances ; consisting
of a beam, tooth, sliding socket, and the shoulder of the
socket.

Tide Gage, an instrument used for determining the
height of the tides, by Mr. Bayly, in the course of a voy-
age towards the south pole, &c, in the Resolution and Ad-
venture, in the years 1772, 1773, 1774, and 1775. This
instrument consisted of a glass tube, whose internal dia-
meter was 7-10ths of an inch, lashed fast to a 10-foot fir
rod, divided into feet, inches, and parts ; the rod being
fastened to a strong post fixed firm and upright in the
water. At the lower end of the tube was an exceeding
small aperture, through which the water was admitted.
In consequence of this construction, the surface of the
water in the tube was so little affected by the agitation of
the sea, that its height was not altered the 10th part of an
inch, when the swell of the sea was 2 feet; and Mr. Bayly
was certain, that with this instrument he could discern a
difference of the 10th of an inch in the height of the tide.

Water Gage. See Altitude, and Hydrometer.

Wind Gage, an instrument for measuring the force of
the wind upon any given surface. Several of these gages
arc of former invention ; but there is one lately invented
by Dr. Lind, which is described in the Philos. Trans, vol.
65- See several also under the article Anemometer.

GAGER. See Gauger.

GAGING. See Gauging.

GALAXY, or Milky-Way, or Via Lactea, in Astronomy,
that long, whitish, luminous track, which appears to en-
compass the heavens like a swath, scarf, or girdle ; and
which is easily seen in a clear night, especially when the

moon is not up. It is of a considerable, though unequal
breadth ; being also in some parts double, but in others
single. The galaxy passes through many of the constel-
lations in its circuit round the heavens, and keeps its ex-
act place or position with respect to them.

There have been various strange and fabulous stories
and opinions concerning the galaxy. The ancient poets,
and even some of the philosophers, speak of it as the road
or way by which the heroes went to heaven. But the
Egyptians called it the Way of Straw, from the story of
its rising from burning straw, thrown behind the goddess
Isis in her flightfrom the giant Typhon. While the Greeks,
who affect to derive every thing in the heavens from some
of their own fables, have two origins for it ; the one, that
.luno, without perceiving it, accidentally gave suck to
Mercury when an infant, but that as soon as she turned
her eyes upon him, she forced him from her, and as the
nipple was drawn from his mouth, the milk streamed about
for a moment and formed the galaxy. The other relates,
that the infant Hercules being laid by the side of Juno
when asleep, on waking she gave him the breast; but soon
perceiving who it was, she threw him from her, and the
heavens were marked by the wasted milk.

Other philosophers however gave it a different turn, and
different origin : these esteemed it to be a tract of liquid
fire, spread in this manner along the skies : and others
again, supposing a celestial region beyond all that was vi-
sible, and imagining that fire, at some time let loose from
thence, was to consume the world, made this a part of
thatcelestial fire, and appealed to it as a presage of what
would surely happen. This diffused brightness they con-
sidered as a crack in the vault or wall of heaven, and fan-
cied this aglimrnering of the celestial fire through it, and
that there required nothing more than the undoing of this
crack by some accident in nature, or by the will of the
gods, to make the whole frame start, and let out the fire
of destruction.

Aristotle makes the galaxy a kind of meteor, formed
from a collection of vapours, drawn into that part by cer-
tain large stars disposed in the region of the heavens an-
swering to it. Others, finding that the galaxy was per-
ceived in most parts of the globe, that it always corre-
sponded to the same fixed stars, and that it was situated
far above the highest planets, set Aristotle's opinion aside,
and placed the galaxy in the firmament or region of the
fixed stars; thence concluding that it was nothing else but
an assemblage of an infinite number of minute stars. And
since the. invention of telescopes, this opinion has been
abundantly confirmed. For, by directing a good telescope
to any part of the milky way, we perceive an innumerable
multitude of very small stars, where before was only ob-
served a confused whiteness, arising from the assemblage
and union of their joint light ; like as an}' thing powdered
with fine white powder, at a distance we only observe the
confused whiteness, but on examining it very near we
perceive all the small particles of the powder separately ;
as Milton finely expresses it,

" A broad and ample road, whose dust is gold,
And pavement stars, as stars to thee appear,
Seen in the galaxy, that milky way,
Which nightly, as a circling zone thou seest
Powder'd with stars."
There are other similar appearances in the heavens; as
the nebula;, or nebulous stars, and certain whitish parts
about the south pole, called Magellanic clouds, which are

GAL

[ 565- ]

GAL

all of the same nature, appearing to be vast clusters of
small stars when viewed through a telescope, which are
too faint to affect the eye singly.

M. le Monnier however, not being able to discover more
stars in this space, than in other parts of the heavens, dis-
putes the opinion above recited, as to the reason of the
whiteness, and supposes that this and the nebulous stars
are occasioned by some other kind of matter. Inst. Ast.
pa. 60. And Dr. Herschel has lately announced some-
thing of a similar opinion. See Philos. Trans, vol. 81, or
my Abridg. vol. 17, pa. 18.

GALILiil (Galileo), a most excellent philosopher,
mathematician and astronomer, was born at Pisa in Italy,
in 1564. From his infancy he had a strong propensity to
philosophy and mathematics, which he indulged with such
success that in 1592 he was chosen professor of mathe-
matics at Padua. While he was professor there, visiting
Venice, then famous for the art of glass-making, he heard
that in Holland a glass had been invented, through which
very distant objects were seen distinctly as if near at hand.
This was sufficient for Galileo ; his curiosity was raised,
and his attention immediately engaged in considering
what must be the form of such a glass, and the manner
of making it. The result of his inquiry was the invention
of the telescope, produced merely from this hint, without
having seen the Dutch glass. All the discoveries he made
in astronomy were easy and natural consequences of this
invention, which opening a way, till then unknown, into
the heavens, thence brought to light the finest discoveries.
One of the first of these, was that of 4 of Jupiter's satel-
lites, which he called the Medicean stars or planets, in
honour of Cosmo the 2d, grand-duke of Tuscany, who
was of that family. Cosmo sent for our astronomer from
Padua, and made him professor of mathematics at Pisa in
l6ll ; and soon after inviting him to Florence, gave him
the office and title of principal philosopher and mathema-
tician to his highness.

Our author had been but a few years atFIorence, before
the Inquisition began to be very troublesome concerning his
opinions. Having observed some solar spots in 1612, he
printed that discovery the following year at Rome ; in
which, and in some other pieces, he ventured to assert
the truth of the Copernican system, and brought several
new arguments to confirm it. For these he was cited be-
fore the Inquisition at Pome, in l6l5: after some months
imprisonment, he was released, and sentence pronounced
against him, that he should renounce his heretical opinions,
and not defend them by word or writing, or insinuate
them into the minds of any persons. But having after-
wards, in 1632, published at Florence his Dialogues of
the two Great Systems of the World, the Ptolemaic and
Copernican, he was again summoned before the holy-office,
and committed to the prison of that ecclesiastical court at
Rome. The inquisitors convened in June that year; and
in his presence pronounced sentence against him and his
books, obliging him to abjure his errors in the most so-
lemn manner; committed him to the prison of their office
during pleasure; and enjoined him, as a saving penance,
for three years to come, to repeat once a week the seven
penitential psalms; reserving to themselves, however, the
power of moderating, changing, or taking away, altogether
or in part, the said punishment and penance. On this
sentence he was detained in prison till l6'34; and his
, Dialogues of the System of thoWorld were burnt at Rome.
Galileo lived ten years after this period; 7 of which were

employed in making still further discoveries with his te-
lescope. But by the continual application to that instru-
ment, added to an injury of sight from the nocturnal air,
his eyes grew gradually weaker, till he became totally
blind in 16'39- He bore this calamity with patience and
resignation, worthy of a great philosopher. The loss nei-
ther broke his spirit, nor stopped the course of his studies.
He supplied the defect by constant meditation; by which
means he prepared a large quantity of materials, and be-
gan to arrange them by dictating his ideas; when, by a
distemper of three months continuance, wasting away by
degrees, he expired at Arcetri near Florence, in January
l642, being the 78th year of his age.

Galileo was in his person of small stature, though of a
venerable aspect, and vigorous constitution. His -conver-
sation was affable and free, and full of pleasantry. He
took great delight in architecture and painting, and de-
signed extremely well. He played exquisitely on the lute;
and whenever he spent any time in the country, he took
great pleasure in husbandry. His learning was very ex-
tensive ; and he possessed in a high degree a clearness and
acuteness of wit. From the time of Archimedes, nothing
had been done in mechanical geometry, till Galileo, who
being possessed of an excellent judgment, and great skill
in the most abtruse points of geometry, first extended the
boundaries of that science, and began to reduce the resist-
ance of solid bodies to its laws. Besides applying geome-
try to the doctrine of motion, by which philosophy be-
came established on a sure foundation, he made surpris-
ing discoveries in the heavens, by means of his telescope.
He made the evidence of the Copernican system more sen-
sible, when he showed, from the phases of Venus, like to
those of the moon, that Venus actually revolves about the
sun. He proved the rotation of the sun on his axis, from
many observations on the spots contained on its surface;
and thence the diurnal rotation of the earth became more
credible. The satellites that attend Jupiter in his revolu-
tion about the sun, represented, in Jupiter's smaller sys-
tem, a just image of the great solar system; and rendered
it more easy to conceive how the moon might attend the
earth, as a satellite, in her annual revolution. By dis-
covering hills and cavities in the moon, and spots in the
sun constantly varying, he showed that there was not so
great a difference between the celestial bodies and the
earth, as had been vainly imagined.

Our author likewise rendered no less service to science
by treating, in a clear and geometrical manner, the doc-
trine of motion, which has justly been called the key of
nature. The rational part of mechanics had been so
much neglected, that hardly any improvement was made
in it for almost 2000 years. But Galileo has given us
fully the theory of equable motions, and of such as are
uniformly accelerated or retarded, and of these two com-
pounded together. He, first of any, demonstrated that
the spaces described by heavy bodies, from the beginning
of their descent, are as the squares of the times ; and that
a body, projected in any direction not perpendicular to
the horizon, describes a parabola. These were the be-
ginnings of the doctrine of the motion of heavy bodies,
which has been since carried to so great a height by New-
ton. In geometry, he invented the cycloid, or trochoid ;
though the properties of this curve were afterwards chiefly
demonstrated by his pupil Torricelli. He invented (he
simple pendulum, and made use of it in his astronomical
experiments : he had also thoughts of applying it to clocks;

GAL

[ 566 ]

GAL

but did not execute that design : the glory of that inven-
tion was reserved for his son Vicenzio, who made the ex-
periment at Venice in l6±9; andHuygens afterwards car-
ried the same to perfection. Our author was likewise pro-
jector of the machine, with which the Venetians render
their Laguna fluid and navigable. He also discovered the
gravity of the air, and endeavoured to compare it with that
of water ; besides laying the foundation for several other
inquiries in natural philosophy. In short, such was the
character of this memorable genius, that he was not only
esteemed and followed by philosophers, but was honoured
by persons of the greatest distinction of all nations.

Galileo had scholars that were worthy of so great a
master, by whom the gravitation of the atmosphere was
fully established, and its varying pressure accurately and
conveniently measured, by the column of quicksilver of
equal weight sustained thereby in the barometrical tube.
The elasticity of the air,' by which it perpetually endea-
vours to expand itself, and, while it admits of condensa-
tion, resists in proportion to its density, was a phenomenon
of a new kind (the common fluids having no such pro-
perty), and was of the utmost importance to philosophy.
These principles opened a vast field of new and useful
knowledge, and explained a great variety of phenomena,
which had been accounted for before that time in a very
absurd manner. It seemed as if the air, the. fluid in which
men lived from the beginning, had been then but first dis-
covered. Philosophers were every where busy inquiring
into its various properties and their effects : and valuable
discoveries rewarded their industry. Of the great num-
ber who distinguished themselves on this occasion, may
be mentioned Torricelli and Viviani in Italy, Pascal in
France, Otto Guerick in Germany, and Boyle in Eng-
land.

Our author wrote a number of treatises, many of which
were published in his life-time. Most of them were also
collected after his death, and published by Mendessi in
2 vols. 4to, under the title of L'Opere di Galileo Galilei
Lynceo, in \6d6. Some of these, with others of his pieces,
were translated into English and published by Thomas
Salisbury, in his Mathematical Collections, in 2 vols folio.
A volume also of his letters to several learned men, and
solutions of several problems, were printed at Bologna in
4to. His last disciple, Vincenzo Viviani, who proved a
very eminent mathematician, methodized a work of his
master's, and published it under this title, Quinto libro
dc gli Element! d'Euclidi, &c ; at Florence in 1674, 4to.
Viviani published also other writings of Galileo, being ex-
tracts from his letters to a learned Frenchman, where he
gives an account of the works which he intended to have
published, and a passage from a letter of Galileo dated at
Arcetri, Oct. 30, 1035, to JohnCamillo, a mathematician
of Naples, concerning the angle of contact. Besides all
these, he wrote many other pieces, which were unfortu-
nately lost through his wife's devotion ; who, solicited by
her confessor, gave him leave to peruse her husband's ma-
nuscripts; of which he destroyed and took away as many
as he said were unfit for publication.

GALLERY, in Architecture, a covered place in a
building, much longer than broad; which is usually placed
in the wings of the building, and serving to walk in as well
as to place pictures. It denotes a small aisle, or walk,
serving as a common passage to several rooms placed in a
line, or row.

Gallery, in Fortification, a covered walk, or passage,

made across the ditch of a besieged town, with timbers
fastened in the ground and covered over.

GALLON, an English measure of capacity, for things
both liquid and dry, containing 2 pottles, or 4 quarts, or
8 pints. But those pints and quarts, and consequently the
gallon itself, are different, according to the quality of the
things measured : the wine gallon, for instance, contains
231 cubic inches, and holds Sib, 5-foz, avoirdupois., of
pure water; the beer and ale gallon contains 282 cubic
inches, and holds 101b, 3-jOZ of water; and the gallon dry
measure, for grain, meal, &c, contains 26"Sf cubic inches,
and holds C)]b, lljoz of water.

GALLOPER, in Artillery, the name of a carriage serv-
ing for the very small guns, and having shafts so as to be
drawn without a limber.

GALVANISM, one of the most interesting branches of
modern philosophy, which derives its name from its illus-
trious discoverer, Professor Galvani, late of Bologna: it
comprises all those electrical phenomena arising from the
chemical agency of certain metals with different fluids.

The following circumstance is said to have given rise to
this very curious branch of philosophical inquiry. Pro-
fessor Galvani had long been considered as one of the most
able lecturers in the university to which he belonged; a
character that he owed to his accurate knowledge of ana-
tomy, chemistry, and other philosophical subjects: one
day, while engaged in some experiments in his laboratory,
a frog, which had been prepared for the purpose of making
soup for his wife, who was sick, was laid on the same table
where his electrical apparatus were placed ; when one of
his assistants in the experiments, by accident or design,
brought the point of a scalpel near the crural nerves of the
frog, lying not far from the conductor. In an instant the
muscles of its limbs were agitated with strong convulsions.
The experiment was repeated, the fact ascertained, and a
great many ingenious experiments were put in execution,
by which he investigated the law of nature, of which ac-
cident had thus given him a glimpse; and the success of
his inquiries has immortalized his name.

His publication on the subject was printed for the in-
stitute at Bologna in 17.91, entitled, Aloysii Galvani de
veribus Electricitatis in motu musculari Commentarius;
which work immediately excited the attention of philoso-
phers both in Italy and in other countries; and the ex-
periments were repeated and extended. Galvani, soon af-
ter he had ascertained the power of electricity on the
limbs of dead frogs, and other animals, was led to a fur-
ther discovery, that a similar effect might be produced
without the assistance of any visible electrical agent; that
.is, by the mere action of a metallic substance; and after
making a number of experiments on this head, he ascer-
tained, that the convulsions were produced only when dis-
similar metals were employed. These experiments, as be-
fore observed, were repeated by many philosophers ; none,
however, added any thing new to what Galvani had dis-
covered, excepting the celebrated Volra, of whom we shall
have frequent occasion to speak in the following pages:
thisgreat improvement took place in the year 1800; after
which time, many other philosophers pursued this subject
with very great success and applause; and none with more
than Professor' Davy, of the Royal Institution: Dr.
Wpllaston ami Mr. Nicholson likewise have done them-
selves considerable honour, by the number and accuracy
of their experiments, and the consequent developement of
many new principles connected with this science; and

GAL [ 5C7 ]

Mr. Wm.Cruickshank, a celebrated English chemist, by
his invention of the galvanic trough, seems to have fur-
nished the means of increasing the power of the galvanic
fluid to its greatest possible extent.

Having thus given a slight view of the discoverer and
subsequent improvers of this interesting science, we shall
proceed to explain, as much at length as our limits will
admit, the nature of its action, and the principal experi-
ments that are necessary, in order to form a connected idea
of its several effects.

The discoveries of Galvani, as before observed, were
made chiefly with dead frogs; the limbs of which, by

GAL

zinc seem to be rather the best; though silver and tin, or
copper and zinc, and some other combinations, are not
much inferior. If part of the nerve proceeding from a
prepared limb be wrapped up in a piece of tinfoil, or only
laid upon zinc; and a piece of silver laid with one end
upon the bare muscle, and with the other on the above-
mentioned tin or zinc ; the motion of the prepared limb
will be very vigorous. The two metals may be so placed,
as not to be in contact with the preparation; but in any
part of the circuit; which may be completed by means of
other conductors, as water, &c.

In a similar manner, the action of the calvanic fluid

means of electricity, acquired a tremulous motion, and may be made sensible on the human body.& Thus, let 1
generally a certain degree of extension; but this effect is man lay a piece of metal, as zinc, upon his tongue;' and (
likewise, sensible, in a greater or less degree, on any other
recently dead and skinned animal; though it is much more
perceptible in cold-blooded creatures. The manner in

which Dr. Galvani made his first experiments, was as fol-
lows : he skinned the legs of a frog recently dead, and left
them attached to a small part of the spine, but separated
from the rest of the body; and any other limb may be
prepared in a similar manner ; that is, the limb must be
deprived of its integuments, and the nerve which belongs
to it must be partly laid bare.

If the limbs thus prepared, for instance, (he legs of a
fro", be so situated, tl^it a little electricity may pass
through them, whether it be by the immediate contact of
an electrical body, or by the action of an electric atmo-
sphere ; as when the preparation is placed within a cer-
tain distance of an electrical machine, and a spark is taken

piece of some other metal, as silver, under his tongue;
then by forming the communication between these two
metals, either by bringing their outer edges in contact, or
by the interposition of some other piece of metal, he will
perceive a peculiar sensation, a kind of irritation, ac-
companied with a cool and subacid taste, not much unlike
that which is produced by artificial electricity. The sen-
sation seems to be more distinct, when the metals are of
the usual temperature of the tongue; and on making the
communication between the two pieces, the taste will be
perceived more sensibly. The effect is rather more re-
markable when the zinc touches the tongue in a small
part, and the silver jn a great portion of its surface, than
when the contrary have place; that is, when ihe zinc
touches more than the silver. Instead of the tongue, the
two metals also may be placed in contact with the3 roof of

from the prime conductor; the prepared legs will be in- the mouth, as far back as possible, and on completing the
slantly affected with a kind of spasmodic contraction, and communication, a similar effect will be produced

sometimes so strong as to jump a considerable way.

The like movements may be produced in the prepared
animal, without the aid of any apparent electrical agent.
In an animal recently dead, detach one end of a nerve
from the surrounding parts, taking care to cut it not too
near its insertion, into the muscle; remove the integu-
ments from over the muscles that depend on that nerve;
then take a piece of metal, as a wire, and touch the nerve
with one extremity of it, and the muscles with the other
extremity; on doing which, the prepared limbs will move
in the same manner, as when electricity is made to pass
through them. This, however, is not the most effectual
wayof formingthecommunication, though it will generally
succeed ; and the experiment will answer, whether the

Different persons are variously affected with this appli-
cation of metals; with some, the sensation of taste is so
slight, as scarcely to be perceptible ; while with others it is
very strong, and even disagreeable.

The above experiment, which is one of the most simple
that can be devised, was accidentally made about 6'0 years
back, by a person of the name of Sultzer, who stated the
fact in the following terms. If a piece of lead, and a similar
piece of silver, be laid together; and the edges of both be
brought in contact with the tongue, a taste is perceived
similar to that of vitriol of iron, at the same time that the
metals applied separately produce no effect. The ob-
server of this experiment docs not appear to have been at
all surprised with its result: for at that time the doctrine

preparation be laid upon conductors, or on electrics. But of vibrations was employed to explain all natural pheno-
if the communication between the nerve and the muscle mena ; and he therefore concluded, that some peculiar

be formed by nonconducting bodies, such as glass, wax, silk,
&c, then no motion will ensue.

The conducting communication between the muscles
and the nerve may consist of one or more pieces; and of
the same, or, which is much better, of different bodies,
connected together, as metals, water, a number of persons,
and even of wood : but it must be observed, that the va-
rious bodies which form this circuit must be placed with
full and perfect contact with each other, which is done
by pressure, or by the interposition of water, &c. The
less perfect conductors will answer only at first, when the
prepared animal retains its vigour; but when the power
begins to diminish, as it will after a few hours, then the
more perfect conductors only will answer, and even these
will produce various effects.

The most effectual way of producing those; movements,
is. by the application of two metals, of which silver and

vibration took place on the application of the metals; and
as every body was satisfied with this explanation, no'thiii"
further was thought of the experiment ; and thus a promi-
nent fact had slept in obscurity, from the time of Sultzer to,
the time of Galvani.

Having seen how the galvanic operation may be made
sensible to the taste, by a similar application it may be
made visible to the sight; thus, let a person in a dark
place, put a piece of tinfoil on the bulb of one of his eyes,
and a piece of silver, as a spoon, or the like, in his mouth;
then on completing the communication between the spoon'
and the tinfoil, a faint flash of white light will appear be-
fore his eyes : but this experiment may bo performed in a
more convenient manner, by placing a piece of zinc be-
tween the upper lip aud the gums, as high up as possible,
and a silver piece of money on the tongue; for when tlio
two metals are made to communicate,, either by the inume-

GAL

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GAL

diate contact of their ed^es, or by the interposition of good often as a communication was made between each end of
conductors, the flash of li«ht will be visible as before; but the pile; and as long as the pieces of cloth remained moist,
by continuing the contact of the two metals, the appear- An account of this discovery was communicated to the
ance of lioht^is not continued, it being only visible at the Royal Society, and published in the Philosophical Trans-
moment the contact takes place ; and sometimes, though actions ; but we do not hear of this celebrated philosopher
but rarely, at the instant of separation: it may therefore making any farther discovery, after the invention of the
be repeated at pleasure, by disjoining, and again Connect- pile, and ascertaining the nature and extent of its eflects
ing the two metals. on animals.

For a considerable time after the first discovery made A Voltaic pile similar to that we have been describing,

by Galvani, little more was known of its nature and its
eflects, than what is above stated ; and it was doubtful,
whether the convulsions in the animal's limbs, and the
sensations produced upon the human body, were owing to
an electrical property, peculiar to the animal parts, which
it was thought might be conducted through the metals
from one part to the other ; or to a small quantity of elec-
tricity, which might be supplied by the metals them-
selves : Galvani himself was of the former opinion, and en-
thusiastic in the application of the electrical theory to
animal economy; and particularly when he found the
metallic substances were capable of exciting muscular
motion, he was confirmed in his opinion, that the inherent
electricity of the animal was transmitted from the nerve to
the muscles by the metals employed.

About this time, however, Signior Volta took up the
subject, adopting a theory of a different nature; which,
though not strictly true, has contributed principally to its
rapid advancement. Instead of supposing, as Galvani
did, that the electricity belonged to the animal, he con-
sidered it as inherent in the different metals ; which idea
was truly fortunate for the cause of philosophy ; for the
very hypothesis of Galvani prevented him from deriving
any farther knowledge of the science ; because if the elec-
tricity belonged to the animal, no greater power was likely
to be produced, than what two pieces of metal could ef-
fect ; but on Volta's hypothesis, since he conceived the
action to be in the metals; it followed, or at least it was
highly probable, that by increasing their number, thoir
effect would also be increased in the same proportion.

He therefore repeated the experiments of Galvani, and
found that when two pieces of metal of different kinds were
placed in different parts of an animal ; at the same time
that the metals were brought in contact, or were con-
nected by a metallic arc ; as often as the contact was made,
convulsions were observed : he also found that the greatest
effect was produced, when the metals were zinc and silver;
and when several pieces of metals were employed, having
pieces of wet cloth between them, the effect appeared to
be increased as the number of pairs ; and hence his idea
of the pile.

This important discovery of accumulating the effects of
this species of electricity, was made by Volta in 1800,
and has hence been denominated the Voltaic pile. The
apparatus first made by Volta, consisted of a number of
pairs of zinc and silver plates, separated from each other
by pieces of wet cloth, the arrangement being as follows ;
viz, zinc, silver, wet cloth ; zinc, silver, wet cloth, and so
on : the silver plates were chiefly silver coins, the plates
of zinc, and pieces of wet cloth being of the same size ;
and he found the action of these combined pairs, to be
much more powerful when the pieces of cloth were moist-
ened with a solution of common salt, instead of pure wa-
ter : a pile consisting of forty pairs, he found to possess
the power of giving a pretty smart shock, similar to that
of a small electric jar , and that this effect took place, as

is represented, plate vii,fig. 6; it consists of pieces of silver,
about the size of half-crowns ; pieces of zinc of an equal
size; and pieces of cloth, or leather, or other bibulous
substance, a little less in diameter than the metallic-
plates ; these last being soaked in water, or any other
proper fluid. These pieces are disposed in the order, silver,
zinc, wet cloth; as indicated by the letters s, z, w; the
pieces of cloth, or leather, must be well soaked in the fluid;
but before- they are applied, they should be squeezed, in
order that the superfluous fluid may not run down the
outside of the pile, or insinuate itself between the contigu-
ous pieces of silver and zinc. Those pieces of cloth, par-
ticularly if soaked in plain water, lose their moisture very
soon, so that they can seldom last longer than for a day
or two, after which the pile must be decomposed, the
metallic pieces cleaned, those of cloth soaked again; and
the whole arranged as before.

The three rods r, it, n, are of glass, or of baked wood,
the piece o, slides freely up and down the rods, which
serves to prevent the Killing of the plates : but when such
a pile is to be very powerful, viz, to consist of many plates,
the best method is to form two, three, or more piles;
and to join them by pieces of metal, as c, c, fig. o ; where
two piles are j her, so that a is the negative ex-

tremity, ami b is the?' oilier, or positive extremity of the
whole arrangement, or of 'the two piles, considered as one;
that is, a is a silver plate, and b the zinc. It may also
be observed, that copper may be substituted for the silver,
without much decrease of the effect.

It was with piles similar to what we have been descri-
bing, but consisting of a greater or less number of plates
that galvanic experiments were first made in this country
by Messrs. Nicholson and Carlisle, and many important
chemical truths were developed by thisfroeans. At length
Mr. Cruickshank, after having employed the Voltaic pile
for various experiments, fell upon the happy idea of the
galvanic trough. This consists of a box of baked wood,
of a width and depth agreeable to the size of the plates,
and of convenient length for handing about ; the inside
being furnished With grooves, passing across the side and
bottom, for the reception of the metallic plates, and at
such a distance, that when these ate cemented into the
grooves, the trough may be divided into a number of
cells, about |- of an inch wide ; each of the plates being
compound; viz, consisting of two plates, one of zinc, and
the other of silver or copper, soldered together ; and the
order in which they were placed was such, that all the
zinc sides of the plates faced one way, and the. copper
sides the other : in fact, the trough was the pile placid
horizontally, the cells being for the reception of a fluid, to
answer the purpose of the pieces of wet cloth. The ce-
mentation of the metallic pieces into the sides and bottom
of the vessel, must be so accurate, as not to permit the
fluid to pass from one cell to another; the proper cement
for this purpose being made, by melting together 5 parts
of resin, 4 of bees-wax, and 2 parts of powdered red ochre :

GAL

[ 569 ]

GAL.

two or more such batteries may be joined together, in or-
der to increase the effect, as in the foregoing case of the
double pile. See fig. 7, pi. vii, which represents a single
battery; with its communicating rods.

The action of all these batteries is the greatest, when
they are first completed, or filled with the fluid ; and it
declines in proportion as the metal is oxidated, orthe lluid
loses its power : therefore, after a certain time the fluid
must be changed, and the metallic pieces cleaned, by re-
moving the oxidated surface ; which may be done, either
by filing or by rubbing them with sand, or sand paper ;
or by immersing them for a short time in diluted mu-
riatic acid, ana then wiping them with a coarse cloth :
the last method is to be preferred for the trough; wiping
them by means of a stick, which may be introduced be-
tween them, with a piece of cloth upon it.

Having thus described the construction of the best
and most simple galvanic arrangements, we shall proceed
to state the effects of these combinations, in a popular
manner; but omitting as much as possible those experi-
ments that are purely of a chemical nature; as these are
not only too numerous to be inserted in the space we have
assigned for this article, but they are also incongruous
with the plan of the present work ; and we must therefore
refer the curious reader, who wishes for particular infor-
mation on this /Very interesting branch of philosophical in-
quiry, to the following works, viz, Rees's Cyclopcedia, and
the professed chemical treatises, for more ample accounts.

The various powers of different simple galvanic circles
may be ascertained, by applying them to such animal pre-
parations as have their irritability more or less exhausted.
Thus, M. Volta in his letter to Gren, says, " If you take
a frog, the bead of which has been cut off, and which has
been deprived of all life, by thrusting a needle into the
spinal marrow, and immerse it without skinning, or taking
out the bowels, or any other preparation, into twoglasses
of water, the rump into one and the legs into the other,
it will be strongly agitated and violently convulsed, when
you connect the water in both glasses, by a bow formed
of very different metals, such as silver and lead ; or, what
is better, silver and zinc ; but this will not be the case
when the two metals are less different in regard to their
powers, such as gold and silver, silver and copper, copper
and iron, tin and lead ; but what is more, the effect will
be fully produced on this so little prepared frog, when
you immerse in one of the two glasses, the end of a bow
merely of tin or zinc ; and into the other glass, the other
end of this bow, which lias been rubbed over with a little
alkali : and you may perform the experiment still better
with an iron bow, one end of which has been covered with
a drop or thin coat of nitrous acid ; but it will succeed
beyond all expectation, when you take a silver bow, hav-
ing a little sulphurate of potass adhering to its extremity."
It had long been asserted, that porter, and some other
liquors, when drank out of pewter pots, had a different
taste, from what they had when drank out of a glass or
earthenware; but it was little expected, that this was a
truth intimately connected with so interesting a branch of
philosophy; though, now, there is no doubt of the fact.
A similar, but more evident effect, may be made sensible
by the following experiment. If a tin bason be filled with
soap-suds, lime water, or a strong ley (which last is the
best) ; and if you then lay hold of the bason with both
hands, having first moistened them with pure water, and
Vol. I.

apply the tip of your tongue to the fluid in the bason, you
will immediately be sensible of an acid taste on your
tongue, which is in contact with the alkaline liquor.
This taste is very perceptible, and for the moment pretty
strong; but it is afterwards changed into a different one,
less acid, but more saline and pungent, until at iast, it
becomes alkaline, and sharp, in proportion as the fluid
acts more upon the tongue.

In the foregoing part of this article, other experiments
have been noticed, made by single combinations of dif*-
ferent metals ; we shall therefore now attend to those that
are made by means of the more powerful apparatus, such
as we have already described ; viz, the Voltaic pile ; and
the trough or battery, invented by Mr. Cruickshank.
These powerful combinations not only convulse the pre-
pared limbs of a frog, or produce the appearance, of a
flash of light before the human eye, but they exhibit all
the phenomena of electricity, in a very considerable de-
gree : an arrangement of the kind we have described, will
give the shock ; it affects the electrometer, shows a lu-
minous spark, accompanied with an audible report; it
burns metallic, and other bodies ; and continues in action
for a very considerable time ; viz, until the chemical ac-
tion between the component parts of the battery be quite
exhausted. Some of which effects are enumerated in the
following paragraph.

If you take a Voltaic pile, consisting only of 20 repeti-
tions of simple combinations, and touch with one hand
one extremity of the pile, as at b, fig. 5, and apply the
other hand to the other extremity, as at a; a very slight
shock will be perceived, like that which is communicated
by a Leyden phial weakly charged; and it will be scarcely
felt beyond the fingers, or at most the wrists; but the
shock is felt as often as the contact is renewed ; and if you
continue the hands in contact with the extremities b and
a, a slight but continued irritation will be perceived :
and when the hand, or other part of the body, which
touches the extremity of the battery, is excoriated or
wounded, this sensation becomes disagreeable, and rather
painful; but as the dry skin of the human body is seldom
capable of conducting this shock, the touching fingers
should be well moistened with water ; and it will be still
better to immerse a wire, that proceeds from one extre-
mity of the battery, in a bason of water, wherein may be
plunged one hand ; grasping, at the same time, with the
other hand well moistened, a large piece of metal, as a
large silver spoon ; by which means the shock will be
felt more distinctly. Instead of one person, several may
join hands, (which must be all well moistened,) and on
completing the circuit, they will all feel the shock at the
same time, as with an electrical discharge : but the
strength of it is thus much diminished by passing through
the several persons, as it is in general by passing through
bodies that are not perfect conductors.

The shock of a battery, containing 50 or 60 pairs of1
combinations of silver or copper and zinc will be felt
as high as the elbows ; and the combined effect of five or
six such batteries will of course be much greater, and
such as few men would be willing to receive.

From the foregoing experiment it appears, that between
the effects resulting from the common electrical machines,
and those produced with the pile, there is this difference ;
that in the forrncr, the effects are annihilated at the instant
of contact ; while in the other, they remain as long as the
4 D

GAL

[ 670 ]

G A It

contact is continued. The pile once charged, becomes
thus a reservoir of electricity, which, without the aid, and
as it were without the knowledge, of the operator, fills it-
self spontaneously; regains continually what is taken
from it ; and would be inexhaustible, if the humid bodies,
of which the pile is composed, could be prevented from
losing their moisture.

If a wire, proceeding from one extremity of a pretty
strong galvanic pile, be made to communicate with the in-
side coating of a common Leyden phial or electrical bat-
tery ; and a wire which proceeds from the other ex tre-.
mity of the pile, be made to communicate with' the outside
coating of the same jar or battery ; the latter will become
weakly, but almost instantaneously charged, in the same
manner as would have been done with a few turns of the
electrical machine; and with that charge you may either
give the shock, or effect an electrometer, &c. In fact,
every experiment convinces, more and more, that a gal-
vanic battery produces a vast quantity of electric fluid,
which is but little condensed ; and indeed it would be im-
possible to suppose that the electric fluid could proceed
in a very condensed state, from an arrangement of bodies,
which, whether more or less, are however all good con-
ductors of electricity; for if the fluid was much con-
densed at one extremity of the battery, and much rarefied
at the other extremity, the condensation would soon be
made through the pile itself: and indeed it is difficult to
comprehend, why this compensation does not take place
in all cases.

Having mentioned above, that the charge of a galvanic
pile or battery may be communicated to a common
electric battery, it is scarcely necessary to observe, that
the same may be communicated to a condenser or a mul-
tiplier; and from hence to the electrometer ; and if the
pile consists of 200 repetitions, the electrometer will be
affected by simple contact.

Tiie spark or the discharge of an electric battery, when
sent through thin inflammable bodies that are in contact
with the common or oxygen air, sets them on fire, and
consumes them with wonderful activity: it also fires gun-
powder, hydrogen gas, phosphorus, and other combustible
matter: it also renders red hot, fuses, and consumes, very
slender metallic wires, and metallic leaves.

The method of employing the power of the battery for
such purposes, is shown at fig. 7, where ab represents a
powerful galvanic battery ; acdf is a wire which commu-
nicates with the first plate of the battery a ; bkiiig is
another wire, which communicates with the last plate at
b ; de and hi are two glass tubes, through which those
wires pass, and into which they are fastened sufficiently
steady : these tubes serve to move the wires by ; or if the
operator applies his fingers to the middle part ot those
tubes, he may move the wires wherever he pleases, with-
out the fear of receiving a shock, If the two extremities
v, g, are brought sufficiently near to each other, the spark
will be seen between them ; and it is between those extre-
mities, that the combustible substances, or metallic leaf,
is to be placed, in order to be fired or consumed ; and in
the figure is shown the situations of the wires, as they must
be posited, in order to accomplish the firing of gunpowder,
or the fusion of metals, &c, &c. A battery, consisting
of 200 puirs of metallic plates, viz, copper and zinc,
each five inches square, will melt 3 inches of very
fine iron wire; and a platina wire about T-fT of an inch

in diameter, may be melted into a globule by the same ap-
paratus ; and more powerful effects will of course be pro-
duced from the combination of two or more such batteries.
Under the exhausted receiver of an air-pump, the gal-
vanic battery acts less powerfully than in the open air;
but its effect is increased in oxygen air. The Hash of light
that appears before the eye ot the experimenter, when the
eye itself, or some other part not very remote from it, is
put in the circle of a galvanic combination, does not ap-
pear much greater when a. battery is employed, than
when two plates are applied in the same manner; which
experiment has been already mentioned ; but when the
battery is used, the sensation of a flash may be produced
in various ways. If one hand or both be placed in per-
fect contact with one extremity of the battery, and al-
most any part of the face be brought into contact with the
other extremity of the battery ; the flash will appear very
distinctly, the operator being in the dark, or keeping his
eyes shut. This flash appears veiy strong, when a wire,
that proceeds from one extremity of the battery, is held
between the teeth and rests on the tongue, while the
other wire is held in the hand ; and in this case, the lips
and tongue are convulsed, the flash appears before the
eyes, and a very pungent taste is perceived in the mouth.
If any part of the human body forming part of the
circuit of a galvanic battery be kept some time in that si-
tuation, the irritation or numbness is more or less distinct,
and more or less painful, according to the sensibility of
the parts concerned.

The above are some of the most popular experiments
that are made by means of a galvanic battery ; but by far
the greater number of those that have been made have
been directed to chemical purposes, and the results that
have thus been obtained are truly curious and interest-
ing: an extensive field has thus been opened for chemical
research, too vast for human conception, by the intro-
duction of this powerful agent ; and more discoveries
have through its means been made in one year than per-
haps any one century can boast. But these experiments,
as we before observed, do not properly belong to a work
of this nature ; we have therefore endeavoured, without
introducing them, to given familiar idea of the nature and
power of this wonderful agent.

GARDliCAUT, or Guard du Cord, in a watch, is
that which stops the fusee, when wound up, and for that
end is driven up by the spring. Some call it guard-cock ;
others guard du Gut.

GARRISON Guns, ^uch as arc mounted and used in
a garrison, consisting of the following weights, viz, the 42,
32, 24, 18, 12, Q> and 6 pounders; being made either of
brass or iron.

Table of the Weight and Dimensions of Garrison Guns.

B.

ass Garrison Guns.

i.

in Garrison Guns.

Shot

Length

Wim«1i'.

Shot

Lengili

Weight,

II,.

f. in.

Cw. qr. Il>.

lb.

f. in.

Civ. qr. lb.

42

10 0

()-l. 0 0

82

i) 8

56' 0 0

32

1) 2

49 2 18

24

9 8

48 0 0

24

S 4

37 0 0

18

9 o

30' 0 0

18

7 a

27 3 0

12

7 8

24 0 0

12

6 7

18 2 0

9

7 0

18 0 0

.9

a o

13 3 0

6

G l

12 0 0

6

5 3

9 i o

4

5 4

8 0 0

GAS

C 571 ]

GAS

GAS. The term gas (with the exception of the common the same with the fluid called lixed air; and some other

■air we breathe) is used to signify any permanently elastic factitious airs; and from him has been employed by

•e\halation afforded in chemical processes. It was ori- modern philosophers as a general name for all the fluids

ginally given by Van Helmont to the vapour of charcoal, about which areology is conversant.

Table, showing the absolute weights and specific gravities of gases, and the quantity of each absorbed by water.
Temperature 60° Fahrenheit, Haromeler 29-8.
B. denotes Brisson ; Cr.Cruickshank; D. Davy; Dal. Dalton; Dei. Deiman ; H. Henry; K. Kirwan; S.Shuckburgh;!
T. Thomson; Th. Thonard ; Tr. Tromsdorff.

Nitric acid --------

Sulphurous --------

Vapour of ether -------

Muriatic acid --------

Vapour of alcohol ----,--

Nitrous oxide -------

Carbonic acid -------

Ditto ditto

Muriatic acid --------

Sulphuretted hydrogen - - - - -

Nitric oxide --------

Ditto ditto --------

Oxygen gas --------

Sulphuretted hydrogen

Oxygen gas --------

Atmospheric air - - - - - - -

Azotic gas --------

Ditto

Carbonic oxide -------

defiant gas .--,---.---
Hydro carburet from stagnant water -
Ditto from coal -------

Ditto from ether -------

Ammonia --------.-

Ditto

Arsenicated hydrogen gas - - - -

Hydro carburet from alcohol - - -
Ditto from water over ignited charcoal
Hydrogen gas --------

Phosphuretted hydrogen ------

Weight of

1 00 cubic

inches in

grains.

76

70215

62-1*

59-8

51-5

50-1

46-5

45-5

44-7

38-17

37

34-3

34'74

34-286

34

31

30-535

30-45

30

28-18

20-66

20-2

20

18-16

18

16

14-5

26-13

Specific Gravity.

Water, the Air the

Standard at 1000. Slandard. Authorlty-

3-
2-75

1-985
1-84
1-802
1-765

1 -465

1-36

1-39

1-36

1-35

1-2279

1-21

1-20

1-185

0-715

0-713
0-6499

2425
2240
2250
1929
2100
1615
1500
1470
1430
1231
1193
1105
1127
1142
1103
1000
985
980
967
905
666
650
645
535
5S0
529
516
468
84

D.

K.

Dal.

K.

Dal.

K. "

D.

B.

T.

K.

D.

D.

K.

K.

S.

K.

D.

Cr.

Dei.

Dal.

Dal.

Cr.

K.

D.

Tr.

Cr.

Cr.

K.

Number of cubic inches

tbSorbed by 100 inches of

water.

86

108

5I5CO
5

108
' 3-7

1-53

201
12.5
1.40

47500

1-61
2-14

T.

H.
H.

H.

H.

Dal.

H.

The following 'are the experiments made by professor
Jacquin of Vienna on the different gases, as the vehicle
of sounds. — A glass-bell being furnished with a metallic
stopper cemented to a neck at the top ; and in the bore
of this cock within the glass, a small flute or pewter (etain)
about six inches in length was fixed. The glass being
then placed on the shelf of the pneumatic vessel, and filled
with any particular kind of gas, a bladder also filled with
the same gas, and provided with a cock, was adapted to
the external aperture of the cock belonging to the bell-
glass. In this disposition of the apparatus the flute was
made to sound by gently pressing the bladder. Compara-
tive experiments were also made with atmospheric air,
oxygen,, hydrogen, carbonic acid, and nitrous gas. The
intensity of the sound did not vary; but when compared
with that produced by atmospheric air, the oxygen gas af-
forded a sound half a tone lower ; azotic gas, prepared by
diffrrent methods, constantly gave a sound half a tone
lower; hydrogen gas gave nine or eleven tones higher;
carbonic acid gas gave one third lower; and nitrous gas

also very nearly a third lower. A mixture of oxygen gas
and azote, in the proportion contained in atmospheric air»
afforded the tone of this last ; that is to say, it was half a
tone higher than each of the component parts alone.
When the two gases were not uniformly mixed, the sound
was abominably harsh. Chladni intends to give a fuller
account of these interesting experiments. Journal de
Physique, vol. 4, N. S. p. 57.

GASC01GNE, an ingenious young astronomer of Lan-
cashire, who, about the year 1640, made many curious
observations on some astronomical objects, and greatly
improved that science, by the invention of the microme-
ter, and by applying telescopic sights to astronomical
quadrants. See accounts of his inventions in the article
Micrometer. See also my Abridg. of the Philos. Trans,
vol. i, p. 169, 1<)5, and vol. x, p. 369.

Mr. Gascoigne was slain in the civil wars in the time
of Charles I.

GASSENDI (Peter), one of the most febrated phi-
losophers France has produced, was .born at Cl •
4 D 2

GAS

[ 57a ]

G A U

about 3 miles from Digne in Provence, in the year 1592.
When a child, he took great delight in gazing at the nioon
and stars whenever they appeared: This pleasure often
drew him into bye-places, that he might feast his eyes
freely and undisturbed ; by which means his parents had
him often to seek, with many anxious fears and apprehen-
sions. In consequence of this promising disposition, he
was sent to the best schools, to cultivate it with the in-
structions of the first masters. He profited so well by
these aids, that he was invited to be professor of rhetoric
at Digne, before he was quite 16 years of age. After fill-
ing this office three years, upon the death of his master at
Aix, he was appointed to succeed him as professor of phi-
losophy. After a few years residence here, he composed
his Paradoxical Exercitations ; which coming to the hands
of Nicholas Peircsc, that great patron of learning joined
with Joseph Walter, prior of Valette, in promoting him ;
and, having entered into holy orders, he was first made
canon of the church of Digne and doctor of divinity, and
then warden or rector of the same church.

Our author's fondness for astronomy grew up with his
years ; and his reputation daily increasing, he was ap-
pointed the king's professor of mathematics at Paris in
1645. . This institution being chiefly intended for astro-
nomy, Gassendi read lectures on that science to a crowded
audience. However, he did not long enjoy this situation ;
for a dangerous cough and inflammation of the lungs ob-
liged him, in l6±7, to return to Digne for the benefit of
his native air. Having thus, and by the intermission of
his studies, recovered his health, he again returned to Pa-
ris in l6'53 ; where, after first writing and publishing the
lives of Tycho Brahe, Copernicus, Purbach, and Regio-
montanus, in 1654 he again renewed his astronomical la-
bours, with the design of completing the system of the
heavens. But while he was thus employed, too intensely
for the feeble state of his health, he relapsed into his
former disorder, under which, with the aid of too copious
and numerous bleedings, by order of three physicians, he
expired in the year 1655, at 63 years of age.

Gassendi wrote against the metaphysical meditations of
Descartes; and divided with that great man the philoso-
phers of his time, almost all of whom were either Carte-
sians or Gassendists. To his knowledge in philosophy and
mathematics, he joined profound erudition and deep skill
in the languages. He wrote, 1. Three volumes on Epi-
curus's philosophy; and six others, which contain his
own philosophy. — 2. Astronomical Works. — 3. The lives
of Nicholas de Pciresc, Epicurus, Copernicus, Tycho
Brahe, Purbach, and Regiomontanus. — 4. Epistles, and
other treatises. All his works were collected together,
and printed at Lyons in 1658, in 6' volumes folio.

Gassendi was the first person who saw the transit of
Mercury over the sun, viz, Nov. 7, 1031 ; as Ilorrox first
predicted and showed the transit of Venus. — His library
was large and valuable: to which he added an astronomi-
cal and philosophical apparatus, which, lor their accuracy
and magnitude, were purchased by the emperor Ferdi-
nand ni. — it appears by his letters, printed in the 6th
volume of his works, that he was often consulted by the
most celebrated astronomers of his time, as Kepler, Lon-
gomontanus, Snell, Ilevelius, Galileo, Kircher, Bulliald,
and others: and he lias generally been esteemed one of
the founders of the reformed philosophy, in opposition to
the grOutldless hypotheses and empty subtleties of Ari-
stotle and the schoolmen.

GATE, in Architecture, a large door leading or giving,
entrance into a city, town, castle, palace, or other consi-
derable building.

Gate, in Fortification, is made of strong planks with
iron bars to oppose an enemy. They are generally made
in the middle of the curtain, whence they are seen and
defended by the two flanks of the bastions. They should
be covered with a good ravelin, that they may not be seen
or infiladed by the enemy. These gates belonging to a
fortified place, are passages through the rampart, which
may be shut and opened by means of doors and a port-
cullis. They are either private or public.

Private Gates, are those passages by which the troops
can go out of the town unseen by the enemy, when they
pass to and fro on the relief of the duty in the outworks,
or on any other occasion which is to be concealed from
the besiegers.

Public Gates, are those passages through the middle of
such curtains to which the great roads or public ways
lead. The dimensions of these are usually about 13 or 14
feet high, and 9 or 10 feet wide, continued through the
rampart, with proper recesses for foot passengers to stand
in out of the way of wheel carriages.

GAUGE-Line, a line on the common gauging-rod, used
for the purpose of gauging liquids. See GA.UGlNG-.Rorf.

GAVGn-Point, of a solid measure, is the diameter of a
circle, whose area is expressed by the same number as the
solid content of that measure. Or it is the diameter of a
cylinder, whose altitude is 1, and its content the same as
of that measure. Thus, the solid content of a wine gal-
lon being 231 cubic inches; if a circle be conceived to
contain so many square inches, its diameter will be 17'15 ;
which is therefore the gauge-point for wine measure. And
an ale-gallon containing 282 cubic inches ; by the same
rule, the gauge-point for ale measure will be found to
be 18'95. In like manner may the gauge-point for any
other measure be determined. Hence it follows, that when
the diameter of a cylinder in inches is equal to the gauge-
point in any measure, given likewise in inches, every inch
in its length will contain an integer of the same measure.
So in a- cylinder whose diameter is 17*1 5 inches, every
inch in height contains one entire gallon in wine measure ;
and in another, whose diameter is 18'9o, every inch in
length contains one ale gallon.

GAUGER, an officer appointed by the commissioners
of excise, to gauge, measure, or examine, all casks, tuns,
pipes, barrels, hogsheads, of beer, wine, oil, &c.

GAUGING, the art or act of measuring the capacities
or contents of all kinds of vessels, and determining the
quantity of fluids, or other matters contained' in them.
These are principally pipes, tuns, barrels, rundlets, and
other casks; also backs, coolers, vats, &c.

As to the solid contents of all prismatical vessels, as
cubes, parallelopipedons, cylinders, &c, they are found
by multiplying the area of the base by their altitude. And
the contents of all pyramidal bodies, and Cones, are equal
to l-3d of the same.

In gauging, it has been usual to divide casks into four
varieties or forms, denominated as follows, from the sup-
posed resemblance they bear to the frustums of solids of
the same names: viz, •

1. The middle frustum of a spheroid,

2. The middle frustum of a parabolic spindle,

3. The two equal frustums ofa paraboloid,

4. The two equal frustums of a cone.

G A U

573 ]

GEL

And particular rules, adapted to each of these forms,
may be found in most books of gauging, and in my Men-
suration, p. 436, &c. But as the form is imaginary, and
only guessed at, it seldom happens that a true solution is
obtained in this way ; besides which, it is very troublesome
and inconvenient to have so many rules to put in prac-
Mice. I shall therefore give here one rule only, from
pa. 451 of that book, which is not only general for all
casks that are commonly met with, but quite easy, and
very accurate, as .having been often verified and proved
by filling the casks with a true gallon measure.

General Rule. Add into one sum,
39 times the square of the bung diameter,

25 times the square of the head diameter, and

26 times the product of those diameters ;

multiply the sum by the length of the cask, and the pro-
duct by the number '00034; then this last product di-
vided by 9 will gwe the wine gallons, and divided by 11
will give the ale gallons.

Or,(39Bz ■+■ 25rr -+- 26bii) x —'-is thecontentin inches;

which being divided by 231 for wine gallons, or by 282
"for ale gallons, will be the content.

For Example. If the length of a cask be 40 inches, the
bung diameter 32, and the head diameter 24.
Here - 321 x 39 = 39936
and - 24? x 25 = 14400

and - 32 x 24 x 26 = 1996S

the sum - 74304

multiplied by - 40

and divid. by 1 14)2972160
gives - - 26071 cubic inches ;
this divided by 231 gives 112 wine gallons,
or divided by 282 gives 92 ale gallons.

But the common practice of gauging is performed me-
chanically, by means of the gauging or diagonal rod, or
the gauging sliding rule, the description and use of which
here follow.

Gauging, or Diagonal, Rod, is a rod or rule adapted
for determining the contents of casks, by measuring the
diagonal only, viz, the diagonal from the bung to the ex-
tremity of the opposite stave next the head. It is a square
rule, having 4 sides or faces, being usually 4 feet long, and
folding together by means of joints. On one face of the
rule is a scale of inches, for taking the measure of the dia-
gonal; to these are adapted the areas, in ale gallons, of
circles to the corresponding diameters, like the lines on
the under sides of the three slides in the sliding rule, de-
scribed below. And on the opposite face are two scales,
of ale and wine gallons, expressing the contents of casks
havi .^ the corresponding diagonals; and these are the
lines which chiefly constitute the difference between this
■instrument and the sliding rule ; for all the other lines
upon it are the same with those on that instrument, and
are to be used in the same manner.

To use the Diagonal Rod. Unfold the rod quite straight
and put it in at the bung hole of the cask to be gauged,
till its end arrive at the intersection of the head and oppo-
site stave, or to the farthest possible distance from the
bung-hole, and note the inches and parts cut by the mid-
dle of the bung; then draw out the rod, and look for the
same inches and parts on the opposite face of it, and an-
nexed to them arc the contents of the cask, both in ale and
wine gallons.

For Example. Let it be required to find, by this rod, the

content of a cask whose diagonal measures 34'4 inches ;
which answers to the cask in the foregoing example, whose
head and bung diameters are 32 and 24, and length 40
inches; for if to the square of 20, half the length, be added
the square of 28, half the sum of the diameters, the square
root of the sum will be 34'4 nearly. Now, to this dia-
gonal 34-4, corresponds, on the rule, the content 91 ale
gallons, or 111 wine gallons; which are but I less than
the content found by the former general rule above given. '
Gauging Rule, or Sliding Ride, is a sliding rule parti-
cularly adapted to the purposes of gauging. It is a square
rule, of four faces or sides, three of which are furnished
with sliding pieces running in grooves. .The lines on these
faces are mostly logarithmic ones, or distances which are
proportional to the logarithms of the numbers placed at
the ends of them ; which kind of lines was disposed first
upon rulers, by Mr. Edmund Gunter, for expeditiously
performing arithmetical operations, using a pair of com-
passes for taking off and applying the several logarithmic
distances. Instead of the compasses, sliding pieces were
afterwards added, by Mr. Thomas Everard, as more cer-
tain and convenient in practice, from whom this sliding
rule is often called Everard's Rule. For the more parti-
cular description and uses of this rule, see my Mensura-
tion, p. 429, 4th ed.

The writers on gauging arc, Beyer, 'Kepler, Dechales,
Hunt, Everard, Dougherty, Shettleworth, Shirtcliffe, Lead-
better, Moss, &c.

GAZONS, in Fortification, turfs, or pieces of fresh
earth covered with grass, cut in form of a wedge, about a
foot long and half a foot thick, to line or face the outside
of works made of earth, to keep them up, and prevent
their mouldering.

GELLIBRAND (Henry), professor of astronomy at
Gresham-college, was born in London the 27th of Nov.
1597. He was sent to Trinity-college, Oxford, in 1615,
and took his degree jn arts 1619- He then entered into
orders, and became curate of Chiddingstonc in Kent.
Afterwards, taking a great fancy to mathematics, by hap-
pening to hear one of Sir Henry Saville's lectures in that
science, he immediately applied himself to the close study
of that noble science, and relinquished his fair prospects
in the church. Content with his private patrimony, which
was now come to his inheritance by the death of his fa-
ther, the same year he entered again a student at Oxford,
making mathematics his sole employment. He made such
proficiency in this science before he proceeded a. m.,
which was in l6'23, that he drew the attention and inti-
mate friendship of Mr. Henry Briggs, then lately removed
from the geometry professorship in Gresham-college to
that of Savilian professor of geometry at Oxford, by the
founder Sir Henry Saville, and who, upon the death of
Mr. Gunter, procured for our author the professorship of
astronomy in Gresham-college, to wdiich he was elected
in the beginning of the year 16"27- His friend Mr. Briggs
dying in 1030, before he had finished the introduction to
his Trigonomctria Britannica, he recommended the com-
pleting and publishing of that work to our author. Gclli-
brand accordingly added a preface, and the application
of the logarithms to plane and spherical trigonometry, &c,
and the whole was printed at Gouda, under the care of
Adrian Vlacq, in 16*33.

While Mr. Gellibrand was preparing that work, he was
brought into trouble in the high-commission court, by Dr.
Laud, then bishop of London, on account of an almanac,

GEN

[ 374 ]

GEN

published by William Beale, servant to Mr. Gellibrand,
for ibe year l631, with the approbation of his master.
In this almanac, the popish saints, then usually put into
calendars, were omitted, and the names of other saints and
martyrs, mentioned in the Book of Martyrs, were placed
hi their stead, as they stand in Fox's calendars. This, it
seems, gave much offence to the bishop, and occasioned
the prosecution. But when the cause came to be heard,
it appeared that other almanacs of the same kind had
formerly been printed ; upon which, both master and man
were acquitted by Abp. Abbot and the whole court, Laud
only excepted : which was afterward made one of the
articles against him on his own trial.

It seems Gellibrand was strongly attached to the old
Ptolemaic system. For when he went over to Holland,
about the printing of Briggs's book above mentioned, he
had some discourse with Lansberg, an eminent brother
astronomer in Zealand, who affirming that he was fully
persuaded of the truth of the Copernican system ; ourau-
ther observes, " that this so styled -a truth he should re-
ceive as an hypothesis ; and so be easily led on to the con-
sideration of the imbecility of man's apprehension, as not
able rightly to conceive of this admirable opifice of God,
or frame of the world, without falling foul of so great an
absurdity :" so firmly was he fixed in his adherence to the
Ptolemaic system. Gellibrand wrote several things after
this, chiefly tending to the improvement of navigation,
which would probably have been further advanced by him,
had his life been continued longer; but he was untimely
carried off by a fever, in 1636, at 39 years of age.

The character of Mr. Gellibrand is that of a plain, plod-
ding, industrious, well-intentioned man, with little inven-
tion or genius. His writings are chiefly as below.

1. Trigonometria Britannica; or the Doctrine of Tri-
angles, being the 2d part of Briggs's work above mentioned.
— 2. A small tract concerning the longitude' — 3, A dis-
course on the Variation of the Magnetic Needle; annexed
to Wright's Errors in Navigation detected. — 4. Institution
Trigonometrical, with its application to astronomy and
navigation; 8vo, 1635. — 5. Epitome of Navigation, with
the necessary tables; Svo. — 6. Several manuscripts never
published; as, The Doctrine of Eclipses, A Treatise of
Lunar Astronomy, A Treatise of Ship-building, &c.

GEMINI, in Astronomy, the Twins ; a constellation
or sign of the zodiac, the third in order, representing Cas-
tor and Pollux; and it is marked thus, n. The stars in
the sign Gemini are. in Ptolemy's catalogue 25, in Tycho's
25, in Mevelius's 38, and in the Britannic catalogue 85.

GEMMA (Renier), commonly called Gemma Fri-
sius, from his native country Friezland, a Dutch physi-
cian and mathematician, who became medical professor at
Lnuvain, where he died in 1558, at 50 years of age. lie
wrote, 1. Methodus Arithmeticae : 2. De Usu Annuli
Astronomici: 3. De Locorum describendorum Rationc,
deque Distantiis corum inveniendis : 4. Demonstrationes
Geometrifte de usu Radii Astronomici: 5. De Astrolabio
Catholico Liber, ecc. See further under Frisi.

Uemma (Cornelius), son of the former, a good astro-
nomer, was born at Louvain in 1535, and died 1579- He
wrote several tracts, as, De Prodigiosa Specie Naturaque
Cometas: and one on the star which appeared in 1577,
Antw. 1578, in Svo.

GENERATE, in Music, is used to signify thcopeia-
tinii of that mechanical power in nature, which every
sound has in producing one or more general sounds. Thus

any given sound, however simple, produces along with
itself, its octave and two other sounds extremely sharp,
viz, its 12th above, that is to say, the octave of its fifth;
and the other the 17th above, or, in other words, the double
octave of its third major. #

Whether we suppose this procreation of sounds to result
from an aptitude in the texture and magnitude of certain
panicles in the air, for conveying to our- ears vibrations
that bear those proportions one to another, as beino de-
termined at once by the partial and total oscillations of
any musical string; or from whatever oeconomy of nature
we choose to trace it; the power of one sound thus to
produce another, when in action, is said to generate. The
same word is applied by Signor Tartini and his followers,
to any two sounds which, simultaneously heard, produce
a third.

GENERATED, is used by some mathematical writers
for whatever is produced by arithmetical operation, or in
geometry by the motion of other magnitudes. Thus 20 is
the product generated of 4 and 5 ; ab that of a and b, 4,
8, \6, &c, the powers generated of or from the root 2,
and a1, a3; aA, &c, those from the root a. So also, a cir-
cle is generated by the revolution of a line about one of
its extremities ; aconeby the rotation of a right-angled
triangle about its perpendicular ; a cylinder by the rotation
of a rectangle about one of its sides, or, otherwise, by the
motion of a circle in the direction of a right line, and keep-
ing always parallel to itself.

GENERATING Line or Figure, in Geometry, is that
which, by any kind of supposed motion, may generate, or
produce, some other figure, plane, or solid. Thus a line,
according to Euclid, generates a circle; or a right-angle,
triangle, a cone, &c ; and thus' also Archimedes supposes
his spirals to be generated by the motions of generating points
and lines. The figure thus generated, is called theGenerant.

It is a general theorem in geometry, that the measure of
any genorant, or figure produced by any kind of motion of
^any other figure, or generating quantity, is equal to the
product of this generating quantity drawn into the length
of the path described by its centre of gravity, whatever the
kind of motion may be, whether rotatory, or direct, &c. —
In the modern analysis, or fluxions, all sorts of quantities
are considered as generated by some such motion, and the
quantity hereby generated is called a Fluent.

GENERATION, In .Mathematics, is used for the forma-
tion or production of any geometrical figure, or other quan-
tities. Such as of the figures mentioned in the foregoing
articles, or the generation of equations, curves, solids, &c.

Generator, in Music, signifies the principal sound or
sounds by which others are produced. Thus, the lowest
c fur the treble of the. harpsichord, besides the octave, will
strike an attentive ear with its 12th above, or G in alt. and
with its 17th above, or E in alt. The c, therefore is called
their generator, the G and E its products, or harmonies.
But in the approximation of chords, for o, its octave he-
low is substituted, which constitutes a filth from the gene-
rator or lowest c; and for e, is likewise substituted its
15th below, which, with the above-mentioned c, forms a
third major. To the lowest notes, therefore, exchanged
for these in alt. by substitution, the denominations of pro-
ducts or harmonies are likewise given, while the c retains
the name of their generator. But still, according to the
system of Tartini, two notes in concord, which when
sounded, produce a third, may he termed the concurring
generators of that third. See Generation Ilarmonique,

GEO

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GEO

par M. Rameau ; see also that delineation of Tartini's
system called The Power and Principles of Harmony.

GENESIS, in Geometry, means much the same as ge-
neration mentioned above, being the formation of a line,
surface, or solid, by the motion or flux of a point, line, or
surface ; as of a globe by the rotation of a semi-circle
about its diameter, &c. In the genesis of figures, the line
or surface which moves, is called the Dcscribent; and the
line round which, or according to which, the revolution
or other motion is made, the Dirigent.

GEOCENTRIC, is said of a planet or its orbit, to de-
note its having the earth for its centre. The moon alone
is properly geocentric. And yet the motions of all the
planets may be considered in respect of the earth, or as
they appear from the earth, and thence called their geo-
centric motions. — Hence also the terms geocentric place,
or latituJe, or longitude, &c, as explained below.

Geocentric Place, of a planet, is the placewhere it ap-
pears to us, from the earth ; or it is a point in the ecliptic,
to which a planet, seen from the earth, is referred.

Geocentric Latitude, of a planet, is its latitude as
seen from the earth; or the inclination of a line, connect-
ing the planet and the earth, to the plane of the earth's
(or true), ecliptic. Or it is the angle which the said line,
(connecting the planet and the earth) makes with a line
drawn to meet a perpendicular let fall from the planet to
the plane of the ecliptic.

Geocentric Longitude, of a planet, is the distance
measured on the ecliptic, in the order of the signs, between
the geocentric place and the first point of Aries.

GEODESIA, is properly that part of practical geo-
metry that teaches how to divide or lay out lands and
fields, among several owners.

Geodesia is also applied, by some writers, to all mea-
surements in the field, and as synonymous with surveying.

Geodesia is defined by Vitalis, as the art of measuring
surfaces and solids, not by imaginary lines, as is done in
geometry, but by sensible and visible things ; or by the
sun's rays, &c.

Geodetic al, something relating to the art of measuring
the earth or land.

GEOGRAPHER, a person skilled in geography.

GEOGRAPHICAL, relating to geography, as

Geographical Mile, the sea-mile or minute, being
the 6'0lh part of a degree of a great circle.

Geographical Table. See Map.

GEOGRAPHY, the science that teaches and explains
the nature and properties of the earth, as to its figure, place,
magnitude, motions, celestial appearances, ccc, with the
various lines, real or imaginary, on its surface. — Geogra-
phy is distinguished from Cosmography, as a part from the
whole ; this latter considering the whole visible world, both
heaven and earth. And from Topography and Chorogra-
phy, it is distinguished, as the whole from a part. Golnitz
considers geography as either exterior or interior: but
Varenius more justly divides it into general and special;
or universal and particular.

General or Universal Geography, is that which consi-
ders the earth in general, without regard to particular
countries, or the affections common to the whole globe:
as its figure, magnitude, motion, land, sea, &c.

Special QX Particular Geography, is that which con-
templates the constitution of the several particular regions,
or countries; their bounds, figure, climate, seasons, wea-
ther^ inhabitants, arts, customs, language, &c.

History ^Geography. The study and practice of
geography must have commenced at a very early period
of the world. By the accounts we have remaining, it ap-
pears this science was in use among the Babylonians, Ara-
bians, and Egyptians, from whom it passed to the Greeks
first of any Europeans, and from these successively to
the Romans, and th4e western nations of Europe. Hero-
dotus relates that the Greeks first learned the pole, the
gnomon, and the 12 divisions of the day, from the Ba-
bylonians. But Pliny and Diogenes Laertius assert, that
Thales of Miletus, in the 6th century before Christ, first
discovered the passage of the sun from tropic to tropic, and'
it is said was the author of two books, the one on the
tropic and the other on the equinox; both probably de-
termined by means of the gnomon ; whence he was led
to the discovery of the four seasons of the year, which are
determined by the equinoxes and solstices. It is however
probable that he learned these things of the Egyptians, as
well as his division of the year into 36"5 days. This it is
said was invented by the second Mercury, surnamed Tris-
megistus, who, according to Eusebius, lived about 50 years
after the Exodus. Pliny expressly informs us that this
discovery was made by observing when the shadow return-
ed to its marks ; a sufficient proof that it was by the gno-
mon. It is also said that Thales constructed a globe, and
represented the land and sea upon a table of brass. Fur-
ther, that Anaximander, a disciple of Thales, first drew the-
figure of the earth upon a globe; and that Hecataeus, De-
mocritus, Eudoxus, and others, formed geographical maps,
and brought them into common use in Greece. — Meton
and Euctemon observed the summer solstice at Athens, on
the 27th of June 432 years before Christ, by watching nar-
rowly the shadow of the gnomon, with the design of fixing
the beginning of their cycle of 19 years.

Timocharis and Aristillus, who began their observations
about 295 B. C, it seems first attempted to fix the lati-
tudes and longitudes of the fixed stars, by considering their
distances from the equator, &c. One of their observations
gave rise to the discovery of the precession of the equi-
noxes, which was first remarked by Hipparchus about 150
years after; who also made use of their method, for deli-
neating the parallels of latitude and the meridians, on the
surface of the earth ; thus laying the foundation of this
science as it now appears. The latitudes and longitudes,
thus introduced by Hipparchus, were not however much
attended to till Ptolemy's time. Strabo, Vitruvius, and
Pliny, have all of them entered into a minute geographical
description of the situation of places, according to the
length of the shadows of the gnomon, without noticing the
longitudes and latitudes.

Maps at first were little more than rude outlines, and
topographical sketches of different countries. The earliest
on record were those of Sesostris, mentioned by Eusta-
thius ; who observes, that " this Egyptian king, having tra-
versed great part of the earth, recorded his march in maps,
and gave copies of them not only to the Egyptians, but to
the Scythians, to their great astonishment." Some have
imagined with much probability, that the Jews made a map
of the Holy Land, when they gave the different portions to
the nine tribes at Shiloh: for Joshua tells us that they
were sent to walk through the land, and that they described
it in seven parts in a book ; and Josephus relates that when
Joshua sent out people from the different tribes to measure,
the land, he gave them, as companions, persons well skill-
ed in geometry, who could not be mistaken in the truth.

GEO

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GEO

The first Grecian map on record, was that of Anacsiman-
der, mentioned by, Strabo, lib. 1, p. 7, supposed to be the
one referred to by Hipparchus under the designation of the
ancient map. Herodotus minutely describes a map made
by Aristagoras tyrant of Miletus, which will serve to give
some idea of the maps of those times. He relates, that
Aristagoras showed it to Cleomencs kwig of Sparta, to in-
duce him to attack the king of Persia at Susa, in order to re-
store the Ionians to their ancient liberty. It was traced
upon brass or copper, and seems to have been a mere itine-
rary, containing the route through the intermediate coun-
tries which were to be traversed in that march, with the
rivers Halys, the Euphrates, and Tigris, which Herodotus
mentions as necessary to be crossed in that expedition. .It
contained one straight line called the Royal Road or High-
way, which took in all the stations or places of encamp-
ment, from Sardis to Susa; being 111 in the whole jour-
ney, and containing 13,500 stadia, or l6S7t Roman miles
of 5000 feet each.

These itinerary maps of the places of encampment were
indispensably necessary in'all armies and marches; and
indeed war and navigation seem to be the two grand causes
of the improvements, both in geography and astronomy.
Athenasus cites BaHon as author of a work entitled The
encampments of Alexander's march ; and likewise Amyntas
to the same purpose. Pliny observes that Diognetus and
Beeton were the surveyors of Alexander's marches, and
then quotes the exact number of miles according to their
mensuration; which he afterwards confirms by the letters
of Alexander himself. The sameauthor also remarks, that
a copy of this great monarch's surveys was given by Xe-
nocles his treasurer to Patrocles the geographer, who was
admiral of the fleets of Selcucus and Antiochus. His
book on geography is often cited both by Strabo and Pliny;
and it appears that this author furnished Eratosthenes with
the principal materials for constructing his map of the
oriental part of the world.

The first attempt to reduce geography to a regular sys-
tem, was made by Eratosthenes, who introduced a regular
parallel of latitude, commencing at the straits of Gibraltar,
and passing eastwards through the isle of Rhodes, and so
on to the mountains of India, noting all the intermediate
places through which it passed. In drawing this line, he
was not regulated by the same latitude, but by observ-
ing where the longest day was 14 hours and a half, which
Hipparchus afterwards determined was the latitude of 36
degrees. This first parallel through Rhodes was ever after
considered with a degree of preference, in constructing all
the ancient maps; and the longitude of the then known
world was often attempted to be measured in stadia and
miles, according to the extent of that line, by many suc-
ceeding geographers. Eratosthenes soon. after attempted
not only to draw other parallels of latitude, but also to trace.
a meridian at right angles to these, passing through Rhodes
and Alexandria, down to Syene and Meroe; and at length
he undertook the arduous task of determining the circum-
ference of the globe, by an actual measurement of a seg-
ment of one of its great circles. To find the magnitude of
the. earth, is indeed a problem which has engaged the at-
tention of astronomers and geographers ever since the sphe-
rical figure of it was known. It seems Anaximander was
the first among the Greeks who wrote upon this subject.
Archytas of Tarentum, a Pythagorean, famous for his skill
in mathematics and mechanics, also made some attempts
in this way; arid Dr. Long conjectures that these are the

authors of the most ancient opinion that the circumference
of the earth is 400,000 stadia : Archimedes makes men-
tion of the antients who estimated the circumference of
the earth at only 50,003 stadia.

With respect to the methods of measuring the circumfe-
rence of the earth, it would appear from what Aristotle savs
in his treat iseDeCcelo, that they were much the same as those
used by the moderns, deficient only in the accuracy of the
instruments. For in that work he says, that different stars
puss through our zenith, according as our situation is more
or less northerly; and that in the southern parts of the
earth stars come above our horizon, which are no longer
visible if we go northward. Hence it appears that there
are two ways of measuring the circumference of the earth ;
one by observing stars which pass through the zenith of
one place, and do not pass through that of another; the
other, by observing some stars which come-above the ho-
rizon of one place, and are observed at the same time to
be in the horizon of another. The former of these methods,
which is the best, was followed by Eratosthenes at Alex-
andria in Egypt, 250 years before Christ. He knew that
at the summer solstice, the sun was vertical to the inhabi-
tants of Syeue, a town on the confines of Ethiopia, uuder
the tropic of Cancer, where they had a well made to ob-
serve it, at the botlom of which the rays of the sun fell per-
pendicularly the day of the summer solstice: he observed by
the shadow of a wireset perpendicularly in an hemispherical
bason, how far the sun was on that day at noon distant from
the zenith of Alexandria; and found that distance to be
equal to the 50th part of a great circle in the heavens. Then
supposing Syene and Alexandria under the same meridian,
he inferred that the distance between them was the 50th
part of a great circle upon the earth ; and this distance
being by measure 5000 stadia, he concluded that the whole
circumference of the earth was 250,000 stadia. But as
this number divided by 360 would give 6[)i± stadia to a
degree, cither Eratosthenes himself, or some of his fol-
lowers assigned the round number 700 stadia to a degree;
which multiplied by 360, makes the circumference of the
earth 252,000 stadia ; whence both these measures are
given by different authors, as that of Eratosthenes.

In the time of Pompey the Great, Posidonius determined
the measure of the circumference of the earth by the 2d
method above hinted by Aristotle, viz, the horizontal ob-
servations. Knowing that the star called Canopns was
but just visible in the horizon of Rhodes, and at Alex-'
andria finding its meridian height was the 4Sth part of a
great circle in the heavens, or 7i degrees, answering to the
like quantity of a circle on the earth : he by considering
these two places under the meridian, and the distance be-
tween them 5000 stadia, concluded the circumference of
the earth to be 210,000 stadia ; which is the first measure
of Posidonius. But, according to Strabo, Posidonius made
the measure of the earth to be 180,000 stadia, al the rate
of 500 stadia to a degree. The reason of this difference is
thought to be, that Eratosthenes measured the distance
between Rhodes and Alexandria, and found it only 3750
stadia : taking this for a 48th part el' the earth's circum-
ference, which is the measure of Posidonius, the whole cir-
cumference will be 180,000 stadia. This measure wits
received by Marinus of Tyre, and is usually ascribed to
Ptolemy. It is to be observed, however, that this me-
thod of determining the circumference of the earth is sub-
ject to great uncertainty, both on account of the great
refraction of the stars near the horizon, the difficulty of

GEO

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G E G

measuring the distance at sea between Rhodes and Alex-
andria, and by supposing those, places under the same
meridian, when they are really very different.

Several geographers afterwards made use of the different
heights of the pole in distant places under the same me-
ridian, to find the dimensions of the earth. About the
year 800, the khalif Almamun had the distance measured
between two places that were 2 degrees asunder, and under
the same meridian in the plains of Sinjar near the Red
Sea. And the result was, that the degree at one time was
found equal to 56 miles, and at another 56j or 56f miles.
The next attempt to ascertain the circumference of the
earth was in 1525, by Fernelius, a learned philosopher
of France. For this purpose, he took the height of the
pole at Paris, going from thence directly northwards, till
he came to the place where the height of the pole was one
degree more than at that city. The length of the way
was measured by the number of revolutions made by one
of the wheels of his carriage; and after proper allowances
for the declivities and turnings of the road, he concluded
that CS Italian miles were equal to a degree on the earth.
According to these methods many other measurements of
the earth's circumference have since that time been made,
with much greater accuracy : a particular account of
which is given under the article Degree.

Though the maps of Eratosthenes were the best of his
time, they were yet very imperfect and inaccurate. They
contained little more than the states of Greece, and the
dominions of the successors of Alexander, digested ac-
cording to the surveys abovementioned. He had indeed
seen, and has quoted, the voyages of Pythias into the
great Atlantic ocean, which gave him some faint idea of
the western parts of Europe; but so imperfect, that they
could not be realised into theoutlines of a chart. Strabo
says he was very ignorant of Gaul, Spain, Germany, and
Britain ; and he was equally ignorant of Italy, the coasts
of the Adriatic, Pontus, and all the countries towards the
north. Such was the state of geography, and the nature
of the maps, before the time of Hipparchus.

War has usually been the occasion of making or impro-
ving the maps of countries; and accordingly geography
made great advances from the progress of the Roman arms.
In all the provinces occupied by that people, camps were
every where constructed at proper intervals, and good
roads made for communication between them; and thus
were civilisation and surveying carried on according to
system, through' the whole extent of that large empire.
Every new war produced a new survey and itinerary of
the countries where the scenes of action passed ; so that
the materials of geography were accumulated by every ad-
ditional conquest. Polybius observes, that at the begin-
ning of the second Punic war, when Hannibal was pre-
paring his expedition against Rome, the countriesthrough
which he was to march were carefully measured by the
Romans. And Julius Cassar caused a general survey of
the Roman empire to be made, by a decree of the senate.
Three surveyors had this task assigned them, which they
completed in 25 years. The Roman itineraries that are
still extant, also show what care and pains they had em-
ployed in making surveys in all the different provinces of
their empire; and Pliny has filled the 3d, 4th, and 5th
books of his Natural History with the Geographical dis-
tances that were thus measured. Other maps are also still
preserved, known by the name of the Pentingerian Tables,
published by Welscr and Bertius, which give a good spe-
Vol. I.

cimen of what Vegetius calls the Itinera Picta, for the bet-
ter direction of their armies in their march.

The Roman empire had been enlarged to its greatest
extent, and all its provinces well known and surveyed,
when Ptolemy, about 150 years after Christ, composed his
system of geography. The chief materials he employed
in composing this work, were the proportions of the gnomon
to its shadow, taken by different astronomers at the times
of the equinoxes and solstices; calculations founded on
the length of the longest days ; the measured or computed
distances of the principal roads contained in their surveys
and itineraries ; and the various reports of travellers and
navigators. All these were compared together, and di-
gested into one uniform body or system ; and afterwards
were translated by him into a new mathematical language,
expressing the different degrees of latitude and longitude,
after the invention of Hipparchus, which had continued
to be neglected for 250 years. Ptolemy's system of geo-
graphy, notwithstanding it was still very imperfect, con-
tinued in vogue til! the last three or four centuries, within
which time the great improvements in astronomy, the many
discoveries of new countries by voyagers, and the pro-
gress of war and arms, have contributed to bring it to a
very considerable degree of perfection; the particulars of
which will be found treated under their respective articles
in this work.

Among the moderns, the chief authors on the subject of
geography are Johannes de Sacrobosco, or John Halifax,
who wrote a treatise on the sphere; Sebastian Minister,
in his Cosmographia Universalis, in 1559 > Clavius, on
the sphere of Sacrobosco; Piccioli's Geographiaet Hydro-
graphia Reformata ; Weigelms's Speculum Terras; De-
chales's Geography, in his Mundus Mathematicus ; Cel-
larius's Geography ;CIuverius's lntroductio in Universani
Geographiam ; Leibnecht's Elementa Geographic gene-
ralis ; Stevenius's Compendium Geographicum ; Wolfius's
Geographia, in his Elementa Matheseos ; Busching's New
System of Geography ; Gordon's, Salmon's, Guthrie's and
Myers's Grammars ; Adams's Geography, ancient and
modern ; and Pinkerton's Geography, lately published in
two volumes 4to, with an introduction by Professor Vince
of Cambridge. But the most scientific and systematical
work of all is Varenius's Gographia Generalis, with Ju*
rin's additions. And it is much to be wished that some
gentleman of proper qualifications would publish anew
edition of this admirable performance suited to the present
state of the science. Dr. Playfair has recently published
a System of Geography, which is in many respects a very
valuable performance; and perhaps best of all, in French,
Fresnoy's most excellent work.

GEOLOGICAL, relating to the subject of Geology.

GEOLOGY, is that part of natural philosophy which
treats of the structure of the earth, in regard to the origin,
constitution, and the composition of its solid contents.

GEOMETER, or Geometrician j which see.

GEOMETRICAL, something that has a relation to
geometry, or done after the manner, or by the means of
geometry. As, a geometrical construction, a geometrical
curve, a geometrical demonstration, genius, line, method,
geometrical strictness, &c.

Geometrical Construction, of an equation, is thedraw-
ing of lines and figures, so as to express by them the same,
general property and relation, as are denoted by the al-
gebraical equation. See Construction of Equations.

Geometrical Curve or Lint?, called also an Algebraical
4E

GEO

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GEO

one, is that in which the relations between the abscisses
and ordinates may. be expressed by a finite algebraical
equation. See Algebraical Curves.

Geometrical Lines, as observed by Newton, are
distinguished into classes, orders, or genera, according to
the number of the dimensions of the equation that expresses
the relation between the ordinates and abscisses ; or,
which comes to the same thing, according to the number
of points in which they may be cut by a right line. Thus,
a line of the first order, is a right line, since it can be only
once cut by another right line, and is expressed by the
simple equation y ■+■ ax -+- b = 0 : those of the 2d, or
quadratic order, will be the circle, and the conic sections,
since all of these may be cut in two points by a right line,
and expressed by the equation y'! -+- {ax -+- b) y -t- m' ■+-
dx -+- e = 0 : those of the 3d or cubic order, will be such
as may be cut in 3 points by a right line, whose most
general equation is y" -+- {ax -t- b) y' -t- (ex1 •+■ dx ■+■ ?) y
-*-fx3 ■+■ gx'1 ■+■ hx -+• i = 0 ; as the cubical and Neilian
parabola, thecissoid, &c. And a line of an infinite order,
is that which a right line may cut in infinite points: as
the spiral, the cycloid, the quadratrix, and every line
that is generated by the infinite revolutions of a radius, or
circle, or wheel, &c. In each of the above equations,
x is the absciss, y its corresponding ordinate, making any
given angle with it ; and a, b, c, &c, are given or con-
stant quantities, affected with their signs -+- and — , of
which one or more may vanish, be wanting or equal to
nothing, provided that by such defect the line or equation
does not become one of an inferior order.

It is to be noted that a curve of any kind is denominated
by a number next less than the line of the same kind :
thus, a curve of the 1st order, (because the right line
cannot be reckoned among curves) is the same with a line
of the 2d order ;' and a curve of the 2d kind, the same
with a line of the 3d order, &c. We ought likewise to ob-
serve, that it is not so much the equation, as the construc-
tion or description, that makes any curve, geometrical
or not. Thus, the circle is a geometrical line, not
because it may be expressed by an equation, but be-
cause its description is a postulate: and it is not the
simplicity of the equation, but the easiness of the descrip-
tion, that is to determine the choice of the lines for the
construction of a problem. The equation that expresses
a parabola, is more simple than that which expresses a
circle; and yet the circle, by reason of its more simple
construction, is admitted in preference. Again, the circle
and the conic sections, with respect to the dimensions of
the equations, arc of the same order; and yet the circle
is not numbered with them in the construction of problems,
but by reason of its simple description is depressed to a
lower order, viz, that of a right line ; so that it is not im-
proper to express that by a circle, which may be expressed
by a right line; but it is a fault to construct that by the
conic sections, which may be constructed by a circle.
Either, therefore, the law must be taken from the dimen-
sions of equations, as observed in a circle, and so the dis-
tinction betaken away between plane and solid problems;
or the law must be allowed not to be strictly observed in
lines of superior kinds, but that some, by reason of .their
more simple description, may be preferred to others of the
same order, and be numbered with lines of inferior orders.
In constructions that arc equally geometrical) the most
simple arc always to be preferred : and this law is so uni-
versal as to be without exception. But algebraical ex-

pressions add nothing to the simplicity of the construction ;
the bare descriptions of the lines here are only to be con-
sidered ; and these alone were considered by those geome-
tricians who joined a circle with a right line. And as
these are easy or hard, the construction becomes easy or
hard : and therefore it is foreign to the nature of the thing,
from any other circumstance to establish laws relating to
constructions.

Either, therefore, with the ancients, we must exclude
all lines besides the circle, and perhaps the conic sections,
out of geometry ; or admit all, according to the simplicity
of the description. If the trochoid were admitted into
geometry, we might by means of it divide an angle in anv
given ratio; would it be right therefore to blame those
who would make use of this line to divide an angle in the
ratio of one number to another; and contend, that you
must make use only of such lines as are defined by equa-
tions, and therefore not of this line, which is not so de-
fined ? If, when an angle is proposed to be divided, for
instance, into 10001 parts, we should be obliged to bring
a curve defined by an equation of more than 100 dimen-
sions to perform the same ; which no body could describe,
much less understand ; and should prefer this to the tro-
choid, which is a line well known, and easily described
by the motion of a wheel, or circle : who would not see
the absurdity ?

The trochoid therefore is either not to be admitted at
all in geometry, or else, in the construction of problems,
it is to be preferred to all lines of a more difficult descrip-
' tion; and the reason is the same for other curves. Hence
the trisection of an angle by a conchoid, which Archi-
medes in his Lemmas, and Pappus in his Collections,
have preferred to the inventions of all others in this case,
must be allowed as good ; because we must either exclude
all lines, beside the circle and right line, out of geometry,
or admit them according to the simplicity of their de-
scriptions ; in which case the conchoid yields to none, ex-
cept the circle. Equations are expressions of arithmetical
computation, and properly have no place in geometry,
excepting so far as quantities truly geometrical (that is,
lines, surfaces, solids, and proportions) may be said to
be some equal to others. Multiplications, divisions, and
such like computations, are. newly received into geometry,
and that unwaiily, and contrary to the first design of this
science. For whoever considers the construction of pro-
blems by a right line and a circle, discovered by the first
geometricians, will easily perceive that geometry was in-
troduced for the purpose of avoiding the tediousness of
computation. For which reason the two sciences ought
not to be confounded together: the ancients so carefully
distinguished between them, that they never introduced
arithmetical terms into geometry ; and the moderns, by
blending them, have lost the simplicity in which all the
elegance of geometry consists. In short, that is arithme-
tically more simple, which is determined by the more sim-
ple equations ; but that is geometrically tnoresimple, which
is determined by the mote simple drawing of lines; and
in geometry that ought to be reckoned best which is geo-
metrically most simple. Newton's Arith. Uuivei's. appen-
dix. See Curves.

Geometrical Locus, <m Placet called also simply Lo-
cus, is the path or track of some certain geometrical de-
termination, in which it always falls. See Locus. ,

Geometrical Medium. See Medium.

Geometrical Method of the Ancients. The ancients

GEO

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G E G

established the higher parts of their geometry on the same way of finding the quadrature of the circle, the duplica-

principl.es as the elements of that science, by dernonstra
tions of the same kind : and they were careful not to sup-
pose any thing done, till by a previous problem they had
shown that it could be done by actual operation. Much
less did they suppose any thing to be done that cannot be
conceived ; such as a line or series to be actually con-
tinued to infinity, or a magnitude diminished till it b

ture of the cube, or two mean proportionals; though
there are mechanical ways, and others, by infinite series, &c.
Pappus informs us, that the ancients endeavoured in
vain to trisect an angle, and to find out two mean propor-
tionals, 03' means of the right line and circle. Afterwards
they began to consider the properties of several other lines ;
as the conchoid, the cissoid, and the conic sections ; and

come infinitely less than what it is. The elements into which by some of these they endeavoured to resolve some of those
they resolved magnitudes were finite, and such as might problems. At length, having more thoroughly examined

be conceived to be real. Unbounded liberties have of late
been introduced ; by which geometry, which ought to be
perfectly clear, is tilled with mysteries. Maclauriu's Flux-
ions, Introd. pa. 39.

Geometrical Pace, is a measure of 5 feet long.

Geometrical Plan, in Architecture. See Plan.

Geometrical Plane. See Plane.

Geometrical Progression, a progression in which the
terms have all successively the same ratio : as 1, 2, 4, 8,
1(5, &c, where the common ratio is 2.-^The general and

the matter, and the conic sections being received into geo-
metry, they distinguished geometrical problems and solu-
tions into three kinds; viz,

1 . Plane ones, which, deriving their origin from lines on
a plane, may be properly resolved by a right line and a
circle. — 2. Solid ones, which are resolved by lines deriv-
ing their original from the consideration of a solid ; that is,
of a cone. — 3. Linear ones, to the solution of which are re-
quired lines more compounded. According to this dis-
tinction/we are not to resolve solid problems by other

common property of a geometrical progression is, that the lines than the conic sections; especially if no other lines
product of any two terms, or the square of any one single besides the right line, circle, and the conic sections, must
term, is equal to the product of every other two terms be received into geometry. But the moderns, advancing
that are taken at an equal distance on both sides from much farther, have received into geometry all lines that
the former. So of these terms, 1, 2, 4, 8, 16, 32, 64, &c, can be expressed by equations; and have distinguished,

according to the dimensions of the equations, those lines
into classes or orders ; and have laid it down as a law, not
to construct a problem by a line of a higher order, that
may be constructed by one of a lower.

GEOMETRICIAN, or Geometer, a person skilled
in, or professing geometry.

GEOMETRY, the science or doctrine of local exten-
sion, as of lines, surfaces, and solids, with that of ratios,
&c. — The name geometry literally signifies measuring of
the earth, as it was the necessity of measuring the land that
first gave occasion to contemplate the principles and rules
of this art; which has since been extended to numberless
other speculations ; insomuch that, together with arith-
metic, geometry forms now the chief foundation of all the
mathematics.

The invention of geometry, like many other things of
antiquity, is variously ascribed.

Herodotus (lib. 2), Diodorus (lib 1), Strabo (lib. 17),
and Proclus assign it to the Egyptians, and assert that the
annual inundations of the Nile gave occasion to it; for
those waters bearing away the bounds and land-marks of
estates and farms, covering the face of the ground uni-
formly with mud, the people, say they, were obliged every
year to distinguish and lay out their lands by the consi-

1 x 64 = 2 x 32 = 4 x 16 = S x S = 6"4.
In any geometrical progression, if
a denotes the least term,
2 the greatest term,
r the common ratio,
n the number of the terms,
s the sum of the series, or all the terms ;
then any of these quantities may be found from the others
by means of these general values, or equations, viz,

r = -V-,

a = z -r r1'

i rz
log- T

log. r + log. x — log. a
log. r.

z —a ~ z — a

When the series is infinite, then the least term a is nothing,

and the sum s =

See Progression.

Geometrical Proportion, called also simply Propor
tion, is the similitude ur equality of ratios.

Thus, if a : b : : c : d, or a : b — c : d, the terms a, b, c, d deration of their figure and quantity;, and thus by ex

are in geometrical proportion; also 6, 3, 14, 7, are in geo-
metrical proportion, because6 : 3 : : 14: 7,or 6:3=14:7.

In a geometrical proportion, the product of the ex-
tremes, or 1st and 4th terms, is equal to the product of the
means, or the 2d and 3d terms : so ad = be, and
6 x 7 = 3 x 14=42. See Proportion.

Geometrical Solution, of a problem, is when the pro-
blem is directly resolved according to the strict rules and
principles of geometry, and by lines that are truly geome-
trical. This expression is used in contradistinction to an
arithmetical, or a mechanical, or instrumental solution ;
the problem being resolved only by a ruler and compasses

peritnee and habit they formed a method or art, which-
was the origin of geometry. A farther contemplation of
the draught of figures of fields thus laid down, andplotted
in proportion, might naturally lead them to the discovery
of some of their excellent and wonderful properties; which
speculation continually improving, the art successively
gained ground, and made advances more and more to-
wards perfection.

Josephus however seems to ascribe the invention to the
Hebrews: and others of the ancients make Mercury the
inventor. Polyd. Virgil, de Invent. Rer. lib. 1, cap. 18.

From Egypt, this science passed into Greece, being

—The same term is also employed in opposition to all in- carried thither by Thales; where it was much cultivated

direct and inadequate kinds of solutions, as by approxi- and improved by himself, as also by Pythagoras, Anaxa-

mation, infinite scries, &c. So, wc have no geometrical goras ol Clazomene, Hippocrates oi Chios, and Pluto, who

& 4 E 2

GEO

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GEO

testified his conviction of the necessity and importance of
geometry lo the successful, study of philosophy, by this in-
scription over the door of his academy, " Let no one igno-
. ff geometry enter here." Plato thought the word
geometry too mean a name for this science, and sub-
stituted instead of it the more extensive name of Mensu-
ration ; and after him others gave it the title of Panto-
met ry. But even these are now become too scanty in
their import, fully to comprehend its extent; for it not
only inquires into, and demonstrates thequantitics of mag-
nitudes, but also their qualities, as the species, figures,
ratios, positions, transformations, descriptions, divisions,
the finding of their centres, diameters, tangents, asym-
ptotes, curvature, ixt. Some again define it as the science
of inquiring, inventing, and demonstrating all the affec-
tions of magnitude. And Proclus calls it the knowledge of
magnitudes and ri^uies, with their limitations; as also of
their ratios, afft ctions, positions, and motions of every kind.
About 50 years after Plato, lived Euclid, who collected
together all ilu se theorems which had been invented by
his predecessors in Egypt and Greece, and digested them
into 15 books, called the Elements of Geometry ; demon-
strating and arranging the whole in a very accurate and
perfect manner. The next to Euclid, of those ancient
writers whose works are extant, is Apollonius Pergaeus,
who flourished in the time of Ptolemy Euergetes. about
230 years before Christ, and about 100 years after Euclid.
He was author of t lie first and principal work on Conic
Sections ; on account of which, and his other accurate
and ingenious geometrical writings, he acquired from his
patron the emphatical appellation of " The Great Geome-
trician." Conti mporarv with Apollonius, or perhaps a
few years before him, flourished Archimedes, celebrated
tor his mechanical inventions at the siese of Syracuse, and
not less so for his very ingenious geometrical compositions.
We can only mention Eudoxus of Cnidus, Archytas of
Tarenium, Philolaus, Eratosthenes, Aristarchus of Samos,
Dinostratus, the inventor of the quadratrix, Monechmus,
his brother and the disciple id" Plato, the two Aristeus*s,
Conon, Thracidius, Nicotoles, Leon, Theudius. Hermo-
•ilnus. Hero, and Nicomedes the inventor of the conchoid :
besides whom, there are many other ancient geometri-
cians, to whom this science has been indebted.

The Greeks continued their attention to it, even after
they were subdued by the Romans. Whereas the Romans
themselves were so little acquainted with it, even in the
most flourishing time of their republic, that Tacitus in-
torms us they gave the name of mathematicians to those
who pursued the chimeras of divination and judicial astro-
logy. Nor does it appear they were more disposed to cul-
tivate geometry during the decline, and after the fall of
the Roman empire. But the case was different with the
Greeks; among whom are found many excellent geome-
tricians since the commencement of the Christian era, and
after the translation of the Roman empire. Ptolemy lived
under Marcus Aurelius; and we have still extant the
works of Pappus of Alexandria, who lived in the time of
1 hcodosius : the commentary of Eutocius, the Ascalonite,
who lived ab-iit the year of Christ 540, on Archimedes*s
mensuration of the circle ; and the commentary on Euclid,
by Proclus, who lived under the empire of Anastasius.

Fhe consequent inundation of ignorance and barbarism
was unfavourable to geometry, as well as to the other
sciences; and the few who applied themselves lo this
science, were calumniated as magicians. However, in

those times of European darkness, the Arabians were dis-
tinguished as the guardians and prom iters oi science ; and
from the lull to the 14th century, the) produced many
astronomers, geometricians, geographers, &t ; from whom
the mathematical sciences were again received into Spain,
Italy, and the rest of Europe, somewhat before the vear
1400. Some oi the earliest writers after this period, are
Leonardus Pisanus, Lucas Paciolus or De Burgo, and
others between 1400 and 1500. And after this appeared
many editions of Euclid, or commentaries upon him: thus,
Orontius Finams, in 1500. published a commentary on the
first I) books; as did James Peletarius, in 1557; and about
the same time Nicholas Tartaglia published a commen-
tary on the whole 15 books. There have been also the
editions, or commentaries, of Commandlne, Clavius, Bil-
lingslv, Seheuhelius, Herlinus, Dasypodius, Ramus, Heri-
gon, Stevinus, Saville, Barrow, Taquet, Dechales, Fur-
nier, Scarborough, Keill, Stone, Playfair, and many others;
but the completest edition of all the works of Euclid, is
that of Dr. Gregory, printed at Oxford 1703, in Greek
and Latin : the edition of Euclid, by Dr. Robert Simson
of Glasgow, containing the first b" books, with the 1 1th and
12th, is much esteemed for its correctness. The principal
other elementary writers, besides the editors of Euclid, are
Borelli, Pardies, Marchetti, Wolrius, Simpson, Leslie, &c.
And among those who have gone beyond Euclid in the
nature of the Elementary parts of Geometry, may be
chiefly reckoned, Apollonius, in his Conies, his Loci Plani,
De Sectione Determinate, his Tangencies, Inclinations,
Section ot a Ratio, Section of a Space, &c; Archimedes,
in his treatises of the Sphere and Cylinder, the Dimension
of the Circle, of Conoids and Spheroids, of Spirals, and
the Quadrature of the Parabola; Theodosius, in his
Spherics ; Serenus, in his Sections of the Cone and Cy-
linder ; Kepler's Nova Srereometrh ; Cavallerius's Geo-
mctria Indivisibilium ; Torricelli's Opera Geometrica ;
Viviani, in his Divinationes Geometrica', Exercitatio Ma-
thcmatica,De Locis Solidis, De Maximis et Minimis, &c;
Yieta, in his Effectio Geometrica, Supplement. Geometric,
Sectiones Angnlares, Responsum ad Problema, Apollonius
Gallus, &c; Gregory St. Vincent's Quadratura Circuli;
Format's Yaria Opera Mathematica ; Dr. Barrow's Lec-
tiones Geometrica?; Bulliald de Lineis Spiralibus; Ca-
valerius ; Schooten and Gregory's Exercitationes Geo-
metrica-, and Gregory's Pars Universalis, &c ; De Billy's
treatise De Proportione Harmonica ; La Lovera's Geo-
metria veterum promota; Slusius's Mesolabium, Pivhle-
mata Solida, <xc ; Wallis, in his treatises De Cyeloide,
Cissoide, &c; De Proportionibus, De Sect ioni bus Conicis,
Arithmetics Intinitorum, De Centro Grav uatis, De Secti-
onibus Angulai ibus, De Angulo Contactfis, Cuno-Cuneus,
ixc: Hugo De Omcrique, in his Analysis Geometrica;
Pascal on the Cycloid j Step. Angeli's Probleniata Geo-
metrica; Alex. Anderson's Suppl. Apollonii Redivivi,
Variorum Problematum Practice, Sic; Baronius's Geo-
met. Prob. &c ; Guido Giandi Geometr. Demonstr. &c;
Ghvtaldi Apollonius Redivivus, &c; Ludolph van Colen
or a Ceulen, de Circnlo et Adscriptis, &c ; Snell's Apol-
lonius Batavus, Cyclometricus, Stc ; Herbersjein's Dio-
tome Circulorum ; "Palma's Exercit. in Geometriam ; Gul-
ilmi Centro-Barica; with several others equally eminent,
of more modern date, as Dr. Rob. Simson, Dr. Mat.
Stewart, Tho. Simpson, John Leslie, Legendre, Sec, Since
the introduction of the new geometry, or the geometry of
curve lilies, as impressed by algebraical equations, in this

G E O

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GEO

part of geometry, the following names, among many others,
are more especially to be respected ; viz, Descartes,
Schooten, Newton, Maclaurin, Brackenridge, Cramer,
Cotes, Waring, &c, &c. — As to the subject of practical
geometry, the chief writers are Beyer, Kepler, Ramus,
Clavius, Mallet, Tacquet, Ozanam, Wolfius, Gregory, with
innumerable others.

Geometry is distinguished into theoretical or specula-
tive, and practical.

Theoretical or Speculative Geometry, treats of the va-
rious properties and relations in magnitudes, demonstra-
ting the theorems, &c And

Practical Geometry, is that which applies those spe-
culations and theorems to particular uses in the solution
of problems, and in the measurements in the ordinary con-
cerns of life.

Speculative geometry again may be divided into ele-
mentary and sublime;

Elementary or Common Geometry, is that which is em-
ployed in the consideration of right lines and plane sur-
faces, with the solids generated from them.

Higher or Subline Geometry, is that which is em-
ployed in the consideration of curve lines, conic sections,
and the bodies formed of them. This part has been chiefly
cultivated by the moderns, by help of the improved state
of algebra, and the modern analysis or fluxions.

Descriptive Geometry, a name given to a new species
of geometry, chiefly treated on by M. Monge. When one
surface penetrates another, there often result from their
intersection curves of double curvature, the description of
which is necessary in some of the arts; as, in groined
vault-work, and in cutting arch-stones, or wood-cutting,
for ornamental work, &c, the form of which is often very
fantastical ami complicated: it is in. the solution of pro-
blems relating to these subjects that descriptive geometry
is chiefly useful.

Some architects, more versed in geometry than usual
with persons of that profession, have long since thrown
some light on the first principles of this kind of geometry.
There is, for instance, a work by a Jesuit named Father
Courcier, showing how to describe the curves resulting
from the mutual penetration of cylindrical, spherical, and
conical surfaces: this work was published at Paris in
166.3. P. Derand. M. Jousse, Frazier, &c, had also con-
tributed a little towards the promotion of this branch of
geometry. But Monge has given it very great extension,
not only by proposing and resolving various problems both
curious and difficult, but by the invention of several new
and interesting theorems. We can only mention in this
place one or two of these. As, 1st, Two right lines being
given in space, which are neith< r parallel nor in the same
plane, to find in both of them the points of their least dis-
tance, and the position of the line joining these points.
2d, Three spheres being given in space, to determine the
position of the plane which touches them. There are also
curious problems relative to lines of double curvature,
and to surfaces resulting from the application of a right
line that leans continually upon two or three others given
in position in space. Among the theorems, the following
may be mentioned : If a plane surface, given in space, be
projected on three plalnes, the one Horizontal, and the two
others vertical and perpendicular to each other, the square
of that surface will be equal to the sum of the squares of
the three surfaces of projection. This theorem is as in-
teresting in the geometry of solids, as that of Pythagoras

(Eucl. i, 47) is iii plane geometry. But for more on this
subject, reference may be made to Monge's and Lacroix's
ingenious works on Geometrie Descriptive.

Geometry of the Compasses, a new species of geometry,
more. ingenious than profound or useful, lately proposed
by the Abbe Mascheroni. Hitherto both the ruler and
the compasses have been employed in the solution of pro-
blems in plane geometry; and it had not been imagined
that such problems could be solved or constructed with-
out the combined use of these two instruments. But that
author has shown a great number of problems, which are
rendered very piquant and amusing, by the new condition
of employing only the compasses in their solution. Thus:
Two points terminating aright line -being given, to find
either between or without those two points, any number
of points which shall be in the same right line with the
former, and which shall make the intervals between them
in a given ratio : 2d, To a given line to draw parallels, or
perpendiculars, or lines making given angles with them :
3d, To inscribe or circumscribe within or about a circle,
the various problems which are constructible by plane
geometry : 4th, To determine the mean proportional, be-
tween two given distances; or to find third or fourth pro-
portionals. All the problems indeed of the Euclidean
geometry are thus resolved by the mere intersection of
arcs of circles, without drawing a single right line. The
author also resolves, by ingenious approximations, many
problems which lie beyond the limits of common geo-
metry; such as those relating to duplications, trisections,
&c, still employing only the compasses. The work above
alluded to is entitled, Geometria del Compassa, and of
which a French translation was published in 1798, in
1 vol. 8vo. A chapter on the same practice has also been
lately given in Leslie's Geometry.

The following problem, proposed by the Emperor Na-
poleon to the French mathematicians, with several others
in this geometry, are here constructed to show the method.

1. Imperial Problem. To divide the circumference of
a circle into four equal parts by the compasses only, let
adeb be the given circle, whose centre is c. From any
point a in its circum-

;*c

ference, with a distance
equal Jo the given radius
ac, lay off the three suc-
cessive divisions ad, de,
eb, and ab will be a di-
ameter. With the centres
a and b, and radius ae or
bd, the chord of two divi-
sions, describe arcs inter-
secting each other in p;
and PC will be the chord of
a quadrant of the given
circle; which being applied
four times to its circumference will divide it as required.

The truth of this is readily shown thus: Because
ab* = 4ac2 = ae2 -1- ac2 (the angle aeb in a semicircle
being a right angle) ; ae2 = 3ac2. But ap2 = ae' by
construction ; therefore at'2 = 3ac* : and since if ac and
cp were drawn, the angle acp would be a right angle,
ap2 = ac1 ■+■ cp*: hence AC1 -+- CP" = 3ac! ; and con-
sequently cp2 = 2a c2 which is the square of the chord of ■
90°. Therefore, &c, q.e.d.

From the above solution the division of the circum-
ference into 8 equal parts is readily derived. For from

G 1 L

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G L O

the intersection p with the radius ac of the given circle,
descrihe a-n arc intersecting the circumference in d. This
intersection will bisect the arc of 90° above determined.

2. To divide the circumference of a circle into five
equal parts, or to construct a regular pentagon by means
of the compasses only: Let ab be the given circle; then
having found the diameter ab and ai the chord of a
quadrant or 90° as before ; with centre I and radius ac,
set off the arcs il, Oi. From the points h and m, with
the radius ai, describe arcs intersecting at n, on that side
of the chord lm as the centre c of the circle ; and the
distance of this intersection from either extremity of the
diameter ab will be the side of the pentagon required.
The demonstration is simple.

Nearly to this construction of the pentagon is that
given by Ptolemy in his Almagest, for finding the chord of
the fifth part of the circumference.

An ingenious friend (Mr. J. B. Wise, ofBoyn Hill, near
Maidenhead) has favoured us with a construction, in one
simple diagram, for all the regular polygons inscribed in a
circle, of the following number of sides, viz, 3, 4, 5,6, 7,
8, 9, 10, 12, 15, 16, is, 24, 30, 40, 60, &c.

GEORG1UM SIDUS. Sec Herschel, and Uranus.
GERBERT, Pope Silvester the Second, was born of
indigent parents at Auvergne, in the early part of the 10th
century. Scarcely had he acquired the first principles of
the sciences, when he left a country then immersed in ig-
norance, and travelled intp Spain, where he remained se-
veral years. Having made great progress in the mathe-
matics, he returned into France, where he taught the
sciences, which he had learned from the Arabians, parti-
cularly arithmetic, which he thus had the honour of com-
municating to the western Christians, about the year 96O
or 970. His merit and wisdom afterwards elevated him
to the papal chair; which he filled respectably till the
time of his death, in the year 1003.

GHETALDUS(MARiNus),a Ragusan nobleman, and
very respectable geometrician, who flourished about the
beginning of the 17th century. He was author of several
ingenious works on geometry, after the manner of the
ancients; of which the titles and dates of those that are in
my possession are the following: 1. Promotus Archime-
des, &c, 4to, Romso, l603. — 2. Problematum Collcctio,
4to, Venet. 1607. — 3. Apollonius Redivivus, &c, 4to,
Venet. 1607 and lfjl3. — 4. Supplementum Apollonii
Galli, Venet. 1607, 4to. — 5. De Resolut. et Composit.
Mathemat. in folio, Roma, 1630. And there might be
others.

GIBBOUS, is used for the shape of one state of the en-
lightened part of the moon, being that in which she ap-
pears raorc than half full or enlightened, which is the time
between the first quarter and the full moon, and from the
full moon to the last quarter; appearing then gibbous,
that is, bunched out, or convex on both sides of the en-
lightened part; as contradistinguished from the state when
she is less than half full, when she is said to be horned, or
a crescent.

GILBERT (Win.), a physician, who was born at Col-
chester in 1540, and educated at Cambridge, but took his
degree of M.n. abroad. On his return he was elected a
fellow of the college of physicians, and became physi-
cian to queen Elizabeth. Dr. Gilbert discovered several
of the properties of the loadstone; and in lG6'0 he pub-
lished a work in 4to, entitled, De Magnete, magneticisque
Corporibus, et de magno Magnete Tellure, Physiologia

nova ; in which are many important suggestions for the im-
provement of navigation. He died in 1603, consequently
at 63 years of age.

GIMBOLS, are the brass rings by which a sea-compass
is suspended in its box that usually stands in the binacle.

GIN, in Artillery and Mechanics, is a machine for rais-
ing great weights, usually composed of three long legs, &c.

GIRARD (Albert). See an account of him in the ar-
ticle Algebra.

GIRDERS, in Architecture, are the largest beams or
pieces of timber supporting the floors. Their ends are
usuall}7 fastened into the summers, or breast-summers; and
the joists are framed in at one end to the girders. By the
statute for rebuilding London, no girder is to lie less than
10 inches into the wall, and their ends to be always laid
in loam, &c. The shorter bearings a girder has, and the
oftener it is supported by the internal or partition walls,
so much the better. The established breadth and depth
of a girder, according, to its length of bearing, are as in
the following tablet :

From

to

must be in

Breadth

Depth

■Jj to

10 feet

15 feet

1 1 inches

S inches

1~

15

IS

13

9

« .«

IS

21

14

10

T

21

24

16

12

O

24

2fj

17

14

GIRT, in Timber-measuring, is the circumference of a
tree ; though some use this word for the quarter or 4th
part of the circumference only, on account of the great
use that is made of it ; for the square of this 4th part is
esteemed and used as equal to the area of the section of
the tree ; which square therefore multiplied by the length
of the tree, is accounted the solid content. This content
however is always about one-fourth part less than the true
quantity; being nearly equal to what this will be after the
tree is hewed square in the usual way : so that it seems in-
tended to make an allowance for the squaring of the tree.

GiKT-Line, is a line on the common or carpenter's sli-
ding rule, employed in computing the contents of trees by
means of their girt.

GIVEN, Datum, a term often used in mathematics, and
signifies something that is supposed to be known.

Thus, if a magnitude be known, or if we can find an-
other equal to it, it is said to be given in magnitude. Or
when the position of any thing is known, it is said to be
given in position. And when the diameter of a circle is
known, the circle is given in magnitude. Or the circle is
given in position when its centre is given in position. When
the kind or species of a figure is known, or remains the
same, it is given in specie, &c.

Euclid wrote a book of Data, or concerningthings given,
in 95 propositions, usually accompanying his Elements, in
the best editions, and which Pappus reckons as one of the
best specimens of the analytical works of the ancients.

GLACIS, in Fortification, a sloping bank reaching from
the parapet of the counterscarp, or covered-way, to the
level side of the field, commonly at the distance of about
40 yards.

GLOBE, a round or spherical body, more usually called
a sphere, bounded by one uniform convex surface, every
point of which is equally distant from a point within called

G L O

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G L O

its centre. Euclid de_fines the globe, or sphere, to be a
solid figure described by the revolution of a semi-circle
about its diameter, which remains unmoved. Also, its
axis is the fixed line or diameter about which the semi-
circle revolves ; and its centre is the same with that of the
revolving semi-circles a diameter of it being any right line
that passes through the centre, and terminated both ways
by the superficies of the sphere. Elem.l 1. def. 14, 15,1 6, 17.
Euclid, at the end. of the 12th book, shows that spheres
are to one another in the triplicate ratio of their diameters,
that is, their solidities are to one another as the cubes of
their diameters. And Archimedes determines the real
magnitudes and measures of the surfaces and solidities of
spheres and their segments, in his treatise de Spha;ia et
Cylindro : viz, 1, That the superficies of any globe is
equal to 4 times a great circle of it. — 2, That any sphere
is equal to -J of its circumscribing cylinder, or of the cy-
linder of the same diameter and altitude. — 3, That the
curve surface of the segment of a globe, is equal to the
circle whose radius is the line drawn from the vertex of
the segment to the circumference of the base. — 4, That
the content of a solid sector of the globe is equal to a
cone whose altitude is the radius of the globe, audits
base equal to the curve superficies or base of the sector.
With many other properties. And from hence are easily
deduced these practical rules for the surfaces and solidities
of globes and their segments ; viz,

1. For the Sutface of a Globe, multiply the square of
the diameter by 3'14l6' ; or multiply the diameter by the
circumference.

2. For the Solidity of a Globe, multiply the cube of the
diameter by '5236" (viz ^-of 31416) ; or multiply the sur-
face by -^ of the diameter.

3. For the Surface of a Segment, multiply the diameter
of the globe by the altitude of the segment, and the pro-
duct again by 3'14lC.

4. For the Solidity of a Segment, multiply the squareof
the diameter of the globe by the difference between 3 times
that diameter and 2 times the altitude of the segment, and
the product again by '5236, or|- of 3-14l6.

Hence, if d denote the diameter of the globe,
c the circumference,
a the altitude of any segment, and
p = 3'14l6; then

The Surface | The Solidity

In the Globe pd2 = cd \pd}

In the Segt. ' pad \ \pu.i x (3d — 2«) j

See the art. Sphere, and my Mensuration, pa. 197,
&c, 2d edit.

The Globe, Terraqueous, is the body or mass of the
earth and water together, which is nearly globular.

Globe, or .irtificial Globe, is more particularly used
for a globe of metal, plaster, paper, pasteboard, &c, on
the surface of which is drawn a map, or representation of
either the heavens or the earth, with the several circles
conceived upon them. And hence

Globes are of two kinds, terrestrial, and celestial ;
which arc of considerable use in geography and astronomy,
by serving to give a lively representation of their principal
objects, and for performing and illustrating many of their
operations in a manner easy to be perceived by the senses,
and so as to be conceived even without any knowledge of
the mathematical grounds of those sciences.
Description of the Globes.

The fundamental parts that are common to both globes,

are an axis, representing the axis of the world, passing
'through the two poles of a spherical shell, representing
those of the world, which shell makes the body of the
globe, upon the external surface of which is drawn the
representation of the whole surface of the earth, sea, rivers,
islands, &c, for the terrestrial globe, and the stars and
constellations of the heavens, for the celestial one; be-
sides the equinoctial and ecliptic lines, the zodiac, the
tropics and polar circles, and a number of meridian lines.
There is also a brazen meridian, being a strong circle of
brass, circumscribing the globe, at a small distance from
it, quite around, in which the globe is hung by its two
poles, upon which it turns round within this circle, which
is divided into 4 times GO degrees, beginning at the equa-
tor on both sides, and ending with 90 at the two poles.
There are also two small hour circles, of brass, divided
into twice 12 hours, and fitted on the meridian round the
poles, which carry an index pointing to the hour. The
whole is set in a wooden ling, placed parallel to, and re-
presenting the. horizon, in which the globe slides by the
brass meridian, elevating or depressing the pole accord-
ing to any proposed latitude. There is also a thin slip of
brass, called a Quadrant of Altitude, made to fit on oc-
casionally upon the brass meridian, at the highest or ver-
tical point, to measure the altitude of any object above
the horizon. A magnetic compass is sometimes set under-
neath. See the figure of the globes so mounted, at fig. 1,
plate xiv. '

Such is the plain and simple construction of the arti-
ficial globe, whether celestial or terrestrial, as adapted
to the time only for which it is made. But as the angle
formed by the equator and ecliptic, as well as their points
of intersection, is always changing ; to remedy these in-
conveniences, several contrivances have been made, so as
to adapt the same globes to any other time, either past
or to come ; as well as other contrivances to answer par-
ticular purposes.

Thus, Mr. Senex, a celebrated maker of globes, had a
contrivance which, by means of a nut and screw, caused
the pole of the equator to revolve about the pole of the
ecliptic, by any quantity answering to the precession of
the equinoxes, since the time for which the globe was made.
Philos. Trans. No. 447, or my Abr. vol. 8, pa. 176, also
Philos. Trans, vol. 46, pa. 2fl0.

Mr. Joseph Harris, late assay-master of the Mint, made
some contrivances to show the effects of the earth's mo-
tions. He fixed two horary circles under the brass meri-
dian, to the axis, one at each pole, so as to turn round
with the globe, and that meridian served as :>n index
to cut the horary divisions. The globe in this state serve.)
equally for resolving problems in both north and south
latitudes, as also in places near the' equator; whereas,
in the common construction, the axis and horary circle
prevent the brass meridian from being moveable quite
round in the horizon. This globe is also adapted for show-
ing how the vicissitudes of day and night, and the alteration
of their lengths, are really occasioned by the motion of the
earth: for this purpose, he divides the brass meridian, at
one of the poles, into months and days, according to the
sun's declination, reckoning from the pole. Therefore, by
bringing the day of the month to the horizon, and rectify-
* ing the globe according to the time of the day, the hori-
zon will represent the circle separating light and darkness,
and the upper half of theglobe theilluminated hemisphere,
the sun being in the zenith. Mr. Harris also gives an ac-

G L O

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count of a cheap machine for showing how the annual
motion of the earth in its orbit causes the change of the
sun's declination, without the great expense of an orrery.
Philos. Trans. No. 456'.

The late Mr. George Adams made also some useful im-
provements in the construction of the globes. Besides what
is usual, his globes have a thin brass semi-circle move-
able about the poles, with asmall thin sliding circle upon it.
On the terrestrial globe, the former of these is a moveable
meridian, and the latter is the visible horizon of any par-
ticular place to which it is set. But on the celestial globe,
the semi-circle is a moveable circle of declination, and its
small annexed circle an artificial sun or planet. Each
globe has a brass wire circle, placed at the limits of the
twilight. The terrestrial globe has many additional cir-
cles, as well as the rhumb-lines, for resolving all the ne-
cessary geographical and nautical problems : and on the
celestial globe are drawn, on each side of the ecliptic, 8
parallel circles, at the distance of one degree from each
other, including the zodiac; which are crossed at right-
angles by segments of great circles at every 5th degree of
the ecliptic, for the more readily noting the place of the
moon or of any planet on the globe. On the strong brass
circle of the terrestrial globe, and about 23£ degrees on
each side of the north pole, the days of each month are
laid down according to the sun's declination : and this
brass circle is so contrived, thatthc globe may be placed
with the north and south poles in the plane of the horizon,
and with the south pole elevated above it. The equator,
on the surface of either globe, serves the purpose of the
horary circle, by means of a semi-circular wire placed in
the plane of the equator, carrying two indices, one of
which is occasionally to be used to point out the time. For
a farther account of these globes, with the method of using
them, see Mr. Adams's Treatise on their construction and
use.

There are also what are called Patent Globes, made
by Mr. Neale; by means of which he resolves several astro-
nomical problems, which do not admit of solution by the
common globes.

Mr. Ferguson also made several improvements of the
globes, particularly one for constructing dials, and ano-
ther called a planetary globe. SeePhilos. Trans, vol. 44,
pa. 535, and Ferguson's Astron. pa. 291 and 292.

Lastly, in the Philos. Trans, for 1789, vol. 79> Pa- !>
Mr. Smeaton has proposed some improvements of the ce-
lestial globe, especially with respect to the quadrant of
altitude, for the resolution of problems relating to the azi-
muth and altitude. The difficulty, he observes, that has
Recurred in fixing a semi-circle, so as to have a centre in
the zenith and nadir points of the globe, at the same time
that the meridian is left at liberty to raise the pole to its
desired elevation, I suppose, has induced the globe-makers
to be contented with the strip of thin flexible brass, called
the quadrant of altitude ; and it is well known how im-
perfectly it performs its office. The improvement I have
attempted, is in the application of a quadrant of altitude
of a more solid construction; which being affixed to a
brass socket of 6ome length, and this ground, and made to
turn upon an upright steel spindle, fixed in the zenith,
steadily directs the quadrant, or rather arc of altitude, to
its true azimuth, without being at liberty to deviate from
a vertical circle to the right hand or left : by which means
the azimuth and altitude are given with the same exact-
ness as the measure of any other of the great circles. For

a more particular description of this improvement, illus-
trated with figures, see. the reference above.

For the practical uses of the globes, see any of the little
books on this subject.

GLOBULAR, relating to, or partaking of the property
or shape of the globe. As globular chart, globular pro-
jection, or globular sailing, &c.

Globular Chart, is a representation of the surface, or
part of the surface, of the terraqueous globe on a plane:
in which the parallels of latitude are circles nearlv con-
centric ; and the meridians are curves bending towards
the poles ; the rhumb-lines being curves also.

The merits of this chart consist in these particulars,
viz, that the distances between places on the same
rhumb are all measured by the same scale of equal parts ;
and the distance of any two places in the arch of a great
circle, is nearly represented in this chart by a straight
line.

Land maps also, made according to this projection,
would have great advantages over those made in any other
way. But for sea charts for the use of navigation, Mer-
cator's are preferable, as both the meridians and parallels,
as also the rhumbs, are all straight lines.

This projection is not new, though not much noticed
till of late. It is mentioned by Ptolemy, in his Geogra-
phv; and also by Blundeville, in his Exercises.

For globular projection of maps or charts, see Mat.
Globular Sailing, is the method of resolving the
cases of sailing on principles deduced from the spherical
figure of the earth. Such as Mercator's sailing, or great-
circle sailing; which see.

GLOSSOCOMON, in Mechanics, is a name given by
Heron to a machine, composed of divers dented wheels
with pinions, serving to raise huge weights.

GNOMON, in Astronomy, is an instrument or appa-
ratus for measuring the altitudes, declinations, &c, of the
sun and stars. The gnomon is usually a pillar, or column,
or pyramid, erected upon level ground, or a pavement.
For making the more considerable observations, both the
ancients and moderns have made great use of it, especially
the former; and many have preferred it to the smaller
quadrants, both as more accurate, easier made, and more
easily applied.

The most ancient observation of this kind extant, is that
made by Pytheas, in the time of Alexander the Great, at
Marseilles,' where he found the height of the gnomon was
in proportion to the meridian shadow at the summer sol-
stice, as 213-g- to 600 ; just the same as Gassendi found it
to be, by an observation made at the same place, almost
2000 years after, viz, in the year l636\ Ricciol. Alinag.
vol.1, lib. 3, cap. 14.

Ulugh Beigh, king of Parthia, &c, used a gnomon in the
year 1437, which was 180 Roman feet high. That erected
by Ignatius Dante, in the church of St. Petronius, at Bo-
logna, in the year 1576', was 67 feet high. M. C'assini
erected another of 20 feet high, in the same church, in
the year 1655.

The Egyptian obelisks were also used as gnomons; and
it is thought by some modem travellers that this was the
very use thev were designed and built for; it has also been
found that their four sides stand exactly facing the four
cardinal points of the compass. It may be added, that
the Spaniards in their conquest of Peru, found pillars of
curious and costly workmanship, set up in several places,
by the meridian shadows of which their amatas or pnilo-

G N O

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sophers had, by long experience and repeated observations,
learned to determine the times of the equinoxes ; which
seasons of the year were celebrated with great festivity
and rich offerings, in honour of the sun. Garcillasso de
la Vega, Hist. Peru, lib. 2, cap. 22.

height of which above the floor let be accurately measured
with a plummet. Let the floor, or stage, be perfectly
plane and horizontal, and coloured over with some white
substance, to show the sun more distinctly. Upon this
horizontal plane draw a meridian line passing through the

Use of the Gnomon, in taking the meridian altitude of foot or centre of the gnomon, i. e. the point upon which

the Sun, and thence finding the Latitude of the place.— A the plummet falls from the centre of the hole; and on

meridian line being drawn through the centre of the gno- this line note the extreme points I and k of the sun's image

mon, note the point where the shadow of the gnomon ter- or diameter, and from each end subtract the image of

minates when projected along the meridian line, and mea- half the diameter of the aperture, viz kh and n : then

sure the-distance of that extreme point from the centre of will hl be the image of the sun's diameter, which, when

the gnomon, which will be the length of its shadow. Then bisected in b, gives the point on which the rays fall from

having the height of the gnomon, and the length of the the centre of the sun.
shadow, the sun's altitude is thence easily found. G — .

Suppose, ex. gr. ab the gnomon, and ac the length of
the shadow. Here, in the right-angled triangle abc, are
given the base ac, and the perpendicular a b, to find the
angle c, or the sun's altitude, which will be found by this
analogy, as ca : ab : : radius : the tang, of Z.C, that is,
as the length of the shadow is to the height of the gnomon,
so is the radius to the tangent of the sun's altitude above
the horizon. — The following example will serve to illus-
trate this proposition : Pliny says, Nat. Hist. lib. 2, cap. 72,
that at Rome, at the time of the equinoxes, the shadow is
to the gnomon as S to 9 > therefore, as 8 : 9 : : 1 or ra-
dius : £ = 1" 125 a tangent, to which answers the angle
48° 22, which is the height of the equator at Rome, and
its complement 41° 38' is therefore the height of the pole,
or the latitude of the place.

Riccioli remarks the following defects in the observa-
tions of the sun's height, made with the gnomon by the
ancients, and some of the moderns; viz, that they ne-
glected the sun's parallax, which makes his apparent al-
titude less, by the quantity of the parallax, than it would
be, if the gnomon were placed at the centre of the earth :
2d, they neglected also the refraction, by which the ap-
parent height of the sun is a little increased : and 3dly,
they made the calculations from the length of the shadow,
as if it were terminated by a ray coming from the centre
of the sun's disc, whereas the shadow is really terminated
by a ray coming from the upper edge of the sun's disc ; so
that, instead of the height of the sun's centre, their calcu-
lations gave the height of the upper edge of his disc. And
therefore, to the altitude of the sun found by the gnomon,
the sun's parallax must be added, and from the sum must
be subtracted the sun's semidiameter, and refraction, which
is different at different altitudes; which being done, the
correct height of the equator at Rome will be 48° 4' 13",
the complement of which is the latitude, or 41° So 46".
Ricciol. Geogr. Refor. lib. 7, cap. 4.

Thcpreccding problem may be resolved more accurately
by means of a ray of light let in through a small hole,
than by a shadow, thus: Make a circular perforation in a
brass plate, to transmit enough of the sun's rays to exhibit
his image on the floor, or a stage ; fix the plate parallel to
the horizon in a high place, proper for observation, the

Vol. I.

nr ^tt: — tj — r~ir

Now having given the line ab, and the altitude of the
gnomon ag, besides the right angle a, the angle b, or the
apparent altitude of the sun's centre, is easily found,
thus : as ab : ag : : radius : tang, angle b.

Gnomon, in Dialling, is the style, pin, or cock of a
dial, the shadow of which points out the hours. This is
always supposed to represent the axis of the world, to
which it is therefore parallel, or coincident, the two ends
of it pointing straight to the north and south poles of the
world.

Gnomon, in Geometry, is a figure formed of the two
complements, in a parallelogram, together with either of
the parallelograms about the diameter. Thus the paral-
lclogram ac being divided into four
parallelograms by the two lines dg,
ef parallel to the sides, forming the
two complements a Band bc, with the
two de, fg about the diameter in :
then the two gnomons are ab ■+- BC
h- de, and AB +IC + FG.

Gnomonic Projection of the Sphere, is the representa-
tion of the circles of an hemisphere on a plane touching
it in the vertex, by the eye in the centre, or by lines or
rays issuing from the centre of the hemisphere, to all the
points in the surface. — In this projection of the sphere,
all the great circles are projected into right lines, on the
plane, of an indefinite length ; and all lesser circles that
are parallel to the plane, into circles; but if oblique to
the plane, then are they projected either into ellipses or
hyperbolas, according to their different obliquity. It
has its name from gnomonics, or dialling, because the
lines on the face of every dial are from a projection of this
kind : for if the sphere be projected on any plane, and
upon that side of it on which the sun is to shine ; also
the projected pole be made the centre of the dial, and
the axis of the globe the style or gnomon, and the ra-
dius of projection its height; you will have a dial drawn
with all its furniture. See Emerson's Projection of the
Sphere.

GNOMONICS, the same as Dialling ; or the art of
drawing sun and moon dials, on any given plane ; being
so called, because it shows how to find the hour of the
day or night by the shadow of a gnomon or style.
4F

G O N

£ 586 ]

G O R

20
3)420
Quot. 1404. or 71.

GOLDEN Number, is the particular year of the Me-
tonic or Lunar Cycle. See Lunar Cycle.

To find the Golden Number:
Add 1 lo the given year, and di-
vide the sura by 19, and what re-
mains is the golden number ; unless
0 remain, for then 19 is the golden
number. Thus, the golden num-
ber for the year 1812 is 8 ; as by
the operation in the margin.

Golden Rule, a rule so called on account of its ex-
cellent use, in arithmetic, and especially in ordinary cal-
culations, by which numbers are found in certain propor-
tions, viz, having three numbers given, to find a 4th
number in proportion. On this account, it is otherwise
called The Rule of Three, and The Rule of Proportion.
See Rule of Three.

Having stated, or set down in a line, the three terms,
in the order in which they are proportional, multiply
the 2d and 3d together, and divide the product by the
1st, so shall the quotient be the answer, or the 4th term
sought. — Thus, if 3 yards of cloth cost a guinea Or 21
shillings, what will 20 yards cost. Here the two prices
or values must bear the same proportion to each other as
the two quantities, or number of yards of cloth, i.e. 3
must bear the same proportion to 20, as 21s, the value of
the former, must bear to the value 3 ; 20 : : 21 ■ 140s
of the latter : and therefore the
stating and operation of the num-
bers will be thus. Then multiply-
ing the 2d and 3d together, and
dividing the product by the 1st, it gives 140s. or 71. for
the answer, being the cost of 20 yards.

GONIOMETRICALLwes, arelines used for measuring
or determining the quantity of angles : such as sines, tan-
gents, secants, versed sines, &c. Mr. Jones, in the Philos.
Trans. No. 483, sect. 26", gave a paper, containing a
commodious disposition of equations for exhibiting the
relations of goniometrical lines ; from which a multitude
of curious theorems may be derived. See also Robert-
son's Elem. of Navigation, vol.1, p. 1S1, edit. 4.

GON10MKTRY, a method of measuring angles, so
called by M. De Lagny, who gave several papers, on this
method, in the Memoircs of the Royal Acad. an. 1724,
1725, 1729- M. De Lagny 's method ofgoniomctry con-
sists in measuring the angles with a pair of compasses,
and that without any scale whatever, except an undivided
semicircle. Thus, having any angle drawn upon paper,
to be measured ; produce one of the sides of the angle
backwards; then wilh a pair of fine compasses describe a
pretty large semicircle, from the angular point as a centre,
cutting the sides of the proposed angle, which will inter-
cept a part of the semicircle. Take then this intercepted
part very exactly between the points of the compasses, and
turn them successively over upon the arc of the semicircle,
to find how often it is contained in it, after which there is
commonly some remainder: then take this remainder in
the compasses, and in like manner find how often it is Con-
tained in the last of the integral parts of the 1st arc, with
again some remainder : find in like manner how often this
last remainder is contained in the former ; and so on con-
tinually, till the remainder become too small to be taken
and applied as a measure. By this means he obtains a
series of quotients, or fractional parts, one of another,
which being properly reduced iuto one fraction, give the

ratio of the first arc to the. semicircle, or of the proposed
angle to two right angles, or 180 degrees, and conse-
quently that angle itself in degrees and minutes.

Thus, suppose the angle bac be proposed to be mea-
sured. Produce b a out towards/; and from the centre a
describe the semicircle abef, in which ab is the measure
of the proposed angle. Take ab in the compasses, and
apply it 4 times on thevsemicircle, as at b, c, d, and e ;
then take the remainder/, and apply it back upon ed,
which is but once, viz at g ; again take the remainder^
and apply it 5 times on ge, as at //, i, k, I, and m ; lastly
take the remainder me, and it is contained just 2 times in
ml. Hence the series of quotients is 4, 1,5, 2 ; conse-
quently the 4th or last arc em is \ the third ml or gd, and

therefore the 3d arc gd is — or -A of the 2d arc ef; there-
of

fore again this 2d arc ef is -7-or \^ of the 1st arc ab;
and consequently this 1st arc ab is -— - or -J-| of the whole

4-j-y

semicircle of. But if of 180° are 37 j- degrees, or 37°
8' 34"f, which therefore is the measure of the ancle
sought. When the operation is nicely performed, this an-
gle may be within 2 or 3 minutes of the truth ; though M.
De Lagny pretends to measure much nearer than that.

It may be added, that the series effractions forms what
is called a continued fraction. ■ Thus, in the example
above, the continued fraction, and its reduction, will be
as follow :

1 1 _I j _1 13

4 + T +i~4+ TT ~ 4Ji ~~ 63" '
5| 1_rr

the quotients being the successive denominators, and 1
always for each numerator.

GORGE, or Neck, in Architecture, the narrowest part
of the Tuscan or Doric capitals, lying above the shaft of
the pillar, between the astragal and annulets.

It is also a kind of concave moulding, serving for com-
partments Sec, larger than ascotia, but not so deep.

Gorge, in Fortification, is the entrance into a bastion,
or a ravelin, or other out-work.

The Gouge of a Bastion, is what remains of the sides of
the polygon of a place, after cutting off the curtains; in
which case it makes an angle in the centre of the bastion,
viz, the angle made by two adjacent curtains produced to
meet within the bastion. — In fiat bastions, the gorge is a
right line on the curtain, reaching between the two flanks.

Gouge of a Half- moon, or of a Ravelin, is the space be-
tween the two ends of their faces next the place.

Gouge of the other out-works, is the interval between
their sides next the great ditch. All the gorges are to be
made without parapets : otherwise the besiegers, having
taken possession of a work, might make use of them to de-
fend themselves from the shot of the place. So that they
are only fortified with pallisadocs, to prevent a surprise.

The Dcmi-GoKGys,, or Halfihc Gouge, is that part of

GRA

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GRA

the polygon between the flank and the centre of the bas-
tion.

GOTHIC Architecture, is that which deviates from the
manner, character, proportions, &c, of the antique;
having its ornaments wild and chimerical, and its profiles
incorrect. This manner of building came originally from
the North, whence it was brought, in the 5th century, by
the Goths into Germany, and has since been introduced
into other countries. The first or most ancient style of
Gothic building was very solid, heavy, massive and simple,
with semicircular arches, &c: but the more modern style
of the Gothic is exceedingly rich, light, and delicate ;
having an abundance of little whimsical ornaments, with
sharp-pointed arches formed by the intersections of dif-
ferent circular segments ; also lofty and light spires and
steeples, large ramified windows, clustered pillars, &c.
Of this kind are our English cathedrals, and many other
old buildings.

GRADUATION, is used for the act of graduating, or
dividing any thing into degrees. — For an account of the
various methods of graduating mathematical and astrono-
mical instruments, by straight and circular diagonals,
and by concentric arcs, &c ; see Plain Scale, Nonius,
and Vernier. And for an accountof Mr. Bird's improved
method of dividing astronomical instruments, see Mural
Arch.

Mr. Ramsden, an ingenious mathematical instrument-
maker of London, has lately published, by encouragement
of the commissioners of longitude, an explanation and
description of an engine contrived by him for dividing ma-
thematical instruments, accompanied with proper draw-
ings ; in consideration of which, the said commissioners
granted him the sum of 6l5l. See his book, 4to, 1777.

On the subject of dividing a foot into many thousand
parts, for mathematical purposes, see Philos. Trans, vol. 2,
p. 4.57, 459, 541, or my Abr. vol. 1, pa. l6l, 195, &c.
And for an account of various other methods and gradu-
ations, see a paper of Mr. Smeaton's in the Philos. Trans.
vol. 76, for the year 17S6,pa. 1 ; being " Observations on
the graduation of astronomical instruments; with an ex-
planation of the method invented by the late Mr. Henry
Hindley, of York, clock-maker, to divide circles into any
given number of parts." Also Mr. Troughton's method
and history of such operations, in the vol. for 1809.

GRAHAM (George), clock- and watch-maker, the
most ingenious and accurate artist in his time, was born at
Gratwick, a village in the north of Cumberland, in 1675.
He came up to London in l688, and was put appren-
tice to a person in that profession ; but after being some
time with his master, he was received, purely on ac-
count of his merit, into the family of the celebrated Mr.
Tompion, who treated him with a kind of parental affec-
tion as long as he lived. That Mr. Graham was, with-
out competition, the most eminent of his profession, is but
a small part of his character : he was the best general me-
chanic of his time, and had a complete knowledge of prac-
tical astronomy ; so that he not only gave to various move-
ments for measuring time a degree of perfection which
had never before been attained, but invented several astro-
nomical instruments, by which considerable advances have
been made in that science : he made great improvements
in those which had before been in use; and, by a wonder-
ful manual dexterity, constructed them with greater pre-
cision and accuracy than perhaps any other person in the

Will']

A great mural arch in the observatory at Greenwich
was madefor Dr. Halley, under Mr. Graham's immediate in-
spection, and divided by his own hand : and from this in-
comparable original, the best foreign instruments of the
kind are copies made by English artists. The sector by
which Dr. Bradley first discovered two new motions in the
fixed stars, was of our author's invention and fabric. He
comprised the whole planetary system within the compass
of a small cabinet ; from which, as a model, all the modern
orreries have been constructed. And when the French
academicians were sent to the North, to make observations
for ascertaining the figure of the earth, Mr. Graham was
thought the fittest person in Europe to supply them with
instruments; by which means they finished their opera-
tions in one year; while those who went to the South, not
being so well furnished, were very much embarrassed and
retarded in their operations.

Mr. Graham was many years a member of the Royal
Society, to which he communicated several ingenious and
important discoveries, viz, from the 31st to the 42d volume
of the Philos. Transactions, chiefly on astronomical and
philosophical subjects ; particular!)' a kind of horary alte-
ration of the magnetic needle; a quicksilver pendulum,
and many curious particulars relating to the true length of
the simple pendulum, on which he continued to make ex-
periments till almost the time of his death, which hap-
pened in 1751, at 76' years of age.

His temper was not less communicative than his genius
was penetrating; and his principal view was the advance-
ment of science, and the benefit of mankind. As he was
perfectly sincere, he was above suspicion ; and as he was
above envy, he was candid.

GRANADO, in Artillery, is a small shell or hollow
globe of iron, or other matter, which, being filled with
powder, is fired by means of a small fusee, and thrown
cither by the hand, or a piece of ordnance. As soon as it
is kindled, the case flies in pieces, to the great danger of
all that stand near it. Granadoes serve to set fire to close
and narrow passages, and are often thrown with the hand
among the soldiers, to disorder their ranks; more especi-
ally in those posts where they stand thickest, as in trenches,
redoubts, lodgments, &c.

GRANDI (Guido), a learned Italian mathematician,
was born at Cremona in l6"71 ; appointed professor of
mathematics at Pisa in 1714; and died in 1742. In 1690
he published, in 4to, the demonstration of Viviani's won-
derful or quadrable dome, under the title of Geometrica
Divinatio Yivianiorum Problematum; a work which con-
tains more than its title would lead us to expect, and in
which the author remarks many other curiosities in geo-
metry of the same kind, and among others, a portion of the
surface of a right cone, which is perfectly quadrable, and
to which he gives the name of Velum Camaldulense, he
being a friar of the order of Camaldunes; from which it
seems he did not know the same thing, in a more general
form, had before been given by John Bernoulli, in the Leip-
sic Acts. In J 70 1 , he published his demonstration of
Huygens's theorems on the Logistic Curve, which that au-
thor had simply announced without demonstration; being
an excellent specimen of the ancient geometrical method ;
in which piece also, as well as in his letter to the Jesuit
Ceva, which follows it, are found several other curious and
novel particulars. — Another paper of Grandi's is also in-
serted in the 33d volume of the Philos. Trans., called a
Handful or a Bouquet bf Geometrical Hosts, being a di ■
4 F a

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G R A

scrtation on certain curves geometrically described in a
circle. This he afterwards enlarged in another treatise,
published in 1/28, entitled Flores Geometrici ex Rhodo-
nearum, &c. He was also author of several other miscel-
laneous pieces, on the ancient and modern geometry ; as,
his Quadrature of the Circle and Hyperbola by Infinite
Parabolas, in 1/03 and 1710; his Dissertation on Infinites
of Infinites, &c, in 1/10; an Italian edition of Euclid's
Elements; and a posthumous treatise on Conic Sections,
also in the Italian language, in 1 7-14. Grandi it seems was
of a turbulent and quarrelsome disposition, being almost
always engaged in disputes on various subjects, geometri-
cal, theological, metaphysical, or philological.

GRAPE-Shot, in Artillery, is a combination of small
shot, put into a thick canvass bag, and corded strongly to-
gether, so as to form a kind of cylinder, of a diameter
suited to the intended piece of ordnance, from the smallest
to the largest. The number of shot in a grape varies ac-
cording to the service, or size of the guns. Of late the
grape-shot have been superseded by canister-shot, and by
shrapnel shells, &c, as more convenient and efficacious.

GRAPHOMETER, a mathematical instrument, used
in land-surveying &c, and is otherwise called Semicircle ;
which see

GRAVE, in Music, is applied to a sound which is in a
low or deep tone. The thicker the chord or string, the
more grave the tone or note; and the smaller, the acuter.
Notes are supposed to be the more grave, in proportion as
the vibrations of the chord are less quick.

GRAVESANDE (William James), a very celebrated
Dutch mathematician and philosopher, was born at Bois-
le-duc, Sept. 27, l688. He studied the civil law at Ley-
den, but mathematical learning was his favourite amuse-
ment. When he had taken his doctor's degree, in 1707,
he went and settled at the Hague, where he practised at
the bar, and cultivated an acquaintance with learned men;
with a society of whom he published a periodical review,
entitled Le Journal Litteraire, which was continued with-
out interruption from the year 1713 to the year 17-2.
The parts of it written or extracted by Gravesande were
chiefly those relating to geometry and physics. But he
enriched it also with several original pieces entirely of his
own composition ; viz, Remarks on the Construction of
Pneumatical Engines: A Moral Essay on Lying: and a
celebrated Essay on the Collision of Bodies; which, as it
opposed the Newtonian philosophy, was attacked by Dr.
Clarke, and many other learned men. In 1715, when the
States sent to congratulate George the 1st, on his acces-
sion to the throne, Dr. Gravesande was appointed secre-
tary to the embassy. During his stay in England he was
admitted a member of the Royal Society, and became in-
timately acquainted with Sir Isaac Newton. On his return
to Holland, he was chosen professor of mathematics and
astronomy at Leydeti; where he had the honour, of first
teaching the Newtonian philosophy, which was then in its
infancy. He died in 1742, at 54 years of age.

Gravesande was a man amiable in his private character,
and respectable in his public one; for few men of letters
have rendered more eminent services to their country. The
ministers of the republic consulted him on all occasions
when his talents were requisite to assist them, which liis

skill in calculation often enabled him io df> in financial
concerns. He was of great service as a decipherer, in de-
tecting the secret correspondence of their enemies. And,
in his own profession, none ever applied the powers of na-

ture with more success, or to more useful purposes. — Of
his publications, the principal are,

1. An Introduction to the Newtonian Philosophy; or,
a Treatise on the Elements of Physics, confirmed by Ex-
periments. This performance, being only a more perfect
copy of his public lectures, was first printed in 1720 ; and
went through many editions, with considerable improve-
ments: the 6th edit, is in English, in 2 large vols. 4to, by
Dr. Desaguliers, in 1747, under the title of Mathematical
Elements of Natural Philosophy, confirmed by Experi-
ments.

2. A treatise on the Elements of Algebra, for the use
of Young Students; to which is added a Specimen of a
Commentary on Newton's Universal Arithmetic; as also,
A New Rule for determining the Form of an Assumed Infi-
nite Series. — 3. An Essay on Perspective : This was writ-
ten at 19 years of age. — 4. A New Theory of the Colli-
sion of Bodies. — 5. A Course of Logic and Metaphysics.
— With several smaller pieces.

His whole mathematical and philosophical works, ex-
cept the first-mentioned article, were collected and pub-
lished at Amsterdam, in 2 vols. 4to, to which is prefixed a
critical account of his life and writings, by Professor Alla-
mand.

GRAVIMETER, the name given by M. Guyton, to an
instrument for measuring the specific gravities of bodies:
he adopts this name rather than either areometer or hydro-
meter, because these latter terms are grounded on the sup-
position that the liquid is always the thing weighed ;
whereas, with regard to solids, the liquid is the known term
of comparison, to which the unknown weight is referred.

Guyton's gravimeter is executed in glass, and is of a cy-
lindricform, being that which requires the smallest quan-
tity of fluid, and is on that account preferable, except so
far as it is necessary to deviate for the security of a verti-
cal position. It carries two basins, one of them superior,
at the extremity of a thin stem, towards the middle of
which, the fixed point of immersion is marked. The other,
or lower basin, terminates in a point; it contains the bal-
last, and is attached to the cylinder by two branches. The
moveable suspension, by means of a hook, has the inconve-
nience of shortening the lever which is to secure the verti-
cal position.

The cylinder of this instrument is 071 inches diameter;
and (v85 inches in length. It carries in the upper basin an
additional constant weight of 5'1 15 grains. These dimen-
sions might be increased so as to render it capable of re-
ceiving a much more considerable weight ; but this is un-
necessary. M. Guyton has added a piece which he calls
the plonsjeur, because in fact it is placed in the lower ba-
sin when used, and is consequently entirely immersed in
the fluid. It is a bulb of glass loaded with a sufficient
quantity of mercury, in order that its total weight may be
equal to the constant additional weight, added to the weight
of the volume of water displaced by this piece. It will be
readily understood that the weight being determined at
the same temperature at which the instrument was origi-
nally adjusted, it will jink to tin- same mark mi the stem,
whether it be loaded with a constant additional weight in
tin- Upper basin, or whether the effect of this weight be
produced by the additional piece in the lower dish,

This instrument may he used lor solids, as well as fluids,
provided the absolute weight of the solid to be examined
be rather less than the constant additional weight in the
upper basin, which is 115 grains-.

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GfiA

For liquids of less specific gravity than water, the gra-
vimeter, without the additional weight above mentioned,
weighs about 459 grains, in the dimensions before laid
down. It would be easy to limit its weight to the greatest
accuracy. We have therefore the range of one-fifth of
buoyancy, and consequently the means of ascertaining all
the intermediate densities, from water to the most highly
rectified spirit, which is known to bear in this respect the
ratio of 8 to 10, with regard to water.

When liquids of greater specific gravity than water are
to be tried, the constant weight being applied below, by
means of the additional piece, which weighs about 138
grains, the instrument can receive in the upper basin more
than 4 times the usual additional weight, without losing
the equilibrium of its vertical position. In this state it is
capable of showing the specific gravity of the most concen-
trated acid. It possesses also another property, which is this ;
nameiy, that it may be employed as a balance to deter-
mine the absolute weight of such bodies as do not exceed
its additional load. And the purity of water being known,
it will indicate the degrees of rarefaction and condensation,
in proportion to its own bulk.

To rind the specific gravity of any solid by the gravime-
ter. Rule : " From the weight in the upper basin, when the
instrument is properly immersed in the unknown fluid, take
the weight which isplaeed with the body in the same scale at
the like adjustment; the remainder is the absolute weight
of the solid. Multiply this by the specific gravity of the
fluid, and reserve the product. From the additional weight
when the body is placed in the lower basin, take the weight
when it was placed in the upper; the remainder will be
the loss of weight by immersion. Divide the reserved by
the loss by immersion, and the quotient will be the spe-
cific gravity of the solid with regard to distilled water, at
the standard temperature and pressure."

To find the specific gravity of a fluid, proceed thus:
" To the weight of the gravimeter add the weight required
in the upper basin, to sink it in the unknown fluid. Again,
To the wcightof the gravimeter add the weight required in
the same manner to sink it in distilled water. Divide the
first sum by the second, and the quotient will be the spe-
cific gravity of the fluid in question."

GRAVITATION, the exercise of gravity, or the pres-
sure a body exerts on another body beneath it by its weight.
This is sometimes distinguished from gravity. Thus, M.
Maupcrtuis, in his Figure dclaTerrc, lakesgravity for that
force by which a body would fall to the earth supposed at
rest; and gravitation for the same, but diminished by the
centrifugal force. It is only gravitation, or gravity thus
blended with the centrifugal force, that we can usually
measure by our experiments. Methods however have been
found to distinguish what remains of the primitive gravity,
and what has been destroyed by the centrifugal force.

It is one of the laws of nature, discovered by Newton,
and now received by all philosophers, that every particle
of matter in nature gravitates towards every other particle ;
which law is the main principle in the Newtonian philoso-
phy. Bat what is called gravitation with respect to the
gravitating bofly1, is usually called attraction with respect
to the body gravitated to. The planets, both primary and
secondary, as afs'd the comets, do all gravitate towards the
sun, and towards each other; as well as the sun towards
them; and that in proportion to the quantity of matter in
each of them.

The Peripatetics &c hold, that bodies only gravitate or
weigh when out of their natural places, and that gravita-

tion ceases when they are restored to the same, the pur
pose of nature being then fulfilled ; and they maintain that
the final cause of this faculty is only to bring elementary
bodies to their proper place, where they may rest. But
the moderns show that bodies exercise gravity even when
at rest, and in their proper places. This is particularly
shown of fluids ; and it is one of the laws of hydrostatics,
demonstrated by Boyle and others, that fluids gravitate in
proprio loco, the upper parts pressing on the lower, &c.
For the laws of gravitation of bodies in fluids specifically
lighter or heavier than themselves, see Specific Gra-
vity. Also for the centre or line or plane of gravitation,
see Centre, Line, or Plane.

GRAVITY, in Physics, the natural tendency or incli-
nation of bodies towards the centre. And in this sense gra-
vity agrees with centripetal force.

Gravity however is, by some, defined more generally as
the natural tendency of one body towards another; and
again by others still more generally as the mutual tendency
of each body, and each particle of a body, towards all
others: in which sense the word answers to what is more
usually called attraction. Indeed the terms gravity, weight,
centripetal force, and attraction, denote in effect all the
same thing, only in different views and relations ; all which
however it is very common to confound, and use promis-
cuously. But, in propriety, when a body is considered as
tending towards the earth, the force with which it so tends
is called Gravity, Force of Gravity, or Gravitating Force;
when the body is considered as immediately tending to the
centre of the earth, it is called Centripetal Force ; but
when we consider the earth, or mass to which the body
tends, it is called Attraction, or Attractive Force; and
when it is considered in respect of an obstacle or another
body in the way of its tendency, on which it acts, it is
called Weight.

Philosophers think differently on the subject of gravity.
Some consider it as an inactive property or innate powerin
bodies, by which they endeavour to unite together. Others
hold gravity in this sense to be an occult quality, and to
be exploded assuch out of all sound philosophy. Newton,
though he often calls it a vis, power, or property in bo-
dies, yet explains himself, that he means nothing more by
the word but the effect or phenomenon : he does not con-
sider the principle, the cause by which bodies tend down-
wards, but the tendency itself, which is no occult quality,
but a sensible phenomenon, be its causes what they may ;
whether a property essential to body, as some make it, or
superadded to it, as others; or even an impulse of some
body from without, as others.

It is a law of nature long observed, that all bodies near
the earth have a gravity or weight, or a tendency towards
its centre, or at least perpendicular to its surface; which
law the moderns, and especially Sir I. Newton, from cer-
tain observations, have found to be much more extensive,
and holding universally with respect to all known bodies
and matter in nature. It is therefore at present acknow-
ledged as a principle or law of nature, that all bodies, and
all the particles of all bodies, mutually gravitate towards
each other: from which single principle it is that Newton
has happily deduced all the great phenomena of nature.
Hence, Gravity may be distinguished into Particular and
General.

Particular Gravity, is thatwhich respects the earth,
or by which bodies descend, or tend towards the centre of
the earth ; the phenomena or properlies of which are as
follow : —

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1. All circumterrestrial bodies do hereby tend to- same distance from the centre, are proportional to the quari-
wards a point, which is either accurately or very nearly tities of matter in the bodies.

the centre of magnitude of the terraqueous globe. Not General or Universal Gravity, is that by which not
that it is meant that there is really any virtue or charm only the planets, but all the bodies and particles of mat-
in the point called the centre, by which it attracts bodies ; ter in the universe tend towards one another. The exist-
but because this is the result of the gravitation of bodies ence of the same principle of gravitation in the superior
towards all the parts of which the earth consists. regions of the heavens, as on the earth, is one of the

2. This point or centre is fixed within the earth, or great discoveries of Newton, who made the proof of it as
at least has been so far as any authentic history reaches, easy as that on earth. At first it would seem this was only
For a consequence of its shifting, though ever so little, conjecture with him: he observed that all bodies near the
would he the overflowing of the low lands on that side earth, and in its atmosphere, had the property of tending
of the globe towards which it should approach. Dr. directly towards it ; he soon conjectured that it probably
Halley suggests, that it would well account for the uni- extended much higher than any distance to which we
versal deluge, to have the centre of gravitation removed could reach, or make experiments; and so on, from one
for a time towards the middle of the then inhabited distance to another, till he at length saw no reason why
world ; for the change of its place but the 2000th part it might not extend as far as the moon, by means of which
of the radius of the earth, or about two miles, would be she might be retained in her orbit, as a stone in a sling is
sufficient to lay the tops of the highest hills under water. retained by the hand ; and if so, he next inferred, why

3. In all places equidistant from the centre of the earth might not a similar principle exist in the otheiygreat bo-
ttle force of gravity is nearly equal. Indeed all parts of dies in the universe, the sun and all the other planets,
the earth's surface are not at equal distances from the cen- both primary and secondary, which might all be retained
tre, because the equatorial parts are higher than the polar in their orbits, and perform their revolutions, by means
parts by about 17 miles; as has been proved by the ne- of the same universal principle of gravitation.

cessity of making the pendulum shorter in those places, These conjectures he soon realized and verified by ma-
Wore it will vibrate seconds. In the new Petersburg thematical proofs. Kepler had discovered by contempla-
Transactions, vol. 6' and 7, M. Krafft gives a formula for ting the motions of the planets about the sun, that the
the proportion of gravity in different latitudes on the area described by a line connecting the sun and planet,
earth's surface, which is this : as this revolved in its orbit, was always proportional to
y = ( I -+- 0'0052S48 sine2A)g; the time of its description, or that it described equal areas
where g denotes the gravity at the equator, and_y fhegra- in equal times, in whatever part of its orbit the planet
vity under any other latitude A. On this subject, see also might be, moving always so much the quicker as its dis-
the articles Degree, and Earth. tance from the sun was less. Audit is also found that

4. Gravity equally affects all bodies, without regard the satellites, or secondary planets, observe the same law
either to their bulk, figure, or matter : so that, abstracting in revolving about their primaries. But it was soon
from the resistance of the medium, the most compact and proved by Newton, that all bodies moving in any curve
loose, the greatest and smallest bodies, would all descend line described on a plane, and which, by radii drawn to
through an equal space in the same time; as appears from any certain point, describe areas about the point propor-
the quick descent of very light bodies in an exhausted re- tional to the times, are impelled or acted on by some
ceiver. The space which bodies do actually fall, in vacuo, power tending towards that point. Consequently the
is 16^ feet in the first second of time, in the latitude of power by which all these planets revolve, and are retained
London ; and for othertimes, either greater orless than that, in their orbits, is directed to the centre about which they
the spaces descended from rest are directly proportional'to move, viz, the primary planets to the sun, and the satel-

the squares of the times, while the falling body is net far
from the earth's surface.

5. This power is the greatest at the earth's surface,
from whence it decreases both upwards and downwards,

lites to their several primaries.

Newton also demonstrated, that if several bodies re-
volve with an equable motion in several circles about the
same centre, and that if the squares of their periodical

but not both ways in the same proportion; for upwards times be in the same proportion as the cubes of their dis-
the force of gravity is less, or decreases, as the square of tances from the common centre, then the centripetal
the distance from the centre increases, so that at a double forces of the revolving bodies, by which they tend to their
distance from the centre, above the surface, the force central body, will be in the reciprocal or inverse ratio of
would be only l-4th o'l what it is at the surface; but be- the squares of the distances. Or if bodies revolve in or-
low the surface, the power decreases in such sort that its bits approaching to circles, and the apses of those orbits
intensity is in the direct ratio of the distancefrom the cen- be at rest, then also the centripetal forces of the revolving
tre ; so that at the distance of half a semidiameter from the bodies will be reciprocally proportional to the squares of
centre, the force would be but half what it is at the surface; the distances. But it had been agreed on by astronomer?,
at 4 of a semidiameter the force would be j, and so on. and particularly Kepler, that both these cases obtain in
0. As all bodies gravitate towards the earth, so does the all the planets. And therefore he inferred, that the fen-
earth equally gravitate towards all bodies; as well as all tripetal forces of all the planets, are reciprocally propor-
bodies towards particular parts of the earth, as hills, &c, tional to the squares of the distances from the centres of
which has been proved by the attraction a hill has upon their motions.

a plumb line, insensibly drawing it aside. — Hence the gra- On the whole it appears, that the planets are retained
vitating force of entire bodies consists of those of all their in their orbits by some power which is continually acting
parts : lor, by adding or taking away any part of the mat- upon them : that this power is directed towards the Cen-
ter ol a body, its gravity is increased or decreased in the tre of their motions: that the intensity or efficacy of this
proportion of the quantity of such portion to tho whole power increases on an approach towards the centre, and
mass. Hence also the gravitating powers of bodies, at the diminishes on receding from the same, and that in the re-

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ciprocal duplicate ratio of the distances : and that, by
comparing this centripetal force of the planets with the
force of gravity on the earth, they are found to be perfectly
alike, as may easily be shown in various instances. For ex-
ample, in the case of the moon, the nearest of all the pla-
nets. The rectilinear spaces described in any given time by a
falling body, urged by any powers, reckoning from the
beginning of its descent, are proportional to those powers.
Consequently the centripetal force of the moon revolving
in her orbit, will be to the force of gravity on the surface
of the earth, as the space which the moon , would describe
in falling during any small time, by her centripetal force
towards the earth, if she had no circular motion at all, to
the space a body near the earth would describe in falling
by its gravity towards the same.

Now by an easy calculation of those two spaces, it ap-
pears that the former force is to the latter, as the square

refernnggravity or weight to a native inclination in heavy
bodies to be in their proper place or sphere, the centre, of
the earth. And Copernicus ascribes it to an innate prin-
ciple in all parts of matter, by which, when separated
from their wholes, thej endeavour to return to them again
the nearest way. Jn answer to Aristotle and his followers,
who considered the centre of the earth as the centre of the
universe, he observed that it was reasonable to think there
was nothing peculiar to the earth in this principle of «ra-
vity: that the parts of the sun, moon, and stars, tended
likewise to each other, and that their spherical figure was
preserved in their various motions by this power. °Copern.
Revol. lib. 1, cap. 9. But neither of these systems assigns
any physical cause of this great effect : they only amount
to this, that bodies descend because they are inclined to
descend.

Kepler, in his preface to the commentaries concerning

of the semi-diameter ofahe earth is to the square of that of the planet Mars, speaks of gravity as of a power that

the moon's orbit. The moon's centripetal force therefore
is equal to the force of gravity ; and consequently these
forces are not different, but they are one and the same : for
if they were two distinctforces, bodies acted on by the two
powers conjointly would fall towards the earth with a velo-
city double to that arising from the sole power of gravity.

It is evident therefore that the moon's centripetal force,
by which she is retained in her orbit, and prevented from
flying off in tangents, is the very power of gravity of the
earth extended thither. See Newton's Pnncip. lib. 1,
prop. 45, cor. 2, and lib. 3, prop. 3 ; where the numeral
calculation may be seen at full length.

The moon therefore gravitates towards the earth, and
reciprocally the earth towards the moon. And this is also
further confirmed by the phenomena of the tides.

The like reasoning may also be applied to the other
planets. For, as the revolutions of the primary planets
round the sun, and those of the satellites of Jupiter and
Saturn round their primaries, are phenomena of the same
kind with the revolution of the moon about the earth; and
as the centripetal powers of the primary are directed to-
wards the centre of the sun, and those of the satellites to-
wards the centres of their primaries ; and lastly, as all
these powers are reciprocally as the squares of the dis-
tances from the centres, it may safely be concluded that
the power and cause are the same in all.

Therefore, as the inoon gravitates towards the earth,
and the earth towards the moon ; so do all the secondaries
to their primaries, and these to their secondaries ; and so
also do the primaries to the sun, and the sun to the pri-
maries. Newton's Princip. lib. 3, prop. 4, 5, 6'; Greg.
Astron. lib. 1, sect. 7, prop. 46 and 47.

The laws of Universal Gravity are the same as those of
bodies gravitating to the earth, before laid down.

Cause of Gravity. Various theories have been ad-
vanced by the philosophers of different ages to account for
this grand principle of gravitation. The ancients, who
were only acquainted with particular gravity, or the ten-
dency of sublunar bodies towards the earth, aimed no fur-
ther than a system that might answer the more obvious
phenomena of it. However, some hints are found con-
cerning the gravitation of celestial bodies, in the account
given of the doctrine of Thales and his successors ; and it
would seem that Pythagoras was still better acquainted
with it, to which it is supposed he had a view in what he
taught concerning the Harmony of the Spheres.

Aristotle and the Peripatetics content themselves with

mutual between bodies, and says that the earth and moon
tend towards each other, and would meet in a point so
many times nearer to the earth than to the moon, as the
earth is greater than the moon, if their motions did not
hinder it. He adds, that the tides arise from the gravity
of the waters towards the moon. To him we also owe the
important discovery of the analogy between the distances
of the several planets from the sun, and the periods in
which they complete their revolutions, viz, that the squares
ot their periodic times, are always in the same proportion
as the cubes of their mean distances from the sun. How-
ever, Kepler, Gassendi, Gilbert, and others, ascribe Gra-
vity to a certain magnetic attraction of the earth ; con-
ceiving the earth to be onegreat magnet, continually emit-
ting effluvia, which take hold of all bodies, and draw
them towards the earth. But this is inconsistent with the
several phenomena.

Descartes and his followers, Rohault, &c, attribute gra-
vity to an external impulse or trusion of some subtile mat-
ter. By the rotation of the earth, say they, all the parts
and appendages of it necessarily endeavour to recede from
the centre of rotation; butwhence they cannotall actually
recede, as there is no vacuum or space to receive them.
But this hypothesis, founded on the supposition of a ple-
num, is overthrown by what has been since Droved of the
existence of a vacuum.

Dr. Hooke inclines to an opinion much like that of Des-
cartes. Gravity he thinks deducible from the action of a
most subtile medium, which easily pervades and penetrates
the most solid bodies ; and which, by some motion it has,
detrudes all earthly bodies from it, towards the centre of
the earth. Vossius too, and many others, give partly into
the Cartesian notion, and suppose gravity to arise from
the diurnal rotation of the earth round its axis. See Birch's
Hist, of the Royal Soc. vol. 2, pp. ~i1 and 91.

Dr. Halley, despairing of any satisfactory theory, chooses
to have immediate recourse to the agency of the Deity.
So Dr. Clarke, from a view of several properties of gravity,
concludes that it is no adventitious effect of any motion,
orsubtile matter, butanoriginal and general law impressed
by God on all matter, and preserved in it by some effi-
cient power penetrating the very solid' and intimate sub-
stance of it; being found always proportional, not to the
surfaces of bodies or corpuscles, but to their solid quan-
tity and contents. It should therefore be no more inquired
why bodies gravitate, than how they came to be first put
in motion. Annot.in Rohault. Pliys. part 1, cap. 11.

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Gravesande, in his Introduct. ad Philos. Newton, con-
tends that the cause of gravity is utterly unknown; and
that we are to consider it no otherwise than as a law of
nature originally and immediately impressed by the Crea-
tor, without any dependence on any second law or cause
at all. Of this he thinks the three following considera-
tions sufficient proof. 1. That gravity requires thepresence
of the gravitating or attracting body : so the satellites of
Jupiter, for ex. gravitate towards Jupiter, wherever he
may be. 2. That the distance being supposed the same,
the velocity with which bodies are moved by the force ot
gravity, depends on the quantity of matter in the attracting
body ; and the velocity is not changed, whatever the hiass
of the gravitating body may be. 3. That if gravity do
depend on any known law of motion, it must be some im-
pulse from an extraneous body ; so that as gravity is con-
tinual, a continual stroke must also be required. Now if
there be any such matter continually striking on bodies,
it must be fluid, and subtile enough to penetrate the sub-
stance of all bodies : but how shall a body subtile enough
to penetrate the substance of the hardest bodies, and so
rare as not sensibly to hinder the motion of bodies, be able
to impel vast masses towards each other with such force ?
how does this force increase the ratio of the mass of the
body, towards which the other body is moved ? whence
is it that all bodies move with the same velocity, the
distance and body gravitated to being the same ? can a
fluid which only acts on the surface, either of the bodies
themselves, or their internal particles, communicate such
& quantity of motion to bodies, which in all bodies shall
exactly follow the proportion of the quantity of matter in
them ?

Mr. Cotes goes yet further. Giving a view of Newton's
philosophy, he asserts that gravity is to be ranked among
the primary qualities of all bodies; and deemed equally
essential to matter as extension, mobility, or impenetra-
bility. Praifat. ad Newt. Princip. But Newton himself
disclaims this notion ; and to show that he does not take
gravity to be essential to bodies, he declares his opinion
of the cause ; choosing to propose it by way of query, not
being yet sufficiently satisfied about it for want of experi-
ments. Thus, after having shown that there is a medium
in nature vastly more subtile than air, by whose vibrations
sound is propagated, by which -light communicates heat
to bodies, and by the different densities of which the re-
fraction and reflection of light are performed ; he pro-
ceeds to inquire: " Js not this medium much rarer within
the dense bodies of the sun, stars, planets, and comets,
than in the empty celestial spaces between them ? and in
passing from them to greater distances, doth it not grow
denser and denser perpetually, and thereby cause the
gravity of those great bodies towards one another, and
of their parts towards the bodies ; every body endeavouring
to recede from the denser parts of the medium towards
the rarer ?

" For if this medium be supposed rarer within the sun's
body than at its surface, and rarer there than at the
hundredth part of an inch from his body, and rarer there
than at the orbit of Saturn ; I see no reason why the. in-
crease of density" should stop anywhere, and not rather
be continued through all distances from the sun to Sa-
turn, and beyond.

" And though this increase of density may at great
distances be exceeding slow : yet if the elastic force of this
medium be exceeding great, it may suffice to impel bodies

from the denser parts of the medium towards the rarer
with all that power which we call gravity.

" That the elastic force of this medium is exceeding
great, may be gathered from theswiftness of its vibrations.
Sounds move about 1140 English feet in a second of time,
and in seven or eight minutes of time, they move about
one hundred English miles: light moves from the sun to
us in about 7 or 8 minutes of time, which distance is
about 70000000 English miles, supposing tin- horizontal
parallax of the sun to be about 12 seconds; and the vi-
brations, or pulses of this medium, that they may cause
the alternate fits of easy transmission, and easy reflection,
must be swifter than light, and by consequence abo\e
7UO00O times swifter than sounds ; and therefore the
elastic force of this medium, in proportion to its den-
sity, must be above 700000 x 700000 (that is, above
4-90000000000) times greater than the clastic force of
the air is, in proportion to its density : for the velocities
of the pulses of elastic mediums, are in a subduplicate
ratio of the elasticities and the rarities of the mediums
conjointly.

" As magnetism is stronger in small loadstones than in
great ones, in proportion to their bulk; and gravity is
stronger on the surfaceof small planets, than on those of
great ones, in proportion to their bulk ; and small bodies
are agitated much more by electric attraction than great
ones : so thesmallness of the rays of light may contribute
very much to the power of the agent by wdiich they are
refracted ; and if any one should suppose, that ether (like
our air) may contain particles which endeavour to recede
from one another (for I do not know what this ether is),
and that its particles are exceedingly smaller than those
of air, or even than those of light ; the exceeding small-
ness of such particles may contribute to the greatness of
the force, by which they recede from one another, and
thereby make that medium exceedingly more rare and
elastic than air, and, of consequence, exceedingly less able
to resist the motions of projectiles, and exceedingly more
able to press upon gross bodies by endeavouring to expand
itself." Optics, Query 21, &c.

IYI. Le Sage attempts to account for the cause of gravity
by the following ingenious hypothesis : " Imagine," says
this author, " that through all space numberless cor-
puscles or atoms, almost infinitely small, are in perpetual
motion: that every corpuscle has its determined direction,
and moves for ever in a straight line with a velocity far
exceeding that of light. It is evident that the directions
of these corpuscles may be so various, they may bethem-
svlves so small, and their velocity so great, that though
they follow each other at vast distances, and leave space,
in reality, almost empty ; yet they may abound every
where in such a manner, that in a portion of lime, almost
infinitely small, a great number of them shall pass through
every point of space whatsoever. On whatever point of
space therefore our attention is fixed, we may consider it
as a centre, to which the motions of an infinite number
of atoms arc referred, either by diverging therefrom', or
converging to it." This constitution ol what Le Sage calls
the gravific fluid being conceived, suppose a solid body to
be plunged into it of any figure whatever, larger than one
of the corpuscles, and in some degree, if not entirely, im-
pervious to llu-se panicles; this body will remain at
rest, or at least without any progressive motion, the im-
pulses of Hie particles that strike against it being equal 111
opposite directions. It may oscillate a little backward

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and forward, but will not be forced any considerable
distance from its place.

Now let there be plunged into the gravific fluid, an-
other body, of any figure, and at any distance from the
first. These two bodies will immediately begin to move
towards each other. For the one serving to protect the
other from a certain quantity of the impulsion of the
corpuscles, the currents thus left without opposition ne-
cessarily produce their effect, and impel the bodies to-
wards each other.

Their motion towards one another will be continually
accelerated ; and the forces producing that acceleration
will increase in proportion as the one body stops more of
the currents from falling on the other ; that is, nearly as
the squares of the distances diminish.

Again, if the solid particles of which the bodies are
made up, be impenetrable to the gravific corpuscles, but
the bodies themselves, on account of their porosity, per-
meable by them in a certain degree, the number of parti-
cles that are stopped by each of the bodies will be, caeteris
paribus, proportional to the number of solid particles,
that is, to the quantities of matter in the bodies ; and
hence, in general, the force urging the bodies towards
each other, will be directly as their masses, and inversely
as the squares of their distances. " Thus, by mechanical
action," says Le Sage, " is the Newtonian law of gravi-
tation explained in all its parts."

In objection to this system, we advance the following,
urged by Boscovich :— " In this system," says Boscovich,
" no particle of the fluid returns to its place, or ever
passes a second time through the same point of space. A
constant supply of new particles is therefore necessary, as
all those that are contained within the limits of the sen-
sible universe, at any instant, must be replaced before they
have entirely escaped from it, and sent forth to traverse
for ever the deserts of uninhabited extension. The
imagination is terrified at this constant exertion of what
cannot be considered as less than creative power employed
in producing existencies that, for a limited time, are to
be useful ; and through all the rest of infinite duration,
are to serve no purpose whatsoever."

About the year 1806 several of the clergy, as well as
other individuals in this country, being alarmed, it was
said, at the efforts which the philosophers on the continent
were making to account for the various operations of na-
ture upon mere mechanical principles, with a view, as
they supposed, to exclude the Deity from any concern in
the government of the world, and thereby to lay a foun-
dation for atheism, Mr. Vince, professor of astronomy in
the university of Cambridge, was requested to examine
the most plausible and generally received hypotheses,
which had been framed to account for gravitation, and
to give the result of his examination. The inquiry, he
tells us, was favourably received, and it was suggested that
it might not be improper to offer it to the Royal Society;
but on being presented by his friend Dr. Maskelyne, the
astronomer-royal, and not meeting that reception which
would have been gratifying to the professor's feelings, and
have reflected credit on the Society, he in consequence
withdrew it, and submitted the paper to the opinion of
the public, in a small pamphlet on this subject, printed
at Cambridge, in 1806.

The systems which he has there considered, are those
of Descartes, Newton, Le Sage, and Bernoulli, with a
slight mention of some others whose inconsistency ren-

VOL. I.

dered any detailed account of them unnecessary, but
omitting that of Boscovich.

Speaking of the system of Saussure, (which, by the bye,
is that of Le Sage, given above,) he observes, that if the
corpuscles there mentioned be admitted to move as the
author describes them, it will follow that the moving
force of each body depends on the surfaces of the bodies,
whereas it ought to be as the quantity of matter in each
respectively : " this consideration, therefore," continues
the professor, " without entering into any further exa-
mination of the hypothesis, is sufficient to show that it
cannot be admitted."

The hypothesis which Mr. Vince has made the subject
of mathematical consideration, is that of our illustrious
countryman Newton; who, as has been observed, sup-
posed, that if the sun and the planets acted on each other,
it must be by some intermediate and invisible substance,
such as that of an elastic fluid. The professor then ad-
mitting that an elastic fluid is the cause of gravity, lays
down, previous to the estimation of its effects on bodies
placed in it, the two following facts, as standards, to which
his results are to be referred for refutation or support :

1. That the gravitation of a planet varies inversely as
the square of its distance from the sun ; that is, whatever
be the magnitude or density of the planet, its acceleration
towards the sun varies in that ratio.

2. That a force varying inversely as the square of the
distance, cannot be compounded of several forces, each of
which does not vary in that ratio.

On these grounds, the professor enters on the investi-
gation of the hypothesis; by supposing that the density
of the medium is measurable by the number of particles,,
uniformly diffused on a given square area ; that the varia-
tion of density at any distance d, from the sun is as dm ;
and consequently the distance of the particles themselves

as -r-. Then, admitting that the elastic force of thepar-

ticles vary as the nth power of their distance, or as -t=j

he at last obtains :

n 'im — mn - -

— x a 1, for the ac-

ue n

celerative force of the planet towards the sun at the dis-
tance a from the same.

" Now it is to be observed," says Mr. Vince, "that this
expression is in terms off, the density of the planet; and
therefore does not agree with the first given fact or law of
gravitation, that law being independent of the density or
magnitude. Therefore in this case, the hypothesis is ren-
dered nugatory."

But besides this, he states that the body will, instead
of being forced towards the sun, be driven from it, even
when the force varies inversely as the square of the dis-
tance. Hence, and from what is afterwards demonstrated,
he concludes, that it is not possible for any law of varia-
tion of density, of the fluid, in terms of the distance from
the sun, combined with any law of variation of the repul-
sive force of the particles of the fluid in terms of their dis-
tance, which can satisfy the law of gravitation. And if
we were to suppose the law of density to vary in terms of
any other quantities besides those expressing the distance
from the sun, such quantities must enter into the law of
force, and thereby make a still greater deviation from the
law of the inverse square of the distance. Considering
the matter therefore, in a mathematical point of view, we
4G

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are justified, he says, in rejecting this hypothesis as the
cause of gravitation.

Mr. Playfair however, in reviewing this examination, ob-
serves that, this conclusion he (Mr. Vince) next endea-
vours to extend to all the laws of variation of density and
of elasticity that can possibly exist, by showing that no
one can be admitted that is not capable of being expressed
by a single term, and consequently by the mth or' wth
power of the distance. It is here, if we mistake not, that
the error lies. For, though a variation of density or of
elasticity, expressed thus, axm -+- bx", cannot take place,
because the force arising from it would also involve two
terms, yet if one of the terms be constant, as if m = 0,
and so the expression = a + bx", then the force would
be expressed by one term only, viz, by the fluxion of ft.c",
that is, by a quantity proportional to xn_I. Thus, for ex-
ample, if a; be any distance from the sun's centre, d the
density of the ether at that distance, and e its elasticity.
Let d increase in the direct ratio of x, or d = ax-, and

suppose e = c =c— — , where c is a constant

rl d , ax

quantity, that would be determined, if we knew at what
distance from the sun's centre the elasticity of the ether is
= 0. If, for example, it is equal to nothing at the dis-
tance s from the sun's centre, or when x = s, c = , so

that e = — , and the differential of this being taken,

as_ ax °

gives e = — ?. Now e is the difference of elasticity for
° oar J

the change of distance x ; and therefore is the force with
which a small spherical body, or a single particle of mat-
ter, impervious to the ether, would be impelled toward

the sun ; and it varies as -3-, or in the inverse ratio of the

squares of the distances from the sun. It is therefore
possible that an elastic fluid may be so constituted as to
produce a tendency of one body to another, varying inversely
as the squares of the distances of these bodies. — For this pur-
pose, there is only required an elastic fluid, of which the
density is as the distance from the sun, and the elasticity
as a certain given magnitude diminished by the reciprocal
of that distance. There are many other hypotheses con-
cerning the density and elasticity of the fluid, which will
give the same result with this ; all indeed in which we

have these equations, d = axm, and e = c -. This

is directly contrary to Mr. Vince's conclusion, " that it
is not possible for any law of variation of the density of the
fluid in terms of the distance from the sun, combined with
any law of variation of the repulsive force of the particles
of the fluid in terms of their distance, to satisfy the law of
gravitation." See the Edinburgh Review, vol. 13, pa.
101, &c.

Gravity, in Mechanics, denotes the conatus or ten-
dency of bodies towards the centre of the earth. That
part of mechanics which considers the equilibrium or mo-
tion of bodies arising from gravity or weight, is particu-
larly called Statics.

Gravity is distinguished into absolute and relative.

Absolute Gravity is that with which a body descends
freely and perpendicularly through an unresisting me-
dium. The laws of which see under Descent of Bo-
dies, Acceleration, Motion, &c.

Relative Gravity is that with which a body descends

on an inclined plane, or through a resisting medium, or
as opposed by some other resistance. The laws of which,
see under the articles Inclined Plane, Descent,
Fluid, Resistance, &c.

Gravity, in Hydrostatics. The laws of bodies gra-
vitating in fluids constitute the doctrine of Hydrostatics.

Gravity is here divided into absolute and specific.

Absolute or True Gravity, is the whole force with
which the body tends downwards.

Specific Gravity, is the relative, comparative, or ap-
parent gravity in any body, in respect of that of an equal
bulk or magnitude of another body ; denoting that gravity
or weight which is peculiar to each species or kind
of body, and by which it is distinguished from all other
kinds.

In this .sense a body is said to be specifically heavier
than another, when under the same bulk it contains a
greater weight than that other ; and reciprocally, the lat-
ter is said to be specifically lighter than the former. Thus,
if there be two equal spheres, each one foot in diameter ;
the one of lead, and the other of wood : since the leaden
one is found heavier than the wooden one, it is said to be
specifically, or in specie, heavier ; and the wooden one
specifically lighter.

This kind of gravity is by some, called relative; in op-
position to absolute gravity, which increases in proportion
to the quantity or mass of the body.

Laws of the Specific Gravity of Bodies.

I. If two bodies be equal in bulk, their specific gravities
are to each other as their weights, or as their densities.

II. If two bodies be of the same specific gravity or
density, their absolute weights will be as their magnitudes
or bulks.

III. In bodies of the same weight, the specific gravities
are reciprocally as their bulks.

IV. The specific gravities of all bodies are in a ratio
compounded of the direct ratio of their weights, and the
reciprocal ratio of their magnitudes. And hence again
the specific gravities are as the densities.

V. The absolute gravities or weights of bodies are in the
compound ratio of their specific gravities and magnitudes
or bulks.

VI. The magnitudes of bodies are directly as then-
weights, and reciprocally as their specific gravities.

VII. A body specifically heavier than a fluid, loses as
much of its weight, when immersed in it, as is equal to
the weight of a quantity of the fluid of the same bulk or
magnitude. — Hence, since the specific gravities are as the
absolute gravities under the same bulk; the specific gra-
vity of the fluid, will be to that of the body immersed, as
the part of the weight lost by the solid, is to the whole
weight. And hence the specific gravities of fluids, are
as the weights lost by the same solid immersed in them.

VIII. To find the' Specific Gravity of a Fluid, or of a
Solid. — On one arm of a balance suspend a globe of lead
by a fine thread, and to the other fasten an equal weight,
which may just balance it in the open air. Immerse the
globe into the fluid, and observe what weight balances it
then, and consequently what weight is lost, which is pro-
portional to the specific gravity as above. And thus the
proportion of the specific gravity of one fluid to another,
is determined by immersing the globe successively in all
the fluids, and observing the weights lost in each, which

, will be the proportions of the specific gravities of the fluids
sought.

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This same operation determines also the specific gravity
of the solid immersed, whether it be a globe or of any
other shape or bulk, supposing that of the fluid known.
For the specific gravity of the fluid is to that of the solid,
as the weight lost is to the whole weight. Hence also may
be found the specific gravity of a body that is lighter than
the fluid, as follows :

IX.' To find the Specific Gravity of a Solid that is lighter
than thejluid, as water, in which it is placed. — Annex to
the lighter body another that is much heavier than the
fluid, so as the compound mass may sink in the fluid.
Weigh the heavier body and the compound mass sepa-
rately, both in water and out of it ; then find how much
each loses in water, by subtracting its weight in water
from its weight in air; and subtract the less of these re-
mainders from the greater. Then say,

As this last remainder,

Is to the weight of the light body in air,

So is the specific gravity of the fluid,

To the specific gravity of that body.

X. The specific gravities of bodies of equal weight, are
reciprocally proportional to the quantities of weight lost
in the same fluid. And hence is found the ratio of the
specific gravities of solids, by weighing in the same fluids,
masses of them that weigh equally in air, and noting the
weights lost by each.

The specific gravities of many kinds of bodies, both
solid and fluid, have been determined by various authors.
Marinus Ghetaldus particularly tried the specific gravities
of various bodies, especially metals ; which were taken
from thence by Oughtred. In the Philos. Trans, are se-
veral ample tables of them, by various authors, particu-
larly those of Mr. Davis, vol.45, pa. 41 6. Some tables
of them were also published by P. Mersenne, Muschen-
broeck, Ward, Cotes, Emerson, Martin, &c.

The following table, taken chiefly from Gregory's
excellent book of Mechanics, contains the specific
gravities of most of the principal bodies, both solid and
fluid. The numbers express the avoirdupois ounces in a
cubic foot of each body, that of distilled water being just
1000 ounces, which is here made the standard of com-
parison of all the rest.

Table of the Specific Gravities of Different Bodies.
I. Metals.

Antimony, crude - - - - 4064

. . '. . glass of - - - - 4946

.... molten - 6702

Arsenic, glass of, natural ... 3504,

. . . molten .... 5763

. . . native orpiment ... 5452

Bismuth, molten .... 9823

. . . native .... 9020

. . ; ore of, in plumes - - - 4371

Brass, cast, not hammered ... 8396

. . . ditto, wire-drawn ... 8544

. .cast, common ... 7824

Cobalt, molten - - - 7812

. . . blue, glass of ... 2441

Copper, not hammered ... 7788

. . . the same wire-drawn - - 8878

. . '. ore of soft copper, or natural verdigris 3572
Gold, pure, of 24 carats -

. . melted, but not hammered - - 19258

. . the same hammered - - 19362

. . Parisian standard, 22 car. not hammered 17486

Gold, Parisian standard, 22 car. hammered - 17589

guinea, of Geo. 2. - - 17150

guinea, of Geo. 3. - - - 17629

Spanish gold coin - - - 17655

Holland ducats ... 19352

trinket standard, 20 car. not hammered 15709

the same hammered - - 15775

Iron, cast ..... 7 207

bar, either hardened or not - - 7788

Steel, neither tempered nor hardened - 78331

. hardened, but not tempered ,- - 784Q

. tempered and hardened - - 7SI8

. ditto, not hardened - - - 7816

Iron, ore prismatic ... _ 7355

ditto spicular - - . 5218

ditto lenticular ... 5012

Lead, molten .... 11352

. ore of cubic - - . . 7587

. ditto horned - - - - 6072

. ore of black lead ... 6745

. ditto, white lead ... 4059

. ditto, ditto, vitreous ... 6558

. ditto, red lead ... 6Q27

. ditto, saturnite ... 5925

Manganese, striated ... 4756

Molybdena ..... 473S

Mercury, solid or congealed - - 15632

. . fluent .... i3563

. . natural calx of - . - 9230

. . precipitate, per se - - 10871

. . precipitate, red - - - 8399

. . brown cinnabar - - : - 10218

. . red cinnabar - - 6902

Nickel, molten. .... 7807

. ore of, called Kupfernickel of Saxe - 6648

Platina, crude, in grains ... 15602

. purified, not hammered - - ] 9500

. ditto hammered ... 20337

. ditto wire-drawn ... 21042

. ditto rolled ... 22069

Silver, virgin, 1 2 deniers, fine, not hammered 10474

. ditto hammered ... 10511

. Paris standard ... 10175

. shilling of Geo. 2. ... 10000

. shilling of Geo. 3. ... 10534

. French coin - , - 1 0408

Tin, pure Cornish, melted, and not hardened 7291

. the same hardened ... 7299

. of Malacca, not hardened - - 7296

. the same hardened ... 7307

. ore of, red .... 6935

. ore of, black - - - ■ - - 6901

. ore of, white .... 6008

. common .... 7920

Tungsten ..... <5o66

Uranium ..... 6440
Wolfram - - - - -7119

Zinc, molten - - - - 7191

II. Precious Stones..

Beryl, or aqua-marine oriental - - 3549

. . . ditto occidental ... 2723

Chrysolite, of the jeweller - - 2782

.... of Brazil ... 2692

Crystal, pure rock of Madagascar - - 2653

.... of Brazil .... 2653

.... European - - - 2655
4, G 2

GRA

Crystal, ros-ecoloured
. . yellow

. . -violet, or amethyst
. . white amethyst ■ •
. . Carthaginian
. . black
Diamond, white oriental

. . rose-coloured oriental
. . orange ditto
. green ditto
. blue ditto
. . Brazilian
. . yellow
Emerald, of Peru
Garnet, of Bohemia

. of Syria
. . . dodecaedral -

. . volcanic 24 faces
Girasol --.-.■

Hyacinth, common
Jargon of Ceylon
Quartz, crystallized
. in the mass
. brown crystallized
. fragile
. milky
. fat or greasy
Ruby, oriental
. Spinell
. Ballas
. Brazillian
Sapphire, oriental

. . ditto white
i . of Puys
. . Brazilian
Spar, white sparkling
. red ditto

green ditto
, bine sparkling
, green and white ditto
, transparent ditto
, adamantine
Topaz, oriental

. pistachio ditto
.Brazilian
. of Saxe
. white ditto
Vermilion - -

III. Silicious Stones.
Agate, oriental
, onyx
. cloudy
. speckled
. veined
. stained
Chalcedony, common
. . transparent
. . veined
. . reddish
. . blueish
. . onyx
Carnelian, pale

. . speckled
. . veined

. onyx
. . stalactite
. . simple

[ £>9<> 1

GRA

2670 Flint, white

.

2654 .

. . black

.

2654 .

. . veined

.

2651 .

. . Egyptian

•

2657 .

. . olive

-

2654 .

. . spotted

-

3521 .

. . onyx

-

3531

. . of Rennes

-

3550 .

. . of England

-.

3524 .

. . variegated of Limosin

3525 Jade, white"

-

3444

. . green

.

3519

. . olive

_

2775

. . from the East Indies

.

4189

. . ef Swisserland

.

4000 Jasper, clear green

-

4063

. brownish green

-

2468

. . red

.

4000

. brown

. .

3687

. . yellow

-

4416

. . violet

.

2655

. cloudy

-

2647

. . veined

-

2647

. . blackish

.

2640

. . blood-coloured

.

2652

. . helostrope -

-

2Q46

. . onyx

-

4283

. . flowered, red and white

3760

■ . . red and yellow

.

3646

. . green and yellow

.

3531

. . . red, green, and grey

3994

. . red, green, and yellow

3991

. . universal

-

4077

. . . agate

-

3131

Opal

-

2595

Pearl, virgin oriental

-

243S

Pebble onyx

.

2704

. . . of Rennes

.

2693

. . English

-

3105

. . veined

.

2564

. . stained

-

3873

Prasium

-

4011

Sardonyx, pure

-

4061

. . pale

.

3536

. . . speckled

-

3564

. . . veined

.

3554

. . . onyx

-

4230

. . . blackish

.

Schorl, black prismatic, hexaedral

2590

. . octaedral

-

2638

. . tourmalin of Ceylon

2625

. . . antique basaltes

-

2607

. Brazilian emerald

-

2667

. . cruciform

-

2632

. . . green

-

2616

, . . violet of Dauphiny

2664

. . . common

-

2606

Stone, paving

-

26G5

. . cutlers

-

2587

. grind

-

2615

. . . mill

-

2630

IV. Various Stones, E

2612

Alabaster, oriental white

-

2623

, . . . ditto semi-transparent

2623

. . . yellow

-

2598

. . . stained brown

-

2613

. . . veined

-

2594
2532
2612
2565
2606
2587
2664
2654
2603
2243
2950
2906
2983
2977
3389
2359
2681
2661
2691
2710
2711
2735
2696
2672
262S
2633
2816
2623
2750
2684
2732
2749
2563
2661
2114
2684
2664
2654
2609
2612
2587
2581
2603
2606
2622
2595
2595
2628
3364
3226
3054
2923
3156
32S6
3453
3296
3092
2416
2111
2143
2484

2730
2762
2699
2744
2691

G R A

Alabaster, of Piedmont
. of Malta
. Spanish saline
. of Valencia
. of Malaga
. ofDalias
Amber, yellow, transparent
yellow, opaque
red
green
Ambergris -

Amianthus, long
. . . • short
Asbestos, ripe
.... starry

Basaltes, from Giant's Causeway
Bitumen, of Judea
Brick -

Chalk, Spanish

. coarse Briancon
. British
Gypsum, opaque

. semi-transparent
. fine ditto
. rhomboidal
. ditto 10 faces
. cuneiform crystallized
Glass, green
. white
. bottle

. Leith crystal
. fluid -

. of Bohemia
. of Cherbourgh
. of St. Cloud
Granite, red Egyptian
grey Egyptian
beautiful red
red of Dauphiny
ofGirardnor
violet of Gyromagny
green
radiated
red of Semur
grey of Bretagne
yellowish
of Carinthia, blue
Hone, white razor
Lapis, nephriticus
lazuli,
haematites
calaminaris
Judaicus
Manati
Limestone

. white fluor
. green
Marble, green Campanian
red

white Cassara
white Parian
Pyrenean
black Biscayan,
Brocatelle
Castilian

[ 597 ]

GRA

2693 Marble, Valencian
2699 . . . white Grenadan
2713 . . . Siennian -
2638 . . . Roman violet
2876 . . . African
2611 . . . violet Italian
1078 . . . Norwegian

1086 . . . Siberian

1083 . . . green Egyptian

1083 . . . Swisserland

926 .. . French

909 . . . yellow of Florence -

2313 Obsidian stone -

2578 Peat, hard ....

3073 Phosphorus - - . '

2864 Porcelain, Seves

1104 .... Limoges -

2000 .... China

2790 Porphyry, red ...

2727 .... green

2784 ... . red, from Dauphiny

2168 • • • . red, from Cordova

2306 .... green, from ditto

2274 Pyrites, coppery ...

2311 cubical

2312 • • . ferruginous cubic

2306 . • • ditto round - -

2642 • • . ditto of St. Domingo -
2892 Serpentine, opake, green Italian

2733 .... ditto, veined black and olive

3189 • • • . ditto, red and black

3329 .... semi-transparent grained

2395 • . . . ditto, fibrous

2560 ••.. ditto, from Dauphiny,

3255 • • , • opake penetrated with water

2654 Slate, common ...

2728 • •' new ....
2761 • • black stone ...

2643 • • fresh polished - - -

2716 • • common, penetrated with water
2685 Stalactite, transparent

2684 .... opake ...

2668 Stone, pumice ...

2638 • • prismatic basaltes

2738 • • touch ....

2614 • • Siberian blue ...

2956 . • oriental ditto - - -

2876 . . common ...

2894 . . Bristol ...

3054 . . Burford

4360 • . Portland ...

5000 • . rag ....

2500 • . rotten - -

2270 . . hard paring ...

3179 . . rock of Chatillon

3156 . . clicard, from Brachet

3182 . . ditto, from Auchain

2742 . . Noire Dame ...

2724 . • St. Maur

2717 . . St. Cloud -
2838 Sulphur, native

2726 . . . molten

2695 Talc, of Muscovy ...

2650 . . black crayon " -

2700 . . ditto German

2710
2705
2678
2755
2708
2858
2728
2718
2668
2714
2649
2516
2348
1329
17.14
2146
2341
2385
2765
2676
2793
2754
2728
, 4954
4702
3900
4101
3440
2430
2594
2627
2586
3000
2669
2473
2672
2854
2186
2766
2691
2324
2478
915
2722
2415
2945
2771
2520
2510
2049
2496
2470
1981
2460
2122
2357
2274
2378
2034
2201
2033
1991
2792
2089
2246

G R A

[ 598 ]

G R A

Talc,

yellow

.

.

_

2655

3il of cod-fish

.

.

.

923

black -

.

.

.

2900

. . of olives

.

.

.

915

•

white -

.

.

.

2704.

. . of almonds, sweet

.

.

.

917

. • volatile of mint, common

.

„

898

V. Liquors

Oils,

Sfc.

.... of sage,

-

.

.

902

Acid

, sulphuric

.

-

-

1841

. of thyme

-

-

. •

902

ditto, highly concentrated

-

-

2125

. of rosemary

-

-

-

906

nitric

.

-

-

1271

. of calamint

-

.

.

912

ditto, highly concentrated

-

-

1580

. of cochlearia

-

-

-

943

muriatic

-

-

-

1194

. of wormwood

-

-

-

907

. .

red acetous

-

-

-

1025

. of tansy

-

-

-

933

white acetous

-

-

-

1014

of Stragan

-

-

-

995

distilled ditto

-

-

-

1010

. . t

. of Roman camomile

.

.

894

fluoric

-

- ■

-

1500

. ofsabine

-

-

.

929

acetic

.

-

-

1063'

. of fennel

-

.

.

929

phosphoric

-

-

. 1

1558

. of fennel-seed

.

-

-

1008

formic

~-

-

-

994

. of coriander-seed

- •

-

866

citric

-

-

-

1035

. of carraway-seed

- -

- '

905

arsenic

-

-

-

1873

. of dill-seed

-

-

-

913

of oranges

-

-

-

1018

. of anise-seed

-

•-

-

987

of gooseberries

-

-

-

1058

. of juniper-seed

-

-

-

- 858

of grapes

-

-

-'

1024

of cloves

-

-

-

1036

Alcohol, commercial

-

-

-

837

. of cinnamon

-

-

-

1044

highly rectified

-

-

-'

829

. . .

. of turpentine

-

-

-

870

Alcohol, mixed with water.

. . of amber

-

-

-

887

15-16ths alcohol

-

-

-

853

. of the flowers of

orange

-

-

880

]4-l6ths ditto

-

-

■

867

. of lavender

.

-

-

894

13-16ths ditto

-

.

-

882

. of hyssop

-

-

-

8S9

12-16ths ditto

•

-

-

895

Spirit of wine. See Alcohol.

ll-16ths ditto

-

-

'-

908

Turpentine, liquid

-

-

-

991

10-16ths ditto

-

-

-

920 1

Jr'me, human

.

.

.

1011

9-16ths ditto

-

-

-

932

Water, rain - -

-

-

.

1000

8-16ths ditto

-

-

-

943

. . distilled

-

-

.

1000

#

7-lrJths ditto

-

-

-

952

. . sea (average)

-

-

.

1026

6-l6ths ditto

-

-

-

960

. . of Dead-sea

.

.

.

1240

5-l6ths ditto

-

-

-

967

Wine, of Torrins, red -

-

. .

.

•998

4-16ths ditto

-

-

-

973

. white

-

-

.

788

3-16ths ditto

-

-

-

979

. Champagne, white

-

-

.

998

2-16ths ditto

-

-

-

985

. Paharet

.

.

.

1000

l-16ths ditto

-

-

992

. Xeret -

-

-

.

992

Ammoniac, liquid

-

-

-

897

. Malmsey of Madeira

-

.

.

1038

Beer

pale

-'

-

-

1023

. 1 l. <rundy

-

-

•

992

brown

-

-

-

1034

. Jurancon

-

-

.

993

Cider

-

-

-

1018

. Bourdeaux

-

-

.

094

Ether, sulphuric

-

-

-

739

. Malaga

-

-

-

1022

nitric

-

-

-

909

. Constance

.

.

.

1082

muriatic •»

-

-

-

730

. Tokay

-

.

-

1054

acetic

-

-

866

. Canara

-

-

-

1033

Milk

, woman's -
cow's

-

-

-

1020
1032

. Port

-

-

-

997

ass's

.

„

.

1036

VI. Resins, Gums, and Animal Substances, Sfc.

ewe's

.

-

-

1041

Aloes, soGotrine,

-

-

.

1380

goat's

-

-

-

1035

. hepatic

-

-

-

1359

mare's

-

-

-

1034

Asafoetida,

-

-

-

132S

cow's clarified -

-

-

-

1019

Bees-wax, yellow

-

-

-

965

b

1

of filberts

-

.

-

916

. . . . white

.

-

-

969

of walnuts

-

.

.

923

Bone of an ox

-

-

-

1656

of hempseed

-

-

.

926

Butter

-

-

-

942

of poppies

-

.

-

924

Calculus humanus

-

-

-

1700

of rape-seed

-

-

.

919

. . . ditto

-

-

-

12+0

of lint-seed

-

.

.

940

. . . ditto

-

-

-

143 +

of poppy -seed

.

.

,

929

Camphor

-

-

-

989

of whale.

.

.

.

923

Copal, opake

-

-

-

1140

of ben, (a tree in Arab

a)

.

.

912

. Madagascar

-

-

-

1060

of beechmast

-

-

918

. Chinese

-

-

1063

G R A

Crassamentum, human blood
Dragon's blood
Elemi - -

Fat, beef ...

. ' . hog's
. . mutton -

. . veal ...

Galbanum ...
Gamboge
Gum, ammoniac
Arabic
Euphorbia
serapliic
tragncanth
bdellium

scammon y of Smyrna
ditto of Aleppo
Gunpowder, shiken

. . in a loose heap
. solid

Honey ...

Indigo ...

Ivory ...

Juice of liquorice
. . of acacia
Labdanum

Lard ...

Mastic
Myrrh

Opium -

Scammony. See Gum.
Serum of human blood
Spermaceti

Storax . - -

Tallow-
Terra Japonica
Tragacanlh. See Gum.
Wax. See Bee.s-wax.
. . shoemaker's

[ sgg ]

GRE

VII. Woods.

Alder

Apple-tree

Ash, the trunk of

Bay -tree

Beech

Box, French -

. . Dutrh

. Brazilian red
Campechv wood
Cedar, wild
. . . Palestine
. . . Indian
. . . American
Citron

Cocoa-wood -
Cherry-tree
Cork

Cypress, Spanish
Ebony, American
. . . Indian
Elder-tree
Elm, trunk of
Filbert-tree
Fir, male
. . female
Hazel

1126
1205
1018
923
937

934
1212
1222

1207
1452
1124
1201
1316
1372
1274
1235

937

836
1745
1150

769
1826
1723
1515
1186

94S
1074
1360
1336

Jasmin, Spanish
Juniper-tree
Lemon-tree
Lignum vitae
Linden-tree

Logwood. See Cumpechy.
Mastick-tree
Mahogany
Maple

Medlar - /

Mulberry, Spanish
Oak, heart of, 00 years old
Olive-tree
Orange-tree -
Pear-tree
Pomegranate-tree
Poplar

. . . white, Spanish
Plum-tree
Quince-tree
Sassafras
Vine
Walnut
Willow

Yew, Dutch -
Spanish

770
556
703
1333
604

849

1063
750
944
897

1170
927
705
661

1354
383
529.
7 85
705
482

1327
671
585
788
807

For the Weight and Specific Gravities of the different
Gases, see the article Gas.

These numbers being the weight of a cubic foot, or
172S cubic inches, of each of the bodies, in avoirdupois
ounces, by proportion the quantity in any other weight, or
the weight of any other quantity, may be readily known.

For Example. Required the content of an irregular
block of common stone which weighs 1 cwt, or 1121b, or
1792ounces. Here,as2500: 1792 :: 1728: 122Sf cubic
inches the content.

Example 2. To find the weight of a block of granite,
whose length is 63 feet, and breadth and thickness each
12 feet; being the dimensions of one of the stones, of gra-
nite, in the walls of Bal beck. Here, 63 x 12 x 12 =9072
feet is the content of the stone; therefore as 1 : 9072 : :
3000 oz. : 272l6'000 oz. or 759 tons 7 cwt. 2 qrs. the
weight of the stone.

XI. A body descends in a fluid specifically lighter, oi
ascends in a fluid specifically heavier, with a force equal
to the difference between its weight and that of an equal
bulk of the fluid.

XII. A body sinks in a fluid specifically heavier, so far
as that the weight of the body is equal to the weight of a
quantity of the fluid of the same bulk as the part im-
mersed. Hence, as the specific gravity of the fluid is to
that of the body, so is the whole magnitude of the body, to
the magnitude of the part immersed.

XIII. The specific gravities of equal solids, are as their
parts immersed in the same fluid.

The several theorems here delivered, are both demon-
240 strable from the principles of mechanics, and are also
044 equally conformable to experiment, which answers exactly
1331 to the calculation; as is abundantly evident from the
1209 courses of philosophical experiments, so frequently exhi-
695 bited ; where the laws of specific gravitation are well il-
07 1 lustrated.

600 GREAT BEAR, one of the constellations in the north-
550 ern hemisphere. See Ursa Major.
498 Great Chicles, of the Globe or Sphere, are those
OOO whose planes pass through the centre, dividing it into two

1030
943

11 10
942

1398

897

800
793
845
822
852
912

1328

1031
913
596
613

1315
561
726

1040
715

G R E

[ 600 ]

G R E

equal parts or hemispheres, and therefore having the same the ancients may be deduced. He also publishe d severa
centre and diameter with the sphere itself. The principal other curious works concerning antiquities, &c

of these are, the equator, the ecliptic, the horizon, the
meridians, and the two colures.

Great-Circle Sailing, is the art of conducting a
ship along the arc of a great circle. And it is also that
part of the theory of navigation which treats of sailing in
the arc of a great circle. See Navigation.

GREAVES (John), an eminent astronomer, antiquary,
and linguist, was born in 1602, being the eldest son of
John Greaves, rector of Colemore, near Alresford in
Hampshire, and master of a grammar-school, where his
son of course was well grounded in the primary rules of
literature. He then went to Baliol-coliege, Oxford, in
lrJl7 ; but afterwards, on account of his skill in philoso-
phy and polite literature, he was the first of five that were
elected into Merton-college. Having read over all the
ancient Greek and Latin writers, he applied to the study
of natural philosophy and mathematics ; and having con-
tracted an intimacy with Mr. Briggs, Savilian professor of
geometry at Oxford, and Dr. Bainbridge, Savilian profes-
sor of astronomy there, he was animated by their examples
to prosecute that study with the greatest industry. Not
being content, however, with reading the writings of Pur-
bach, Regiomontanus, Copernicus, Tycho Brahe, Kepler,
and other celebrated astronomers of that and the pre-
ceding age, he made the ancient Greek, Arabian, and Per-
sian authors familiar to him, having before gained an ac-
curate skill in the oriental languages. These accomplish-
ments procured him the professorship of geometry in Gre-
sham-college London, in 1630 ; and at the same time he
held his fellowship of Merton-college.

In a journey to the continent, in 1635, he visited the
celebrated Golius, professor of Arabic at Leyden, and
Claud Hardy at Paris, to converse about the Persian lan-
guage. Hence he passed through Italy, and accurately
surveyed the venerable remains of antiquity at Rome, vi-
siting and corresponding everywhere with the most learned
men of every nation. After visiting Padua, Florence, and
Leghorn, he hence embarked for Constantinople, where he
arrived in l638. Thence he passed over to Rhodes, and
Alexandria in Egypt, where he staid four or five months,
and made a great number of curious observations. He
next went to Grand Cairo, and measured the pyramids;
and while there he adjusted the measure of the foot, ob-
served by all nations. Hence he returned again through
Italy, and arrived in England in the year 1640, having
stored his mind with a variety of curious knowledge, and
'collected many valuable oriental manuscripts and an-
cient curiosities. While at Rome he made a particular
inquiry into the true state of the ancient weights and mea-
sures.

On the death of Dr. John Bainbridge, in 1643, he was
chosen Savilian professor of astronomy at Oxford, and
principal reader of Linacre's lecture in Merton-college ; an
appointment for which he was eminently qualified, from
his critical acquaintance with the works of the ancient
and modern astronomers. In Ki-1-5 he proposed a method
of reforming the calendar, by omitting the intercalary day
for 40 years to come : the paper relating to which, was
published by Dr. Thomas' Smith, in the I'hilos. Trans, for
1699. In 16*1*6, he published his Pyrarnidographia, or a
Description of the Pyramids of Egypt; and, in 1647, his
Discourse on tin- Roman Foot and Denarius ; from which,
as from two principles, the measures and weights used by

Soon after publishing the last-mentioned book, he was
ejected, by the parliament visitors, from the professorship
of astronomy and fellowship of Merton-college ; v neu the
soldiers committed many outrages, breaking open his
chests, and destroying many of his manuscripts ; which
greatly affected him. On this occasion he retired to Lon-
don, where he afterwards married, and prosecuted his stu-
dies with great vigour, as appears from several of his phi-
losophical and theological writings. This however proved
but a transient happiness to him ; for he died at London,
the 8th of October l652, before he was quite 50 years of
age ; and left his astronomical instruments to the Savilian
library in Oxford, where they are deposited.

GREEK Orders, in Architecture, are the Doric, Ionic,
and Corinthian; in contradistinction to the two Latin or-
ders, viz the Tuscan and Composite.

GREEN, one of the original colours of the rays of light,
or of the prismatic colours exhibited by the retraction of
the rays of light. This is the pleasantest of all the colours
to the sight. And hence it has been inferred as a proof of
the wisdom and goodness of the Deity, that almost all ve-
getables, clothing the surface of the earth, are green ;
which they are when growing in the open air; though
those in subterraneous places, or places inaccessible to
fresh air, are while or yellow. See Chromatics, and Co-
lours.

GREGORIAN Calendar, so called from Pope Gre-
gory the 13th, is the new or reformed calendar, showing
the new and full moons, with the time of Easter, and the
other moveable feasts depending on it, by means of epacts
disposed through the several months of the Gregorian
year.

Gregorian Epoch, is the epoch or time, from which
the Gregorian calendar, or computation, took place. This
began in the year 1582 ; so that the year 1800 is the 218th
of this epoch.

Gregorian Telescope, a particular kind of telescope,
invented by Mr. James Gregory. See Telescope.

Gregorian Year, the new acdount, or new style, in-
troduced on the reformation of the calendar, by Pope Gre-
gory the 13th, in the year 15S2, and from whom it took
its name. This was introduced to reform the old, or Ju-
lian year, established by Julius Cassar, which consisted of
365 days 6 hours, or 365 days and a quarter, that is three
years of 365 days each, and the fourth year of 366 days.
But as the mean tropical .year consists only of 36"5ds. 5hrs.
4Sm. 57sec.the former lost llmin. 3sec. every year, which
in the time of Pope Gregory had amounted to 10 days,
and who, by adding these 10 days, brought the account of
time to its proper day again, and at the same time ap-
pointed that every century after, a day more should be
added, thereby making tlie years of the complete centu-
ries, viz, the 17lh, 18th, 19th, 21st, which are not ex-
actly divisible by 4, &c, to be common years of 365 days
each, instead of leap-years of 366 days, which makes the
mean Gregorian year equal to 36*5ds. 5brs. 45m. 36sec.

This computation was not introduced into the account
of time in England, till the year 1752, when the Julian
account had lost 1 1 days, and therefore the 3d of Septem-
ber was in that year, by act of parliament, accounted the
14th, thereby restoring the 11 days which had thus been
omitted. See Yiar.

GREGORY (James), one of the most celebrated ma-

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thematicians of the 17th century, was the son of the Rev.
John Gregory, minister of Drumoak in the county of
Aberdeen, and born at Aberdeen in November J.638 or-9-
His mother was a daughter of Mr. David Anderson of
Finzaugh, or Fmshaugh, a gentleman who possessed a
singular turn for mathematical and mechanical knowledge.
This mathematical genius was hereditary in the family of
the Andersons, and from them it appears to have been
transmitted to their descendants of the names of Gregory,
Reid, &c. Alexander Anderson, cousin-german of the
said David, was professor of mathematics at Paris in the
beginning of the 17th century, and published there several
valuable and ingenious works; as may be seen in the me-
moirs of his life and writings, under the article Ander-
son. The mother of James Gregory inherited the genius
of her family; and observing in her son, while yet a child,
a strong propensity to mathematics, she instructed him
herself in the elements of that science. His education in
the languages he received at the grammar-school of Aber-
deen, and went through the usual course of academical
studies in the Marischal college ; but he was chiefly de-
lighted with philosophical researches, into which a new
door had lately been opened by the key of the mathe-
matics. Galileo, Kepler, Descartes, &c, were the great
masters of this new method : their works therefore became
.the principal study of young Gregory, who soon began to
make improvements on their discoveries in Optics. The
first of these improvements was the invention of the re-
flecting telescope; the construction of which instrument
he published in his Optica Promota, in l66"3, at 24 years
of age. This discovery soon attracted1 the attention of the
mathematicians, both of our own and of foreign countries,
who immediately perceived its great importance to the
sciences of optics and astronomy. But the manner of
placing the two specula on the same axis appearing to
Newton to be attended with the disadvantage of losing the
central rays of the larger speculum, he proposed an im-
provement on the instrument, by giving an oblique posi-
tion to the smaller speculum, and placing the eye-glass in
the side of the tube. It is observable, however, that the
Newtonian construction of that instrument was long aban-
doned for the original or Gregorian, which is now always
used when the instrument is of a moderate size ; though
Herschel has preferred the Newtonian form for the con-
struction of those immense telescopes, which he has of late
so successfully employed in observing the heavens.

About the year l664 or 1665, coming to London, he
became acquainted with Mr. John Collins, who recom-
mended him to the best optic glass-grinders there, to have
his telescope executed. But as this could not be done for
want of skill in the artists to grind a plate of metal for the
object-speculum into a true parabolic concave, which the
design required, he was much discouraged with the dis-
appointment ; and after a few imperfect trials made with an
ill-polished spherical one, which did not succeed to his
wish, he gave up the pursuit, and resolved to make the
tour of Italy, then the. mart of mathematical learning, that
he might prosecute his favourite study with greater advan-
tage. And the university of Padua being at that time in
high reputation for mathematical studies, Mr. Gregory
fixed his residence there for some years. Here it was
that he published, in 1667, Vera Circuli et Hyperbolae
Quadratura ; in which he propounded another discovery
of his own, the invention of an infinitely converging series
for the areas of the circle and hyperbola. lie sent to
Vol. I.

England a copy of this work to his friend Mr. Collins, who
communicated it to the Royal Society, where it met with
the commendations of Lord Brounker and Dr. Wallis.
He reprinted it at Venice the year following, to which he
added a new work, entitled Geometria; Pars Universalis,
inservions Quantitatum Curvarum Transmutationi et Men-
suras ; in which he is allowed to have shown, for the first
time, a method for the transmutation of curves. These
works engaged the notice, and procured the author the
correspondence, of the greatest mathematicians of the age,
Newton, Huygens, Wallis, and others. An account of
this piece was also read by Mr. Collins before the Royal
Society, of which Mr. Gregory, being returned from his
travels, was chosen a member the same year, and com-
municated to them an account of a controversy in Italy
about the motion of the earth, which was denied by Ricci-
oli and his followers.— Through this channel, in particu-
lar, he carried on a dispute with Mr. Huygens on the oc-
casion of his treatise on the quadrature of the circle and
hyperbola, to which that great man had started some ob-
jections; in the course of which our author produced
some improvements of his series. But in this controversy
it happened, as it generally does on such occasions, that
the antagonists, though setting out with proper coolness,
yet grew too warm in the combat. This was the case
here, especially on the side of Gregory, whose defence was,
at his own request, inserted in the Philosophical Transac-
tions. It is unnecessary to enter into particulars: suffice
it therefore to say, that, in the opinion of Leibnitz, who
allows Mr. Gregory the highest merit for his genius and
discoveries, M. Huygens has pointed out, though not
errors, some considerable deficiencies in the treatise above
mentioned, and shown a much simpler method of attain-
ing the same end.

In I66S, our author published at London another work,
entitled, Exercitationes Geometrical, which contributed
still much farther to extend his reputation. About this
time he was elected professor of mathematics in the uni-
versity of St. Andrews, an office which he held for six
years. During his residence there, he married, in lr76<),
Mary, the daughter of George Jameson, the celebrated
painter, whom Mr. Walpole has termed the Vandyke of
Scotland, and who was fellow-disciple with that great artist
in the school of Rubens at Antwerp.

In 1672, he published The Great and New Alt of
weighing Vanity: or a Discovery of the Ignorance and
Arrogance of the Great and New Artist, in his Pseudo-
philosophical Writings. By M. Patrick Mathers, arch-
bedal to the University of St. Andrews. To which are
annexed some Tentamina de Motu Penduli et Projec-
torum." Under this fictitious name, our author wrote this
little piece to expose the pretended ignorance of Mr.
Sinclar, professor at Glasgow, in his hydrostatical wri-
tings, and in return for some alleged ill usage of that au-
thor to a colleague of Mr. Gregory's. The same year,
Newton, on his wonderful discoveries in the nature of light,
contrived a new reflecting telescope, and made several
objections to Mr. Gregory's. This gave birth to a dispute
between those two philosophers, which was carried on
during this and the following year, in the most amicable
manner on both sides; Mr. Gregory defending his own
construction, so far, as to give his antagonist the whole
honour of having made the catoptric telescopes preferable
to the dioptric; and showing, that the imperfections in
these instruments were not so much owing to a defect in

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the object-speculum, as to the different refrangibility of Desaguliers; and some series in the ExercitatioGcometrica

the rays of light. In the course of this dispute, our author
described a burning concave mirror, which was approved
by Newton, and is still in good esteem. Several letters
that passed in this dispute, are printed by Dr. Desaguliers,
in an Appendix to the English edition of Dr. David Gre-
gory's Elements of Catoptrics and Dioptrics.

In 16*74, Mr. Gregory was called to Edinburgh, to fill
the chair of mathematics in that university. This place
he had held but little more than a year, when, in October
1675, being employed in showing the satellites of Jupiter
through a telescope to some of his pupils, he was suddenly
struck with total blindness, and died a few days after, to the
great loss of the mathematical world, at only 36 years of age.

As to his character, Mr. James Gregory was a man of a

of the same author, 4to, l6S4, Edinburgh ; as well as in
his little piece on practical geometry. — In different parts
of the foregoing works, of James Gregory, are found seve-
ral ingenious thoughts and inventions, that bavesometimes
been attributed to later periods and authors. Thus, at the
end of the Optica Promota, is a collection of curious as-
tronomical problems, conceived and resolved in a very
masterly manner; in a scholium at the end of one of which
(pa. 130), is remarked the use to be made of the transits of
Venus and Mercury, in determining the sun's parallax;
an idea which, I think, has usually been ascribed to Dr.
Halley.

Gregory (Dr. David), Savilian professor of astro-
nomy at Oxford, was nephew of the above-mentioned Mr

very acute and penetrating genius. His temper was in « James Gregory, being the eldest son of his brother Mr
some degree an irritable one, as appears from his conduct
in the dispute with.Huygens ; and, conscious perhaps of
his own merits as a discoverer, he seems to have been jea-
lous of losing any portion of his reputation by the im-
provements of others upon his inventions. He possessed
one of the most amiable characters of a true philosopher,
that of being content with his fortune in his situation.
But the most brilliant part of his character is that of his
mathematical genius as an inventor, which was of the first
order; as will appear by the following list of his inven-
tions and discoveries. Among many others may be.
reckoned, his Reflecting Telescope; — Burning Concave
Mirror; — Quadrature of the Circle and Hyperbola, by
an infinite converging series; — his method for the Trans-
formation of Curves; — a Geometrical Demonstration of
lord Brounkcr's series for Squaring the Hyperbola; — his
Demonstration that the Meridian Line is analogous to a
scale of Logarithmic Tangents of the Half Complements of
the Latitude; — he also invented and demonstrated geo-
metrically, by help of the hyperbola, a very simple con-
verging series for making the logarithms; — he sent to Mr.
Collins the solution of the famous Keplerian problem by
an infinite series; — he discovered a method of drawing
Tangents to Curves geometrically, without any previous
calculations; — a rule for the Direct and Inverse method
of Tangents, which stands upon the same principle (of ex-
haustions) with that of fluxions, and differs not much from
it in the manner of application ; a Series for the length of
the Arc of a Circle from the Tangent, and vice versa; as
also for the Secant and Logarithmic Tangent and Secant,
and vice versa:. — these, with others, for measuring the
length of the elliptic and hyperbolic curves, were sent to
Mr. Collins, in return for some received from him of New-
ton's, in which he followed the elegant example of this au-
thor, in delivering his series in simple terms, independent
of each other. These and other writings of our author are
mostly contained in the following works, viz,

1. Optica Promota; 4to, London 16"6'3. .

2. VeraCirculi et Hyperbolas Quadrature,; 4to, Padua
ln'o"7 and 1668.

.3. Geometric Pars Universalis ; 4to, Padua I66S.

4. Exercitationes Geometricas; 4to, London 1668.

5. The Great and New Art of weighing Vanity, &c.
8vo, Glasgow lo"72.

The rest of his inventions make the subject of several
letters and papers, printed either in the Philos, Trans,
vol. 3 ; the Commerc. Epistol. Job. Collins et Aliorum,
8vo, 1715; in the Appendix to the English edition of Dr.
David Gregory's Elements of Optics, 8vo, 1735, by Dr.

David Gregory of Kinardie, a gentleman who had the sin-
gular fortune to see three of his sons professors of mathe-
matics, at the same time, in three of the British universi-
ties, viz, our author David at Oxford, the second son
James at Edinburgh, and the third son Charles at St.
Andrews. Our author David, the eldest son, was born at
Aberdeen in 1(56*1, where he received the early parts of his
education, but completed his studies at Edinburgh ; and,
being possessed of the mathematical papers of his uncle,
soon distinguished himself likewise as the heir of his
genius. In the 23d year of his age, he was elected pro-
fessor of mathematics in the university of Edinburgh ;
and, in the same year, he published Exercitatio Geo-
metricade Dimensione Figurarum, sive Specimen Methodi
generalis Dimetiendi quasvis Figuras, Edinb. l6S4, 4to.
He very soon perceived the excellence of the Newtonian
philosophy; and had the merit of being the first who in-
troduced it into the schools, by his public lectures at
Edinburgh. " He had (says Mr. VVhiston, in the Memoirs
of his own Life, i. 32) already caused several of his scho-
lars to keep acts, as we call them, on several branches of
the Newtonian philosophy: while we at Cambridge, poor
wretches, were ignominiously studying the fictitious hypo-
thesis of the Cartesian."

On the report of Dr. Bernard's intention of resigning the
Savilian professorship of astronomy at Oxford, our author
went to London in 1 6*9 1 ; and being patronised by New-
ton, and warmly befriended by Mr. Flamsteed the astro-
nomer-royal, h'e obtained the vacant professorship, for
which Dr. Halley was also a competitor. This rivalship,
however, instead of animosity, laid the foundation of
friendship between these eminent men ; and Halley soon
after became the colleague of Gregory, by obtaining the
professorship of geometry in the same university. Our au-
thor had not long remained in London after his arrival, be-
fore he was elected a fellow of the Royal Society ; and,
previously to his election into the Savillian professorship,
had the degree of doctor of physic conferred on him by
the university of Oxford.

In 16*93, he published in. the Philos. Trans, a resolution
of the Florentine problem De Testudine vehfonni quadra-
bili; and he continued to communicate to the public,
from time to time, many ingenious mathematical papers
through the same channel.

In 1()*95, he printed at Oxford, Catoptrics et Dioptricae
Sphrcrka; El omenta; a work which, we are informed in
the preface, contains the substance of some of his public
lectures read at Edinburgh, eleven years before. This va-
luable treatise was republished in English, fust with addi-

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GRE

tions ;by Dr. William Brown, with the recommendation of
Mr. Jones and Dr. Desaguliers ; and afterwards by the
latter of these gentlemen, with an appendix containing an
account of the Gregorian and Newtonian telescopes, toge-
ther with Mr. Hadley's tables for the construction of both
those instruments. It may not be unworthy of remark,
that, in the conclusion of this treatise, there is an observa-
tion which shows, that the construction of achromatic te-
lescopes, which Mr_Dollond has carried to such great per-
fection, had occurred to the mind of David Gregory, from
reflecting on the admirable contrivance of nature in com-
bining the different humours of the eye. The passage is
as follows: " Perhaps it would be of service to make the;
object-lens of a different medium, as we see done in the
fabric of the eye; where the crystalline humour (whose
power of refracting the rays of light differs very little from
that of glass) is Jby nature, who never does any thing in
vain, joined with the aqueous and vitreous humours (not
differing from water as to their power of refraction), in or-
der that the image may be painted as distinct as possible
on the bottom of the eye."

In 1702 our author published at Oxford, in folio, As-
tronomiaj Physicae et Geometrical Elementa ; a work which
is accounted his master-piece. It is founded on the New-
tonian doctrines, and was esteemed by Newton himself as
a most excellent explanation and defence of his philosophy.
In the following year he gave to the world an edition, in
folio, of the works of Euclid, in Greek and Latin ; being
done in prosecution of a design begun by his predecessor
Dr. Bernard, of printing the works of all the ancient
mathematicians. In this work, which contains all the
treatises that have been attributed to Euclid, Dr. Gregory
has been careful to point out such as he found reason,
from internal evidence, to believe to be the productions of
some inferior geometrician. In prosecution of the same
plan, our author engaged soon after, with his colleague
Dr. Halley, in the publication of the Conies of Apollo-
nius; but he had proceeded only a little way in this un-
dertaking, when he died at Maidenhead in Berkshire, in
1710, being the 49th year of his age only.

Besides those works published in our anthor's life-time,
as mentioned above, he had several papers inserted in the
Philos. Trans, vol. 18, 19, 21, 24, and 25, particularly a
paper on the catenarian curve, first considered by our au-
thor. He left also in manuscript, A Short Treatise of the
Nature and Arithmetic of Logarithms, which is printed
at the end of Keill's translation of Commandinc's Euclid ;
and a Treatise of Practical Geometry, which was after-
wards translated, and published in 1745, by Mr. Maclaurin.

Dr. David Gregory married, in 1695, Elizabeth, the
daughter of Mr. Oliphant of Langtown in Scotland. By
this lady he had four sons, of whom, the eldest, David,
was appointed regius professor of modern history at Oxford
by King George the 1st., and died at an advanced age in
1767, after enjoying for many years the dignity of dean of
Ghristchurch in that university.

When David Gregory quitted Edinburgh, he was suc-
ceeded in the professorship at that university, by his bro-
ther James, likewise an eminent mathematician ; who held
that office for 33 years, and, retiring in 1725, was suc-
ceeded by the celebrated Maclaurin. A daughter of this
professor James Gregory, a young lady of great beauty
and accomplishments, was the victim of an unfortunate
attachment, that furnished the subject of Mallet's well-
known ballad of William and Margaret.

Another brother, Charles* was created professor of ma-
thematics at St. Andrews by Queen Anne, in 1707. This
office he held with reputation and ability for 32 years;
and, resigning in 1739, was succeeded by his son, who emi-
nently inherited the talents of his family, and died in 1763.
Some farther Particulars of the Family of the Gregorys and
Andersons, communicated by Dr. Thomas Reid, Professor
of Moral Philosophy in the University of Glasgow, a Ne-
phew of the late Dr. David Gregory Saviltian Professor
at Oxford.

Some, account of the family of the Gregorys at Aber-
deen, is given in the Life of the late Dr. John Gregory pre-
fixed to his works, printed at Edinburgh 1788, in four
small Svo volumes.

Who was the author of that Life, or whence he obtain-
ed his information, I do not know. I have heard it ascribed
to Mr. Tytler the younger, whose father was appointed
one of the guardians of Dr. John Gregory's children. Some
additions to what is contained in it, and a few remarks
upon it, is all I can furnish on this subject.

Page 3. 1 know nothing of the education of David An-
derson of Finzaugh. He seems to have been a self-taught
engineer. Every public work which surpassed the skill of
common artists, was committed to the management of Da-
vid. Such a reputation he acquired by his success in
works of this kind, that with the vulgar he got the by-
name of Davie do a' thing, that is, in the Scottish dialect,
David who could do every thing. By this appellation he
is better known than by his proper name. He raised the
great bells into the steeple of the principal church : he cut
a passage for ships of burden through a ridge of rock un-
der water, which crossed the entrance into the harbour of
Aberdeen. In a long picture gallery at Cullen House, the
seat of the earl of Findlater, the wooden ceiling is painted
with several of the fables of Ovid's Metamorphoses. The
colours are still bright, and the representation lively. The
present earl's grandfather told me that this painting was
the work of David Anderson my ancestor, whom be ac-
knowledged as a friend and relation of his family.

Such works, while they gave reputation to David, suited
ill with his proper business, which was that of a merchant
in Aberdeen. In that he succeeded ill; and having given
up mercantile business, from a small remainder of his for-
tune began a trade of making malt; and after instructing
his wife in the management of the same, left it to her care,
and went into England to try his fortune as an engineer;
an employment which in his own country he had practised
gratuitously. Having in that way realised a fortune which
satisfied him, he returned to Aberdeen, where his wife had
also made money by her malting business.

After making such provision for their children as they
thought reasonable, they agreed that the longest liver of
the two should enjoy the remainder, and at death should
bequeath it to certain purposes in the management of the
magistrates of Aberdeen.

The wife happened to be the survivor, and fulfilled what
had been concerted with her husband. Her legacies, well
known in Aberdeen, are called after her name Jane Guild's
Mortifications, a mortification in Scots law signifying a
bequeathment for some charitable purpose. They consist
of sums for different purposes : for orphans, for the edu-
cation of boys and girls, for, unmarried gentlewomen, and
for widows; and they still continue to be useful to many
in indigent circumstances. She was the daughter of Dr.
Guild, a minister of Aberdeen. Besides her money, she
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bequeathed a piece of tapestry, wrought by her own hand,
and representing the history of queen Esther, from a draw-
ing made by her husband. The tapestry continues to
ornament the wall of the principal church.

In the same page it is said -that Alexander Anderson,
professor of mathematics at Paris, was the cousin-german
of David above-mentioned. I know not the writer's au-
thority for this: I have always understood that they were
brothers ; but for this I have only family tradition.

Page4. Itishere related that JamesGregory wasinstruct-
ed in the Elements of Euclid by his mother, the daughter of
David Anderson. The account I have heard differs from
this. It is, that his brother David, being ten or eleven
years older, had the direction of his education after their
father's death, and, when James had finished his course of
philosoph}', was at a loss to what literary profession he
should direct him. After some unsuccessful trials, he put
Euclid's Elements into his hand, and finding that he ap-
plied to Euclid with great avidity and success, he encou-
raged and assisted him in his mathematical studies. This
tradition agrees with what James Gregory says in the pre-
face to his Optica Promota; where after mentioning his
advance to the 26'ih proposition, he adds, Ubi diu hjesi
omne spe progrediendi orbatus, sed continuis hortatibus et
auxiliis fratris mei Davidis Gregorii, in Mathematicis non
parum versati (cui si quid in hisce Scientiis praestitero, me
illud debere non inficias ibo) animatus, tandem incidi &c.
Whether David had been instructed in mathematics by his
mother, or had any living instructor, I know not.

Page 5, 6. In these two pages I think the merit of Gre-
gory compared with that of Newton in the invention of the
catoptric telescope, is put in a light more unfavourable to
Newton than is just. Gregory, believing that the imper-
fection of the dioptric telescope arose solely from the sphe-
rical figure of the glasses, invented his telescope to remedy
that imperfection. Being less conversant in the practice
of mechanics, he did not attempt to make any model. The
specula of his telescope required a degree of polish and a
figure which the best opticians of that age were unable to
execute. Newton demonstrated that the imperfection of
the dioptric telescope arose chiefly from the different re-
frangibility of the rays of light ; he demonstrated also that
the catoptric telescope required a degree of polish far be-
yond what was necessary for the dioptric. He made a mo-
del of his telescope ; and finding that the best polish which
the opticians could give, was insufficient, he improved the
polish with his own hand, so as to make it answer the in-
tended purpose, and has described most accurately the
manner in which he did this. And, had he not given this
example of the practicability of making a reflecting tele-
scope, it is probable that it would have passed as an im-
practicable idea to this day.

Page 11. To what is said of this JamesGregory might have
been added, that he was led by analogy to the true law of
Refraction, not knowing that it was discovered by Des-
cartes before (sec Preface to Optica Promota) ; and that
in 1670 having received, in a letter from Collins, a series,
for the area of the zone of a circle, and as Newton had
invented an universal method by which he could square
all curves geometrical and mechanical by infinite series of
that kind ; Gregory after much thought discovered this
universal method, or an equivalent one. Of this he per-
fectly satisfied Newton and the other mathematicians of
that time, by a letter to Collins in Feb. 167 1. He was
strongly solicited by his brother David to publish his Uni-

versal Method of Series without delay, but he excused
himself on a point of honour; that as Newton was the first
inventor, and as he had been led to it by an account of
Newton's having such a method, he thought himself bound
to wait till Newton should publish his method. I have
seen the letters that passed between the brothers on this
subject.

With regard to the controversy between James Gregory
and Huygens, 1 take the subject of that controversy to
have been, not whether J. Gregory's quadrature of the cir
cle by a converging series was just, but whether he had
demonstrated, as in one of his propositions he pretended
to-do, That it is impossible to express perfectly the Area
of a Circle in any known algebraical form, besides that of
an infinite converging series. Huygens excepted to the
demonstration of this proposition, and Gregory defended
it; neither of them convinced his antagonist, nor do I
know that Leibnitz improved on what Gregory had done.

Page 12. D. Gregory of Kinardie deserved a more par-
ticular account than is here given. It is true that he
served an apprenticeship to a mercantile house in Hol-
land, but he followed that profession no longer than he
was under authority, having a stronger passion for know-
ledge than for money. He returned to his own country
in l6"55, being about 28 years of age, and from thattime
led the life of a philosopher. Having succeeded to the
estate of Kinardie by the death of an elder brother, he
lived there to the end of that century. There all his chil-
dren were born, of whom he had thirty-two by two wives.
Kinardie is above 40 English miles north from Aberdeen,
and a few miles from Bamf, upon the river Diveron. He
was a jest among the neighbouring gentlemen for his ig-
norance of what was doing about his own farm, but an ora-
cle in matters of learning and philosophy, and particularly
in medicine, which he had studied for his amusement,
and begun to practise among his poor neighbours. He ac-
quired such a reputation in that science, that he was after-
wards employed by the nobility and gentlemen of that
county, but took no fees. His hours of study were sin-
gular. Being much occupied through the day with those
who applied to him as a physician, he went early to bed,
rose about two or three in the morning, and, after stu-
dying some hours, went to bed again and slept an hour or
two before breakfast.

He was the first man in that country who had a baro-
meter; and by some old letters which 1 have seen, it ap-
peared, that he had corresponded with some philosophers
on the continent about the changes in the barometer and
in the weather, particularly with Mariotte the Fiench phi-
losopher. He was once in danger of being prosecuted as
a conjurer by the Presbytery on account of his barometer.
A deputation of that body having waited upon him to en-
quire into the ground of certain reports that had come to
their ears, he satisfied them so far as to prevent the pro-
secution of a man known to be so extensively useful by his
knowledge of medicine. — About the beginning of the laso
century he removed with his family to Aberdeen, and in
the time of Queen Anne's waremployed his thoughts upon'
an improvement in artillery, in order to make the shot of
great guns more destructive to the enemy, and executed
a model of the engine he had conceived. I have conversed
with a clock-maker in Aberdeen who was employed in,
making this model-; but having made many different pieces
by direction, without knowing their intention, of how
they were to be put together, he could give 110 account

ORE

[

]

GRE

of the whole. After making some experiments with this
model, which satisfied him, the old gentleman was so san-
guine in the hope of being useful to the allies in the war
against France, that he set about preparing a field equi-
page with a view to make a campaign in Flanders, and
in the mean time sent his model to his son the Savillian
professor, that he might have his and Sir Isaac Newton's
opinion of it. His son showed it to Newton, without let-
ting him know that his own father was the inventor. Sir
Isaac was much displeased with it, saying, that if it tended
as much to the preservation of mankind as to their de-
struction, the inventor would have deserved a great re-
ward; but as it was contrived solely for destruction, and
would soon be known by the enemy, he rather deserved
to be punished; and urged the professor very strongly to
destroy it, and if possible to suppress the invention. It is
probable theprofessorfollowed this advice; for at his death,
which happened soon after, the model was not to be found.

When the rebellion broke out in 1715, the old gentle-
man went a second time to Holland, and returned when
it was over to Aberdeen, where he died about 1720, aged
93. He left an historical manuscript of the transactions
of his own time and country, which my father told me
he had read. I was well acquainted with two of this gen-
tleman's sons, and with several of his daughters, besides
my own mother. The facts abovementionedaretakenfrom
what 1 have occasionally heard from them, and from other
persons of his acquaintance.

' Page 14. In confirmation of what is said in this page,
that the two brothers David and James were the first who
taught the Newtonian philosophy in the Scotch Univer-
sities; I have by me a Thesis, printed at Edinburgh in
169O, by James Gregory, who was at that time a professor
of philosophy at St. Andrews, and succeeded his brother
David in the profession of mathematics at Edinburgh. In
this Thesis, after a dedication to viscount Tarbet, follow
the names of twenty-one of his scholars who were candi-
dates for the degree of a. m., then 25 positions or Theses.
The first three relate to logic, and the abuse of it in the
Aristotelian and Cartesian philosophy. He defines logic
to be the art of making a proper use of things granted, in
order to find what is sought, and therefore admits only two
Categories 111 logic, viz, Data and Quaesita. The remain-
ing 22 positions are a compend of Newton's Principia.
This Thesis, as was the custom at that time in the Scotch
universities, was to be defended in a public disputation,
by the candidates, previous to their taking their de»ree.

The famous Dr. Pitcairn was a fellow-student and in-
timate companion of these two Gregorys, and during the
vacation of the college was accustomed to go home with
them to Kinardie, to their father's house.

David Gregory was appointed a preceptor to the duke
of Gloucester, Queen Anne's son ; but his eHtering on that
office was prevented by the death of that prince in the 1 lth
year of his age.

Page ly. D. Gregory's Euclid is said to have been written
in prosecution of a design of his predecessor Dr. Bernard,
of printing the works of all the ancient mathematicians.
This design ought to have been ascribed to Saville, who
left in charge to the two professors of his foundation, to
print the mathematical works of the ancients, and I think
left also a fund for defraying the expense. Wallis did
something in. consequence of this charge; Gregory and
Halley performed a great deal ; but I think nothing has
been done in this design by the Savillian professors since
their time.

Page 20. Besides what is above mentioned,, Dr. Gregory
left in manuscript a Commentary on Newton's Principia,
which Newton valued, and kept by him for many years
after the author's death. It is probable that in what re-
lates to astronomy, this commentary may coincide in a
great measure with the author's astronomy, which indeed
is an excellent commentary upon that part of the Principia.
Page 24. This David Gregory published in Latin, a very
good compendium of arithmetic and algebra, with the title
Arithmetics et Algebra; Compendium, in Usum Juventu-
tis Academies, Edinb. 1736". He had a design of pub-
lishing his uncle's Commentary on the Principia, with ex-
tracts from the. papers left by James Gregory his grand-
uncle ; but the expense being too great for his fortune,
and he too gentle a solicitor of the assistance of others, the
design was dropped. His son David, yet alive, was mas-
ter of an East-India ship.

Page 40. To the projectors of the society at Aberdeen,
ought to have been added John Stewart, professor of ma-
thematics in the Marischal college at Aberdeen. He pub-
lished an explanation of two treatises of Sir Isaac Newttfn,
viz, his Quadrature of Curves, and his Analysis by Equa-
tions of an infinite number of terms. He was an intimate
friend of Dr. Reid's.

Another of the first members of that society was Dr.
David Skene, who, besides his eminence in the practice
of medicine, had applied much to all parts of natural his-
tory, particularly to botany, and was a correspondent of
the celebrated Linnaeus.

Dr. John Gregory and Dr. David Skene were the first
who attempted a college of medicine at Aberdeen. The
first gave lectures to his pupils in the theory and practice
of medicine, and in chemistry ; the last, in anatomy, ma-
teria medica, and midwifery, in order to prepare them for
attending the medical college at Edinburgh. T. R.

Tltc following additional lines hy Mr. James Millar, Pro-
fessor of Mathematics, Glasgow.
Another instance of the prevalence of mathematical
genius in the family of Gregory or Anderson, whether pro-
duced by an original and inexplicable determination of
the mind, or communicated by the force of example, and
the consciousness of an intimate connexion with a repu-
tation already acquired in a particular line, is the cele-
brated Dr. Reid, professor of moral philosophy in the uni-
versity of Glasgow ; a nephew, by his mother, of the, late
Dr. David Gregory, Savillian professor at Oxford.

This gentleman, well known to the public by his moral
and metaphysical writings, and remarkable for that libe-
rality, and that ardent spirit of inquiry, which neither
overlooks nor undervalues any bra-nch of science, is pe-
culiarly distinguished -by his abilities and proficiency in
mathematical learning. The objects of literary pursuit are
often directed by accidental occurrences. An appre-
hension of the bad consequences which might result from
the philosophy of the late Mr. Hume, induced Dr. Reid
to combat the doctrines of that eminent author; and pro-
duced a work, which has excited universal attention, and
has given a new turn to speculations on that subject. But
it is well known to Dr. Reid's literary acquaintance, that
these exertions have not diminished the original bent of
his genius, nor blunted the edge of his inclination for ma-
thematical researches ; which, at a very advanced age,,
he still continues to prosecute with a youthful attachment,
and with unremitting assiduity.

It may further be observed, of the extraordinary family
above mentioned, that Dr. James Gregory, the present

G R I

60(5 ]

GUG

learned professor of physic and medicine in tbe university his Society, he would have boldly opposed the dogmas of

of Edinburgh, is the son of the late Dr. John Gregory,
upon the memoirs of whose life the above remarks have
been written by Dr. Reid. The said James has lately pub-
lished a most ingenious work, entitled, Philosophical and
Literary Essays, in 2 volumes Svo, Edinb. 1792; and he
appears to be another worthy inheritan: of the singular
genius of his family.

Gregory (St. Vincent), a very respectable Flemish
geometrician, was born at Bruges in 1584, and became a
Jesuit at Romeat '20 years of age. He studied mathematics
under the learned Jesuit Clavius. He afterwards became
a reputable professor of those sciences himself, and his
instructions were solicited by several princes : he was
called to Prague by the emperor Ferdinand the 2d ; and
Philip the 4th, kins of Spain, was desirous of having him
to teach mathematics to his son the young "prince John of
Asturia. He was not less estimable for his virtues, than
his skill in the sciences. His weli-meant endeavours were
very commendable, when his holy zeal, though for a false
religion, led him to follow the'army in Flanders one cam-
paign, to confess the wounded and dying soldiers, in
which he received several wounds himself.- He died of an
apoplexy at Ghent, in l66'7, at 83 years of age.

As a writer, Gregory St. Vincent was very diffuse and
voluminous, but he was an excellent geometrician. He
published, in Latin, three mathematical works, the prin-
cipal of which was his Opus Geometricum Quadrature
Circuli, et Sectionum Coni, Antwerp. 164-7, 2 vol. folio.
Though he has not demonstrated, in this work, the qua-
drature of the circle, as he pretends to have done, the
book nevertheless contains a great number of truths and
important discoveries ; one of which is this, viz, that if
one asymptote of an hyperbola be divided into parts in geo-
metrical progression, and from the points of division ordi-
nates be drawn parallel to the other asymptote, they will
divide the space between the asymptote and curve into
equal portions ; whence it was shown by Mersenne, that,
by taking the continual sums of those parts, there would
be obtained areas in arithmetical progression, adapted to
abscisses in geometrical progression, and which therefore
were analogous to a system of logarithms.

GRENADE, orGRENADO. See Granade.

GRIMALDI (Auth. Franc. Maria), was a learned
Italian Jesuit, and one of those philosophers of the 17th
century to whom the sciences of optics and astronomy
owe great obligations ; sciences which be cultivated in com-
pact with his compatriot Riccioli, by diligent experiments
and observations. He noticed the spots in the sun, and gave
to those in the moon names which are still in use, denomi-
nating them after the most eminent astronomers and phi-
losophers. He made numerous experiments on optics,
and some discoveries, which were afterwards confirmed and
much farther extended by the brilliantdiscoveries of New-
ton. Those were chiefly contained in his work, Physico
Mathesis de Lumine, Coloribus, et Iride, &c : Bononia?,
1 665, in 4to ; a work which exhibits a great quan-
tity of curious experiments and remarks on light and co-
lours; a particular account of which may be 9ecn in my
Abridg. of the Philos. Trans, vol. 1, pa. 6'75. This indeed
was Grimaldi's chief merit : for his philosophy was but too
much after the manner of his country, which, though it
has given to the world a Galileo, a Torricelli, &C, was not
the first in shake oil' the yoke of Aristotle. To his honour
however it must he acknowledged, that had he lived in
another country, and under other authority than that of

the ancient philosophy. — Grimaldi was cut off at a very
early period, in 1663, being only 44 years of age.

GRINDING Telescope Glasses, &c.

GROIN, with builders, is the angle made by the inter-
section of two arches. It is of two kinds, regular and ir-
regular; viz, regular when both the archishave the same
diameter, but an irregular groin when one arch is a semi-
circle and the othera semiellipsis. Groins are chiefly used
in forming arched roofs, where one hollow arched vault
intersects with another; as in the roofs of most churches,
and in some cellars in large houses.

GRUS, the Crane, one of the new constellations, in the.
southern hemisphere; containing, according to Mr. Sharp's
catalogue, 13 stars.

Grus is also one of the Arabian constellations, and
answers to our Ophiucus, to which they changed this
constellation, their religion prohibiting them from draw-
ing any human figures.

GRY, a measure containing Tr^ of a line. A line is-j%
of a digit, and a digit is -^ of a foot, and a philosophical
foot -f of a pendulum, whose vibrations, in the latitude of
45 degrees, are each performed in one second of time, or
.gtj of a minute.

GUARDS, a name that has been sometimes applied to
the two stars nearest the north pole ; being in the hind
part of the chariot, at the tail of Ursa Minor, or little
bear ; one of them being also called the pole-star.

GUERICKE (Otto or Oriio), counsellor to theelector
of Brandenbourg and burgomaster of Magdebourg, was
born in l6'02, and died in l6'S6 at Hambourg. He-was
one of the greatest philosophers" of his time. It was Guc-
ricke that invented the air-pump ; also the two brass he-
mispheres, which being applied to each other, and the air
exhausted, it is said several horses were not able to draw
them asunder ; and the marmouset or virunculus of glass
which descended in a tube against rain, and rose again on
the return of serene weather. This last machine fell into
disuse on the invention of the barometer, especially after
Huygens and Anmntons gave theirs to the world. Gue-
ricke made use of his marmouset to foretell storms ; whence
he was looked upon as a sorcerer by the people; so that
the thunder having one day fallen upon his house, and
shivered to pieces several machines which he had em-
ployed in his experiments, they failed not to construe the
event into a punishment from heaven that was angry with
him. — Gucricke was author of several works in natural
philosophy, the principal of which was hisExperimenta
Magdeburgica, in folio, which contains his experiments
on a vacuum.

GUERITE, in Fortification, a sentry-box ; being a
small tower of wood, or stone, usually placed on the
point of a bastion, or on the angles of the shoulder, to hold
a sentinel, who is to take care of the ditch, and watch
against a surprise.
"GUFULE, in Architecture. See Gula.

GUGLIELMINl (Dominic), an eminent Italian ma-
thematician and civil engineer, was born at Bologna in
1655. His favourite studies were the mathematics and
medicine, of which hitter faculty he was admitted to the
degree of doctor by the university of Bologna, in 10*78.
On the appearance of the remarkable comet in the years
I 680 and l6.81; he published a treatise Qts Comelarum
Nattira et Ortu, cVc, 1 681, in which he proposed a new
system on the subject, which did not, however, meet with
the approbation of the scientific world. But his next

G U I

[ 607 ]

GUN

astronomical treatise, containing remarks on the solar
eclipse of July 12, l684, reflected greater credit on his
knowledge and accuracy of observation. Soon afterwards,
the senate of Bologna appointed him principal professor of
mathematics in iheir university, and, in the year loKo",
intendant-general of the rivers of the Bolognese. In con-
sequence of his attention to the duties of this office, in the"
year 169O he published the first part, and in the following
year the 2d part, of his hydrostatical treatise, entitled
Aquarum Fluentium Mensura, Novo Methodo Inquisita.
Some of the observations in this work were attacked by
M. Papin, who also entered into a contest with the author
on the subject of syphons. Their difference in opinion
gave rise to two letters by Gugliehnini, which were printed
under the title of Epistolce Duae Hydrostatics.

Sig. G. was engaged in settling the differences which
arose between the cities of Bologna and Ferrara, respect-
ing the management of the embankment and sluices in
their contiguous districts ; and received, in reward of his
services from his native city, the appointment to a new
office in the university, which was that of professor of hy-
drometry. In the year l6\95 he assisted M. Cassini in re-
pairing the meridian line which he had drawn 40 years
before in the church of St. Petronius, at Bologna; on
which occasion our author publisheda memoir, descriptive
of the method of laying it down, and establishing its
claims to correctness and accuracy. In the year 1697
he published his grand physico- mathematical treatise on
the nature of rivers, entitled Delia Natura de Fiume,
which raised his reputation to the highest pitch for hy-
draulic science. This reputation procured him the honour
of being employed by the dukes of Mantua, of Parma,
and Modena, the grand-duke of Tuscany, Pope Clement
the 11th, the republics of Venice and Lucca, &c, in the
invention and construction of the proper hydraulic works
in their respective territories.

In the year 1(598, our author was induced, by the re-
public of Venice, to accept of the mathematical chair in
the university of Padua ; and the senate of Bologna de-
creed that he should still retain the title of professor in
their university, and the emoluments annexed to it. In
1702 he exchanged his mathematical chair at Padua, for
the more lucrative one of medicine ; after which he pub-
lished different tracts on chemical and medical subjects.
He died at Padua in 1710, in the 55th year of his age.
He had been admitted a member of the Academy of
Sciences at Paris in 1696*, and was also associate, or cor-
respondent member of the academies of Berlin and Vien-
na, and of the Royal Society at London. — The whole of
his works were printed in a collective form at Geneva, in
1719, in 2 vols. 4to ; but the best edition of his treatise on
the nature of rivers has been published at Bologna in
17.56, with the notes of Manfredi.

GU1NLA, a gold coin struck in England. — The value
or rate of the guinea has varied. It was at first equalto
20 shillings ; but by the scarcity of gold it was afterwards
advanced io 21s. W.; and then reduced to 21s. For
some years past (1812) it has disappeared from common
circulation, and clandestinely sold at 27*. or 28*.

The pound weight troy of gold is cut into 44 parts and
a half, and each part makes a guinea, which is therefore
equal to Tyb, or f-$'>z, Or i dwts. 9|§ gr.

This coin took its name, Guinea, from the circumstance
of thegold of which it was first struck being brought from
I hat part of Africa called Guinea, for which reason also
it bore the impression of an elephant.

GULA, GuEtn.E, or Gola, in Architecfure, a wavy
member whose contour resembles the letter s, commonly
called an ogee.

GULBE, in Architecture, the same as Gorge.
GULD1N (Father), was born at St. Gall in !577,.en-
teied into the society of the Jesuits in 1597, after abjuring
the protestant religion, and died in ]6'43. Having been
sent to Rome to improve his talents, he became professor
of philosophy and mathematics in that city. He also
publicly taught the same sciences at Gratz, and at Vienna,
where, in 16'35, Jo40, and 1 6*4 1 , his chief work was
printed in a large folio volume, on the Centre of Gravity,
or the Centrobaric Method, being a very elaborate geome-
trical work.

GULF, or Gulph, in Geography, a part of the ocean
running up into the land through a narrow passage, or
strait, and forming a bay within. As, the Gulf of Venice,
or Adriatic Sea ; the Gulf of Arabia, or of Persia, which
is the Red Sea; the Gulf of Constantinople, or the Black
Sea ; the Gulf of Mexico, &c.

GUN, a fire-arm, or weapon of offence, which forcibly
discharges a ball or other matter through a cylindrical
tube, by means of inflamed gunpowder.

The term gun now includes most of the species of fire-
arms ; mortars and pistols being almost the only ones ex-
cepted from this denomination. They are divided into
great? and small guns : the former including all that are
usually called cannon, ordnance, or artillery ; and the
latter musquets, firelocks, carbines, musquetoons, blun-
derbusses, fowling-pieces, &c.

It is not certainly known at what time these weapons
were first invented. And though the introduction of guns
into the western part of the world is but' of modern date,
comparatively speaking, yet it is certain that in some parts
of Asia they had been used for many ages before, though
in a very rude and imperfect manner. Philostratus speaks
of a city near the river Hyphasis in the Indies, which was
said to be impregnable, and that its inhabitants were re-
lations of the gods, because they threw thunder and light-
ning upon their enemies; and other Greek authors, as
also Quintus Curtius, mention the same thing having
happened to Alexander the Great". Hence some have
imagined that guns were used by the eastern nations in
his time, while others suppose the thunder and lightning
alluded to by those authors, were only certain artificial
fire-works, or rockets, such as we know are used in ilie
wars by the Indians even in the present day against the
Europeans. Be ihis however as it may, it is asserted -by
many modern travellers, that guns were used in China as
far back as the year of Christ 85, and have continued in
use ever since.

The fust hint of the invention of guns in Europe, is in
the works of Roger Bacon, who flourished in the 13th
century. In a treatise written by him about the year 1280,
he proposes to apply the violent explosive force of gun-
powder to the destruction of armies. And though it is cer-
tainly known that the composition of gun-powder is de-
scribed by Bacon in the said work, yet the invention has
usually, though improperly, been ascribed to Bartholdus
Schwartz, a German inonK, who it is said discovered it
only in the year 1320 ; and the invention is related in the
following manner. Schwartz having, fur sonic purpose,
pounded nitre; sulphur, and charcoal together, in a mor-
tar, which he afterwards covered imperfectly with a stone,
a spark of fire accidentally fell into the mortar, which
setting the mixture on fire, the explosion blew the stone

GUN

[ 608 ]

GUN

to a considerable distance. • Hence it is probable that
Schwartz might be taught the simplest method of applying
it in war; for it rather seems that Bacon conceived the
manner of using it to be by the violent effort of the flame
unconfined, and which is indeed capable of producing
astonishing effects. (See Gunpowder) Andthefigure
'and name of mortars, given to a species of old artillery,
and their employment, in throwing large stone bullets at
an elevation, very much favour this conjecture.

Soon after the time of Schwartz, we find guns com-
monly employed as instruments ofwar. Great guns were
first used. These were originally made of iron bars solder-
ed together, and fortified with strong iron hoops or rings ;
several of which are still to be seen in the Tower of Lon-
don, and in the Arsenal at Woolwich. Others were made
of thin sheets of iron rolled together and hooped : and on
particular emergencies some have been made of leather,
and of lead, with plates of iron or copper. These first
pieces were executed in a rude and imperfect manner, like
the first essays of most new inventions. Stone balls were
thrown out of them, and a small quantity of powder used
on account of their weakness. They were of a cylindri-
cal form, without ornaments, and were placed on their
carriages by rings.

When, or by whom they were first made, is uncertain.
It is known however that the Venetians used cannon at the
siege of Claudia Jcssa, now called Chioggia, in 1366,
which were brought thither by two Germans, with some
powder and leaden balls; as also in their wars with the
Genoese in 1379. But before that period King Edward
the 3d made use of cannon at the battle ofCressy in 134-6",
and at the siege of Calais in 1347. Cannon were em-
ployed by theTurks at the siege of Constantinople, then in
possession of the Christians, in 1394, and in that of 1452,
which threw a stone ball of 2001b. weight ; but they
commonly burst at the first two or three firings. Louis
the 12th had one cast at Tours, of the same size, which
threw a ball from the Bastile to Charenton ; one of these
extraordinary cannon was taken at the siege of Dieu in
1546, by Don John dc Castro, and is now in the castle
of St. Julian da Barra, 10 miles from Lisbon : the length
•of it is 20 feet 7 inches, its diameter at the middle 6 feet
•3 inches, and it threw a ball of 1001b. weight. It has
neither dolphins, rings, nor button ; is of an unusual kind
of metal ; and it has a jlprge Indostan inscription on it,
which says it was cast in 1400.

Formerly, it was customary to dignify cannon with
uncommon names. Thus, Lewis the 12th, in 1503, had
12 brass cannon cast, of an extraordinary size, called after
the names of the 12 peers, of France. The Spanish and
Portuguese called them after their saints. The Emperor
■ Charles the 5th, when he marched against Tunis, founded
the 12 Apostles. At Milan there is a 70-pounder, called
the Pimontelle ; and one at Bois-le-duc, called the Devil.
A 60-pounder at Dover-castle, called Queen Elizabeth's
pocket-pistol. An 80-pounder in the Tower of London,
brought there from Edinburgh-castle, called Mount s-mog.
An 80-pounder in the royal arsenal at Berlin, called the
Thunderer. An 80-pounder at Malaga, called the Terri-
ble. Two curious 60-pounders in the arsenal at Bremen,
called the Messenger of bad news. And lastly an uncom-
mon 70-pounder in the castle of St. Angelo at Rome, made
of the nails that fastened the copper-plates which covered
the ancient Pantheon, with this inscription on it, Ex clavis
tcabalibus porticus Agrippte.

In the beginning of the 15th century these uncommon
names were generally abolished, and the following more
universal ones adopted in their stead, viz,

Wt

. of ball, Wt.

of piece in

Names.

Pounders. ewts

about Gwt.

Cannon royal, or earth

oun

48

90

Bastard cannon, or| carthoun

36

79

Demi-carthoun

-

24

60

Whole culverins

-

IS

50

Demi-culverins

-

9

30

Falcon

-

6 - .

2.5

f largest size

-

8

IS

Sacker ■? ordinary

-

6

15

(, lowest sort

-

5

13

Basilisk

-

48

85

Serpentine

'-

.4

S

Aspic

-

2

7

Dragon

.-

6

12

Syren

-

60

81

Falconet

-

3, 2, and 1

15,10,5

Rabinet

-

1

Moyens

-

10 or 12 oe.

These curious names, of beasts and birds of prey, were
adopted on account of their swiftness in motion, or of their
cruelty ; as the falconet, falcon, sacker, and culverin, &c,
for their swiftness in flying; the basilisk, serpentine, aspic,
dragon, syren, &c, for their cruelty. But, at present,
cannon take their names from the weight of their proper
ball only. Thus, a piece that discharges a cast-iron ball
of 24 pounds, is called a 24-pounder ; one that carries a
ball of 12 pounds, is called-a 12-pounder: and so of the
rest, divided into the following kinds, viz,

Ship-guns, consisting in 42, 36, 32, 24, 18, 12,9,6,
and 3-pounders. Garrison-guns, in 42, 32, 24, 18, 12,
9, and 6 pounders.

Battering-guns, in 24, 18, and 12-poundcrs.
Field-pieces, in 12, 9> 6, 3, 2, If, 1, and f-pounders.
Mortars, it is thought, have been at least as ancient as
cannon. They were employed in the wars of Italy, to
throw balls of red-hot iron, stones, &c, long before the
invention of shells. These last, it is supposed, were of
German invention, and the use of them in war shown by
the following accident. ; viz, a citizen of Venlo, at a fes-
tival celebrated in honour of the duke of Cleves, throwing
a number of shells, one of them fell on a house and set it
on fire, by which misfortune the greatest part of the town
was reduced to ashes. The first account of shells used
for military purposes, is in 1435, when Naples was be-
sieged by Charles the 8th. History informs us, with more
certainty, that shells were projected from mortars at the
siege of Wachtcndonk, in Gueldcrland, in 1588, by the
earl of Mansfield ; and Cyprian Lucar wrote upon the
method of filling and throwing such shells, in his Appendix
to the Colloquies of Tartaglia, printed at London in 158S ;
where also the compounding and throwing of carcases and
various kinds of fire-works is shown.

Mr. Malter, an English engineer, first taught the
French the art of throwing shells, which they practised
at the siege of Motte in l6'34. The method of firing red-
hot balls out of mortars was first certainly put in practice
at the siege of Stralsund in 16'75 by the elector of Bran-
denburg: though some say in ]6.'>3 at the siege of Bre-
men.

Another species of ordnance has been long in use, by
the name of howitzer, which is a kind of medium as to Jts

GUN

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GUN

length, between the cannon and the mortar, and is a very
useful piece, for discharging either shells or large balls,
which is done either at point-blanc, or at a small eleva-
tion, -A new species of ordnance has lately been introduced
by the Carron company, and thence called a Carronade,
which is only a very short howitzer, and which possesses
the advantage of being very light and easy to work. The
species of guns before mentioned, are now made chiefly of
cast iron ; except the howitzer, which is of brass, as well
as some cannon and mortars.

Muskets were first used at the siege of Rhege in the
year 1-521. The Spaniards were the first who armed part
of their foot with these weapons. At first they were very
heavy, and could not be used without a rest. They had
match-locks, and did execution at a great distance. On
their march the soldiers carried only the rests and am-
munition, having boys to bear their muskets after them.
They were very slow in loading, not only by reason of the
unwieldjness of their pieces, and because they carried the
powder and ball separate, but from the time it took to
prepare and adjust the match; so that their fire was not
near so brisk as ours is now. Afterwards a lighter match-
lock musket came in use : and they carried their ammu-
nition in bandeiiers, to which were hung several little cases
of wood covered with leather, each containing a charge of
powder. The muskets with rests were used as late as the
beginning of the civil wars in the time of Charles the 1st.
The lighter kind succeeded them, and continued till the
beginning of the present century, when thev also were
disused, and the troops throughout Europe armed with fire-
locks. These are usually made of hammered iron. For
the dimensions, construction, and practice of every species
of gun, &c, see the several articles Cannon, Mortar, &c.
Sec also Gunnery.

GUNNERY, the art of charging, directing, and explo-
ding fire-.arms, as cannon, mortars, muskets, &c, to the
best advantage. Gunnery is sometimes considered as a
part of the military art, and sometimes as a part of pyro-
techny. To the art of gunnery belongs the knowledge of
the force and effect of gunpowder, the dimensions of the
pieces, and the proportions of the powder and ball they
carry, with the methods of managing, charging, pointing,
spunging. &c. Also some parts of gunnery are subject to
mathematical consideration, which among mathematicians
are called absolutely by the name gunnery, viz, the rules
and method of computing the range, elevation, quantity
of powder, &c, so as to hit a mark or object proposed, and
is more particularly called Projectiles; which see.

H I STO R Y (/GUKSE RY.

Long before the invention of gunpowder, and of gunnery,
properly so called, the art of artillery, or projectiles, was
actually in practice. For, not to mention the use of spears,
javelins, or stones thrown with the hand, or of bows and
arrows, all which are found among the most barbarous and
ignorant people, accounts of the larger machines for throw-
ing stones, darts, &c, are recorded by the most ancient
writers. Thus, one of the kings of Judah, 800 years bo-
fore the Christian aera, erected engines of war on the towers
and bulwarks of Jerusalem, for shooting arrows and great
stones forthe defence of that city. 2 Chron.'xxvi. 15. Such
machines were afterwards known among the Greeks and
Romans by the names of Ballista, Cutapulta, &c, which
produced effects by the action of a spring of a strongly
twisted cordage, formed of tough and elastic animal sub-
stances, no less terrible than the artillery of the moderns.

Vol. I.

Such warlike instruments continued in use down to the
12th and 13th centuries, and the use of bows still longer ;
nor is it probable that they were totally laid aside till they
were superseded by gunpowder and the modern ord-
nance. •

The first application of gunpowder to military affairs, it
seems, was made soon after the year 1 300, for which the
proposal of friar Bacon, about the year 1280, for. applying
its enormous explosion to the destruction of armies, might
give the_ first hint ; and Schwartz, to whom the invention
of gunpowder has been erroneously ascribed, on account
of the accident abovementioned under the article Gun,
might have been the first who actually applied it in this
way, that is in Europe ; for as to Asia, it is probable that
the Chinese and Indians had something of the kind many
ages before. Thus, only to mention the prohibition of fire-
arms in the code of Gentoo laws, printed by the East-In-
dia Company. in 1776, which seems to confirm the suspi-
cion suggested by a passage in Quintus Curtius, that Alex-
ander the Great found some weapons of that kind in India :
Cannon in the Shanscrit idiom is called shet-aghnee, or the
weapon that kills a hundred men at once.

However, the first pieces of artillery, which were charged
with gunpowder and stone bullets of a prodigious size,
were of very clumsy and inconvenient structure and
weight. Thus, when Mahomet the 2d besieged Constan-
tinople in 14-53, he battered the walls with stones of this
kind, and with pieces of the calibre of 1200 pounds ; which
could not be fired more than four times a day. It was
however soon discovered that iron bullets, of much less
weight than stone ones, would be more efficacious if im-
pelled by greater quantities of stronger powder. This oc-
casioned an alteration in the matter and form of the can-
non, which were now cast of brass. These were lighter
and more manageable than the former, at the same time
that they were- stronger in proportion to their bore. This
change took place about the close of the 15th century.

By this means came first into use such powder as is now
employed over all Europe, by varying the proportion of
the materials. But this change of the proportion was not
the only improvement it received. The practice of grain-
ing it is doubtless of considerable advantage. At first the
powder had been always used in the form of fine meal, such
as it was reduced to by grinding the materials together.
And it is doubtful whether the first graining of powder
was intended to increase its strength, or only to render it
more convenient for filling into small charges and the
loading of small arms, to which alone it was applied for
many years, while meal-powder was still used for cannon.
But at last the additional strength which the grained pow-
der was found to possess, doubtless from the free passage
of the air between the grains, occasioned the meal-powder
to be entirely laid aside.

For the last 200 years, the formation of cannon has been
very little improved; the best pieces of modern artillery
differing little in their proportions from those used in the
time of Charles the 5th. Indeed lighter and shorter pieces
have been often proposed and tried ; but though they have
their advantages in particular cases, it is agreed they are
not so efficacious for general service. Yet the size of the
pieces has been much diminished ; the same purposes being
now accomplished by smaller pieces than what were for-
merly thought necessary. Thus the battering cannon now
approved, arc those that formerly were called demi can-
non, carrying a ball of 24 pounds weight; this weight
I I

GUN

[ 610 ]

GUN

having been found fully sufficient. The method also of
makinc a breach, by first cutting off the whole wall as low
as possible before its upper part is attempted to be beaten
down, seems to be a considerable modern improvement in
the practical part of gunnery. But the most considerable
improvement in the practice of artillery, is the ricochet
firing, or the method of firing with small quantities of
powder, and elevating the piece but a little, so that the
bullet may just go clear of the parapet of the enemy, and
drop into their works: for by thismeans the ball, coming
to the ground at a small angle, and with a small velocity,
does not bury itself, but bounds or rolls along a great way,
destroying all before it. This method was first practised
by M. Vauban at the siege of Aeth, in the year 1692. A
practice of this kind was successfully used by the king of
Prussia at the battle of Rosbach in 1757. He had several
six-inch mortars, made with trunnions, Mid mounted on
travelling carriages, which were fired obliquely on the
enemy's lines, and among their horse. These being charged
with only S ounces of powder, and elevated at one degree
and a quarter, did great execution: for the_ shells rolling
along the lines with burning fuses made the stoutest of the
enemy to fly with precipitation.

The use of fire-arms was however long known before any
theory of projectiles was formed. The Italians were the
first people who made any attempts at the theory, which
they did about the beginning of the l6th century; and
among them it appears the first who wrote professedly on
the flight of cannon shot, was Nicholas Tartalia, of Brescia,
the same author who had so great a share in the invention
of the rules for cubic equations. In 1537 he published, at
Venice, his Nova Scientia, and in 1546 his Quesiti et In-
ventioni diversi, in both which he treats professedly on
these motions, as well as in another work, translated into
English with additions by Cyprian Lucar, under the title
of Colloquies concerning the Art of Shooting in great and
small Pieces of Artillery, and published at London in 1588.
Tartalia determined, that the greatest range of a shot was
when discharged at an elevation of 45°: and he asserted,
contrary to the opinion of his contemporaries, that no part
of the path described by a ball is a right line ; though the
curvature in the first part of it is so small, that it need not
be attended to. He compared it to the surface of the sea ;
which, though it appears to be a plane, is yet doubtless
incurvated round the centre of the earth. The same au-
thor observes, that he invented the gunner's quadrant, for
laying a piece of ordnance at any point or degree of ele-
vation; and though he had but little opportunity of acqui-
ring any practical knowledge by experiments, he yet gave
shrewd guesses at the event of some-untried methods.

The philosophers of those times also took part in the
questions arising on this subject; and many disputes on
motion were held, especially in Italy, which continued till
the time of Galileo, and probably gave rise to his cele-
brated Dialogues on Motion. These were not published
till the year 1638 •, and in the interval there were publish-
ed many theories of the motion of military projectiles, as
well as many tables of their comparative ranges; though
for the most part very fallacious, and inconsistent with the
motions of these bodies.

It is remarkablehowever that, during those contests, so
few of those who were intrusted with the care of artillery,
thought it necessary to bring their theories to the test of
experiment. Mr. Robins informs us, in the preface to his
New Principles of Gunnery, that he had met with no more

Elevation at

Points.

Deg.

0

or

0

1

-

7\

2

-.

15

3

-

22 1

4

-

30

5

-

37i

6

-

45

7

-

52|

s

-

60

9

-

67f

10

-

75

11

-

S2|

than four authors who had treated experimentally on this
subject. The first of these is Collado, in 164-2, who has
given the ranges of a falconet, carrying a three-pound shot,
to every point of the gunner's quadrant, each point being
the 12th part, or 7i°. But from his numbers it is mani-
fest that the piece was not charged with its usual allot-
ment of powder. The result of his trials shows the ranges-
at the point-blanc, and the several points of elevation, as
below.

Collado's Experiments.

Range in
paces.
268
594
794
954

1.01O *
1040
- . 1053
between the 3d and 4th
between the 2d and 3d
between the 1st and 2d
between the 0 and 1st
fell very near the piece.
The next treatise was by Win, Bourne, in 1643, in his
Art of Shooting in Great Ordnance. His elevations were
not regulated by the points of the gunner's quadrant, but
by degrees; and he gives the proportions between the
ranges at different elevations and the extent of the point-
blanc shot, thus: If the extent of the point-blanc shot be
represented by 1, then the proportions of the ranges at se-
veral, elevations will be as below, viz.

Bourne's Proportion of Ranges.

Elevation. Range.

0° - - 1

5 2|

10 3|

15 - - 4f

20 . .. , .,4|

and the greatest random 5| ;

which greatest random, he says, in a calm day is at 42°

elevation ; but according to the strength of the wind, and

as it favours or opposes the flight of the shot, the elevation

may be from 45° to 36°. — He does not inform us with

what piece he made his trials ; though from his proportions

it seems to have beenasraall one. This however ought to

have been mentioned, as the relation between the extent ot

different ranges varies extremely according to the velocity

and density of the bullet.

After him, Lldrcd and Anderson, both Englishmen also,
published treatises on this subject. The former of these
was many years gunner of Dover Castle, where most ot his
experiments were made, the earliest of which are dated in
16-11, though his book was not published till 1 646, and
was entitled The Gunner's Glass. His principles were
sufficiently simple, and within certain limits very near the
truth, though they were not rigorously so. He has given
the actual ranges of different puces of artillery at small
elevations, till under 10 degrees. His experiments are
numerous, and appear to be made with great care and
caution ; and he has honestly set down some, which were
not reconcilable to his method : on the whole he seems
to have taken more pains, and to have had a juster know-
ledge of his business, than is to be found in most of his
practical brethren.

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Galileo's Dialogues on Motion were printed in the year
1646. In these he pointed out the general laws observed
by nature in the production and composition of motion,
and was the first who described the action and effects of
gravity on falling bodies : on these principles he deter-
mined, that the flight of a cannon-shot, Or of any other
projectile, would be in the curve of a parabola, unless so
far as it should be diverted from that track by the resist-
ance of the air. He also proposed the means of exami-
ning the inequalities which arise from thence, and of dis-
covering what sensible effects that resistance would pro-
duce in the motion of a bullet at some given distance from
the piece.

Notwithstanding these determinations and hints of Ga-
lileo, it seems that those who came after him, never ima-
gined that it was necessary to consider how far the opera-
tions of gunnery were affected by this resistance. Instead
of this, they boldly asserted, without making the experi-
ment, that no great variation could arise from the resist-
ance of the air in the flight of shells or cannon shot. In
this persuasion they supported themselves chiefly by con-
sidering the extreme rarity of the air, compared with those
dense and ponderous bodies ; and at last it became an al-
most generally established maxim, that the flight of these
bodies was nearly in the curve of a parabola.

Thus, Robert Anderson, in his Genuine Use and Effects
of the Gunne, published in 1()74, and again in his book,
To hit a Mark, in 169O, relates a great many experi-
ments ; but proceeding on the principles of Galileo, he
strenuously asserts that the flight of all bullets is in the
curve of a parabola; undertaking to answer all objections
that could be brought to the contrary. The same thing was
also undertaken hy-Blondel in his Art de jetter les Bombes,
published in lo'S.j ; where, after long discussion, he con-
cludes, that the variations from the-air's resistance are so
slight, as not to deserve any notice. The same subject is
treated of in the Philos. Trans No. 216", pa. 68, by Dr.
Halley ; .who also, swayed by 'the very great dispropor-
tion between the density of the air and that of iron or lead,
thought it reasonable to believe that the opposition of the
air to large metal-shot is scarcely discernible; though in
small and light shot he owns that it must be accounted for.

Though this hypothesis went on smoothly in specula-
tion ; yet Anderson, who made a great number of trials,
found it impossible to support it without some new modi-
fication. For though it does not appear that he ever
examined the comparative ranges of either cannon or
musket-shot when fired with their usual velocities, yet his
experiments on the ranges of shells thrown with velocities
that were but small, in comparison of those above men-
tioned, convinced him that their whole track was not
parabolical. But instead of making the proper inferences
from hence, and concluding that the resistance of the air
was of considerable efficacy, he framed a new hypothesis;
which was, that the shell or bullet at its first discharge
flew to a certain distance in a right line, from the end of
which line only it began to describe a parabola: and this
right line, which he calls the line of the impulse of the
fire, he supposes is the same for all elevations. So that, by
assigning a proper length to this line of impulse, it was al-
ways in his power to reconcile any two shots made at any
two different angles ; though the same method could not
succeed with three shots; nor indeed does he ever inform
us of the event of his experiments when three ranges were
tried at one time.

But after the publication of Newton's Principia, it
might have been expected, that the defects of the theory
would be ascribed to their true cause, which is the great
resistance of the air to such swift motions ; as in that
work he particularly considered the subject of such mo-
tions, and related the result of experiments, made on slow
motions at least; by which it appeared, that in such mo-
tions the resistance increases as the square of the veloci-
ties, and he even hints a suspicion that it will increase
above that law in swifter motions, as is now known to be
the case. So far however were those who treated this
subject scientifically, from making a proper allowance for
the resistance of the atmosphere, that they still neglected
it, or rather opposed it, and their theories still differed
most egregiously from the truth. Huygens alone seems
to have attended to this principle: for in the year 1690
he published a treatise on gravity, in which he gave an ac-
count of some experiments tending to prove that the track
of all projectiles, moving with very swift motions, wag
widely different from that of a parabola. The rest of the
learned generally acquiesced in the justness and sufficiency
of Galileo's doctrine, and accordingly very erroneous cal-
culations concerning the ranges of cannon were given.
Nor was any farther notice taken, of these errors till the
year 17 lfj, when Mr. Ressonp, a French officer of artil-
lery, of great merit and experience, gave in a memoir to
the Royal Academy, importing that, " though it was
agreed that theory joined with practice did constitute the
perfection of every art ; yet experience had taught him
that theory was of very little service in the. use of mortars :
that the works of M. Blondel had justly enough described
the several parabolic lines, according to the different de-
grees of the elevation of the piece; but that practice had
convinced him there was no theory in the effect of gun-
powder; for having endeavoured, with the greatest preci-
sion, to point a mortar according to these calculations, he
had never been able to establish any solid foundation upon
them." — One instance only occurs in which D. Bernoulli
applies the doctrine of Newton to the motions of pro-
jectiles, in the Com. Acad. Petrop. torn. 2, pa. 338, &c.
Besides which nothing farther was done in this science till
the time of Mr. Benjamin Robins, who published a trea-
tise in 1742, intitled New Principles of Gunnery, in
which he treated particularly, not only of the resistance
of the atmosphere, but also of the force of gunpowder,
the nature and effects of different guns, and almost every
thing else relating to the flight of military projectiles; and
indeed he carried the theory of gunnery nearly to its ut-
most perfection.

The first important consideration which engaged the
attention of Mr. Robins, and which is indeed the founda-
tion of all other particulars relating to gunnery, is the ex-
plosive force of gunpowder. M. Lahire, in the Hist.
of the Acad, of Sciences for the year 1702, supposed that
this force may be owing to the increased elasticity of the
air contained in, and between the grains, in consequence
of the heat and fire produced at the time of the explosion :
a cause not adequate to the 200th part of the effect. On
the other hand, Mr. Robins determined, by irrefragable
experiments, that this force was owing to an elastic fluid,
similar to our atmosphere, existing in the powder in an ex-
tremely condensed state, which being suddenly disengaged
from the powder by the combustion, expanded with an
amazing force, and violently impelled the bullet, or what-
ever may oppose its expansion.

4 I 2

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In ascertaining the intensity of this force of exploded
gunpowder. Mr. Robins had recourse to several different
ways, after the example of Mr. Hawksbee, related in the
Fhilos. Trans. No. 295, and his Physico-Mechah. Exper.
pa. 81. One of these is by firing the powder in the air
thus: A small quantity of the powder is placed in the
upper part of a glass tube, and the lower part of the tube
is immersed in water, the water being made to rise so near
the top, that only a small portion of air is left in that part
where the powder is placed : then in this situation the com-
munication between the upper part of the tube and the
external air being closed, the powder is fired by means
of a burning glass, or otherwise ; the water descends
upon the explosion, and stands lower in the tube than
before, by a space proportioned to the quantity of powder
fired.

Another method was by firing the powder in vacuo, viz,
in an exhausted receiver, by dropping the grains of pow-
der upon a hot iron included in the receiver. By this
means a permanent elastic fluid was generated from the
fired gunpowder, and the quantity of it was always in pro-
portion to the quantity of powder that was used, as was
found by the proportional sinking of the mercurial gage
annexed to the air pump. The result of these experi-
ments was, that the weight of the elastic air- thus gene-
rated, was equal to T3W of the compound mass of the gun-
powder which yielded it; and that its bulk, when cold and
expanded to the rarity of common atmospheric air, was
about 240 times the bulk of the powder ; and consequently
in the same proportion would such fluid at first, if it were
cold, exceed the force or elasticity of the atmosphere. But
as Mr. Robins found, by another ingenious experiment,
that air heated to the extreme degree of the white heat of
iron, has its elasticity quadrupled, or is 4 times as strong ;
he thence inferred that the force of the elastic air generated
as above, at the moment of the explosion, is at least 4
times 240, or 960, or in round numbers about 1000 times
as strong as the elasticity or pressure of the atmosphere, on
the same space.

Having thus determined the force of the gunpowder, or
intensity of the agent by which the projectile is to be
urged, Mr. Robins next proceeds to determine the effects
it will produce, or the velocity with which it will impel a
shot of a given weight from a piece of ordnance of given
dimensions; which is a problem strictly limited, and per-
fectly soluble by mathematical rules, and is'in general
this: Given the first force, and the law of its variation, to
determine the velocity with which it will impel a given
body in passing through a given space, which is the length
of (he bore of the gun.

In the solution of this problem, Mr. Robins assumes
these two postulates, viz, 1, That the action of the pow-
der on the bullet ceases as soon as the bullet is out of the
piece; and 2d, That all the powder of the charge is fired
and converted into elastic fluid before the bullet is. sensibly
moved from its place: assumptions which for good rea-
sons are found to- be in many cases very near the truth.
It is to be noted also, that the law by which the force of
the elastic fluid varies, is this, viz, that its intensity is di-
rectly as its density, or reciprocally proportional to the
space it occupies, beingso'much the stronger as the space
is less; a principle well known, anil common to all elastic
fluids. On these principles then Mr. Robins resolves this
problem, by means of the 3£)ili prop, of Newton's Principia.
in a direct way, anil the result is equivalent to this theo-

rem, when the quantities are expressed by algebraic sym-
bols ; viz, the velocity of the ball

.= 27130 ^ x log. \) = 190 y^- * logi);

where » is the velocity of the ball,

a the length of the charge of powder,
b the whole length of the bore,
c the spec. grav. of the ball, or wt. of a cubic foot of
the same matter in ounces,
d the diameter of the bore,
iv the weight of the ball in ounces.
For Exu?nple, Suppose a = 2^- inc., b = 45 inches,
c = 11345 oz, foraballof lead, and,d = finches; then

v = 27130 */(—— x log. — ) = i674 feet per second,
v \226g a 7

the velocity of the ball.

Or, if the wt. of the bullet be to = l/5 oz. =-§#oz.

Then v = lOOvX'"5 * '89 * log. — ) = 1,674 feet, as

v s 29 x a-2 ° 7

before.

" Having in this proposition, "says Mr. Robins, "shown
how the velocity, which any bullet acquires from the
force of powder, may be computed upon the principles of
the theory laid down in the preceding propositions; we
shall next show, that the actual velocities, with which
bullets of different magnitudes are impelled from dif-
ferent pieces, with different quantities of powder, are really
the same with the velocities assigned by these computa-
tions ; and consequently that this theory of the force of
powder, here delivered, does unquestionably. ascertain the
true action and modification of this enormous power.

" But in order to compare the velocities communicated
to bullets by the explosion with the. velocities resulting
from the theory b}' computation ; it is necessary that the
actual velocities with which bullets move, should be ca-
pable of being discovered, which yet is impossible to be
done by any methods hitherto made public. The only
means hitherto practised by. others for that purpose, have
been either by observing the time of the flight of the shot
through a given space, or by measuring the range of the
shot at a given elevation ; and thpnee computing, on the
parabolic hypothesis, what velocity would produce this
range. The first method labours under this insurmount-
able,difficulty, that the velocities' of these bodies are often
so swift, and consequently the time observed is so sbort,
that an imperceptible error in that time may occasion an
error in the velocity thus found, of 2, 3, 4, 5, or 600 feet
in a second. The other method is so -fallacious, by reason
of the resistance of the air (to which inequality the first is
also liable), that the velocities thus assigned may not be
perhaps the 10th part of the actual velocities sought.

" To remedy then these inconveniences, I have in-
vented a new method of finding the real velocities of bul-
lets of all kinds; and this to such a decree of exactness
(which may be augmented too at pleasure), that in a
bullet moving with a velocity of. 17,00 I'eet ill l", the er-
ror in the estimation of it need never amount to itsSOOth
part; and this without any extraordinary nicety in the
construction of the machine."

Mr. Robins then gives an account of the machine by
which he measures the velocities of the balls, which ma-
chine is simply this, viz, a pendulous block'of wood sus-
pended freely by a horizontal axis, against which block
are to be fired the balls whose velocities are to be deter-
mined.

GUN

[ 613 ]

GUN

" This instrument thus fitted, if the weight of the pen-
dulum be known, and likewise the respective distances of
its centre of gravity, and of its centre of oscillation, from
its axis of suspension, it will thence be known what motion
will be communicated to this pendulum by the percussion
of a. body of a known weight moving with a known degree
of celerity, and striking it in a given point; that is, if the
pendulum be supposed at rest before the percussion, it
will be known what vibration it ought to make in conse-
quence of such a determined blow ; and, on the contrary,
if the pendulum, being at rest, is struck by a body of a
known weight, and the vibration, which the pendulum
makes after the blow, is known, the velocity of the .striking
body may from thence be determined.

" Hence then, if a bullet of a known weight strikes the
pendulum, and the vibration, which the pendulum makes
in consequence of the stroke, be ascertained ; the velocity
with which the ball moved, is thence to be known."

Mr. Robins then explains his method of computing ve-
locities from experiments with this machine ; which me-
thod is rather troublesome and perplexed, as well as the
rule's of Euler and Antoni, who followed him in these in-
vestigations; but a much simpler rule is given in my
Tracts, vol.2, pa. 325, where such experiments are ex-
plained at full length, and this rule is expressed by either
of the two following formulas,

P + b p -+- b

v = 5'6727c? * ~T—s/° = 6l4"58ct' x - — , the velo-
° bir v ° bim

city ; where v denotes the velocity of the ball when it strikes
the pendulum, p the weight of the pendulum, b the weight
of the ball, c the chord of the arc described by the vibra-
tion to the radius r, g the distance below the axis of mo-
tion to the centre of gravity, o the distance to the centre
of oscillation, i the distance to the point of impact, and n
the number of oscillations the pendulum will perform in
one minute, when made to oscillate in small arcs. 'The
latter of these two theorems is much the easiest, both be-
cause it is free of radicals, and because the value of the
radical y'o, in the former, is to be first computed from
the number re, or number of oscillations the pendulum is
observed to make.

With such machines Mr. Robins made a great number
of experiments, with musket barrels of different lengths,
with balls of various weights, arid with different charges
or quantities of powder. He has set down the results of
6l of these experiments, which nearly agree with the cor-
responding velocities as computed by his theory of the
force of powder, and which therefore establish that theory
on a sure foundation.

From these experiments, as well as from the preceding
theory, many important conclusions were deduced by Mr.
Robins; and indeed by means of these it is obvious that
every thing may be determined relative both to the true
theory of projectiles, and to the practice of artillery.
For, by. firing, a piece, of ordnance, charged in a similar -
manner, against such a ballistic pendulum from different
distances, the velocity lost by passing through such spaces
of air will be found, and consequently the resistance of
the air, the only circumstance that was wanting to com-
plete the theory of gujinery, or military projectiles; and
of (Ids kind 1 have since made a great number of experi-
ments vvith cannon balls, and have thereby obtained the
whole series of resistances to such a ball when moving with
every degree of velocity, from 0 up to '2200 feet per se-
cond of tune. In the structure of artillery, they may

likewise be of the greatest use : for hence may be deter-
mined the best lengths of guns; the proportions of the shot
and powder to the several lengths; the thickness of a piece;
soas.it may be able to confine, without bursting, any
given charge of powder; as also the effect of wads, cham-
bers, placing of the vent, ramming the powder, &c. For
the many other curious circumstances relating to this sub-
ject, and the various other improvements in the theory
and practice of gunnery, made by Mr. Robins, consult
the first vol. of his Tracts, collected and published by Dr.
Wilson, in the year 176I, and republished with many im-
portant notes, and observations, by the author of this
work, in 1805, where ample information may be obtained.
Soon after the first publication of Robins's New Prin-
ciples of Gunnery, in 1742, the learned in several other
nations, treading in his- steps, repeated and farther ex-
tended the same subject, sometimes varying and enlarging
the machinery; particularly Euler in Germany, D'Antoni
in Italy, and Messrs. D'Arcy and LeRoy in France.- But
most of these, like Mr. Robins, with small fire-arms, such
as muskets and fusils.

But in the year 1775, in conjunction with several able
officers of the royal artillery, and- other ingenious gentle-
men, was undertaken a course of experiments with the
ballistic pendulum, in -which we ventured to extend the
machinery to cannon shot of 1,2, and 3 pounds weight.
An account of these experiments was published in the
Philos. Trans, for 1778, and for which the Royal Society
honoured me with the prize, of the gold medal" " These
were the only experiments that I know of which had been
made with cannon balls for this purpose, though the con-
clusions to be deduced from such, are of the greatest im-
portance to those parts of natural philosophy which are
dependent on the effects of fired gunpowder; nor do I
know of any other practical method of ascertaining the
initial velocities within any tolerable degree of the truth.
The knowledge of this velocity is of the utmost conse-
quence in gunnery: by means of it, together with the
law of the resistance of the medium, every thing is deter-
minable relative to that science, for, besides its being an
excellent method of trying the strength of different kinds
of powder, it gives us the law relative to the different
quantities of powder, to the different weights of shot, and
to the different lengths and sizes of guns. Besides these
there does not appear to be any thing wanting to answer
any inquiry that can be made concerning the flight and
ranges of shot, except the effects arising from the resistance
■of the medium. In these experiments the weights of the
pendulums employed were from 300 to near 6'00 pounds.
In that paper is described the method of constructing the
machinery, of finding the centres of gravity and. oscillation
of the pendulum, and of making the experiments, which
are all set down in the form of a journal, with all the mi-
nute and concomitant circumstances ; as also the investi-
gation of the new and> easy rule, given'abuve, for com-
puting the velocity of the ball from the experiments. The
charges of powder were varied from 2 to 8 ounces, and
the shot from 1 to near 3 pounds. And from the whole
were clearly deduced these principal inferences, viz,

" 1. First, That gunpowder fires almost instantaneous-
ly.— 2. That the velocities communicated to balls or shot,
of the same weight, by different quantities of powder, are
nearly in the subduplicate ratio of those quantities: a
small variation, in defect, taking place when the quanti-
ties of powder became great.— 3. And when shot of dif-

G tJ N

[ 614 ]

GUN

fereht weights arc employed, with the same quantity of
powder, the velocities communicated to them, are nearly
in the reciprocal subduplicate ratio of their weights. —
4. So that, universally, shot which are of different weights,
and impelled by die flame of different quantities of pow-
de"r, acquire velocities which are directly as the square
roots of the quantities of powder, and inversely as the
square roots of the weights of the shot, nearly. — 5. It
would therefore be a great improvement in artillery, to
make use of shot of a long form, or of heavier matter ; for
thus the momentum of a shot, when fired with the same
weight of powder, would be increased in the ratio of the
square root of the weight of the shot. — 6. It would also be
an improvement to diminish the windage; for by so doing,
one-third or more of the quantity of powder might be
saved. — 7. When the improvements mentioned in the last
two articles are considered as both taking place, it is evi-
dent that about half the quantity of powder might be
saved, which is a very considerable object. But im-
portant as this saving may be, it seems to be still exceeded
by that of the article of the guns ; for thus a small gun
may be made to have the effect and execution of another
of two or three times its size in the present mode,, by dis-
charging a shot of two or three times the weight of its na-
tural ball or round shot. And thus a small ship might
discharge shot as heavy as those of the greatest now made
use of.

" Finally, as the above experiments exhibit the regu-
lations with regard to the weights of powder and balls,
when fired from the same piece of ordnance, &c ; so by
making similar experiments with a gun, varied in its
length, by cutting off from it a certain part before each
course of experiments, the effects and general rules for the
different lengths of guns may be certainly determined by
them. In short, the principles on which these experi-
ments were made, are so fruitful in consequences, that,
in conjunction with the effects resulting from the resist-
ance of the medium, they seem to be sufficient for answer-
ing all the inquiries of the speculative philosopher, as
well as those of the practical artillerist.

In the year 1786 was published the first volume of my
Tracts, in which is detailed, at great length, another very
extensive course of experiments which were carried on at
Woolwich in the years 1783, 1784, and 1785, by 'order
of the Duke of Richmond, master-general of the ordnance.
The objects of this course were very numerous, but the
principal of them were the following:

" 1. The velocities with which balls are projected by
equal charges of powder, from pieces of the same weight
and caliber, but of different lengths.

" 2. The velocities with different charges of pdwder,
the weight and length of the gun being the same.

" 3. The greatest velocity due to guns of different
lengths, to be obtained by increasing the charge as far as
the resistance of the piece is capable of sustaining.

" 4. The (fleet of varying the weight of the piece ; every
thing else being the same.

" 5. The penetration of balls into blocks of wood.

*' 6" The ranges and times of flight of balls; to com-
pare them with their initial velocities, for determining the
resistance of the medium.

" 7. The effect of wads ;

of different degrees of ramming;
of different degrees of windage;
of different positions of the vent ;

of chambers, and trunnions, and every other cir"
cumstance necessary to be known for the im"
provement of artillery." v

All these objects were obtained in a very perfect and
accurate manner ; excepting only the article of ranges,
which were not quite so regular and uniform as might be
wished. The balls were most of them of one pound
weight; but the powder was increased from 1 ounce, up
till the bore was quite full; and the p'endulum was from
600 to SOOlb. weight. The conclusions from the whole
were as follow :

" 1. That the former law, between the charge and ve-
locity of ball, is again confirmed, viz, that the velocity is
directly as the square root of the weight of powder, as far
as to about the charge of S ounces : and so it would con-
tinue for all charges, were the guns of an indefinite length.
But as the length of the charge is increased, and bears a
more considerable proportion to the length of the bore,
the velocity falls the more short of that proportion.

" 2. That the velocity of the ball increases with the
charge to a certain point, which is peculiar to each gun,
where it is greatest ; and that by farther increasing
the charge, the velocity gradually diminishes, till the
bore is quite full of powder. That this charge for the
greatest velocity is greater as the gun is longer, but not
greater however in so high a proportion as the length of
the gun is ; so that the part of the bore filled with powder
bears a less proportion to the whole in the long guns,
than it does in the short ones; the part of the whole which
is filled being indeed nearly in the reciprocal subduplicate
ratio of the length of the empty part. And the other cir-
cumstances as in this table.

Charges producing the Greatest Velocity.

Gun

Lentil of

Length

Part of the

Wt; of the

Number

the bore.

filled.

whole.

powder.

inches.

inches.

oz.

1

28-5

8-2

_4_

12

2

38-4

9"5

A

14

3

57-7

10-7

TT

16

4

80-2

12-1

IT

18

" 3. It appears that the velocity continually increases
as the gun is longer, though the increase in velocity is but
very small in respect of the increase in length, the velo-
cities being in a ratio somewhat less than that of the square
roots of the length of the bore, but somewhat greater than
that of the cube roots of the length, and is indeed nearly
in the middle of the ratio between the two.

" 4. The range increases in a much less ratio than the
velocity, and indeed is nearly as the square root of the
velocity, the gun and elevation being the same. And
when this is compared with the property of the velocity
and length of gun in the foregoing paragraph, we perceive
that very little is gained in the range by a great increase
in the length of the gun, the charge being the same. And
indeed the range is nearly as the 5th root of the length of
the bore ; which is so small an increase, as to amount only
to about yth part more range for a double length of gun.

" 5. It appears that the time of the ball's flight is nearly
as the range; the gun and elevation being the same.

" 6. It appears also that there is no sensible difference
caused in the velocity or range, by varying the weight of
the gun, nor by the use of wads, nor by different degrees
of ramming, nor by firing the charge of powder in different
parts of it.

GUN

t 615 ]

GUN

" 7. But a great difference in the velocity arises from a
small degree of windage. Indeed with the usual established
windage only, namely, about -^th of the caliber, no less
than between j and -| of the powder escapes and is lost.
And as the balls are often smaller than that size, it fre-
quently happens that half thepowder is lost by unnecessary
windage.

" 8. It appears that the resisting force of wood,' to balls
fired into it, is not constant. And that the depths pene-
trated by different velocities or charges, are nearly as the
logarithms of the charges, instead of being as the charges
themselves, or, which is the same thing, as the square of
the velocity.

" 9- These, and most other experiments, show that balls
are greatly deflected from the direction they are projected
in ; and that so much as 300 or 400 yards in a range of a
mile, or almost i of the range, which is nearly a deflection
t>f an angle of 15 degrees.

" 10. l-'inally, these experiments furnish us with the
following concomitant data, to a tolerable degree of accu-
racy; namely, the dimensions and elevation of the gun,
the weight and dimensions of the powder ami shot, with
the range and time of flight, and the first velocity of the
ball. From which it is to be hoped that the measure of the
resistance of the air to projectiles, may be determined,
and thereby lay the foundation for a true and practical
system of gunnery, which may be as well useful in service
as in theory."

Since the publication of thoseTracts, we have prosecuted
the experiments still farther, from year to year, gradually
extending o-ur aim to more objects, and enlarging the guns
and machinery, till we have arrived at experiments with
the6-poun<Wr guns, and pendulums of 1800 pounds weight.
One of the new objects of inquiry, was the resistance the
atmosphere makes to military projectiles ; to obtain which,
the guns have been placed at many different distances from
the pendulum, against which the\ are fired, to get the ve-
locity'lost m passing th rough' those spaces of air ; by which,
and the use of the whirling machine, described in my
new Tracts, vol. 3, tor the slower motions, I have inves-
tigated the resistance of the air to given balls moving with
all degrees of velocity, from 0 up to 2200 feet per se-
cond ; as well as the resistance for many degrees of ve-
locity, to planes and figures of other shapes, and in-
clined to their path in all varieties of angles; from which
1 have deduced general laws and formulas for all such
motions.

Mr. Robins made also similar experiments on the re-
sistance of the air; but being only with musket bullets, on
account of their smallness, and of their change of figure
by the explosion of the powder, I find they are very in-
accurate, and considerably different from those above
mentioned, which were accurately made with pretty con-
siderable cannon balls, of iron. For which reason we may
omit here the rules and theory deduced from them by Mr.
Robins, till others more correct shall have been established.
All these experiments indeed agree in evincing the very
enormous resistance the air makes to the swift motions of
military projectiles-, amounting in some cases to 20 or 30
times the weight of the ball itself; on which account the
common rules for projectiles, deduced from the parabolic
theory, are of little or no use in real practice ; for, from
these experiments it is clearly proved, that the track de-
scribed by the flight even of the heaviest shot, is neither a
parabola, nor yet approaching any thing near it, except

when they are projected with very small velocities ; in so
much that some balls, which in the air range only to the
distance of one mile, would in vacuo, when projected with
the same velocity, range above 10 or 20 times as far. For
the common rules of the parabolic theory, see Projec-
tiles. See also the articles Force, and Resistance,
in this Dictionary.

Mr. Benjamin Thompson (now Count Rumford), insti-
tuted a very considerable course of experiments of the
same kind as those of Mr. Robins, with musket barrels,
which was published in the Philos. Trans, vol. 71, for the
year 1781. In these experiments, the conclusions of Mr.
Robins are generally confirmed, and several other curious
circumstances in this business are at the same time re-
marked by Mr. Thompson. This gentleman also pursues
a hint thrown out by Mr. Robins relative to the determi-
ning the velocity of a ball from the recoil of the pendulous
gun itself. Mr. Robins, in prop. 11, remarks that the
effect of the exploded powder upon the recoil of the gun,
is the same, whether the gun is charged with a ball, or with-
out one ; and that the chord, or velocity, of recoil with
the powder alone, being subtracted from that of the recoil
when charged with both powder and ball, leaves the velo-
city which is due to the ball alone. Hence Mr. Thomp-
son observes, that the inference is obvious, viz, that the
momentum thus communicated to the gun by the ball
alone, being equal to the momentum of the ball, this be-
comes known; and therefore being divided by the known
weight of the ball, the quotient will be its velocity. Mr.
Thompson sets a great value on this new rule, the velo-
cities by means of which, he found to agree nearly with
several of those deduced from the motion of the pendulum ;
and in the other cases in which they differed greatly from
thi se, he very inconsistently supposes that these latter
ones are erroneous. In the experiments however contain-
ed in my Tracts, a great multitude of those cases are
compared together, and the inaccuracy of that new rule
is fully proved

Having in the 9th prop, compared together a number
of computed and experimental .velocitiesof balls, to verify
his theory ; in the 10th prop. Mr. Robins assigns the
changes in the force of powder, which arise from the dif-
ferent state of the atmosphere, as to heat and moisture,
both which he finds have some effect on it, but especially
the latter. In prop. 1 1 he investigates the velocity which
the flame of gunpowder acquires by expanding itself, sup-
posing it fired in a given piece of artillery, without either
a bullet or any other body placed before it. This velocity
he finds is upwards of 7000 feet per second. But the ce-
lebrated Euler, in his commentary on this part of'Mr.
Robins's book, thinks it may be still much greater. And
in this prop, it is that Mr. Robins declares his opinion,
above alluded to, viz, that the effect of the powder upon
the recoil of the gun is the same, in all cases, whether
fired with a ball, or without one. — In prop. 12 he ascer-
tains the manner in which the flame of powder impels a
ball which is laid at a considerable distance from the
charge; showing here that the sudden accumulation and
density of the fluid against the ball, is the reason that the
barrel is so often burst in those cases. — In prop. 13 he
enumerates the various kinds of powder, and describes the
properest methods of examining its goodness. He here
shows that the best proportion of the ingredients, is when
the saltpetre is -| of the whole compound mass of thepow-
der, and the sulphur and charcoal the other ^ between

GUN

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GUN

them, in equal quantities. In this prop. Mr. Robins takes
occasion to remark upon the use of eprouvettes, or me-
thods of trying powder ; condemning the praciice of the
English in using what is called the vertical eprouyette ; as
well as that of the French, in employing a small mortar,
with a very large ball, and a small charge of powder : and
instead of these, he strongly recommends the use of his
ballistic pendulum, fur its great accuracy: but for still
greater dispatch, he tells us he should use another me-
thod, which however he reserves to himself, without giv-
ing any particular description of it. From whathas been ■
done by Mr. Robins upon this head, several persons have
introduced his method of suspending the gun as a pendu-
lum, and noting the quantity of its oscillating recoil when
fired with a certain quantity of powder; and of this kind
I have contrived a machine, which possesses several ad-
vantages overall others, being extremely simple, accurate,
and expeditious ; so much so indeed, that the weighing
out of the powder is the chief part of the trouble. See
Gunpowder, and Eprouvette.

The other or 2d chapter of Mr. Robins's work, in 8
propositions, treats " of the resistance of the air, and of the
track described by the flight of shot and shells." ^And of
these, prop. 1 describes the general principles of the re-
sistance of fluids to solid bodies moving in them. Here
Mr. Robins discriminates between continued and com-
pressed fluids, which immediately rush into the space
quitted by a body moving in them, and whose parts yield
to the impulse of the body without condensing and accu-
mulating before it ; and such fluids as are imperfectly
compressed, rushing into a void space with a limited ve-
locity, as in the case of our atmosphere, which condenses
more and more before the ball as this' moves quicker, and
also presses the less behind it,, by following it always with
only a given velocity ; hence it happens that the former
fluid will resist moving bodies in proportion to the square
of the velocity, while the latter resists in a higher propor-
tion.— Proposition 2 is " to determine the resistance of
the air to projectiles by experiments." One of the me-
thods for this purpose, is by the ballistic pendulum, pla-
cing the gun at different distances from it, by which he
finds the velocity lost in passing through certain spaces of
air, and consequently the force of resistance to such ve-
locities as the body moves with in the several parts of its
path. And another way was by firing balls, with a given
velocity, over a large piece of water, in which the fall and
plunge of the ball could be seen, and consequently the
space it passed over in a given time. By these means
Mr. Robins determined the resistances of the air to several
different velocities, all which showed that there was a gra-
dual increase of the resistance, over the law of the square
of the velocity, as the body moved quicker. In the remain-
ing propositions of this chapter, he proceeds a little fur-
ther in this subject of the resistance of the air; in which
he lays clown a rule for the proportion of the resistance be-
tween two assigned velocities; and he shows that when a
24-pound ball, fired with its full charge of powder, first
issues from the piece, the resistance it meets with from
the air is more than 20 times its weight. Me further proves
that " the track described by the flight of shot or shells is
neither a parabola, nor nearly a parabola, unless they are
projected with small velocities ;" and that " bullets in
their flight are not only depressed beneath their original
direction by the action of gravity, but are also frequently
driven to the right or left of that direction by the action

of some other force: and in the Sth or last proposition,
he pretends to show that the depths of penetration of balls
into firm substances, are as the squares of the velocities.
But this is a mistake ; for neither does it appear that his
trials were sufficiently numerous or various, nor were his
small leaden balls fit for this purpose ; and 1 have found,
from a number of trials with iron cannon balls, that the
penetrations are in a much lower proportion, and that the
resisting force of wood is not uniform. See my Tracts.
In the following small' tracts, added to the principles, in
this volume, Mr. Robins prosecutes the subject of the re-
sistance of the air much further, and lays down rules for
computing ranges made in the air. But these must be far
from accurate, as they are' founded on the two following
principles, which I know, from numerous experiments, are
erroneous: viz, 1st, " That till the velocity of the projec-
tile surpasses that of 1 100 feet in,a second, the resistance
may be esteemed to be in the duplicate proportion of the*
velocity. 2d, That if the velocity be greater than that of
11 or 1200 feet in a second, then the absolute quantity of
that resistance in these greater velocities will be near 3
times as great, as it should be by a comparison with the
smaller velocities." For, instead of passing at once from
the law of the square of the velocities, and ever after being
about 3 times as much, my experiments prove that the in-
crease of the resistance above the law of the square of the
velocity, takes place at first in the smallest motions, and
increases gradually more and more, to a certain point, but
never rises so high as to be 3 times that quantity, after
which ifdecreases again. All which fully appears in the
3d vol. of my new Tracts, lately published. •

The additional tracts of Mr. Robins, in the latter part
of this volume, which contain many useful and important
matters, are numbered and titled as follows, viz. Number 1,
" Of the resistance of the air. Number 2, Of the resist-
ance of the air; together with the method of computing
the motions of bodies projected in that medium. Number 3,
An account of the experiments, relating to the resistance
of the air, exhibited at different times before the Royal So-
ciety, in the year 1746. Number 4, Ot the force of fired
gunpowder, together with the computation of the veloci-
ties thereby communicated to military projectiles. Num-
ber 5, A comparison of the experimental ranges of cannon
and mortars with the theory contained in the preceding
papers. — Practical Maxims relating to the effects and ma-
nagement of artillery, and the flight of shells and shot. —
A proposal for increasing the strength of the British navy,
by changing all the guns, from the 18- pounders down-
wards, into others of equal weight, but of a greater bore."
With several letters, and other papers, " On pointing, or
the directing of cannon to strike distant objects ; Of the
nature and advantage of rifled barrel-pieces," &c.

1 have dwelt thus long on Mr. Robins's New Principles
of Gunnery, because it is the first work that can be consi-
dered as attempting to establish a practical system of gun-
nery, and projocliles, on good experiments, on the (force
of gunpowder, on the resistance of the air, and on the ef-
fects of different pieces of artillery. Those, experiments
are however not sufficiently perfect, both on account of
the smallness of the bullets, and for want of good ranges,
to form a proper theory upon. I have supplied some of
the necessary desiderata for this purpose, viz, the. resist-
ance of the air to cannon balls moving with all degrees of
velocity, and the velocities communicated by given charges
of powder to different balls, and from different pieces of

GUN

C 617 1

GUN

artillery. But there are still wanting good experiments
with different pieces of ordnance, giving the ranges and
times of flight, with all varieties of charges, and at all dif-
ferent angles of elevation. A few however of tliose I have
ohtained, as in the following small table, which are de-
rived from -experiments made with a medium one-poun-
der gun, the iron ball being nearly 2 inches in diameter.

Powder.

Elevation
of gun.

Velocity of
ball.

Range.

Time
of flight.

oz

0

feet

feet

r>

2

15

860

4100

9

4

1 5

1230

5100

12

8

I 5

1(540

6000

14|

12

15

l6'S0

6700

i5i

2

45

S6o

5100

21

The celebrated Euler added many excellent disserta-
tions on the subject of gunnery, in his translation of Ro-
bins's gunnery into the German language; which were
again farther improved in Brown's translation of the same
into English in the year 1777. See also Antoui's Examen
de la Poudre; the experiments of MM. d'Arcy and Le
Roy, in the Memoirs of the Academy in 1751 ; and
D'Arcy's Essai d'une theorie d'artillerie in 1760: my
Tracts ; and paper on the force of fired gunpowder in the
Philos. Trans, for 1778 : and Thompson's paper on the
same subject in 1781 : but above all, vol. 2 and S of my
new Tracts, containing the concluding deductions drawn
from all the experiments made at Woolwich.

Of the common or parabolic theory of gunnery, Mr.
Simpson gave a very neat and concise treatise in his Select
Exercises. And a very ingenious treatise was also given
by Mr. Reuben Burrough, in 1779; containing, not only
the problems that are common on this subject, and the
method of determining the maximum range on an inclined
plane when the piece is situated in that plane; but also
when it is placed above or below the given plane; a pro-
blem of much greater difficulty, and which had not before
been publicly given by any other author. Other writers
on this part, are Starrat, Gray, Williams, Glenie, &c.

GUNPOWDER, a composition of nitre, sulphur, and
charcoal, mixed together, and usually granulated. This
easily takes fire ; and when fired, it rarefies or expands
with great vehemence, by means of its elastic force. — It
is to this powder that we owe all the effect and action of
nuns, and ordnance of all sorts. So that fortification,
with the modern military art, &c, in a great measure de-
pend upon it. The above definition however is not gene-
ral, for instead of the nitre, it has lately been discovered
that the marine acid answers much better, as to strength,
but is dangerous in its use, as too liable to accidental ex-
plosion.

The invention of gunpowder is ascribed, by Polydore
Virgil, to a chemist ; who having accidentajly put some of
this composition in a mortar, and covered it with a stone, it
happened to take fire, and blew up the stone. The'vet in-
forms us that the person here spoken of was a monk of
Fribourg, named Constantine Anelzen ; but Bellcforet
and other authors, with more probability, hold it to be
Bartholdus Schwartz, or Black, who discovered it, as
some say, about the year 1320; and the first use of it is
ascribed to the Venetians, in the year 1380, during the
war with the Genoese. But there arc earlier accounts of
its use, after the accident of Schwartz, as well as before it.

Vol. I.

For Peter Mexia, in his Various Readings, mentions that
the Moors being besieged in 1343, by Alphonsus the 11th,
king of Castile, discharged a kind of iron mortars upon
them, which made a noise like thunder; and this is se-
conded by what is related by Don Pedro, bishop of Leon,
in his chronicle of King Alphonsus, who reduced Toledo,
viz, that in a sea-combat between the king of Tunis and
the Moorish king of Seville, about that time, those of Tu-
nis had certain iron tubs or barrels, with which they threw
thunderbolts of fire, Ducange adds, that there is men-
tion riiade of gunpowder in the registers of the chambers
of accounts in France as early as the year 1338.

But it appears that Roger Bacon knew of gunpowder
near 100 years before Schwartz was born. He tells us, in
his Treatise De Secretis Operibus Artis et Naturae, et de
Nullitate Magise, cap. 6, (which is supposed by some to
have been published at Oxford in 1216", and which was
undoubtedly written before his Opus Majus, in 1267),
" that from saltpetre, and other ingredients, we are able
to make a fire that shall burn at what distance we please."
Dr. Plolt also in his History of Oxfordshire, pa. 236, assures
us that these " other ingredients were explained in a MS.
copy of the same treatise, in the hands of Dr. G. Langbain,
and seen by Dr. Wallis, to be sulphur and wood coal."
In addition to which we are told in the life of Friar Ba-
con in Biographia Britannica, vol. 1, that Bacon himself
has divulged the secret of this composition in a cipher, by
transposing the letters of the two words in chap. xi. of the
said treatise ; where it is thus expressed : sed ramen salis
petras Lura mope can ubre (i. e. carbonum pulvere)
et sulphuris ; et sic facies tonitrum et corruscationem, si
scias artificium : and hence the biographer apprehends the
words carbonum pulvere were transferred to the 6th chap-
ter of Langbain's Ms. In this same chapter Bacon ex-
pressly says that sounds like thunder, and corruscations,
may be formed in the air, much more horrible than those
that happen naturally. And farther adds, that there are
many ways of doing this, by which a city or an army
might be destroyed : and he supposes that by an artifice
of this kind Gideon defeated the Midianites with only 300
men: Judges, chap. 7- There is also another passage to
the same purpose, in the treatise De Scientia Experimen-
tal!. See Dr. Jebb's edition of the Opus Majus, pa. 474.
See also the life ofR. Bacon in this Dictionary.

Mr. Robins, in the preface to his gunnery, apprehends
that Bacon describes gunpowder not as a new composi-
tion first proposed by himself, but as the application of an
old one to military purposes, and that it was known long
before his time.

But M. Dutens carries the antiquity of gunpowder still
much higher, and refers to the writings of the ancients
themselves for the proof of it. " Virgil," says he, " and
his Commentator Servius (/Eneid, lib. 6, v. 585), Hygi-
nus (Fabul. 6l and 650), Eustathius (ad Odyss, A 234,
pa. 1682, lib. 1), La Cerda (in Virgil, loc. cit!), Valerius
Flaccus (lib. i. 662), and many other authors (as Raphael
Volatarran. in Commentar. Cornelius Agrippa poster.Opcr.
de Verbo Dei, c. 100, pal 237. — Gruteri Fax Artium Li-
beral, torn. 2, pa. 1236), speak in such a manner of Sal-
moneus's attempts to imitate thunder, as suggest to us
that this prince used for that purpose a composition of the
nature of gunpowder. Eustathius in particular mentions
him on this occasion, as being so very expert in mecha-
nics, that he formed machines, which imitated the noise of
thunder; and the writers of fable, whose surprise in this
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respect may be compared to that of the Mexicans when
they first beheld the fire-arms of the Spaniards, give out
that Jupiter, incensed at the audacity of this prince, slew
him with lightning, as he was employing himself in launch-
ing his thunder. But it is much more natural to suppose
that this unfortunate prince, the inventor of gunpowder,
gave rise to these fables, by having accidentally fallen a
victim to his own experiments. Dion (Hist. Rom. in Ca-
ligula, pa. 662) and Joannes Antiochenus (in Chronico,
&c. a Valesio edita, Paris l6'34, pa. 804), report the very
same thing of Caligula, assuring us that this emperor imi-
tated thunder and lightning by means of certain machines,
which at the same time emitted stones. Themistius also
informs us that the Brachmans encountered one another
with thunder and lightning, which they had the art of
launchingfromonhighat a considerable distance; (Themist.
Oratio 27, pa. 337). And in another place he relates,
that Hercules and Bacchus, attempting to assail them in a
fort where they were entrenched, were so roughly receiv-
ed by reiterated strokes of thunder and lightning, that
they were obliged to retire, leaving behind them an ever-
lasting monument of the rashness of their enterprise. Aga-
thias the historian reports of Anthemius Traliensis, that
having disagreed with his neighbour Zeno the rhetorician,
he set fire to his hduse with thunder and lightning. It
appears from all these passages, that the effects ascribed to
these engines of war., especially those of Caligula, Anthe-
mius, and the Indians, could be only brought about by
gunpowder. And what is still more, we find in Julius
Africanus a receipt for an ingenious composition to be
thrown upon an enemy, which very nearly resembles that
powder. But what places this beyond all doubt, is a clear
and positive passage of an author called Marcus Grascus,
whose work in manuscript is in the royal library at Paris,
entitled Liber Ignium. Dr. Mead had the same also in
manuscript, and a copy of that is now in my hands. (See
above.) The author describes several ways of encounter-
ing an enemy, by launching fire upon him ; and among
others gives the following. Mix together one pound of
live sulphur, two of charcoal of willow, and 6 of saltpetre ;
reducing them to a very fine powder in a marble mortar.
He adds, that a certain quantity of this ii to be put into
a long, narrow, and well compacted cover, and so dis-
charged into the air. Here we have the description of a
rocket. The cover with which thunder is imitated, he
represents as short, thick, but half-filled, and strongly
bound with packthread; which is exactly the form
of a cracker. He then treats of different methods' of
preparing the match, and how one squib may set fire
to another in the air, by having it inclosed within it.
In short, he speaks as clearly of the composition and effects
of gunpowder, as any person in our times could do. I
own I have not yet been able precisely to determine when
this author lived, but probably it was before the time of
the Arabian physician Mesne, who mentions him, and who
lluurished in the beginning of the ,0th century. Nay, there
is reason to believe that he is the same of whom Galen
speaks ; in which case he will be of antiquity sufficient to
support what I advance." It appears also from many au-
thors, and many circumstances, that this composition has
been known to the Chinese and Indians for thousands of
years. Ste what is said on this head under the article
Gun.

To this history of gunpowder it may be added, that it
has lately been discovered that saltpetre or nitre is not es-

sential to this composition, but that its place may be sup-
plied by other substances; for new gunpowder, of double
the strength of the old, has lately been made in France, by
the chemists in that country, without any nitre at all ;
and in the year 1 790 I tried some of this new powder,
that was made at Woolwich, with my eprouvette, when I
found it about double the strength of the ordinary sort.
This is effected by substituting, instead of the nitre, the like
quantity of the marine acid.

But perhaps this new composition ma}' not come into
common aud general use ; both because of the great ex-
pense in procuring or making the acid, and of the trouble
and danger of preventing it from taking fire by the heat
in the operation ; for it is found to catch fire and explode
from a very small degree of heat, and without the aid of a
spark.

As to the Preparation of Gunpowder ; there are diver9
compositions of it, with respect to the proportions of the
three ingredients, to be met with in pyrotechmcal writings ;
but the process of making it up is much the sameln all.

For some time after the invention of artillery, gunpow-
der was of a much weaker composition than that now in
use, or that described by Marcus Grascus; which was
chiefly owing to the weakness of their first pieces. See
Gun and Cannon. Of 23 different compositions, used
at different times, and mentioned by Tartaglia in hisQues.
and Inv. lib. 3, ques. 5, the first, which was the oldest,
contained equal parts of the three ingredients. But when
guns of modern structure were introduced, gunpowder of
the same composition as the present came also into use.
In the time of Tartaglia the cannon powder was made of
4 parts of nitre, one of sulphur, and one of charcoal ; and
the musket powder of 48 parts of nitre, 7 parts of sulphur,
and S parts of charcoal ; or of ] S parts of nitre, 2 parts
of sulphur, and 3 parts of charcoal. But the modern
composition is 6 parts of nitre, to one of each of the other
two ingredients. Though Mr. Napier says, he finds the
strength commonly to be greatest when the proportions
are, nitre 3lb, charcoal about £)oz, and sulphur about 3oz.
See his paper on Gunpowder in the Transactions of the
Royal Irish Academy, vol. 2. The cannon powder was in
meal, and the musket powder grained. And it is certain
that the graining of powder, which is a very considerable
advantage, is a modern improvement. Seethe preface to
Robins's Math. Tracts, pa. 32.

In making gunpowder, regard is to be had to the purity
or goodness of the ingredients, as well as the proportions of
them; for the strength of the powder depends much on
that circumstance, and also on the proper working or
mixing them together.

To purify the nitre, by taking away the fixt or common
salt, and earthy part. Dissolve it in a quantity of hot
water over the fire; then filtrate it through a flannel bag,
into an open vessel, and set it aside to cool, and to crystal-
lize. These crystals may in like manner be dissolved and
crystallized again ; and so on, till they become quite pure
and white. Then put these crystals into a dry kettle over
a moderate fire, which gradually increase till it begins to
smoke, evaporate, lose its humidity, and grow very white:
it must be kept continually stirring with a ladle, lest it
should return to its former figure, by which its greasiness
would be taken away: after that, so much water is to be
poured into the kettle as will cover the nitre; and when
it is dissolved, and reduced to the consistency of a thick
liquor, it must be continually stirred with a ladle till all

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the moisture is again evaporated, and it he reduced to a dry
and white meal. • ^

The like regard is to be had to the sulphur; choosing
that which is in large lumps, clear and perfectly yellow ;
not very hard, nor compact, but porous; nor yet too
much shining; and if, when set on fire, it freely burns all
away, it is a sign of its goodness: so likewise, if it be
pressed between two iron plates that are hot enough to
make it run, and in the running it appears yellow, and
that which remains of a reddish colour, it is then fit for
the purpose. But in case it be foul, it may be purified in
this manner: melt the sulphur in a large iron ladle, or
pot, over a very gentle coal fire, well kindled, but not
flaming; then scum off all that rises on the top, and swims
upon the sulphur ; take it presently after from the fire,
and strain it through a double linen cloth, letting it pass
leisurely ; so will it be pure, the gross matter remaining
behind in the cloth.

For the charcoal, the third ingredient, such should be
chosen as is large, clear, and free from knots, well burnt,
and cleaving. The charcoal of light woods is mostly pre-
ferred, as of willow, and that of the branches or twigs of
a moderate thickness, as of an inch or two in diameter.
Dogwood is now much esteemed for this purpose. And a
method of charring- the wood in a large iron cylinder has
lately been recommended, and indeed proved, as yielding
better charcoal than formerly. The charcoal not only
concurs with the sulphur in supplying the inflammable
matter, which causes the detonation of the nitre, but also
greatly adds to the explosive power of it by the quantity
of elastic vapour expelled during its combustion.

These three ingredients, in their purest state, being pro-
cured, long experience has shown that they are then to be
mixed together in the proportion before mentioned, to
have the best effect, viz, three-quarters of the composition
to be nitre, and the other quarter made up ofequal parts of
the other two ingredients; or, which is the same thing, 6
parts nitre, 1 part sulphur, and 1 part charcoal.

But it is not the due proportion of the materials only,
which is necessary to the making of good powder ; another
circumstance, not less essential, is the mixing them well
together : if this be not effectually done, some parts of the
composition will have too much nitre in them, and others
too little; and in either case there will be a defect of
strength in the powder. Robins, pa. 119.

After the materials have been reduced to fine dust, they
are mixed together, and moistened with water, or vinegar,
or urine, or spirit of wine, &c, and then beaten together
with wooden pestles for 24 hours, either by hand, or by
mills, and afterwards pressed into a hard, firm, and solid
cake. When dry, it is grained or corned ; which is done
by breaking the cake of powder into small pieces, and so
running it through a sieve; by which means the grains
may have any size given them, according to the nature of
the sieve employed, either finer or coarser; and thus also
the dust is separated from the grains, and again mixed
with other manufacturing powder, or worked again into
rakes.

Powder is smoothed, or glazed, as it is called, for small
arms, by the following operation: a hollow cylinder or cask
is mounted on an axis, turned by a wheel; this cask is half
filled with powder, and turned for 6 hours; and thus by
the mutual friction of the grains of powder it is smoothed,
or glazed. The fine mealy part, thus separated or worn
off from the rest, is again granulated.

The Nature, Effects, fye, ofPoxvder. — When the powVler
is prepared as above, if the least spark be struck upon it
from a steel and flint, the whole will immediately inflame,
and burst forth with extreme violence. — The effect is by
no means difficult to account for : the charcoal part of the
grain upon which the spark falls, catching fire like tinder,
the sulphur and nitre are readily melted, and the former
also breaks into flame; the contiguous grains at the same
time undergoing the same decomposition.

Upon this point Sir isaac Newton reasons thus: The
charcoal and sulphur in gunpowder easily take fire, and
kindle the nitre; and the spirit of the nitre, being thereby
rarefied into vapour, rushes out with an explosion much
after the manner that the vapour of water rushes out of an
eolipile; the sulphur also, being volatile, is converted into
vapour, and augments the explosion: add, that the acid
vapour of the sulphur, namely that which distils under a
bell into oil of sulphur,entering violently into the fixt body
of the nitre, lets loose the spirit of the nitre, and excites a
greater fermentation, by which the heat is farther aug-
mented, and the fixt body of the nitre is also rarefied into
fume; and the explosion in consequence made more vehe-
ment and quick.

For if salt of tartar be mixed with gunpowder, and that
mixture be warmed till it takes fire, the explosion will be
far more violent and rapid than that of gunpowder alone ;
which cannot proceed from any other cause, than the ac-
tion of the vapour of the gunpowder upon the salt of tartar,
by which that salt is rarefied.

The explosion of gunpowder therefore arises from the
violent action, by which all the mixture being quickly and
vehemently heated, is rarefied and converted into fume and
vapour; which vapour, by the violetice of that action be-
coming so hot as to shine, appears in the form of a flame.
M. Lahire, in the History of the French Academy
for 1702, ascribes all the force and effect of gunpowder to
the spring or elasticity of the air inclosed in the several
grains of it, and in the intervals or spaces between the
grains : the powder being kindled, sets the springs of so
many little parcels of air into action, and dilates them all
at once, whence the effect ; the powder itself only serving to
light a fire which may put the air in action ; after which
the whole is performed by the air alone.

But it appears from the experiments and observations
of Mr. Robins, that if this air be in its natural state at the
time when the powder is fired, the greatest addition its
elasticity could acquire from the flame of the explosion,
would not amount to five times its usual quantity, and
therefore could not suffice for the 200th part of the effort
which is exerted by fired powder. To understand the
force of gunpowder, it must be considered that, whether it
be fired in a vacuum or in air, it produces by its explosion
a permanent elastic fluid. See Philos. Trans. No. 295;
also Hauksbee's Phys. Mechan. Exp. pa. SI. It also ap-
pears from experiment, that the elasticity or pressure of the
fluid produced by the firing of gunpowder, is, creteris pa-
ribus, directly as its density. ,

In order to determine the elasticity and quantity of this
elastic fluid, produced from the explosion of a given quan-
tity of gunpowder, Mr. Robins premises, that the elas-
ticity of this fluid increases by heat, and diminishes by
cold, in the same manner as common air; and that the
density of this fluid, and consequently its weight, is the
same with the weight of an equal bulk of air, having the
same elasticity and the same temperature. From these
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principles, and from the experiments by which they arc
established (for a detail of which we must refer to the
book itself, so often cited in the preceding articles), he con-
cludes that the fluid produced by the firing of gunpowder
is nearly T35 of the weight of the generating powder itself;
and that the volume or bulk of this air or fluid, when ex-
panded to the rarity of common atmospheric air, is about
244 times the bulk of the said generating powder; and
which 1 have proved to be lfJOO times the bulk it occu-
pied in the nitre (Tracts, vol. 3, pa. 303). Count Saluce,
in his Miscel. Phil. Mathem. Soc. Priv. Taurin. pa. 125,
makes the proportion as 222 to 1 ; which he says agrees
with the computation of Mess. Hauksbee, Amontons, and
Belidor.

Hence it appears, that any quantity of powder fired in
arty confined space, which it adequately fills, exerts at the
instant of its explosion against, the sides of the vessel con-
taining it, and the bodies it impels before it, a force at
least 244 times greater than the elasticity of common air,
of, which is the same thing, than the pressure of the atmo-
sphere ; and this, without considering the great addition
arising from the violent degree of heat with. which it is
endued at that time; the quantity of which augmentation
is the next head of Mr. Robins's enquiry. -He determines
that the elasticity of the air is increased in a proportion
somewhat greater than that of 4 to 1, when heated to the
extremestheat of red hot iron ; and supposing that the flame
of fired gunpowder is not of a less degree, of heat,, increa-
sing the former number a little more than 4 times, makes
nearly 1000; which shows that the elasticity of the flame,
at the moment of explosion, is about 1000 times stronger
than the elasticity of common air, or than the pressure of
the atmosphere. But, from the height of the barometer,
it is known that the pressure of the atmosphere upon every
square inch, is on a medium 14jlb ; and therefore 1000
times this, or 147501b, is the force or pressure of the flame
of gunpowder, at the moment of explosion, upon a square
inch, which is very nearly equivalent to 6 tons and a half.
And it is proved in my Tracts, vol. 3, that the force is
more than double of this. This great force however di-
minishes as the fluid dilates itself, and in that proportion,
viz, in proportion to the space it occupies, it being only half
the Strength when it occupies a double space, one third the
strength when triple the space, and so on.

Mr. Robins further supposes the degree of heat above
mentioned to be a kind of medium heat; but that in the
case of large quantities of powder the heat will be higher,
and in very small quantities lower; and that therefore in
the former case the force will be somewhat more, and in
the latter somewhat less, than 1000 times the force or pres-
sure of the atm jsphere.

He further found that the strength of powder is the
same in all variations in the density of the atmosphere.
But that the moisture of the air has a great effect upon it;
for the same quantity which in a dry season would dis-
charge a bullet with a velocity of 1700 feet in one second,
will not in damp weather give it a velocity of more than
12 or 1300 feet in a second, or even less, if the powder
be bad, and improperly kept. Robins's Tracts, vol. 1,
pa. 101, &c. Farther, as there is a certain quantity of
water which, when mixed with powder, will prevent its
firing at all, it cannot be doubted but every degree of
moisture must abate the violence of the explosion ; and
hence the effects of damp powder are not difficult to be
accounted for.

It is to be observed, however, that the moisture imbibed
by powder does not render it less active when dried again.
Indeed, if powder be exposed to very great damps without
any caution, or when common salt abounds in it, as often
happens through negligence in refining the nitre, in such
cases the moisture it imbibes may perhaps be sufficient to
dissolve some part of the nitre: which is a permanent da-
mage that no drying can retrieve. But when tolerable care
is taken in preserving powder, and the nitre it is composed
of has been well purged from common salt, it will retain its
force for a long time; and it is said that powder has been
known to have been preserved for 50 years without any-
apparent damage from its age.

The velocity of expansion of the flame of gunpowder,
when fired in a piece of artillery, without either bullet ot
other body placed before it, is prodigiously great, viz,
7000 feet per second, or upwards, as appears from the ex-
periments of Mr. Robins. But Mr. Bernoulli as well as
Euler suspect it is still much greater ; though I have found
it to be much the same as Mr. Robins. See my Tracts,
vol. 3, pa. 306".

It is this prodigious celerity of expansion of the flame
of fired gunpowder, which is its peculiar excellence, and
the circumstance in which it so eminently surpasses all
other inventions,, either ancient or modern; for as to the
momentum of these projectiles only, many of the warlike
machines of the ancients produced this in a degree far sur-
passing that of our heaviest cannon shot or shells ; but the
great celerity given to these bodies cannot be in the least
approached by any other means than the flame of powder.
To prove Gunpowder. There are several ways of doing
this. 1, By sight: thus, if it be too black, it is a sign that
it is moist, or else that it contains too much charcoal ; so
also, if rubbed upon white paper, it blackens it more than
good powder does : but if it be of a kind of a2ure colour,
somewhat inclining to red, it is a proof of good powder.
2, By touching : for if in crushing it with the fingers'
ends, the grains break easily, and turn into dust, without
feeling hard, it has too much coal in it ; or if, in pressing
it under the fingers upon a smooth hard board, some
grains feel harder than the rest, it denotes that the sul-
phur is not well mixed with the nitre. Also by thrusting
the hand into the parcel of powder, and grasping it, as if
to take out a handful, you will feel whether it be dry and
equally grained, by its e.vading the grasp, and running
mostly out of the hand. 3, By burning : and here the
method most commonly followed for this purpose with us,
says Mr. Robins, is to fire a small heap of it on a clean
board, and to attend nicely to the flame and smoke it pro-
duces, as well as to the marks it leaves behind on the
board : but besides this uncertain method, there are
other contrivances made use of, such as powder-triers
acting by a spring, commonly sold at the shops, and
others again that move a great weight, throwing it up-
wards, which is a very bad sort of epiouvettc. But these
machines, says. Mr. Robins, though more perfect than
the common powder-triers, are yet liable to great irregu-
larities; lor as they are all moved by the instantaneous
stroke of the flame, and not by its continued pressure,
they do not determine the force of the fired powder with
sufficient certainty and uniformity. Another method is.
to judge from the range given to a large solid ball,. thrown
from a very short mortar, charged with a small quantity
of powder; which is also an uncertain way, both on ac-
count of the great disproportion between the weight o£

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the ball and powder, and the unequal resistance of the
air ; not to mention that it is too tedious to prove large
quantities of powder in this way ; for, "if each barrel of
powder was to be proved in this manner, the trouble of
charging the mortar, and bringing back the ball each
time, wouid be intolerable, and the delay so great, that'
no business of this kind could ever be finished ; and if a
number of barrels are received on the merit of a few, it is
great odds, but some bad ones would be among them,
which may prove a great disappointment in time of ser-
vice." These exceptions do " noways hold," continues
Mr. Robins, "against the method by which 1 have tried
the comparative strength of different kinds of powder,
which has been by the actual velocity given to a bullet,
by such a quantity of powder as is usually esteemed a
proper charge for the piece ; and as this velocity, how-
ever great, is easily discovered by the motion which the
pendulum acquires from the stroke of the bullet, it might
seem a good amendment to the method used by the French
(viz, that of the small mortar above-mentioned) to intro-
duce this trial by the pendulum instead of it. But though
I am satisfied, that this would be much more accurate,
less laborious, and readier than the other, yet, as there is
some little attention and caution required in this practice,
which might render ii of less dispatch than might be con-
venient, when a great number of barrels were to be sepa-
rately tried, I should myself choose to practise another
method not less certain, but prodigiously more expedi-
tious; so that 1 could engage, that the weighing out of a
small parcel of powder from each barrel should be the
greatest part of the labour; and doubtless three or four
hands could, by this means, examine 500 barrels in a
morning : besides, the machines for this purpose, as they
might be made of cast-iron, would be so very cheap, that
they might be multiplied at pleasure." Robins, page 123.
It is not certainly known what might be the particular
construction of the eprouvette here hinted at, but it was
probably a piece of ordnance suspended like a pendu-
lum, as he had made several experiments with a barrel in
that manner. Be this however as it may, several persons,
from those ideas and experiments of Mr. Robins, have
made eprouvettes on this principle, which appears to be
the best of any ; and on this idea also I have lately made
a machine for this purpose, which has several peculiar
contrivances, and advantages over all others, both in the
nature of its motion, and the divisions on its arc, &c. It
is a small cannon, the bore of which is about one inch in
diameter, and is usually charged with 2 ounces of pow-
der, and with powder only, as a ball is not necessary, and
the strength of the powder is accurately shown by the arc
of the gun's recoil. The whole machine is so simple, easy,
and expeditious, that, as Mr. Robins observed above, the
weighing of the powder is the chief part of the trouble ;
and so accurate and uniform, that the successive repeti-
tions or firings with the same quantity of the same kind
of powder, seldom gave a difference in the recoil of the
100th part of itself. See the construction and use of this
eprouvette in my Tracts, vol. 3, tract 35.

To recover damaged Powder. The method of the pow-
der merchants is this: they put part of the powder on a
sail-cloth, to which they add an equal weight of what is
really good ; then with a shovel they mingle it well toge-
ther, dry it in the sun, and barrel it up, keeping it in a
dry and proper place.

Others again, if it be wry bad, restore it by moistening

it with vinegar, water, urine, or brandy ; they then beat
it fine, sift it, and to every pound of powder add an ounce,
or an ounce and a half, or two ounces (according as it is
decayed), of melted nitre ; and afterwards these ingredi-
ents are to be moistened and well mixed, so that nothing
may be discerned in the composition, which may be
known by cutting the mass, and then they granulate it as
useful.

In case the powder be quite spoiled, the only way is to
extract the saltpetre with water, in the usual way, by
boiling, filtrating, evaporating, and crystallizing; and then,
with fresh sulphur and charcoal, to make it up again.

On the subject of gunpowder, see also Euler on Ro-
bins's Gunnery, Antoni Examen de la Poudre, Baume's
Chemistry, and Thompson's Experiments in- the Philos.
Trans, for 1781.

GUNTER (Edmund), an excellent English mathe-
matician, was born in Hertfordshire in 1581. He was
educated at Westminster-school under Dr. Busby, and.
from thence was elected to Christ-church college, Oxford,
in 1599, where he took the degree of master of arts in
l6"06", and afterwards entered into holy orders; and in
1615 he took the degree of bachelor of divinity. But be-
ing particularly distinguished for his mathematical ta-
lents, when Mr. Williams resigned the professorship of
astronomy in Gresham-college, London, Mr. Gunter was
chosen to succeed him, the 6th of March, 1619 ; where
he greatly distinguished himself by his lectures and wri-
tings, and where he died in 1626, at only 45 years of age,
to the great loss of the mathematical world.

Mr. Gunter was the author of many useful works and
inventions. About the year 1606, he merited the title of
an inventor, by the new projection of his Sector, which he
then described in a Latin treatise, not printed however till
some time afterwards. — In 161S he had invented a small
portable quadrant, for the more readily finding the hour
and azimuth, and other useful purposes in astronomy. —
And in 16'20 or 1623, he published his Canon Triangu-
lorum, or Table of Artificial Sines and Tangents, to the
radius, 10,000,000 parts, to every minute of the quad-
rant, being the first tables of this kind published ; together
with the first 1000 of Briggs's logarithms of common num-
bers, which were in later editions extended to 10,000 num-
bers.— In 1622, he discovered, byexperiment madeatDept-
ford, the variation or changeable declination of the mag-
netic needle; his experiment showing that the declination
had varied 5 degrees in the space of 42 years; and the
same was confirmed and established by his successor Mr.
Gellibrand. — He applied the logarithms of numbers, and
of sines and tangents, to straight lines drawn on a scaleor
ruler; with which, proportions in common numbers and
trigonometry were resolved, by the mere application of a
pair of compasses; a method founded on this property,
that the logarithms of the terms of equal ratios are equi-
different. This was called Gunter's Proportion, and Gun-
ter's Line ; and the instrument, in the form of a two-foot
scale, is still in common use for navigation and other pur-
poses, and is commonly called the Gunter. He also
greatly improved the sector and other instruments, for the
same uses ; the description of all which he published in
l624- — He introduced the common measuring chain, now
constantly used in land-surveying, which is thence called
Gunter's chain. — Mr. Gunter drew the lines on the dials
in Whitehall-garden, and wrote the description- and use
of them, by the direction of Prince Charles, iu- a small

GUN

[ &n ]

GUN

tract ; which he afterwards printed at the desire of King
James, in l624. — He was the first who used the word co-
sine, for the sine of the complement of an arc. He also
introduced the use of arithmetical complements into the
logarithmical arithmetic, as is witnessed by Briggs, cap.
15, Arith. Log. And it has been said that he first started
the idea of the Logarithmic Curve, which was so called,
because the segments of its axis are the logarithms of the
corresponding ordinates.

His works have been collected, and various editions of
them have been published ; the 5th is by Mr. William
Leybourne, in lo"73, containing the Description and Use
of the Sector, Cross-staff, Bow, Quadrant, and other in-
struments; with several pieces added by Samuel Foster,
Henry Bond, and William Leybourne.

Gunter's Chain, the chain in common use formea-
sui-ing land, according to true or statute measure; so
called from Mr. Gunter its reputed inventor. The length
of the chain is 66 feet, or 22 yards, or 4 poles of b\ yards
each; and it is divided into 100 links, of 7'92 inches each.
This chain is the most convenient of any instrument for
measuring land, because the contents 'hence computed are
so easily reduced to acres. The reason of which is, that
an acre of land is just equal to 10 square chains, or 10
chains in length and 1 in breadth, or equal to 100,000
square links. Hence, the dimensions being taken in chains,
and multiplied together, it gives the content in square
chains; which therefore being divided by 10, or a figure
cut off for decimals, brings trie content to acres ; after
■which the decimals are reduced to roods and perches, by
multiplying by 4 and 40. But the better way is to set the
dimensions down in links as integers, considering each
chain as 100 links; then, having multiplied the dimen-
sions together, producing square links, divide these by
100,000, that is, cut off five places for decimals, the rest
are acres, and the decimals are reduced to roods and
perches, as before.

Example. Suppose in measuring a rectangular piece of
ground, its length be 795 links,
and its breadth 480 links.

Ac.

Ro.

63600
3180
¥8 1600

4

3'26'4

40

• Per. 10 560
So the content is 3 ac. 3 roods 10 perches.

Gunter's Line, a Logarithmic line, usually graduated
upon scales, sectors, &c ; and so called from its inventor
Mr. Gunter. This is otherwise called the line of lines, or
line of numbers, and consists of the logarithms transferred
upon a ruler, &c, from the tables, by means of a scale of
equal parts, which therefore serves to resolve problems in-
strumentally, in the same manner as logarithms do arith-
metically. For, whereas logarithms resolve proportions,
or perform multiplication and division, by only addition
and subtraction, the same are performed on this line by
turning a pair of compasses over this way or that, or by
sliding one slip of wood by the side of another, &c.

A line of this description has been contrived various
ways, for the advantage of having it as long as possible.
As, first, on the two feet ruler or scale, by Gunter. Then,

in 1627 the logarithms were drawn by Wingate, on two
separate rulers, sliding against each other, to save the use
of compasses in resolving proportions. They were also in
1627 applied to concentric circles by Oughtred. Then
in a spiral form by Mr. Milburne of Yorkshire, about the
year 1650. Also, in l657, on the present common sliding
rule, by Seth Partridge.

Lastly, Mr. William Nicholson has proposed another
disposition of them, on concentric circles, in the Philos.
Trans, an. 1787, pa. 251. His instrument is equivalent to
a straight rule of 28f inches long. It consists of three
concentric circles, engraved and graduated on a plate of
about 1J inch in diameter. From the centre proceed two
legs, having right-lined edges in the direction of radii ;
which are moveable cither singly, or together. To use this
instrument ; place the edge of one leg at the antecedent
of any proportion, and the other at the consequent, and
fix them to that angle: the two legs being then moved to-
gether, and the antecedent leg placed at any other num-
ber, the other leg gives its consequent in the like position
or situation on the lines.

The whole length of the line is divided into two equal
intervals, or radii, of 9 larger divisions in each radius,
which are numbered from 1 to 10, the 1 standing at the
the begining of the line, because the logarithm of 1 is 0,
and the 10 at the end of each radius; also each of these
9 spaces is subdivided into 10 other parts, unequal ac-
cording to the logarithms of numbers ; the smaller divi-
sions being always lOths of the larger ; thus, if the large
divisions be units or ones, the smaller are tenth-parts; if
the larger be tens, the smaller are ones; and if the larger
be 100's, the smaller are 10's ; &c.

Use of Gunter's Line. 1. To find the product of two num-
bers. Extend the compasses from 1 to either of the num-
bers, and that extent will reach the same way from the
other number to the product. Thus, tx) mi Itiply 7 and 5
together; extend the compasses from 1 to 5, and that ex-
tent will reach from 7 to 35, the product.

2^ To divide one number by another. Extend the com-
passes from the divisor to 1, and that extent will reach
the same way from the dividend to the quotient. Thus, to
divide 35 by 5 ; extend the compasses from 5 to I, and
that extent will reach from 35 to 7, which is the quotient.

3. To find a ith proportional to three given numbers ; as
suppose to 6, 9> and 10- Extend from 6 to 9, and that
extent will reach from 10 to 15, which is the 4th propor-
tional sought. In like manner, a 3d proportional is found
to two given terms, extending from the 1st to the 2d, and
then from the 2d to the 3d.

4. To find a mean proportional between two giten numbers,
as suppose between 7 and 2S Extend from 7 to 28, and
bisect that extent ; then its half will reach from 7 forward,
or from 28 backward, to 14, the mean proportional be-
tween them. — Also, to ex tract the square root, as of 25, which
is only finding a mean proportional between 1 and the given
square 25, bisect the distance between 1 and 25, and the
half will reach from 1 to 5, the root sought. — Inlikcman-
ner the cube or 3d root, or the 4th, 5th, or any higher
root, is found, by taking the extent between 1 and the
given power ; then take such part of it as is denoted by
the index of the root, viz, the 3A part for the cube root,
the 4th part for the 4th root, and so on, and that part
will reach from ] to the root sought.

If the line on the scale or ruler have a slider, this is to
be used instead of the compasses.

GUN

t

Gunter's Quajjrant, is a quadrant made of wood,
brass, or some other substance ; being a kind of stereo-
graphic projection on the plane of the equinoctial, the eye
being supposed in one of the poles: so that the tropic,
ecliptic, and horizon, form the arches of circles, but the
hour circles other curves, drawn by means of several alti-
tudes of the sun, for some particular latitude every day
in the year.

The use of this instrument, is to find the hour of the
the day, the sun's azimuth, &c, and other common pro-
blems of the sphere or globe ; as also to take the altitude
of an object in degrees. See Quadrant.

Gunter's Scale, usually called byseamenThe Gunter,
is a large plain scale, having various lines upon it of great
' use in working the cases or questions in Navigation. This
scale is usually 2 feet long, and about an inch and a half
broad, with various lines upon it, both natural and loga-
rithmic, relating to trigonometry, and navigation, &c.

On the one side are the natural lines, and on the other
the artificial or logarithmic ones. The former side is first
divided into inches and tenths, and numbered from 1 to
24 inches, running the whole length near one edge. One
half the length of this side consists of two plane diagonal
scales, for taking off dimensions to three places of figures.
On the other half or foot of this side,are contained various
lines relating to trigonometry, in the natural numbers, and
marked thus, viz,

Rumb, the rumbs or points of the compass,

Chord, the line of chords,

Sine, the line of sines,

Tang, the tangents,

S. T. the semitangents,
and at the other end of this half are

Leag. leagues, or equal parts,

Rumb. another line of rumbs,

623 ] GYM

M. L. miles of longitude,

Chor. another line of chords.

Also in the middle of this foot are L. and P. two other
lines of equal parts. And all these lines, on this side of
the scale, serve for drawing or laying down the figures to
the cases in trigonometry and navigation.

On the other side of the scale are the following artifi-
cial or logarithmic lines, which serve for working or re-
solving those c^ses ; viz,

S. R. the sine rumbs,

T. R. the tangent rumbs,

Numb, line of numbers,

Sine, Sines,

V. S. the yersed sines,

Tang, the tangents,

Meri. Meridional parts,

E. P. Equal parts.

The late Mr. John Robertson, clerk and librarian to
the Royal Society, greatly improved this scale, both as to
size and accuracy, for the use of mariners. He extended
it to 30 inches long, 2 inches broad, and half an inch
thick ; upon which the several lines are very accurately
laid down by Messrs. Nairne and Blunt, ingenious instru-
ment makers. Mr. Robertson died before his improved
scales were published ; but the account and description of
them were supplied and drawn up by his friend Mr. Wil-
liam Mountaine, and published in 177S.

GUTTiE, or Drops, in Architecture, are ornaments in
form of little bells or cones, used in the Doric order, on
the architrave, below the tryglyphs. There are usually
six of them.

GYMNOSOPHISTS, a set of Indian philosophers, fa-
mous in antiquity ; so denominated from their going bare-
foot ; as that of Peripatetics was given to those who phi-
losophized walking.

H A I

H.

H A I

TIADLEY (John), a distinguished member of theRoyal
Society, was elected a fellow in 1717, and died Feb.
15, 1744. He contributed many useful papers to the
Royal Society, which arc. contained in their Trans, from
vol. 32 to vol. 3y inclusive. But he has been chiefly cele-
brated for the invention of his reflecting instrument for
taking angles, commonly called the Hadley's Quadrant or
Sextant, offered to the Royal Society in 1731, by which
his name has been rendered immortal.

Hadley's Quadrant, Sextant, &c, an excellent instru-
ment so called from its inventor John Hadley, esq. See
its description and use under the article Quadrant.

HAIL, or Hailstones, an aqueous concretion, usually
in form of white or pellucid spherules, descending out of
the atmosphere. Hailstones assume various shapes, being
sometimes round, at other times pyramidal, crenated, an-
gular, thin, and flat, and sometimes stellated, with six
radii like the small crystals of snow.

It is very difficult to accouut for the phenomena of hail
in a satisfactory manner ; and there are various opinions
on this head. It is usually conceived that hail is formed
of drops of rain, frozen in their passage through the mid-

dle region. Others, as the Cartesians, take it for the
fragments of a frozen cloud, half melted, and thus preci-
pitated and congealed again. Signor Beccaria supposes,
that it is formed in the higher regions of the air, where
the cold is intense, and where the electric matter is very
copious. In these circumstances, a great number of par-
ticles of water are brought near together, where they are
frozen, and in their descent they collect other particles ;
so that the density of the substance of the hailstone be-
comes less and less, from the centre ; this being formed
first in the higher regions, and the surface being collected
in the lower. Accordingly, in mountains, hailstones as
well as drops of rain, are very small; and both agree in
this circumstance, that the more intense the electricity is
that forms them, the larger they are. It is frequently ob-
served that hail attends thunder and lightning ; and hence
Beccaria observes, that as motion promotes freezing, so
the rapid motion of the electrified clouds may promote
that effect in the air. Natural histories furnish us with
a great variety of curious instances of extraordinary show-
ers of hail. See the Philos. Trans. No. 203, 229 ; and
Hist, de France, torn. 2, pa. 33 ,

HAL

[ m }

HAL

HALF-Moon, in Fortification, is an outwork having
only two faces, forming together a saliant angle, which is
flanked by some part of the place, and of the other bas-
tions. See Demilune and Ravelin.

Half-Tangents, are the tangents of the half arcs.
.See Scale and Semitangents.

HALLEY (Dr. Edmund), a most eminent English
mathematician, philosopher and astronomer, was born in
the parish of St. Leonard, Shoreditch, near London, Oct.
29, 1656'. His father, a wealthy citizen and soap-boiler,
resolving to improve the promising disposition observed
in his son, put him first to St. Paul's school, where he
soon excelled in all parts of classical learning, and made
besides a considerable advance in the mathematics; so
that, as Wood observes, he had perfectly learnt the use
of the celestial globe, and could make a complete dial ;
and we are informed by Halley himself, that he observed
the change of the variation of the magnetic needle at Lon-
don in 1672, one year before he left school. In lC/3 he
was was sent to Oxford, where he chiefly applied himself
to mathematics and astronomy, in which he was greatly
assisted by a curious apparatus of instruments, which his
father, willing toencourage his son's genius, had purchased
for him. At 19 years of age he began to oblige the world
with new observations and discoveries (which he continued
to do to the end of a very long life), by publishing " A
Directand Geometrical Method of finding the Apheliaand
Excentricity of the planets." Besides various particular
observations, made from time, to time on the celestial phe-
nomena ; he had, from his first admission into college,
pursued a general scheme for ascertaining the true places
of the fixed stars, and by this means to correct the errors
of Tycho Brahe. His original view in this was, to carry
on the design of that first restorer of astronomy, by com-
pleting the catalogue of those stars from his own observa-
tions ; but, on further inquiry, finding this province taken
up by Hevelius and Flamsteed, he dropped that pursuit,
and formed another ; which was, to perfect the whole
scheme of the heavens, by the addition of the stars which
lie so near the south pole, that they could not be observed
by those astronomers,, as never rising above the horizon
'either at Dantzick or at Greenwich. With this intention
he left the University, before he had taken any degree,
and embarked for the island of St. Helena in Nov. 1676,
when he was only 20 years of age, and arrived thereafter
a voyage of three months. He immediately set about his
task with such diligence, that he completed his catalogue,
and, returning home, landed in England in Nov. 1678,
after an absence of two years only. The university of Ox-
ford immediately conferred upon him the degree of a. m.
and the Royal Society of London elected him one of their
members.

In 1679 he was selected by the Royal Society to go to
Dantzick, to endeavour to adjust a dispute between He-
velius and Mr. Hooke, concerning the preference between
plain and glass sights, in astroscopica) instruments. He
arrived at Dantzick the 26th of May, when he immedi-
ately, in conjunction with Hevelius, set about their astro-
nomical observations, which they closely continued till
the 18th of July, when Halley left Dantzick, and returned
to England.

In the year ]680 he undertook what is called the grand
tour, accompanied by his friend the celebrated Mr. Nel-
ROfJ, In the way from Calais to L'aris, Mr. Halley had a
^ight of a remarkable comet, as it then appeared a second

time that year, in its return from the sun. He had, the
November before, seen it in its descent : and he now hasten-
ed to complete his observations upon it, by viewing it from
the royal observatory of France. His design in this part
of his tour was, to settle a friendly correspondence between
the two royal astronomers of Greenwich and Paris ; and
in the mean time to improve himself under so great a mas-
ter as Cassini. From thence he went to Italy, where he
spent a great part of the year l6"Sl ; but his affairs calling
him home, he then returned to England.

Soon after his return, he married the daughter of Mr.
Tooke, auditor of the exchequer, and took up his resi-
dence at Islington, where he set up his tube and sextant,
and eagerly pursued his favourite study : in the society of
this amiable lady he lived happily for five-and-fifty years.
In 1683 he published his Theory of the variation of the
Magnetical Compass ; in which he supposes the whole
globe of the earth to be one great magnet, having four
magnetical poles, or points of attraction, &c. The same
year also he entered upon a new method of finding the
longitude, by an accurate observation of the moon's mo-
tion. His pursuits it seems were now a little interrupted
by the death of his father, who having suffered greatly by
the fire of London, as well as by a second marriage, into
which he had imprudently entered, was found to have
wasted his fortune. Our author however soon resumed
his pursuits; for in the beginning of 1684 he turned his
thoughts to the subject of Kepler's sesqui-alterate pro-
portion ; when, after some meditation, he concluded from
it, that the centripetal force must decrease in proportion
to the square of the distance reciprocally. He found
himself unable to make it out in any geometrical way,
and therefore, after applying in vain for assistance to D.
Hooke and Sir Christopher Wren, he went to Cambridge
to Mr. Newton, who fully supplied him with what he
so ardently sought. But Halley having now found an
immense treasure in Newton, could not rest, till he
had prevailed with the owner to enrich the public with
it; and to this interview the world is in some measure in-
debted for the Principia Mathematics. Philosophic Na-
turaiis. That great work was published in 1686; and
Halley, who had the whole care of the impression, pre-
fixed to it a discourse of his own, giving a general account
of the astronomical part of the book ; and also an elegant
copy of verses in Latin.

In l6"S7 he undertook to explain the cause of a natu-
ral phenomenon, which had till then baffled the researches
of the ablest geographers. It is observed that the Medi-
terranean sea never swells in the least, though there is no
visible discharge of the prodigious quantity of water that
runs into it from nine large rivers, besides several small
ones, and the constant setting in of the current at the
mouth of the Straits. His solution of this difficulty
gave so much satisfaction to the R. Society, that he was
requested to prosecute these enquiries. He did so ; and
having shown, by many accurate experiments, how that
vast accession of water was actually carried off in vapours,
raised by the action of the sun and wind upon its surface,
he proceeded with the like success to point out the me-
thod used by nature to return (he said vapours into the
sea. This circulation he supposes to be carried on by
the winds driving these vapours to the mountains; where
being collected, they form springs, which uniting become
rivulets or brooks, and many of these, again meeting in
the valleys, grow into large rivers, emptying themselves at

HAL

[ fe ]

Hal

last into the sea : thus demonstrating, in the most beauti-
ful and satisfactory manner, the way in which the equili-
brium of receipt and expense is continually preserved in
the universal ocean.

He next ranged in the field of speculative geometry,
where, observing some imperfections in the methods be-
fore given for constructing solid problems, or equations of
the 3d and 4th powers, he furnished new rules, which were
both more easy and more elegant than any of the former;
together with a new method of finding the number of roots
of such equations, and the limits of the same. Mr. Halley
next undertook to publish a more correct Ephemeris for
the year lfjSS. there being then great want of proper
ephemerides of any tolerable exactness, the common ones
being justly complained of by Mr. Flamsteed. — In 1091
he published exact tables of the conjunctions of Venus
and Mcrcur}'; and he afterwards showed one extraordi-
nary use to be made of those tables, viz, for discovering
the sun's parallax, and thence the true distance of the
earth from the sun. — In 1692, our author produced his
tables for showing the value of annuities on lives, calcu-
lated from bills of mortality; and his universal theorem
for finding the foci of optic glasses.

But it would be endless to" enumerate all his valuable
discoveries now communicated to the Royal Society, and
published in the Philos. Trans, of which, for many years,
his pieces were the chief ornament and support. Their
various merit is thrown into one view by the writer of his
eloge in the Paris Memoirs; who, having mentioned his
History of the Trade-winds and Monsoons, proceeds in
these terms: "This was immediately followed by his esti-
mation of the quantity of vapours which the sun raises
from the sea; the circulation of vapours; the origin of
fountains ; questions on the nature of light and transpa-
rent bodies; a determination of the degrees of mortality,
in order to adjust the valuation of annuities on lives; and
many other works in the sciences, relating to astronomy,
geometry, and algebra, optics and dioptrics, ballistics and
artillery, speculative and experimental philosophy, natu-
ral history, antiquities, philology, and criticism ; being
about 25 or 30 dissertations, which he produced during
the 9 or 10 years of his residence at London ; and all
abounding with ideas new, singular, and useful."

In 169I, the Savillian professorship of astronomy at Ox-
ford being vacant, he applied for that office, but without
success. Whiston, in the Memoirs of his own Life, tells
us from Dr. Bentley, that Halley " being thought of for
successor to the mathematical chair at Oxford, bishop
Stillinffleet was desired to recommend him at court; but
hearing that he was a sceptic and a banterer of religion,
the bishop scrupled to be concerned, till his chaplain
Bentley should confer with him on the subject, which he
did. But Halley was so sincere in his infidelity, that he
would not so much as pretend to believe the Christian re-
ligion, though he was likely to lose a professorship: and
it was in consequence given to Dr. David Gregory."

Halley had published his Theory of the Variation of
the Magnetical Compass, as has been already observed, in
1 6'83 ; which, though it was well received both at home
and abroad, he found, upon a review, liable to great and
insuperable objections. Yet the phenomena of the varia-
tion of the needle, upon which it is raised, being so many
certain and indisputable facts, he spared no pains to pos-
sess himself of all the observations relating to it he could
possibly come at. To this end he procured an applica-

VOL. I.

tiontobemade to King William, who appointed him com-
mander of the Paramour pink, with orders to search out
by observations the discovery of the rule of variations, and
to lay down the longitudes and latitudes of" the English
settlements in America. — He set out on this attempt on
the 24th of November, 1698; but having crossed the lines
his men grew sickly, and his fiist-lieuteuant mutinying,
he returned to England in June 16'99. Having got the
lieutenant tried and cashiered, he set sail a second time in
September following, with the same ship, and another .of
less bulk; of which he had also the command. He now
traversed the vast Atlantic ocean from one hemisphere to
the other, as far as the ice would permit him to go; and
having made his observations at St. Helena, Brazil, Cape
Verde, Barbadoes, the Madeiras, the Canaries, the coast
of Barbary, and many other latitudes, he arrived in Eng-
land in September 1 700; and the next year published a
general chart, showing at one view the variation of the
compass in all those places.

Captain Halley, as he was now called, had been at home
little more than half a year, when he was sent by the king,
to observe the course of the tides, with the longitude and
latitude of the principal head-lands in the British chan-
nel ; which having executed with his usual expedition
and accuracy, he published a large map of the channel.
Soon after, the Emperor of Germany resolving to make a
convenient harbour for shipping in the Adriatic, Captain
Halley was sent by Queen Anne to view the two ports on
the coast of Dalmatia. He embarked on the 22d of No-
vember 1702 ; passed over to Holland ; and going through
Germany to Vienna, be proceeded to Istria; but the
Dutch opposing the design, it was laid aside ; yet the em-
peror made him a present of a rich diamond ring from his
finger, and honoured him with a letter of recommendation,
written with his own hand, to Queen Anne. It was not
long after his return that he was sent again on the same
business; when passing through Hanover, he supped with
the Electoral Prince, who was afterwards King George
the 1st, and his sister the queen of Prussia. On his ar-
rival at Vienna, he had likewise the honour the same
evening to be presented to the emperor, who sent his chief
engineer to attend him to Istria, where they repaired the
fortifications of Trieste, and added new ones.

Mr. Halley returned to England in November 1703 ;
and the same year he was made professor of geometry in
the university of Oxford, instead of Dr. Wallis, then just
deceased. On his accession to this office the university
was pleased to honour him with the degree of doctor of
laws. He was scarcely settled in Oxford, when he began
to translate into Latin, from the Arabic, Apollonius de
Sectione Rationis ; and to restore the two books De Sec-
tione Spatii of the same author, which are lost, from the
account given of them by Pappus ; and he published the
whole work in 1706". He afterwards had a share in pre-
paring for the press Apollonius's Conies ; and ventured to
supply the whole 8th book, the original of which is also
lost. To this work he added Serenus on the Section of
the Cylinder and Cone, printed from the original Greek,
with a Latin translation, and published the whole in folio
1710. Besides these, the Miscellanea Curiosa, in 3 vo-
lumes 8vo, had come out under his direction in 1708.

In 1713, he succeeded Doctor, afterwards Sir, Hans
Sloane, in the office of secretary to the Royal Society.
And, on the death of Mr. Flamsteed in 1719, he was ap-
pointed to succeed him at Greenwich as astronomer-royal ;
4 L

HAL

[ 626 1

HAL

on which occasion, that he might be more at leisure to
attend the duties of this important situation, he resigned
the office of secretary to the Royal Society in 1721.
Though he was 63 or 64 years of age when he entered
on his office at Greenwich, yet for the space of IS years
he watched the heavens with the closest attention, hardly
ever missing an observation during all that time, and,
without any assistant, performed the whole business of
the observatory himself.

On the accession of the late king, the queen consort
Caroline made a visit to the Royal Observatory : being
pleased with every thing she saw, and understanding the
smallness of the astronomer's salary (100/. a year), her
majesty very graciously said that she would speak to the
king to have it increased; to which Dr. Halley, alarmed,
replied, " Pray your majesty do no such thing; for
should the salary be increased, it might become an object
of emolument to place there some unqualified needy de-
pendent, to the ruin of the institution." However, un-
derstanding that the Dr. had formerly served the crown
as a captain in the navy, she soon after obtained a grant
of his half-pay for that commission, which he accordingly
enjoyed from that time during his life. An offer was also
made him of being appointed mathematical preceptor to
the Duke of Cumberland ; but he declined that honour,
an account of his advanced age, and the duties of his
office. In 1729 he was chosen a foreign member of the
Academy of Sciences at Paris.

About 1737 he was seized with a paralytic disorder in
his right hand, which, it is said, was the first attack he ever
felt on his constitution: however, he came as usual, once
a week, till within a very short time of his death, to meet
his friends in town on Thursdays, the day of the Royal
Society's meeting, at what is still called Dr.Halley's club.
His paralytical disorder increasing, his strength gradually
wore away, till he expired Jan. 14, 1742, in the Soth year
of his age; and his corps was interred in the church-yard
of Lee near Blackheath.

Besides the works before mentioned, his principal pub-
lications are, I. Catalogus Stellarum Australium. 2. Ta-
bulae Astronomical. 3. The Astronomy of Comets.
With a great multitude of papers in the Philos. Trans.
from vol. 11 to vol.60.

HALIFAX (John). See Holywood.

HALO, or Corona, a coloured circle appearing round
the body of the sun, moon, or the larger -stars.

Naturalists conceive the halo to arise from a refraction
of the rays of light, in passing through the fine rare vesicu-
I32 of a thin vapour, towards the top of the atmosphere.

Descartes observes, that the halo never appears when it
rains ; whence he concludes that this phenomenon is occa-
sioned by the refraction of light in the round particles of
ice, which are then floating in the atmosphere ; and to
the different protuberance of these particles he ascribes
the variation in the diameter of the halo. Gassendi sup-
poses, that a halo is occasioned in the same manner as the
rainbow ; the rays of light being, in both cases, twice re-
fracted and once reflected within each drop of rain or va-
pour, and that the difference ■between them arises entirely
from their different situation with respect to the observer.
Dechales also endeavoured to show that the generation
of the halo is similar to that of the rainbow ; and that the
reason why the colours of the halo arc more dilute than
those of the rainbow, is owing chiefly to their being form-
ed, not in large drops of rain, but in very small vapour.

But the most considerable and generally received theory,
relating to the generation of halos, is that of Huygens.
This celebrated author supposes halos, or circles round
the sun, to be formed by small round grains of hail, com-
posed of two different parts, the one of which is transpa-
rent, inclosing the other, which is opaque : And this is
the general structure actually observed in hail. He fur-
ther supposes that the grains or globules, that form these
halos, consisted at first of soft snow, and that they have
been rounded by a continual agitation in the air, and
thawed on their outside by the heat of the sun, &c. And
he illustrates his ideas of their formation by geometrical
figures. v

Mr. Weidler endeavours to refute Huygens's manner of
accounting for halos, by a vast number of small vapours,
each with a snowy nucleus, coated round with a transpa-
rent covering. He observes, that when the sun paints its
image in the atmosphere, and by the force of its rays puts
the vapours in motion, and drives them toward the sur-
face, till they are collected in such a quantity, and at
such a distance from the sun on each side, that its rays
are twice refracted, and twice reflected, they will when
they reach the eye exhibit the appearance of a halo,adorned
with the colours of the rainbow; which may happen in
globular pellucid vapours without snowy nuclei, as ap-
pears by the experiment of hollow' glass spheres filled with
water: therefore, whenever those spherical vapours are
situated as before mentioned, the refractions and reflec-
tions will happen every where alike, and the figure of a
circular crown, with the usual order of colours, will be
the consequence. Philos. Trans. No. 458.

Newton's theory of halos may be seen in his Optics,
pa. 155. And this curious theory was confirmed by ac-
tual observation in June 1692, when the author saw by
reflection, in a vessel of stagnated water, three halos,
crowns, or rings of colours, about the sun, like three small
rainbows concentric to his body. These crowns inclosed
one another immediately, so that their colours proceeded
in this continual order from the sun outward : blue, white,
red ; purple, blue, green, pale yellow, and red ; pale blue,
pale red. Crowns like these sometimes appear about the
moon. The more equal the globules of water or ice are
to one another, the more crowns of colours will appear,
and the colours will be the more lively. Optics, pa. 288.

There are several ways of exhibiting phenomena similar
to these. The flame of a candle, placed in the midst of
a steam in cold weather, or set at the distance of some feet
from a glass window that has been breathed upon, while
the spectator is also at the distance of some feet from an-
other part of the window, or placed behind a glass receiver,
when air is admitted into the vacuum within it to a cer-
tain density, in each of these circumstances will appear to
be encompassed by a coloured halo. Also, a quantity of
water being thrown up against the sun, will, as it breaks
and disperses into drops, form a kind of halo or iris, ex-
hibiting all the colours of the natural rainbow. Mus-
schenbroek observed, that when the glass windows of his
room were covered with a thin plate of ice on the inside,
the moon seen through it was surrounded with a large
and variously coloured halo; which, upon opening the
window, he found arose entirely from that thin plate of
ice, because none was observed except through this plale.
The same author concludes his account of coronas with
observing, that some density of vapour, or some thickness
of the plates of ice, divides the light in its transmission

HAM

[ 627 ]

H A R

either through the small globules or their interstices, iixo
its separate colours ; but what that density is, or what the
size of the particles which compose the vapour may be,
he does not pretend to determine. Introd. ad Phil. Nat.
pa. 1037.

It has often been observed that a halo about the sun or
moon, does not appear circular aDd concentric to the lu-
minary, but oval and excentric, with its longest diameter
perpendicular to the horizon, and extended from the moon
farther downward than upward. Dr. Smith ascribes this
phenomenon to the apparent concave vault of the sky
being less than a hemisphere. When the angle 'which the
diameter of a halo subtends at the eye is 45° or 46°, and
the bottom of the halo is near the horizon, and conse-
quently its apparent figure is most oval, the apparent ver-
tical diameter is divided by the moon in the proportion of
about 2 to 3 or 4, and is to the horizontal diameter drawn
through the moon, as 4 to 3, nearly. — See farther on the
subject of this article, Priestley's Hist, of Discoveries re-
lating to Vision, pa. 596 — 6l3 ; and Smith's Optics, art.
167, 513, 526, 52", &c.

HAMEL (Johx-Baptiste du), a very learned French
philosopher and writer, of the 17th century, was born in
Lower Normandy in l6l4. At IS years of age he pub-
lished a treatise, in which he explained, in a very simple
manner, and by one or two figures, Theodosius's 3 books
on Spherics; to which he added a tract on trigonometry
extremely perspicuous, and designed as an introduction
to astronomy. He published afterwards various other
works on astronomy and philosophy. Natural philoso-
phy, as it was then taught, was only a collection of vague,
knotty, and barren questions ; when our author under-
took to establish it upon right principles, and published
his Astronomia Physica.

In 1666 M. Colbert proposed to Louis the 14th a
scheme, which was approved of, for establishing a royal
academy of sciences ; and appointed our author secretary
of it. In l678, his Philosophia Vetus et Nova was
printed at Paris in 4 vols, 12mo ; and in 16S1 it was en-
larged and printed there in 6 vols. Hamel wrote several
other pieces; and his works in this way were collected and
published at Nuremberg 1GS1, in 4 volumes 4to, under
the title of Opera Philosophica et Astronomica. These
were highly valued at that period, though the improve-
ments in philosophy since that time have rendered them
of little or no use in the present.

In 1697, our author resigned his place of secretary of
the Royal Academy of Sciences ; in which he was suc-
ceeded by M. Fontenelle. However, he published, in l6"98,
Regiae ScientiarumAcademi8eHistoria,4to, in four books ;
which being much approved, he afterwards augmented
with two books more. This work contains an account of
the foundation of the Royal Academy of Sciences, and its
transactions, from 1666 to 1700, and is now the most
useful of all his works. He was Regius Professor of Phi-
losophy, in which office he was succeeded by M. Varignon,
at his death, which happened Aug. 6, 1706, in the 93d
year of his age.

HAMILTON (Da. Hugh), a very respectable ma-
thematician, and philosopher, was born at Knock, in the
county of Dublin, March 26, IT29- He entered Trinity-
college, Dublin, in November 1742, and obtained a kU
luwship in that college in 1751, when he was only 22
years of age. In the year 1758, he published his well-
known and much-admired Treatise on Conic Sections ;

and the next year was elected Erasmus Smith's professor
of natural philosophy. In the discharge of his duty as
professor, he composed and published four lectures intro-
ductory to the study of natural philosophy ; and in the
year 1762, he wrote three philesophical essays on the
ascent of vapours, the formation of clouds, rain, &c ; on
the nature of the aurora borealis, and the tails of comets ;
and on the principles of mechanics.

Dr. Hamilton resigned his fellowship in 1764, having
accepted the living of Kilmacrean, in the gift of the col-
lege: in 1767, he was collated to the parish ol St. Anne'b,
Dublin. This latter preferment was soon exchanged for
the deanery of Armagh. In 1772, he married; and soon
after published his essay on the existence and attributes of
the Supreme Being. In the year 1792, he made some im-
portant improvements in the construction of the barome-
ter; and in 1794, drew up a paper on the power of fixed
caustic alkaline salts, to preserve the flesh of animals from
putrefaction. All this time he continued to attend sedu-
lously and conscientiously to his duties as a parish priest.
In January 1796, he was consecrated bishop of Clonfert;
and about 3 years afterwards he was translated to the see
of Ossory, of which he continued bishop nearly seven
years. He died Dec. 1, 1805, in the 77th year of his age.

The principal trait in the disposition of Bishop Hamil-
ton, was unaffected humility: his chief intellectual cha-
racteristic, a patient manner of thinking. His theory of
the solution of water in air is now exploded, and his theory
of the aurora borealis has yielded to that of M. Libes.
(See Aurora Borealis). But his mechanical essays may
still be read with pleasure and advantage. Dr. Hamil-
ton's works have been lately collected and published by
his son Alexander Hamilton, in 2 vols. 8vo ; of which,
the first contains the Latin edition of the Conic Sections.

HANCES, Hatches, or Hanses, in Architecture, are
certain intermediate parts of arches between the key or
crown and the spring at the bottom, being perhaps about
one-third of the arch, and situated nearer the bottom than
the top or crown ; and are otherwise called the spandrels.

HANDSPIKE, or Hanspec, a lever or piece of ash,
elm, or other strong wood, for raising by the hand great
weights, &c. It is 5 or 6 feet long, cut thin and crooked
at the lower end, that it may be forced more readily be-
tween things required to be separated, or under any thing
that is to be raised. It is better than a crow of iron, be-
cause its length allows a better poise.

HARD Bodies, are such as are absolutely inflexible to
any pressure or percussion whatever: differing from soft
bodies, whose parts yield and are easily moved among one
another, without restoring themselves again; and from
elastic bodies, the parts of which also yield and give way,
but presently restore themselves again to their former state
and situation. Hence, hard bodies do not bend, or indent,
but break. It is probable however there are no bodies in
nature that are absolutely or perfectly either hard, soft,
or elastic ; but all possessing these qualities, more or less,
in some degree. M. Bernoulli asserts that hardness, in
the common sense, is absolutely impossible, being con-
trary to the law of continuity. The laws of motion for
hard bodies are the same as for soft ones, both being sup-
posed to adhere together on their impact. And these two
kinds of bodies might be comprised under the common
name of Un-elastic.

HARDENING, the giving a greater degree of hardness
to bodies than they had before. There are several ways
4L2

H A R

[ 688 ]

H A R

of hardening iron and steel ; as by hammering them,
quenching them in cold water, &c.

Case-HARDENING, is a superficial conversion of iron
into steel, as if it were casing it, or covering it with a thin
coat of harder matter. It is thus performed: Take cow-
horn or hoof, vvell'dried in an oven, and beat it to powder;
put equal quantities of this powder and of bay-salt into
stale urine, or white-wine vinegar, and mix them well to-
gether; cover the iron or steel with this mixture, and
wrap it up in loam, or plate iron, so as the mixture may
touch every part of the work ; then put it in the fire, and
blow the coals, till the whole lump has acquired a blood-
red heat, but no higher; lastly, take it out, and quench
it. — See Steel, under which article are described other
processes for this purpose.

HARDING, a small new primary planet, called also
Juno, discovered by the astronomer M. Harding.

HARDNESS, or Rigidity, that quality in bodies by
which their parts so cohere as not to yield inward, or give
way to an external impulse, without instantly going be-
yond the distance of their mutual attraction ; and there-
fore are not subject to any motion in respect of each other,
■without breaking the body. There were many fanciful
opinions among the ancients, concerning the cause of hard-
ness ; such as heat, cold, dryness, the hooked figure of the
particles of matter. The Cartesians make the hardness of
bodies to consist in rest, as that of soft and fluid ones in
the motion of their particles.

Newton shows that the primary particles of all bodies,
whether solid or fluid, are perfectly hard ; and are not ca-
pable of being broken or divided by any power in nature.
These particles, he maintains, are connected and kept to-
gether by an attractive power; and according to the cir-
cumstances of this attraction, the body is either hard, or
soft, or even fluid. If the particulars be so disposed or
fitted for each other, as to touch in large surfaces, the
body will be hard ; and the more so as those surfaces are
the larger. If, on the contrary, they only touch in small
surfaces, the body, by the weakness of the attraction, will
remain soft. At present, many philosophers are of opi-
nion, that hardness consists in the absence or want of the
action of the universal fluid, or elementary fire, among
the particles of the bod}', or a deficiency of what is called
latent heat; while on the contrary, fluidity, according to
them, consists in the motion of the particles, in conse-
quence of the action of that elementary fire. *■

Hardness appears to diminish the cohesion of bodies, in
some degree, though their frangibility or brittleness does
not by any means keep pace with their hardness. Thus,
though glass be very hard and very brittle; yet flint is still
harder, though less brittle. Among the metals, these two
properties seem to be more connected, though even here
the connexion is by no means complete : for though steel
be both the hardest and most brittle of all the metals;
yet lead, which is the softest, is not the most ductile.
Neither is hardness connected with the specific gravity of
bodies ; for a diamond, the hardest substance in nature, is
little more than half the weight of the lightest metal. And
as little is it connected with the. coldness, or electrical
properties, or any other quality with which we are ac-
quainted. Some bodies are rendered hard by cold, and
others by different degrees of heat.

Mr. Quist and others have constructed tables of the
hardness of different, substances. And the manner of con-
structing these tables, was by observing the order in which

they were able to cut or make any impression upon one
another. The following table, extracted from Magellan's
edition of Cronstedt's Mineralogy, was taken from Quist,
Bergman, and Kirwan. The first column shows the hard-
ness, and the second the specific gravity.

Hard-

Spec.

ness.

Grav.

Diamond from Ormus

20

3-7

Pink diamond

19

3-4

Blueish diamond

19

3-3

Yellowish diamond

19

33

Cubic diamond

IS

3'2

Ruby

17

4"2

Pale blue sapphire

17

3-S

Pale ruby from Brazil

16

35

Deep blue sapphire

16'

3-8

Topaz

15

4-2

Whitish ditto

14

3-5

Ruby spincll

13

3-4

Emerald

12

2-8

Garnet

12

4-4

Agate

12

2-6

Onyx

12

2-6

Sardonyx

12

2-6

Bohemian topaz

11

2'8

Occid. amethyst

11

2-7

Crystal

11

2-6

Carnelian

11

2-7

Green jasper

11

2-7

Schoerl

10

3-6

Tourmaline

10

30

Quartz

10

2-7

Opal -

10

2-6

Chrysolite

10

37

Reddish yellow jasper

9

2-6

Zeolyte

8

2-1

Fluor

7

3-5

Calcareous spar

6

2-7

Gypsum

5

2-3

Chalk

3

2-7

HARMONICA, Harmonics, a branch or division of
the ancient music; being that part which considers the
differences and proportions of sounds, with respect to acute
and grave; as distinguished from rhythmica, and metrica.
Mr. Malcolm has made a very industrious and learned
inquiry into the Harmonica, or harmonic principles of
the ancients.

Harmonica, the name of a musical instrument in-
vented by Dr. Franklin, consisting of the glasses, called
musical glasses. — We are informed that the first hint of
musical glasses is to be found in an old English book, in
which a number of various amusements were described.
That author directs his pupil to choose half a dozen drink-
ing-glasses; to fill each of them with water in proportion
to the gravity or acuteness of the sound which it is in-
tended to give; and having thus adjusted them to each
other, he might entertain the company with a church
tunc. These were perhaps the rude hints which Mr. Puck-
eridge, an Irish gentleman, afterwards improved, and after
him, Mr. E. Delaval, an ingenious member of the Royal
Society; and finally brought to perfection by ihecelebrated
Franklin, See the history and description in his Letters,
particularly in that to Beccaria.

HARMON1CAL, or Harmonic, something relating
to Harmony. Thus,

H A R

[ 629 J

H A R

Harmonical Arithmetic, is so much of the theory
and doctrine of numbers, as relates to making the compa-
risons and reductions of musical intervals, which are ex-
pressed by numbers, for finding their mutual relations,
compositions, and resolutions.

Harmonical Composition, in its general sense, includes
the composition both of harmony and melody; i. e. of
music, or song, both in a single part, and in several parts.

Harmonical Interval, the difference between two
sounds, in respect of grave and acute: or that imaginary
space terminated with two sounds differing in acuteness or
gravity.

HARMONICAL Curve, an imaginary kind of curve,
into which it has been supposed that a musical chord
would be inflected when put into such a vibratory motion
as excited sound. It has been asserted by Dr. Brook
Taylor, and others, that if a chord be inflected into any
other form than that of the harmonic curve, it will, since
those parts which are without this figure are impelled to-
wards it, by an excess of force, and those within it by a
deficiency, in a very short time arrive at, or very near,
the form of this precise curve. But, as Dr. T. Young
very properly remarks, " it would be easy to prove, if this
reasoning were allowed, that the form of the curve can be
no other than that of the axis, since the tending force is
continually impelling the chord towards this line." Philos.
Trans. 1800, pt. 1. — In theory, a musical chord is con-
ceived to be a mathematical line, flexible by the least
force, and elastic: and when such chord, at the limit of
its vibration, assumes the form of this harmonical curve,
it will vibrate to and fro in curves of the same species. —
According to the investigation of Dr. Smith, the figure con-
tained under the harmonical curve and its base, is of the
same species as the figure of sines. From the investigation
it follows : 1. That the force by which any small particle
of an elastic string is impelled towards the centre of its
curvature, is to the tending force, as the length of the par-
ticle, is to the radius of curvature. 2. The perpendicular
distance of a particle from the string in its quiescent posi-
tion, is to the string's length, as the same length, is to the
radius of curvature into the square of 3*14159. Hence,
if any perpendicular distance dii be put = d, the string's
length=«, and 3' 14 159 = c, then will the radius of curva-
ture d = -pr. 3. Each particle of the string will arrive

at the axis at the same instant of time. But it is all along
supposed that the greatest distance of the string from the
axis, bears no sensible proportion to its length. See
Smith's Harmon, pa. 248, &c; also Notes to the Jesuits'
Newton, torn. 2, pa. 348.

Harmonical Proportion, or Musical Proportion, is
that in which the first term is to the third, as the difference
of the first and second is to the difference of the 2d and
3d; or when the first, the third, and the said two diffe-
rences, are in geometrical proportion. Or, four terms are
in harmonical proportion, when the 1st is to the 4th, as
the difference of the 1st and 2d is to the difference of the
3d and 4th. Thus, 2, 3, 6, are in harmonical proportion,
because 2 : 6 : : 1 : 3. And the four terms 9, 12, l6, 24
are in harmonical proportion, because 9' 24:: 3: 8 — If
the proportional terms be continued in the former case,
they will form an harmonical progression, or series.

1. The reciprocals of an arithmetical progression are in
harmonical progression; and, conversely, the reciprocals

of the harmonicals 2, 3, 6, are f, f, £, which are arith-
meticals; for £• — f = £, arid f - •§. == £ also : and the
reciprocals of the arithmeticals 1, 2, 3, 4, &c, are 4, \, |,
~, &c, which are harmonicals; for-f: ■§■ : : ^- — -'4:f— i;
and so on. And, in general, the reciprocals of the arith-
meticals a, a -+- d, a -t- 2d, a -+- 3d, &c, viz,

3d '

&c, are harmonicals; et e

a' a + d' a + Id'
contra.

2. If three or four numbers in harmonical proportion
be either multiplied or divided by some number, the pro-
ducts, or the quotients, will still be in harmonical propor-
tion. Thus,

the harmonicals - 6, 8, 12,

multiplied by 2 give - 12, 16', 24,

or divided by 2 give • - 3, 4, 6,

which are also harmonicals.

3. To find a harmonical mean proportional between two-
terms : divide double their product by their sum.

4. To find a 3d term in harmonical proportion to two
given terms: divide their product by the difference be-
tween double the 1st term and the 2d term.

5. To find a 4th term in harmonical proportion to three
terms given; divide the product of the 1st and 3d by the
difference between double the 1st and the 2d term.

Hence, of the two terms 6 and 3,
the harmonical mean is 4,
the 3d harmonical proportion is 2,
also to 6, 4, 3, the 4th harmonical is 2.

6. If there be taken an arithmetical mean, and a har-
monical mean, between any two. terms, the four terms will
be in geometrical proportion. Thus, between 0 and 2,

the arithmetical mean is 4, and
the harmonical mean is 3;
and hence 6' : 4 : : 3 : 2.

7- Also the arithmetical mean, the geometrical mean,
and the harmonical mean, between any two terms, are
three terms in continued geometrical progression. So,
between 6 and 2, the arithmetical mean is 4, the geometri-
cal mean is >/ 12, and the harmonical mean is 3 ; then 4 :
■v/12 :: v/12 : 3.

HARMONY, in Music, the agreeable result of an union
of several musical sounds, heard at one and the same time;
or the mixture of divers sounds, which together have an
agreeable effect on the ear. — As a 'continued succession of
musical sounds produces melody, so a continued combina-
tion of them produces harmony. Among the ancients,
however, as sometimes also among the moderns, harmony
is used in the strict sense of consonance; and so is equi-
valent to the symphony.

The words concord and harmony do really signify the
same thing ; though custom has made a little difference
between them. Concord is the agreeable effect of two
sounds in consonance ; harmony the effect of any greater
number of agreeable sounds in consonance. Again, har-
mony always implies consonance ; but concord is also ap-
plied to sounds in succession; though never but where the
terms can stand agreeably in consonance. The effect of an
agreeable succession of several sounds, is called melody; as
that of an agreeable consonance is called harmony.

Harmony is well defined, the sum or result of the com-
bination of two or more concords; that is, of three or more
simple sounds striking the ear altogether; and different
compositions of concords constitute various degrees of

of harmonicals are arithmeticals. Thus, the reciprocals harmony. The ancients seem to have been entirely un,-.

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acquainted with harmony, the soul of the modern music.
In all their explications of the melopoeia, the}1 say not one
word of the concert or harmony of parts. We have in-
stances, indeed, of their joining several voices, or instru-
ments, in consonance; but then these were not so joined,
as that each had a distinct and proper melody, so making
a succession of various concords ; but they were either
unisons, or octaves, in every note; and so all performed
the same individual melody, and constituted one song.
When the parts differ, not in the tension of the whole, but
in the different relations of the successive notes, it is this
that constitutes the modern art of harmony.

Harmony of the Spheres, or Celestial Harmony, a kind
of music much spoken of by many of the ancient philo-
sophers and fathers, supposed to be produced by the
sweetly tuned motions of the stars and planets. This
harmony they attributed to the various proportionate im-
pressions of the heavenly bodies on one another, acting at
proper intervals. They think it impossible that such
prodigious bodies, moving with such rapidity, should be
silent: on the contrary, the atmosphere, continually im-
pelled by them, must yield a set of sounds proportionate
to the impression it receives; and that consequently, as
they run all in different circuits, and with various degrees
of velocity, the different tones arisingfrom the diversity of
motions, directed by the hand of the Almighty, must form
an agreeable symphony or concert. They therefore sup-
posed, that the moon, as being the lowest of the planets,
corresponded to mi; Mercury, to fa; Venus, to sol; the
Sun, to la; Mars, to si; Jupiter, to ut ; Saturn, to re;
and the orb of the fixed stars, as being the highest of all, to
mi, or the octave.

It is thought that Pythagoras had a view to the gravita-
tion of celestial bodies, in what he taught concerning the
harmony of the spheres. A musical chord gives the same
note as one double in length, when the tension or force
with which the latter is stretched is quadruple; and the
gravity of a planet L quadruple of the gravity of a planet
at a double distance. In general, that any musical chord
may become unison to a lesser chord of the same kind, its
tension must be increased in the same proportion as the
square of its length is greater; and that the gravity of a
planet may become equal to the gravity of another planet
nearer the sun, it must be increased in proportion as the
square of its distance from the sun is greater. If there-
fore we should suppose musical chords extended from the
sun to each planet, that all these chords might become
unison, it would be requisite to increase or diminish their
tensions in the same, proportions as would be sufficient to
render the gravities of the planets equal ; and from the
similitude of those proportions, the celebrated doctrine of
the harmony of the spheres is supposed to have been de-
rived.

Kepler wrote a large work, in folio, on the harmonies of
the world, and particularly of that of the celestial bodies.
He first endeavoured to iind out some relation between
the dimensions of the five regular solids, and the intervals
of the planetary spheres; and imagining that a cube, in-
scribed in the sphere of Saturn, would touch by its six
planes the sphere of Jupiter, and that the other four re-
gular solids in like manner lifted the intervals that are be-
tween the spheres of Lhe other planets, he became per-
suaded that this was the true reason why the primary
planets were precisely six in number, and that the author
of the world had determined their distances from the sun,

the centre of the system, from a regard to this analogy.
But afterwards finding that the disposition of the five re-
gular solids among the planetary spheres, was not agree-
able to the intervals between their orbits, he endeavoured
to discover other schemes of harmony. For this purpose,
he compared the motions of the same planet at its great-
est and least distances, and of the different planets in their
several orbits, as they would appear viewed from the sun;
and here he fancied that he found a similitude to the di-
visions of the octave in music. Lastly, he imagined that
if lines were drawn from the earth, to each of the planets,
and the planets appended to them, or stretched by weights
proportional to the planets, these lines would then sound
all the notes in the octave of a musical chord. See his
Harmonics; also Plin. lib. 2, cap. 22; Macrob. in Somn.
Scip. lib. 2, cap 1; Plutarch, de Animal. Procreatione, e
Tiimeo; and Maclaurin's View of Newton's Discov.
book 1, chap. 2.

HARQUEBUSS, a hand-gun, or a fire-arm of a proper
length and weight to be borne in the arm. Hanzelet pre-
scribes its proper length to be 40 calibres, or diameters of
its bore ; and the weight of its ball 1 oz. and f ; its charge
of powder as much.

HARRIOT (Thomas), a very eminent English mathe-
matician and astronomer, was born at Oxford in 1560,
and died at London July 2, 1621, in the fctlst year of his
age. Harriot has hitherto been known to the world only
as an algebraist, though a very eminent one ; but from his
manuscript papers, that have been but lately discovered
by Dr. Zach, astronomer to the duke of Saxe-Gotha, it
appears that he was not less eminent as an astronomer and
geometrician. Dr. Zach has printed an account of those
papers, in the Astronomical Ephemeris of the Royal Aca-
demy of Sciences at Berlin, for the year 17S8; of which,
as it is very curious, and contains a great deal of infor-
mation, I shall here give a translation, to serve as me-
moirs concerning the life and writings of this eminent
man ; afterwards adding only some necessary remarks of
my own.

" I here present to the world," says Dr. Zach, " a short
account of some valuable and curious manuscripts, which
I found in the year 17S4, at the seat of the earl of Egre-
mont, at Petworth in Sussex, in hopes that this learned
and inquisitive age will either think my endeavours about
them worthy of its assistance, or else will be thereby in-
duced to attempt some other means of publishing them.
The only undeniable proof I can now produce, of the use-
fulness of such an undertaking, is by giving a succinct re-
port of the contents of these materials, and briefly showing
what may be effected by them. And though I come to lhe
performance of such an enterprise with much less abilities
than the different parts of it require, yet I trust that my
love for truth, my design and zeal to vindicate the honour
due to an Englishman, the author of these manuscripts,
which arc the chief reasons that have influenced me in this
undertaking, will serve as my excuse.

" A predecessor of the family of Lord Egrernont, viz,
that noble and generous earl of Northumberland, named
Henry Percy, was not only a generous favourer of all good
learning, but also a patron and Maecenas of the literati of
his age. Thomas Harriot, the author of the said manu-
scripts, Robert Hues (well known by his Treatise upon the
Globes), and Walter Warner, all three eminent mathema-
ticians, who were known to the carl, received from him
yearly pensions ; so that when the carl was committed pri-

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r 63* 3

H A R

soner to tlje Tower of London in the year \606, our author,
with Hues and Warner, were his constant companions ; and
were usually called the earl of Northumberland's three
Magi.

" Thomas Harriot is a known and celebrated mathe-
matician among the learned of all nations, by his excellent
work, Artis Analyticae Praxis, ad ajquationes algebraicas
nova expedita et generali methodo, resolvendas, Tractatus
posthumus; Lond. l63i : dedicated to Henry earl of
Northumberland; published after his death by Walter
Warner. It is remarkable, that the fame and the honour
of this truly- great man were constantly attacked by the
French mathematicians; who could not endure that Har-
riot should in any way diminish the fame of their Vieta and
Descartes, especially the latter, who was openly accused
of plagiarism from our author. [See Montucla's Histoire
des Mathematiques, part 3, pa. 4S5 et seq. — Lettres de
M. Descartes, torn. 3, pa. 457, edit. Paris 1667, in 4to. —
Dictionnaire de Morreri, word Harriot. — Encyclopedie,
word Algebra. — Lettres de M. de Voltaire, sur la nation
Angloise, lettie 14. — Memoire de l'Abbe de Guadans les
Mem. de lAcad. des Sciences de Paris pour 1741. — Jer.
Collier's great Historical Dictionary, word Harriot. — Dr.
Wallis's preface to his Algebra. — To which' may be added
the article Algebra, in this dictionary.]

" Descartes published his Geometry 6 years after Har-
riot's work appeared, viz, in the year 1637- Sir Charles
Cavendish, then ambassador at the French court at Paris,
when Descartes's Geometry made its first appearance in
public, observed to the famous geometrician Roberval,
that these improvements in analysis had been already
made these 6 years in England, and showed him after-
wards Harriot's Artis Analyticae Praxis; which, as Ro-
berval was looking over, at every page he cried out, Oui !
oui ! il l'a vu ! Yes ! yes ! he has seen it ! Descartes had
also been in England before Harriot's death, and had heard
of his new improvements and inventions in analysis. A
critical life of this man, which his papers would enable me
to publish, will show more clearly what to think upon this
matter, which I hope may be discussed to the due honour
of our author.

" Now all this relates to Harriot the celebrated ana-
lyst; but it has not hitherto been known that Harriot was
an eminent astronomer, both theoretical and practical,
which first appears by these manuscripts; among which
the most remarkable are 199 observations of the Sun's
Spots, with their drawings, calculations, and determina-
tions of the sun's rotation about his axis. There is the
greatest probability that Harriot was the first discoverer
of these spots, even before either Galileo or Scheiner.
The earliest intelligence we have of the first discovered
solar srjots, is of one Joh. Fabricius Phrysius, who in the
year 16*11 published at Wittemberg a small treatise, in-
titled, De Maculis in Sole observatis et apparente corum
cum Sole conversione narratio. Galileo, who is com-
monly accounted the first discoverer of the solar spots,
published his book, Istoria e Dimonstrazioni intornc alle
Machic Solare eloroaccidenti,at Rome, in the year I6"l3.
His first observation in this work, is dated June 2, lf)12.
Angelo de Filiis, the editor of Galileo's work, who wrote
the dedication and preface to it, mentions, pa. 3, that Ga-
lileo had not only discovered these spots in the month of
April in the year l6ll, at Rome, in the Quirinal Garden,
but had shown them several months before (molti mesi
innanzi) to his friends in Florence; and that the obser-

vations of the disguised Apelles (the Jesuit Scheiner, a pre-
tender to this first discovery) were not later than the
month of October in the same year; by which the epoch
of this discovery was fixed to the beginning of the year

1611. But.a passage in the first letter of Galileo's works,
pa. 11, gives a more precise term to this discovery. Ga-
lileo there says in plain terms, that he had observed the
spots in the sun 18 months before. The date of this letter
is May 24, l6l2; which brings the true epoch of this
discovery to. the month of November, lfilO. However,
Galileo's first produced observations are only from June 2,

1612, and those of Father Scheiner of the month of Oc-
tober, in the same year. But now it appears from Har-
riot's manuscripts, that his first observations of these spots
are of Dec. 8, l6l0. It is not likely that Harriot could
have this notice from Galileo, for I do not find this mathe-
matician's name ever quoted in Harriot's papers. But I
find him mentioning Josephus a Costa's book 1, chap. 2, of
his Natural and Moral History of the West Indies, in
which he relates that in Peru there are spots to be seen in
the sun which are not to be seen in Europe. It rather
seems that Harriot had taken the hint from thence. Be-
sides, it is very likely that Harriot, who lived with so ge-
nerous a patron to all good learning and improvements,
had got the new invention of telescopes in Holland much
sooner in England, than they could reach Galileo, who at
that time lived at Venice. Harriot's very careful and ex-
act observations of these spots, show also that he was in
possession of the best and most improved telescopes of that
time; for it appears he had some with magnifying powers
of 10, 20, and 30 times. At least there are no earlier ob-
servations of the solar spots extant than his : they run
from December 8, l6T0, till January 18, lfjlo. I com-
pared the corresponding ones with these observed by Ga-
lileo, between which I found an exact agreement. Had
Harriot had any notion about Galileo's discoveries, he cer-
tain!)' would have also known something about the phases
of Venus and Mercury, and especially about the singular
shape of Saturn, first discovered by Galileo; but I find not
a word in all his papers concerning the particular figure
of that planet.

" Of Jupiter's Satellites. I found among his papers a
large set of observations, with their drawing, position,
and calculations of their revolutions and periods. His
first observation of those discovered satellites, I find to
be of January IfJ, lfjlO; and they go till February 26,
l6l2. Galileo pretends to have discovered them Ja-
nuary 7, 16'10; so that it is not improbable that Harriot
was likewise the first discoverer of these attendants of Ju-
piter.

" Among his other observations of the moon, of some
eclipses, of the planet Mars, of solstices, of refraction,
of the declination of the needle; there are most remark-
able ones of the noted comets of 1607, and of 161s the
latter, for there were two this year (see Kepler de Cometis,
pa. 49). They were all observed with a cross-staff, by
measuring their distances from fixed stars ; whence these
observations are the more valuable, as comets had before
been but loosely attended to : Kepler himself observed the
comet of 1607 only with the naked eye, pointing out its
place by a rough estimation, without the aid of an instru-
ment ; and the elements of their orbits could, in defect of
better observations, be only calculated by them. The
observations of the comet of the year 1607, are of the
more importance, even now for modern astronomy, as

H A R

this is the same comet that fulfilled Dr. Halley's prediction
of its return in the year 1759- That prediction was only
grounded on the elements afforded him by these loose ob-
servations ; for which reason he only assigned the term of
its return to the space of a year. The very intricate calcu-
lations of the perturbations of this comet, afterwards
made by M. Clairaut, reduced the limits to a month's
space. But a greater light may now be thrown upon this
matter by the "more accurate observations on this comet
by Mr. Harriot. In the month of October 17S5, when I
conversed on the subject of Harriot's papers, and especially
on this comet, with the very celebrated mathematician
M. Lagrange, director of the Royal Academy of Sciences
at Berlin, he then suggested to me an idea, which, if
brought into execution, will clear up an important point
in astronomy. It is well known to astronomers how dif-
ficult a matter it is, to determine the mass, or quantity of
matter, in the planet Saturn ; and how little satisfactory
the notions of it are, that have hitherto been formed.
The whole theory of the perturbations of comets depend-
ing on this uncertain datum, several attempts and trials
have been made towards a more exact determination of it
by the most eminent geometricians of this age, and parti-
larly by Lagrange himself ; but never having been satisfied
with the few and uncertain data heretofore obtained for
the resolution of this problem, he thought that Harriot's
observations on the comet of l607, and the modern ones
of the same comet in 1759, would suggest a way of re-
solving the problem a posteriori ; that of determining by
them the elements of its ellipsis. The retardation of the
comet compared to its period, may clearly be laid to the
account of the attraction and perturbation it has suffered
in the region of Jupiter and Saturn ; and as the part of it
belonging to Jupiter is very well known, the remainder
must be the share which is due to Saturn; from whence
the mass of the latter may be inferred. In consequence
of this consideration I have already begun to reduce most
of Harriot's observations of this comet, in order to calcu-
late by them the true elements of its orbit on an elliptical
hypothesis, to complete M. Lagrange's idea upon this
matter.

" I forbear to mention here any more of Harriot's ana-
lytical papers, which 1 found in a very great number.
They contain several elegant solutions of quadratic, cubic,
and biquadratic equations; with some other solutions and
locageometrica, that show his eminent qualifications, and
will serve to vindicate them against the attacks of several
French writers, who refuse him the justice due to his skill
and accomplishments, merely to save Descartes's honour,
who yet, by some impartial men of his own nation, was
accused of public plagiarism.

" Thomas Harriot was born at Oxford, in the year
1560. After being instructed in the rudiments of lan-
guages, he became a commoner of St. Mary's-Hall, where
he took the degree of bachelor of arts in 1579. He had
then so distinguished himself by his uncommon skill in
mathematics, as to be recommended soon after to Sir
Walter Raleigh, as a proper preceptor to him in that
science. Accordingly that noble knight became his first
patron, took him into his family, and allowed him a hand-
some pension. In 1584 he went over with Sir Walter's
first colony to Virginia; where he was employed in disco-
vering and surveying the country, &c ; maps of which I
have found (says Dr.Zach) very neatly executed among
his papers. After his return he published A Brief and

f 632 J H A R

True Report of the Newfoundland of Virginia, of the Com-
modities there found to be raised, &c ; Lond. 15SS. This
was reprinted in the 3d volume of Hakluyt's Voyages : it
was also translated into Latin, and printed at Frankfort in
the year 1590. Sir Walter introduced him to the ac-
quaintance of the Earl of Northumberland, who allowed
him a yearly pension ; Wood says, of 120/. only ; but by
some of his receipts, which I have found among his papers,
it appears that he had 300/, which indeed was a very
larne sum at that time. Wood, in his Athen. Oxon. men-
lions nothing of Harriot's papers, except a manuscript in
the library at Sion College, London, entitled Ephemeris
Chyrometrica. I got access to this library and manu-
scripts, and was indeed in hopes of finding something
more of Harriot's; for most of his observations are dated
from Sion College ; but I could not find any thing from
Harriot himself. 1 found indeed some other papers of his
friends: he mentions, in his observations, one Mr. Standish,
at Oxford, and Nathaniel Torperly, who also was of the
acquaintance of the Earl of Northumberland, and had a
yearly pension: from the former I found two observations
of the same comet of l6lS, made in Oxford, which he
communicated to Mr. Harriot.

" Thomas Harriot died July 2, l621. His disease was
a cancerous ulcer in the lip, which some pretend he got
by a custom he had of holding the mathematical brass
instruments, when working, in his mouth. I found se-
veral of his letters, and answers to them, from his physi-
cian Dr. Alexander Rhead, who in his treatise mentions
Harriot's disease. His body was conveyed to St. Christo-
pher's church, in London. Over his grave was soon after
erected a monument, with a large and suitable inscrip-
tion, which was destroyed with the church itself by the
dreadful fire of September 1666. He was but 6"0 years
of age."

The peculiar nature and merits of Harriot's Algebra,
we have spoken largely and particularly of, under the ar-
ticle Algebra. As to his manuscripts lately discovered
by Dr.Zach, as above mentioned, it is with pleasure I
can announce, that they are in a fair train to be published :
they have been presented to the university of Oxford, on
condition of their printing them; with a view to which,
they have been lately put into the hands of an ingeni-
ous member of that learned body, to arrange and prepare
them for the press. See Wood's Athena; Oxoniensis, vol. 1.
pa. 390.

HARRIS (John), d. d. and f. n. s. was elected into
the Royal Society about the year 1696* : became one of
the secretaries in 1709, and died Sept. 7, 1719- Dr.
Harris published the first real dictionary of the arts
and sciences, under the title of Lexicon Technicum,
in folio, l6'99, showing the state of the sciences at that
time, a good account of which is given in the Philos.
Trans, for 1704, and in my Abridg. vol.5, pa. 149. —
Some years after, a 2d volume was added, in a new
alphabet, to include the new improvements in science.
A 3d volume was afterwards added, but supposed to be a
compilation by the booksellers. The whole was a very
useful work, and went through several editions. He had
some papers inserted in the Philos. Trans, and published
several other useful books: as, a neat tract on Algebra,
8yo, 1705 ; and another on Geometry and Plane Trigo-
nometry, being chielly a translation from the French of
Pardies, the 2d edit. 1702.

About this time it seems that Dr. Harris " lived and

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taught mathematics at his house in Amen Corner." He
wrote also, A New short Treatise of Algebra ; with the
Geometrical Construction of Equations, as far as the 4th
power or Dimension: together with a specimen of the
nature and algorith of Fluxions.' This tract is an-
nounced to have been written primarily for the use of his
auditors at the public mathematical lecture set up at the
Marine Coffee-house, Birchin-lane, by. Charles Cox, Esq.
m.p. for Southwark. He also published Elements of
Plain and Spherical Trigonometry, together with the prin-
ciples of Spheric Geometry, and the several projections
of the Sphere in Piano. Also The Description and
Uses of the Celestial and Terrestrial Globes, and of Col-
lins's Pocket Quadrant, 4th edition. He was also the
author of a Collection of Voyages, 2 vols, folj 1705 ;
of which useful work there have been; some other edi-
tions, particularly one in 1744, much improved by Dr.
John Campbell.

Dr. Harris appears to have been a patron of the mathe-
matical Wm. Jones, Esq. the friend of Sir I. Newton, and
father of the late learned Sir Wm. Jones, the celebrated
Indian judge, as Mr. Jones dedicated to the Rev. Mr. John
Harris his Treatise on Navigation, published in 1702,
in whose house Mr. Jones says he composed his said
book.

HARRISON (John), a most accurate mechanic, the
celebrated inventor of the famous time-keeper for ascer-
taining the longitude at sea, and also of the compound or
gridiron pendulum ; was born at Foulby, near Pontefract
in Yorkshire, in 1693. His father was a carpenter, in
which profession our author assisted; occasionally also,
according to the miscellaneous practice of country artists,
surveying land, and repairing clocks and watches; and
young Harrison always was, from his earliest years, greatly
attached to any machinery moving by wheels. In 1700
he removed with his father to Barrow in Lincolnshire ;
where, though his opportunities of acquiring knowledge
were very few, he eagerly improved every incident from
which he might collect information ; frequently employ-
ing all or great part of his nights in writing or drawing:
and he always acknowledged his obligations to a clergy-
man who came every Sunday to officiate in the neighbour-
hood, who lent him a ms. copy of Professor Sanderson's
lectures ; which he carefully and neatly transcribed, with
all the diagrams. His native genius exerted itself supe-
rior to these solitary disadvantages ; for in the vear 1726,
he had constructed two clocks, chiefly of wood, in which
he applied the escapement and compound pendulum of
his own invention : these surpassed every thing of the kind
then made, scarcely erring a second in a month. In 1728,
he came up to London with the drawings of a machine for
determining the longitude at sea, in expectation of being
enabled to execute one by the Board of Longitude. On
application to Dr. Halley, the astronomer-royal, he refer-
red him to Mr. George Graham, who advised him to make
his machine, before applying to that board. Accordingly
our author returned home to perform his task; and in
1735 revisited London with his first machine; with which
he was sent to Lisbon the next year, to make trial of it.
In this short voyage, he corrected the dead reckoning
about a degree and a half; a success which procured him
both public and private encouragement. About the year
1739 he completed his second machine, of a construction
much more simple than the former, and which answered
much better : this, though not sent to sea, recommended
Vol. I.

Mr. Harrison yet stronger to the patronage of his friends
and the. public. His third machine, which he produced
in 1749, was still less complicated than the second, and
more accurate, as erring only 3 or 4 seconds in a week.
This he conceived to be the ne plus ultra of his attempts ;
but by endeavouring to improve pocket-watches, he found
the principles he applied to surpass his expectations 'so
much, as to encourage him to make his fourth time-
keeper, which is in the form of a pocket-watch, about 6
inches diameter. With this time-keeper his son made
two voyages, the one to Jamaica, and the other to Barba-
does ; in which experiments it corrected the longitude
within the nearest limits required by the act of the 12th
of Queen Anne; and the inventor had therefore, at dif-
ferent times, more than the proposed reward, receiving
from the Board of Longitude at different times almost.
24,000/. besides a few hundreds from the East India Com-
pany, &c. These four machines were given up to the
Board of Longitude. The three former were not of any
use, as all the advantages gained by making them were
comprehended in the last: being worthy however of pre-
servation, as mechanical curiosities, they are deposited
in the Royal Observatory at Greenwich. The fourth
machine, emphatically distinguished by the name of the
time-keeper, was copied by the ingenious Mr. Kendal ;
and that duplicate, during a three-years circumnavigation
of the globe in the southern hemisphere by Captain Cook,
answered as well as the original.

The latter pa-rt of Mr. Harrison's life was employed in
making a fifth improved time-keeper, on the same princi-
ples with the preceding one; which, after a ten-weeks
trial, in 1772, at the king's private observatory at Rich-
mond, was found to have erred only 4J seconds. Within
a few years of his death, his constitution visibly declined ;
and he had frequent fits of the gout, a disorder that ne-ver
attacked him before his 77th year. His constitution at
last yielding to the infirmities of old age, he died at his
house in Red-Lion Square, in 1776, at 83 years of age.

Our author, like many other mere mechanics, found much
difficulty in delivering his sentiments in writing (at least
in the latter periods of his life, when his faculties were
much impaired), in which he adhered to a peculiar and
uncouth phraseology. This was but too evident in his De-
scription concerning such Mechanism as will afford a nice
or true Mensuration of Time, &c. 8vo, 1775. In this small
work is included also an account of his new musical
scale; beinga mechanical division of the octave, according
to the proportion which the radius and diameter of the
circle have respectively to the circumference. He had in
his youth been the leader of a band of church-singers ; had
a very delicate ear for music; and his experiments on
sound, with a curious monochord of his own improvement,
it has been said were not less accurate than those he was
engaged in for the mensuration of time.

HARTSOEKER (Nicholas), a Dutch philosopher,
born at Gouda, in 1656'. After receiving a liberal edu-
cation under his father, who was a minister under the re-
monstrants, he applied himself assiduously to mathema-
tics and natural philosophy, and became so eminent that
Peter the Great invited him to Moscow, but he declined
the honour. He became professor of philosophy at Hei-
delberg, and mathematician to the Elector Palatine. He
died in 1725. Hartsoeker wrote a Course of Natural Phi-
losophy, in 4to; and some small pieces. '

HAUTEFEUILLE (John), an ingenious mechanic,
4 M

H E A

[ 634 ]

H E A

born at Orleans in 1674. He made a great progress in
mechanics in general, but had a particular taste lor clock-
work, and made several discoveries in it that were of sin-
gular use. It was he, it appears, who determined the me-
thod of moderating the vibration of the balance by means
of a small steel spring, which has since been used. This
discovery he laid before the members of the Academy of
Sciences in lu~94; and these watches are, by way of emi-
nence, called pendulum watches; not that they have real
pendulums, but because they nearly approach to the just-
ness of pendulums. Huygens has the merit of perfecting
this happy invention: but having declared himself the in-
ventor and obtained a patent for making watches with
spiral springs, the abbe Feuille opposed the registering of
it, and published apiece on the subject against Huygens.
He died in 1724, at 50 years of age. Besides the above,
he wrote a great many other pieces, most of which are
small pamphlets, but very curious : as, 1. His Perpetual
Pendulum. 2. New Inventions. 3. The Art of Breathing
under Water, and the means of preserving a Flame shut
up in a Small Place. 4, Reflections on machines for
raising Water. 5. His Opinion on the different Senti-
ments of Mallebranche and Regis, relating to the Appear-
ance of the Moon when seen in the Horizon. 6. The
Magnetic Balance. 7- A Placet to the king on the Longi-
tude. S. Letter on the Secret of the Longitude. 9. A
New System on the Flux and Reflux of the Sea. 10. The
means of making Sensible Experiments that prove the Mo-
tion of the Earth : and many other pieces.

HAYES (Charles, Esq.), a very singular person, whose
great erudition was so concealed by his modesty, that his
name is known but to few, though his publications are
many. He was born in 1678, and died in 176O, at 82
years of age. He became distinguished in 1704 by a Trea-
tise of Fluxions, in folio, being we believe the first treatise
on that science ever published in the English language;
and the only work to which he ever set his name. In
1710 came out a small 4to pamphlet, in ip pages, enti-
tled, A New and easy Method of determining the Longi-
tude from observing the Altitudes of the Celestial Bodies.
Also, in 1723, he published, The Moon, a philosophical
dialogue; tending to show, that the moon is not an opaque
body, but has native light of her own.

To a skill in the Greek and the Latin, as well as ,the
modern languages, he added the knowledge of the He-
brew : and he published several pieces relating to the trans-
lation and chronology of the Scriptures. During a long
course of years he had the chief management of the late
African company, being annually elected sub-governor.
But on the dissolution of that company in 1752, he re-
tired to Down, in Kent, where he spent most ofthis hours
in study ; from whence however he returned, in 1758, to
chambers in Gray's-inn, London, where he died in 1760,
as mentioned above. He left a posthumous work, that
was published in 8vo, under the title of Chronographia
Asiatica et iEgyptiaca, &c.

HEAD, in Architecture. Heads are used as an orna-
ment of sculpture or carved work, frequently serving as
the key of an arch or platband, and on other occasions.

HEAT, theopposite to cold, being a relative term deno-
ting the property of fire, or of those bodies we denominate
hot ; being in us a sensation excited by the action of fire.

Heat, as it exists in the hot body, or that which con-
stitutes and denominates a body hot, and enables it to
produce such effects on our organs, is variously consider-

ed by philosophers : some making it a quality, others a
substance, and others only a mechanical affection. The
former principle is given and maintained by Aristotle and
the Peripatetics. While the Epicureans, and other corpus-
cularians,' define heat not as an accident of fire, but as an
essential power or property of it, the same in reality with
it, and only distinguished from fire in the manner of our
conception. Heat, therefore, on their principles, is no
other than the volatile substance of fire itself, reduced into
atoms, and emitted in a continual stream from ignited
bodies ; so as not only to warm the objects within its
reach, but also, if they be inflammable, to kindle them,
turn them into fire, and conspire with them to make flame.
In effect, these corpuscles, observe they, flying off from
the ignited body, constitutefire while yet contained within
the sphere of its flame; but when fled, or got beyond the
same, and dispersed everyway, so as to escape the appre-
hension of the eye, and only to be perceived by the feeling,
they take the denomination of heat, inasmuch as they ex-
cite in us that sensation.

The Cartesians, improving on this doctrine, assert that
heat consists in a certain motion of the insensible particles
of a body, resembling the motion by which the several
parts of our body are agitated by the motion of the heart
and blood.

Our latest and best writers of mechanical, experimental,
and chemical philosophy, differ very considerably about
heat. The chief difference is, whether it be a peculiar
property of one certain immutable body, called fire, or
phlogiston, or electricity; or whether it may be produced
mechanically in other bodies, by inducing an alteration in
their particles. The former tenet, which is as ancient as
Democritus, and the system of atoms, had yielded to that
of the Cartesians, and other mechanists ; but is now with
great address retrieved, and improved on, by some of the
latest writers, particularly Homberg, the younger Lemery,
Gravesande, Boerhaave in his lectures on fire, Black,
Crawford, and other chemical philosophers.

That which is denominated fire, according to Boer-
haave, is a body sui generis, created such ab origine, un-
alterable in its nature and properties, and not either pro-
ducible de novo from any other body, nor capable of being
reduced into any other body, or of ceasing to be fire. This
fire, he contends, is diffused equably every where, and ex-
ists alike, or in equal quantity, in all the parts of space,
whether void, or possessed by bodies; but that naturally,
and in itself, it rs perfectly latent and imperceptible; be-
ing only discovered by certain effects which it produces,
and which are cognizable by our senses. These effects are
heat, light, colour, rarefaction, and burning, which are
all indications of fire, as being none of them producible
by any other cause ; so that wherever we observe any of
these, we may safely infer the action and presence of fire.
But though the effect cannot be without the cause, yet the
fire ma)' remain without any of these effects ; any, we
mean, gross enough to affect our senses, or become ob-
jects of them: and this, he adds, is the ordinary case ;
there being a concurrence of other circumstances, which
are often wanting, necessary to the production of such
sensible effects. v

Heat, according to the mechanical philosophers, es-
pecially Bacon, Boyle, and Newton, is considered not as
an original inherent property of any particular kind of
body; but as mechanically producible in any body. The
former, in an express treatise De Forma Calidi, from

H E A

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H E A

a particular enumeration of the several phenomena and
effects of heat, deduces several general properties of it;
and hence he defines heat, an expansive undulatory mo-
tion in the minute particles of the body; by which they
tend, with some rapidity, towards the circumference', and
at the same time inclinea little upwards.

Mr. Boyle, in a treatise on the mechanical origin of
heat and cold, strongly supports the doctrine of the pro-
ducibility of heat, with new observations and experiments ;
as in the instance of a smith briskly hammering a small
piece of iron, which, though cold at the commencement,
soon becomes exceedingly hot.

This system is also further supported by Newton, who
docs not conceive fire as any particular species of body,
originally endued with such and such properties. Fire,
according to him, is only a body much ignited, that is
heated hot, so as to emit light copiously : what else, ob-
serves he, is red-hot iron but fire ? and what else is a burn-
ing charcoal but red-hot wood ? or flame itself, but red-
hot smoke ? It is certain that flame is only the volatile
part of the fuel heated red-hot, i. e. so hot as to shine ; and
hence only such bodies as are volatile, that is, such as
emit a copious fume, will flame; nor will they flame longer
than they have fume to burn. In distilling hot spirits, if
the head of the still be taken off, the ascending vapours
will catch fire from a candle, and turn into a flame. And
in the same mannerseveral bodies, much heated by motion,
attrition, fermentation, or the like, will emit lucid fumes,
-which, if they be copious enough, and the heat sufficiently
great, will be flame; and the reason why fused metals do
not flame, is the smallness of their fume; this is evident,
because spilter, which fumes most copiously, does likewise
flame. Add, that all flaming bodies, as oil, tallow, wax,
wood, pitch, sulphur, &c, by flaming, waste and vanish
into burning smoke. And do not all fixed bodies, when
heated beyond a certain degree, emit light, and shine ? and
is not this emission performed by the vibrating motion of
their parts? and do not all bodies, which abound with ter-
restrial and sulphureous parts, emit light as often as those
parts are sufficiently agitated, whether that agitation be
made by external fire, or by friction, or percussion, or
putrefaction, or by any other cause ? Thus, sea-water,
in a storm; quicksilver agitated in vacuo; the back of a
cat, or the neck of a horse, obliquely rubbed in a dark
place; wood, flesh, and fish, while they putrefy; vapours
from putrefying waters, usually called ignes fatui ; stacks
of moist hay or corn ; glow-worms ; amber and diamonds
by rubbing ; fragments of steel struck off with a flint, &c,
all emit light. Are not gross bodies and light convertible
into each other, and may not bodies receive much of their
activity from the particles of light which enter their com-
position ? I know no body less apt to shine than water ;
and yet water, by frequent distillations, changes into fixed
earth, which, by a sufficient heat, may be brought to shine
like other bodies.

Let it further be considered, that the sun and stars, ac-
cording to Newton's conjecture, are no other than great
earths vehemently heated; for large bodies, he observes,
preserve their heat the longest, their parts heating each
other : and why may not great, dense, and fixed bodies,
when heated beyond a certain degree, emit light so copi-
ously, as by the emission and reaction of it, and the reflec-
tions and refractions of the rays within the pores, to grow
still hotter, till they arrive at that degree of heat which
the sun is of? Their parts also may be further preserved

from fuming away, not only by their fixity, but by the
vast weight and density of their atmospheres incumbent
on them, thus strongly compressing them, and condensing
the vapours and exhalations arising from them. Hence
we see warm water, in an exhausted receiver, to boil as
vehemently as the hottest water exposed to the air; the
weight of the incumbent atmosphere, in this latter case,
keeping down the vapours, and hindering the ebullition,
till it has acquired its utmost degree of heat. So also a
mixture of tin and lead, put on a red-hot iron in vacuo,
emits a fumeand flame ; but the same mixture in the open
air, by reason of the incumbent atmosphere, does not emit
the least sensible flame.

Thus much for the system of the producibility of heat.
On the other hand, M. Homberg, in his Essai du Soufre
Principe, holds, that the chemical principle or element,
sulphur, which is supposed one of the simple, primary,
pre-existent ingredients of all natural bodies, is real fire ;
and consequently that fire is co-eval with body. Mem. de
l'Acad. an. 1705.

Dr. Gravesande goes upon much the same principle.
According to him, fire enters the composition of all bodies,
is contained in all bodies, and may be separated or pro-
cured from all bodies, by rubbing them against each other,
and thus putting their fire in motion. But fire, he adds,
is by no means generated by such motion. Elem. Phys.
torn. 2, cap. 1. Heat, in the hot body, he observes, is an
agitation of the parts of the body, made by means of the
fire contained in it; by such agitation a motion is pro-
duced in our bodies, which excites the idea of heat in our
minds: so that heat, in respect of us, is nothing but that
idea, and in the hot body nothing but motion. If such.
motion expel the fire in right lines, it may give us the idea
oi light ; if in a various and irregular motkm, only that
of heat.

Lemery, the younger, agrees with these two authors,
in asserting this absolute and ingenerable nature of fire;
but he extends it farther. Not contented with confining
it as an element to bodies, he endeavours to show, that it
is equably diffused through all space; that it is present in
all places, even in the void spaces between the bodies, as
well as in the insensible interstices between their parts.
And this las;t sentiment falls in with that of Boerhaave
above delivered. Mem. de l'Acad. an. 1713.

Philosophers have lately distinguished heat into abso-
lute and sensible. By absolute heat, or fire, they mean
that power or element which, when it is in a certain de-
gree, excites in animals the sensation of heat; and by
sensible heat, the same power considered in its relation to
the effects it produces^ thus, two bodies are said to have
equal quantities of sensible heat, when they produce equal
effects upon the mercury in the thermometer; but as bo-
dies of different kinds have different capacities for con-
taining heat, the absolute heat in such bodies will be dif-
ferent, though the sensible heat be the same. Thus, a pound
of water and a pound of antimony, of the same tempera-
ture, have equal sensible heat ; but the former contains a
much greater quantity of absolute heat than the latter.
M. DeLuc has evinced, by a variety of experiments, that
the expansions of mercury between the freezing and boil-
ing points of water, correspond precisely to the quantities
of absolute heat applied, and that its contractions are
proportionable to the diminution of this element within
these limits. And hence it may be inferred, that if the
mercury were to retain its fluid form, its contractions would
4 M 2

H E A

[ 636 ]

H E A

be proportionable to the decrements of the absolute heat,

thouoh the diminution were continued to the point of total

privation. But the comparative quantities of absolute heat,

which are communicated to different bodies, or separated

from them, cannot be determined in a direct manner by the

thermometer.

Some philosophers have apprehended that the quantities

of absolute heat in bodies, are in proportion to their den-
sities. While others, as Boerhaave, imagined that heat is

equally diffused through all bodies, the. densest as well as

the rarest, and therefore that the quantities of heat in bo-
dies, are in proportion to their bulk or magnitude: and, at

his desire, Fahrenheit attempted to determine the fact by
experiment. For this purpose, he took equal quantities
of the same fluid, and gave them different degrees of heat;
then upon mixing them intimately -together, he found that
the temperature of the mixture was a just medium, or

arithmetical mean, between the two. But if this experi- the urine -

ment be made with water and mercury, in the same cir- Dr. Hales, the skin - - -
cumstances, viz, in equal bulks, the result will be dif- .... the.urine ...

ferent, as the temperature of the mixture will not be a Mr. John Hunter, under his tongue

mean between the two, but always nearer to that of the . in his rectum

water than to the quicksilver; so that, when the water is his urethra at 1 inch

affect us with the sense of cold ; thus making the human
species a medium between the hot and cold blooded ani-
mals, or at least the lowest order of the hot blooded.

The heat of the human body, in its natural state, ac-
cording to Dr. Boerhaave, is such as to raise the mercury
in the thermometer to 92° or at most to 94° ; and Dr. Pit-
cairn makes the heat of the human skin the same. Indeed
it is evident that different parts of the human body, and its
different states, as well as the different seasons, will make it
show of different temperatures. Thus, by various experi-
ments at different times, the heat of the human bod\' is
made various b)- the following authors :
Boerhaave and Pitcairn ... - go,0
Amontons --•._."_ 91*92, or 93

Sir Isaac Newton

Fahrenheit and Muschenbroek, the blood

Dr. Martine, the skin ...

97 or

the hotter, the temperature of the mixture is above the
mean, and below it when the water is the colder. And
from experiments of this kind it has been inferred, that the
comparative quantities of the absolute heats of these
fluids, are reciprocally proportional to the changes which
arc produced in their sensible heats, when they are mixed
together at different temperatures: and this fact has been
publicly taught, for several years, by Dr. Black and Dr.
Irvine, in the universities of Edinburgh and Glasgow. This
rule however does not apply to those substances which, in
mixture, excite sensible heat by chemical action.

From the experiments and reasoning employed by Dr.
Crawford, it more fully appears, that the quantities of ab-
solute heat in different bodies, are not as their densities ;
or that equal weights of heterogeneous substances, as air
and water, having the same temperature, may contain un-
equal quantities of absolute heat : he also shows, that if
phlogiston be added to a body, a quantity of the absolute
heat of that body will be extricated ; and if the phlo-
giston be separated again, an equal quantity of heat
will be absorbed. So that heat and phlogiston appear to
be two opposite principles in nature. But this ingenious
writer has not presumed absolutely to decide the question
that has been long agitated, whether heat be a substance
or a quality. — He inclines to the former opinion, however,
and observes, that if wc adopt the opinion, that heat is a
distinct substance, or an element sui generis, the pheno-
mena will be found to admit of a simple and obvious inter-
pretation, and to be perfectly agreeable to the analogy of
nature. See Crawford's Experiments and Observations on
Animal Heat and the Inflammation of Combustible Bo-
dies. Also Mr. Leslie's ingenious treatise upon heat ;
Scheele's Experiments on Air and Fire; Thomson's Che-
mistry, vol. 1; Boerhaave's Chemistry ; Davy's Essays;
Humford's Essays, &c; Herschel's in Phil. Trans. ; several
vols, of the Pliilos. Magazine, and in Nicholson's Journal.
Animal Heat. The heat of animals is very various,
both according to the diversity of their kinds, and the
difference of the seasons : accordingly, zoologists have di-
vided them into hot and cold blooded, reckoning those to
be hot that are near or above our own temperature, and
all Cithers cold whose heat is below ours, and consequently

95i
96
98
99
97
103
97
9Si
92
93

at 2 inches
at 4 inches

the ball of the thermom. at the bulb of 7 „
the urethra j ■'"

For the powers of animals to bear various degrees of
heat, see the Philos. Trans, vol. 65, 68, &c. There is
hardly any subject of philosophical investigation that has
afforded a greater variety of hypotheses, conjectures, and
experiments, than the cause of animal heat. The first
opinion which has very generally obtained, is, that the
heat of animal bodies is owing to the attrition between the
arteries and the blood. All the observations and reason-
ing brought in favour of this opinion however, only show
that the heat and the motion of the arteries are generally
proportional to each other; without showing which is the
cause, and which the effect; or indeed that either is the
cause or effect of the other, since both may be the effects
of some other cause. Dr. Douglas, in his Essay on the
Generation of Heat in animals, ascribes it solely to the
friction of the globules of blood in their circulation through
(he capillary vessels. Another opinion is, that the lungs
are the fountain of heat in the. human body : and this opi-
nion is supported by much the same kind of arguments as
the former, and seemingly to little better purpose.

A third opinion is, that the cause of animal heat is
owing to the action of the solid parts upon one another.
And as the heart and arteries have the most motion, it
has been thought natural to expect that the heat should
be owing to this motion. But even this does not seem very
plausible, from the following considerations: 1st, The
moving parts, however we term them solid, are neither
hard nor dry; which two conditions are absolutely re-
quisite to make them lit to generate heat by attrition. 2d,
None of their motions arc swift enough to promise heat in
this way. 3d, They have but little change of surface in
their attritions. And 4-thly, The moveable fibres have fat,
mucilage,or liquors every way surrounding them, to prevent
their being destroyed, or heated by attrition.

Another cause assigned for the heat of our bodies, is that
process by which our aliment and fluids are perpetually
undergoing some alteration. And this opinion is chiefly
supported by Dr. Stevenson, in the Edinburgh Medical

H E A

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H E A

Essays, vol. 5, art. 77. The late ingenious Dr. Franklin
inclines to this opinion, when he says, that the fluid fire,
as well as the fluid air, is attracted by plants in their
growth, and becomes consolidated with the other materials
of which they are formed, and makes a great part of their
substance ; that when they come to be digested, and to
undergo a kind of fermentation in the vessels, part of the
fire, as well as part of the air, recovers its fluid active state
again, and diffuses itself on the body digesting and sepa-
rating it; &c. Exper. and Obs. on Electricity, pa. 346.

Dr. Mortimer thinks the heat of animals explicable from
the phosphorus and air they contain. Phosphorus exists,
at least in a dormant state, in animal fluids; and it is also
known that they all contain air; it is therefore only ne-
cessary to bring the phosphoreal and aerial particles into
contact, and heat must of consequence be generated ; and
were it not for the quantity of aqueous humours in ani-
mals, fatal accensions would frequently happen. Philos.
Trans. No. 476.

Dr. Black supposes, that animal heat is generated en-
tirely in the lungs, by the action of the air on the principle
of inflammability, and is thence diffused over the rest of
the body by means of the circulation. But Dr. Leslie
urges several arguments against this hypothesis, tending to
show that it is repugnant to the known laws of the animal
machine ; and he advances another hypothesis instead of
it, viz, that the subtile principle by chemists termed phlo-
giston, which enters into the composition of natural bo-
dies, is in consequence of the action of the vascular system
gradually evolved through every part of the animal ma-
chine, and that during this evolution heat is generated.
This opinion, he candidly acknowledges, was first delivered
by Dr. Duncan of Edinburgh; and that something simi-
lar to it is to be found in Dr. Franklin's works, and in a
paper of Dr. Mortimer's in the Philos. Trans.

The last hypothesis we shall mention, is the very plau-
sible one of Dr. Crawford, lately published in his Experi-
mentsand Observations on Animal Heat. This ingenious
gentleman has inferred, from a variety of experiments,
that heat and phlogiston, so far from being connected, as
most philosophers have imagined, act in some measure in
opposition to each other. By the action of heat on bodies,
the force of their attraction of phlogiston is diminished, and
by the action of phlogiston, a part of their absolute heat is
expelled. He has also demonstrated, that atmospherical
air contains a greater quantity of absolute heat than the
air which is expired from the lungs of animals : he makes
the proportion ot.the absolute heat of atmospherical air to
that of fixed air, as 67 to 1 ; and the heat of dephlogisti-
cated air to that of atmospherical air, as 4-6 to 1 ; and
observing that Dr. Priestley has proved, that the power of
this dephlogisticated air in supporting animal life, is 5
t.imes as great as that of atmospherical air, he concludes
that the quantity of absolute heat contained in any kind
of air fit for respiration, is very nearly in proportion to its
purity, or to its power of supporting animal life ; and since
the air exhaled by respiration, is found to contain only
the 67th part of the heat which was contained in the at-
mospherical air, previous to inspiration, it is very reason-
ably inferred, that the latter must necessarily deposit a
very great proportion of its absolute heat in the lungs. Dr.
Crawford has also shown, that the blood which passes
from the lungs to the heart, by the pulmonary vein, con-
tains more, absolute heat than that which passes from the
heart to the lungs, by the pulmonary artery; the abso-

lute heat of florid arterial blood being to that of venous
blood, as Hi to 10: therefore, since the blood which is
returned by the pulmonary vein to the heart has the
quantity of its absolute heat increased, it must have ac-
quired this heat in its passage through the lungs ; so that
in the process of respiration a quantity of absolute heat
is separated from the air, and absorbed by the blood. Dr.
Priestley has also proved, that in respiration, phlogiston is
separated from the blood, and combined with air.

This theory however has been contested and disputed,
and, it has been said, Dr. Crawford's experiments repeated,
with contrary results. Though no regular and systemati-
cal theory has yet been formed in its stead.

Heat of Combustible and Inflammable Bodies. Dr.
Crawford's theory with respect to the inflammation of
combustible bodies, is founded on the same principles as
his doctrine concerning the heat of animals. According
to him, the heat which is produced by combustion, is
derived from the air, and not from the inflammable body.
Inflammable bodies, he says, abound with phlogiston, and
contain little absolute heat; the atmosphere, on the con-
trary, abounds with absolute heat, and contains little
phlogiston. In the process of inflammation, the phlogiston
is separated from the inflammable body, and combined
with the air ; the air is phlogisticated, and affords a great
proportion of its absolute heat, which, when extricated
suddenly, bursts forth into flame, and produces an intense
degree of sensible heat. And since it appears by calcu-
lation, that the heat produced by converting atmospherical
into fixed air, is such, if it were not dissipated, as would
be sufficient to raise the air so changed, to more than 12
times the heat of red-hot iron, it follows, that in the "pro-
cess of inflammation a very great quantity of heat is de-
rived from the air. But, on the contrary, no part of the
heat can be derived from the combustible body; because
this body, during the inflammation, being deprived of its
phlogiston, undergoes a change similar to that of the
blood by the process of respiration, in consequence of
which its capacity for heat is increased ; and therefore it
will not evolve any part of its absolute heat, but, like the
blood in its passage through the lungs, it will absorb heat.
A similar theory of heat has lately been published by Mr.
Elliot. See his Philosophical Observations on the senses
of Vision and Hearing; to which is added an Essay on
Combustion and Animal Heat, Svo, 1780. See Caloric,
also Professor Leslie's late work on heat.

Mr. Ezekiel Walker has given a neat account of the
causes and mode of combustion, in the Philos. Magazine,
vol.43, pa. 101, as follows. There are very few bodies
in nature that do not contain either thermogen or photo-
gen. All those bodies which are called combustibles,
contain the element of light, which I call photogen, to
avoid the use of ambiguous terms ; and all those bodies
which are called supporters of combustion, contain the
generator of heat, called thermogen for the same reason.
These elements may be made to unite and produce com-
bustion, or increase the temperature of bodies, by various
means; as: 1st, by friction; 2dly, by percussion; 3dly,
by pressure ; 4thlv, by mixture; 5thly, by increase of
temperature ; and 6'tbly, by the functions of living animals
and vegetables.

Thermogen and photogen in a condensed state, as in a
charged Leyden jar, have so strong an attraction for each
other, that they will pass through any of those bodies
called conductors, to a very great distance, to be united.

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H E A

But when they are united to ponderable matter, they will
remain mixed together for ages without producing any
effect; but as soon as they are brought within a certain
distanceofeach other, which may be called their striking
distance, or their sphere of action, either combustion takes
place, or the temperature of the body, in which they meet,
is increased. The following explanations of some well-
known experiments will tend to show the importance, of a
theory founded on facts.

Experiment 1. In a memoir read to the National Insti-
tute of France, M. Biot announces the important fact,
that a mixture of hydrogen and oxygen gases may be made
to explode by mechanical compression only,independently
of the electric spark. — Explanation. Oxygen and hydrogen
gases would remain mixed together, for any length of time,
without combining, the thermogen and photogen in them
being kept at too great a distance by their bases ; but, by
mechanical compression only, they are forced within their
striking distance, and light and heat are produced on the
same principle as combustion is produced by discharging
a Leyden jar.

Exper. 2. Take 6" grains of oxygenized muriate of pot-
ass, and 3 grains of flour of sulphur, rub them together
in a mortar, and a smart detonating noise will be pro-
duced. Continue to rub the mixture hard, and the re-
port will be frequently repeated, accompanied with vivid
flashes of light. If the same mixture be wrapped in paper,
laid on an anvil, and smartly struck with a hammer, the
report will be as loud as that of a pistol. — Explan. These
experiments may be explained on the same principle as
the last. The two ingredients, though mixed together,
produced no effect until the thermogen in one of them
and the photogen in the other were forced into union, by
friction or percussion. And on the same principle the
effects of many other fulminating substances may be ex-
plained.

Exper. 3. If an ounce of strong nitrous acid be mixed
with about half its weight of sulphuric acid, and poured
into a little oil of turpentine, the whole will immediately
burst into flame. — Explan. The thermogen and photogen
in these fluids coming immediately into contact, flame is
produced, without any mechanical force or increase of
temperature. — Now, from this experiment it appears im-
possible that water can be a compound of oxygen and hy-
drogen; for, if it were, these two principles of combustion
would burstinto flame like the acid and the oil. But no
pressure, no increase of temperature, nor any union with
combustible matter, can produce a single spark of fire
from water. In short, nothing combustible is found in its
composition.

Exper. 4. Conibusfion may also be generated by fric-
tion. Two pieces of hard wood, being smartly rubbed to-
gether with great pressure, will soon burst into flame. —
Explan. The first increment of caloric, by friction, is ex-
cited by a portion of the thermogen of the air being forced
into union with the photogen in the wood ; and by con-
tinuing the operation, the temperature of the inflammable
principle of the wood is increased till flame is produced.

Exper. 5. When a rod of iron is laid upon an anvil, and
hammered with a quick succession of heavy strokes, it
soon becomes red-hot. But it cannot be strongly heated
a second time by the same means, unless it be first intro-
duced into a fire. — Explan. The first stroke of the ham-
mer forces a small portion of the thermogen of the air into
contact with the photogen of the metal, which generates

the first increment of caloric; and by repeated strokes of
the hammer the caloric is increased, till the rod becomes
red-hot. But thephotogeu of the metal becomes exhausted
by this means, and therefore cannot produce caloric a se-
cond time, till it has recovered its combustible property
in the fire. — -The burning of iron wire in oxygen gas, in a
brilliant manner shows how iron loses its photogen, and
becomes oxydized.

Exper. 6. Mr. Thomas Wedgwood has shown, that it has
never yet been explained, how friction produces caloric.
He took a piece of common window-glass, and held the
edge of it against the edge of a revolving grit-stone, when
that part in contact with the stone became red-hot, and
threw off hot particles which fired gunpowder. Thestone
and the glass being both incombustible substances, it re-
mains to be explained how caloric was produced. — Ex-
plan. It has before been shown, that the electric machine
creates nothing, but by friction only collects the two ele-
ments which produce caloric, from the earth and the at-
mosphere. Though the stone and the glass, in Mr. W.'s
experiment, are both incombustible, yet it is almost a
self-evident conclusion, that they collect the two elements
of caloric from the air and the earth by friction, in the
same manner as the electric machine.

Exper. 7. It is well known, that when flint and steel
are. smartly struck against each other, a spark always
ensues, which is capable of setting fire to tinder or to
gunpowder. The spark in this case, as was long since
ascertained by Dr. Hooke, is a small particle of the me-
tal, which is struck off, and takes fire during its passage
through the air. This therefore, and all similar cases,
belongs to the class of combustion. But light often ap-
pears when two bodies aTe struck against each other, when
we are certain that no such thing as combustion can hap-
pen, because both the bodies are incombustible. Thus,
for instance, sparks are emitted when two quartz stones
are struck smartly against each other, and light is emitted
when they are rubbed against each other. Many other
hard stones also emit sparks by such collisions. — If they
be often made to emit sparks above a sheet of white paper,
there are found upon it a number of small black bodies,
somewhat like the eggs of flies. These bodies are hard,
but friable, and leave a black stain when rubbed on the
paper. When viewed with a microscope, they seem to
have been melted. Muriatic acid changes their colour to
a green, as it does that of lavas. These substances evident-
ly produced the sparks by being heated red-hot. — Explan.
It may first be observed, that if the flint be moved over
the stone very slowly, no spark is produced, nor will any
spark appear if the flint barely touch the steel, though
the velocity of the flint be the greatest possible : conse-
quently, caloric cannot be produced by collision or fric-
tion, but by velocity and pressure conjointly. — Sdl}-, By
the collision of the flint and steel, the thermogen of the
atmosphere is forced into union with the photogen of the
metal. The combination thus produced melts the mi-
nute particles of steel, struck oft' by the flint, into small
globules. That this metal contains the generator of light
is well known, by the burning of iron or steel wire in
oxygen gas. — Sparks are emitted when two quartz stones
are struck smartly against each other under water; and
Mr. Kirwan affirms, that sparks are produced by the col-
lision of Hint and steel under common spring water.

Exper. 8. From Count Rumi'ord's experiments it ap-
pears, that by a moderate degree of friction, the same

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H E A

piece of metal afforded so much caloric under water, as
to keep it boiling. — Explan. This experiment may be ex-
plained on the same principle as the last. Water, being
a better conductor of thcrmogen than atmospheric air,
cannot prevent the production of caloric by the collision
• or friction of hard bodies.

Heat,m Geography, is that which relates to the earth.
There is a great variety in the heat of different places and
seasons. Naturalists have commonly assumed it as a fact,
that the nearer any place is to the centre of the earth, the
hotter it is found ; but this does not appear to hold strictly
true. And if it were, the effect might be otherwise ac-
counted for, and more satisfactorily, than from their
imagined central fire.

Mr. Boyle, who had been at the bottom of some mines
himself, with more, probability suspects that this degree of
heat, at least in some of them, may arise from the pecu-
liar nature of the minerals there produced. And he in-
stances a mineral of the vitriolic kind, dug up in large
quantities, in several parts of England, which, by the bare
effusion of common water; will become so hot as almost
to take fire. To which may be added, that such places,
in the bowels of the earth, usually feel hot, from the con-
fined and stagnant state of the air in them, in which the
heat is retained, for want of a current or change of air to
carry the heat off.

On the other hand, on ascending high mountains, the
air feels more and more cold and piercing. Thus, the
tops of the Pike of Teneriffe, the Alps, and several other
mountains, even in the most sultry countries, are found
always invested with snow and ice, which the heat is never
sufficient to thaw. In some of the mountains of Peru there
is no such thing as running water, but all ice : plants
vegetate a little about the bottom of the mountains, but
near the top no vegetable can live, for the intenseness of the
cold. This effect is attributed to the thinness of the air,
and the little surface of the earth there is to reflect the rays,
as well as the great distance of the general surface of the
earth which reflects I he rays back into the atmosphere.

As to the diversity in the heat of different climes and
seasons, it arises from the different angles under which
the sun's rays strike upon the surface of the earth. In the
Philos. Trans. Abr. vol. .'3, p. 576, Dr. Hal ley has given a
computation of this heat, on the principle, that the simple
action of the sun's rays, like other impulses or strokes,
is more or less forcible, according to the sines of the angles
of incidence, or to the sines of the sun's altitudes, at dif-
ferent times or places.

Hence it follows, that the time of continuance, or the
sun's shining on any place, being taken for a basis, and
the sines of the sun's altitudes perpendicularly erected
upon it, and a curve line drawn through the extremities
of those perpendiculars, the area thus comprehended will
be proportional to the collection of all the heat of the
sun's beams in that period of time.

Hence it will likewise follow, that at the pole, the col-
lection of all the heat of a tropical day, is proportional to
the rectangle or product of the sine of 23$ degrees into
24 hours, or the circumference of a circle, or as -fs into
12 hours, the sine of 23| degrees being nearly -,% of ra-
dius. Or the polar heat will be equal to that of the sun
continuing 12 hours above the horizon at 53 degrees
height ; and the sun is not 5 hours more elevated than
this under the equinoctial.

But as it is the nature of heat to remain in the subject,

after the luminary is removed, and particularly in the
air, under the equinoctial, the 12 hours absence of the
sun abates but little from the effect of his heat in the day;
but under the pole, the long absence of the sun for 6
months has so chilled the air, that it is in a manner frozen,
and after the sun has risen upon the pole again, it is long
before his beams can make any impression, being ob-
structed by thick clouds and fogs.

From the foregoing principle Dr. Halley computes the
following table, exhibiting the heat to every 10th degree
of latitude, for the equinoctial and tropical sun, and from
which an estimate may easily be made for the intermediate
degrees.

Latitude.

Sign

that the Sun

is in.

V or c£>

25

y?

0

20000

18341

18341

10

19696

20296

15S34

20

18794

21737

13166

30

17321

22651

10124

40

15321

23048

6944

50

12855

22991

3798

60

10000

22773

1075

70

6840

23543

0

80

3473

24673

0

90

0

25055

0

From the same principles, and table, also are deduced
the following corollaries, viz,
\ 1, That the equatorial heat, when the sun becomes ver-
tical, is as twice the square of the radius. — 2, That at the
equator the heat is as the sine of the sun's declination. —

3. That in the frigid zones, when the sun sets not, the
heat is as the circumference of a circle into the sine of the
altitude at 6: And consequently, that in the same lati-
tude, these aggregates of heat are as the sines of the sun's
declination; and at the same declination of the sun, they
are as the sines of the latitudes ; and generally they are as
the sines of the latitudes into the sines of declination. —

4, That the equatorial day's heat is every where as the
cosine of the latitude. — 5, In all places where the sun sets,
the difference between the summer and winter heats, when
the declinations are contrary, is equal to a circle into the
sine of the altitude at 6, in the summer' parallel; and
consequently those differences are as the rectangles of the
sines of the latitude and declinations. — 6, The tropical '
sun has the least force of any at the equator ; and at the
pole it is greatest of all.

Many objections have been urged against this theory of
Dr. Halley. Some have insisted, that the effect of the
sun's heat is not in the simple, but in the duplicate ratio
of the sines of the angles of incidence ; like the law of the
impulse of fluids. And indeed, the quantity of the sun's
direct rays received at any place, being evidently as the
sine of the angle of incidence, or of the sun's altitude, if
the heat be also proportional to the force with which a
ray strikes, like the mechanical action or impulse of any
body, then it will follow that the heat must be in the
compound ratio of both, that is, as the square of the sine
of the sun's altitude. But this last principle is here only
assumed gratis, as we do not know a priori that the heat
is proportional to the furce of a striking body; and it is
only experiment that can determine this point.

It is certain that heat communicated by the sun to bo-

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H E A

dies on the earth, depends also much upon other circum-
stances beside the direct force of his rays. These must
be modified by our atmosphere, and variously reflected
and combined by the action of the earth's surface itself,
to produce any remarkable effects of heat. So that, if it
were not for these additional circumstances, it is probable
the naked heat of the sun would not be very sensible.

Dr. Halley himself was well apprised, that many other
circumstances, besides the direct force of the sun's rays,
contributed to augment or diminish the effect of this, and
the heat resulting from it, in different climates; and there-
fore no calculation, formed on the preceding theory, can
be supposed to correspond exactly with observation and
experiment. It has.a'lso been objected that, according to
the foregoing theory, the greatest heat in the same place,
should be at the summer' solstice, and the most extreme
cold at the winter solstice ; which is contrary to expe-
rience. To this objection it may be replied, that heat is
not produced in bodies by the sun instantaneously, nor
do the effects of his heat cease immediately when his rays
are withdrawn ; and therefore those parts which are once
heated, retain the heat for some time ; which, with the ad-
ditional heat daily imparted, makes it continue to increase,
though the sun declines from us : and this is the reason
why July is hotter than June, though the sun has with-
drawn from the summer tropic : as we also find it is gene-
rally hotter at one, two, or three in the afternoon, when
the sun has declined towards the west, than at noon, when
he is on the meridian. As long as the heating particles,
which are constantly received, are more numerous than
those which ily away or lose their force, the heat of bodies
must continually increase. So, after the sun has left the
tropic, the number of particles, which heat our atmo-
sphere and earth, constantly increases, because we receive
more in the day than we lose at. night, and therefore our
heat must also increase. But as the days decrease again,'
and the action of the sun becomes weaker, more particles
will be dispersed in the night than are received in the day,
by which means the earth and air will gradually cool.
Further, those places which are well cooled, require time
to be heated again; and therefore January is mostly colder
tlian December, though the sun has withdrawn from the
winter tropic, and begun to emit his rays more directly
upon us.

But the chief cause of the difference between the heat
of summer and winter is, that in summer the rays fall more
directly, and pass through a less dense part of the atmo-
sphere; and therefore with greater force, or at least in
greater number in the same place: and besides, by their
long continuance, a much greater degree of heat is im-
parted by day than can be dispersed by night. For the
calculations and opinions of several other philosophers
on this head, sec Keill's Astron. Lect. 8 ; Ferguson's As-
tron. chap. 10; Long's Astron. § 777; Memo. Acad.
Scienc. 171.9.

As to the temperature or heat of our atmosphere, it
may be observed that the mercury seldom falls under 16°
in Fahrenheit's thermometer; but we are apt to reckon it
very cold at 24°, and it, continues chilly to 40° and a little
above. However, such colds have been often known to
reduce it down to 0°, the beginning of the scale, or nearly
the cold produced by a mixture of snow and salt, often
near it, and in some places below it. Thus, the degree of
the thermometer has been observed at various times and
places as follows :

Places

Latit.

Year

Them

Pensylvania

40° 0'

1732

5'

Paris

48 50

1709 & 1710

S

Ley den

52 10

1729

5

Utrecht

52 8

4

London

51 31

1709 & 1710

0

Copenhagen

55 43

1705

0

Upsal

- 59 56

■ 1732

— 1

Petersburg

59 56

— -2S

Torneo

65 51

1736-7

—33

Hudson's Bay

52 24

1775

—37

The middle temperature of our atmosphere is about
4S°, being nearly a medium of all the seasons. The French
make it somewhat higher, reckoning it equal to the cave
of their royal observatory, or 53°. In cold countries, the
air is found agreeable enough to the inhabitants while it
is between 40 and i0°. But in our climate we are best
pleased with the heat of the air from 50 to 60°; while in
the hot countries the air is generally at a medium about
70°. With us, the air is not reckoned warm till it arrives
about 64°, and it is very warm and sultry at S0°. It. is
to be noted that the foregoing observations are to be under-
stood of the state of the air in the shade; for as to the
heat of bodies acted on by the direct rays of the sun, it is
much greater: thus, Dr. Martine found dry earth heated
to above 120°; but Dr. Hales found a very hot sun-shine
heat'in 1727 to be about 140°; and Muschenbroek once
observed it so high as 150°; but at Montpelier'the sun
was so very hot, on one day in the year 1705, as to raise
M. Amontons' thermometer to the mark of boiling water
itself, which is our 212°. (See Temperature of the Atmo-
sphere under the article Atmosphere.)

It appears from the register of the thermometer kept at
London by Dr. Heberden for 9 years, viz, from the end of
1763 to the end of 1772, that the mean heat at 8 in the
morning was47°'4; and by another register kept at Hawk-
hill near Edinburgh, that the mean heat in that place, du-
ring the same period of time, was 46°. Also by registers
kept in London and at Hawkhill, for the three years 1772,
1773, 1774, it appears, that the mean heat of these three
years in London, at 8 in the morning, was 4S°-5, and at
2 in the afternoon 56°, but the mean of both morning and
afternoon 52°'2 ; while the mean heat at Hawkhill for the
same time,

at 8 in the morning was 450-4

and at 2 in the afternoon 50"1

and the mean of both 47'7-

The mean heal of springs near Edinburgh is found to be

about 47°, and at London 51°. Philos. Trans, vol. 65,

art. 44.

Lastly, from the meteorological journals of the Royal
Society, published in the Philos. Trans, it appears that the
mean heights of the thermometer, for the ivhole years,
kept without and within the house, are as below:
Therm. Without Therm. Within

For 1775 - 51'S - - 527

1776 - 51-1 - - 529

1777 - 51'0 - - 53-0

1778 - 52-0 - - 53-1
mean of all - 51'4 - - 52"9

Heat, Latent, is insensible, concealed, dormant as it
were.

' HEAVEN, an azure transparent orb investing our
earth, where the celestial bodies perform their motions. It
is of divers denominations, as the highest or empyrean

H E I

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H E L

heaven, the etherial or starry heaven, the planetary hea-
ven, &c. Formerly the heavens were considered as solid
substances, or else as spaces full of solid matter; but
Newton has abundantly shown that the heavens are void of
almost all resistance, and consequently of almost all mat-
ter: this he proves from the phenomena of the celestial
bodies; from the planets persisting in their motions, with-
out any sensible diminution of their velocity ; and the
comets freely passing in all directions towards all parts of
the heavens. Heaven, taken in this general sense, or the
whole expanse between our earth and the remotest regions
of the fixed stars, may be divided into two very unequal
-parts, according to the matter occupying them; viz, the
atmosphere or aereal heaven, occupied by air; and the
ethereal heaven, possessed by a thin and unresisting me-
dium, called ether.

Heaven is more particularly used, in Astronomy, for
an orb, or circular region, of the ethereal heaven.

The ancient astronomers assumed as many different
heavens as they observed different celestial motions. All
these they made solid, thinking they could not otherwise
sustain the bodies fixed in them ; and of a spherical form,
as being the most proper for motion. Thus they had seven
heavens for the seven planets; viz, the heavens of the
Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn.
The 8th was for the fixed stars, which they particularly
called the firmament. Ptolemy added a 9th heaven, which
he called the primum mobile. After him two crystalline
heavens were added by King Alphonsus, &c, to account
for some irregularities in the motions of the other heavens.
And lastly an empyrean heaven was drawn over the whole
for the residence of the Deity ; which made the number 12.
But some admitted many other heavens, according as their
different views and hypotheses required. Eudoxus sup-
posed 23, Calippus 30, Regiomontanus 33, Aristotle 47,
and Fracastor no less than 70. The astronomers however
did not much concern themselves whether the heavens
they thus allowed, were real or not ; provided they served
a purpose in accounting for any of the celestial motions,
and agreed with the phenomena.

HEAVINESS, the same as Gravity, which see.

Heavy bodies do not tend precisely to the very centre of
the earth, except at the poles and the equator, on ac-
count of the. spheroidal figure of the earth ; their direction
being every where perpendicular to the surface of the
spheroid.

HEGIRA, a term in Chronology, signifying the epoch,
or account of time, used by the Mahomedans, who begin
their computation from the day that Mahomet was forced
to make his escape from the city of Mecca, which happen-
ed on Friday the l6th of July 622.

The years of the hegira are lunar ones, consisting only
of 354 days. Hence, to reduce these years to the Julian
calendar, that is, to find what Julian year a given year of
the hegira answers to : reduce the year of the hegira into
days, by multiplying by 354, divide the product by 365$ ,
and to the quotient add 622, the year the hegira com-
menced.

HEIGHT, the third dimension of a body, considered
with regard to its elevation above the ground; or any di-
stance above the earth's surface, &c.

Height, of a figure, the same as its altitude, being the
perpendicular from its vertex to the base.

Height, of the pole, he. SoeAltitude of the Pole, &c.
Vol. I.

HELIACAL, something relating to the sun. Thus,

Heliacal Rising, of a star or planet, is when it rises
with, or at the same time as, the sun. And heliacal set-
ting, the same as the setting .with the sun. Or, a star
rises heliacally, when, after it has been in conjunction
with the sun, and so invisible, it gets at such a distance
from him, as to be seen in the morning before the sun's
rising. And it is said to set heliacally, when it approaches
so near the sun as to be hid by his" beams. So that, in
strictness, the heliacal rising and setting are only an appa-
rition and occultation.

HELICE Major and Minor; the same as Ursa Major
and Minor.

HELICOID Parabola, or the Parabolic Spiral, is a curve
arising from the supposition that the common or Apollo-
nian parabola is bent or twisted, till the axis come into
the periphery of a circle, the ordinate still retaining their
places and perpendicular positions with respect to the cir-
cle, all these lines still remainingin the same plane. Thus,
the axis of a parabola being
bent into the circumfer-
ence bcdm, and the ordi-
nates cf, dg, &c, still per-
pendicular to it, then the
parabola itself, passing
through the extremities of
the ordinates, is twisted into
the curve bfg, &c, called
the Helicoid, or Parabolic
Spiral. Hence it is evident
thatall theordinatescF, dg,
&c, tend to the centre of the circle, being perpendicular to
the circumference. — Also, the equation of the curve re-
mains the same as when it was a parabola; viz, putting
x = any circular absciss bc, and y = cf the correspond-
ing ordinate, then ispx = ya, where p is the parameter of
the parabola.

HELIOCENTRIC Place of a Planet, is the place in
which a planet would appear to be when viewed from
the sun ; or the point of the ecliptic, in which a planet
viewed from the sun would appear to be. And therefore
the heliocentric place coincides with the longitude of a
planet viewed from the sun.

Heliocentric Latitude of a Planet, is the inclination ■
of the line drawn between the centre of the sun and the
centre of a planet, to the plane of the ecliptic. The
greatest heliocentric latitude is equal to the inclination of
the planet's orbit to the plane of the ecliptic.

HELIOCOMETE, Comet of the Sun, a phenomenon
sometimes observed at the setting of the sun; thus deno-
minated by Sturmius and Pylen, who had seen it, because
it seems to make a comet of the sun, being a large tail,
or column of light, fixed or attached to that luminary,
and dragging after it, at its setting, like the tail of a comet.

HELIOMETER, or Astuometer ; an instrument for
measuring, with particular exactness, the diameters of
the sun, moon, and stars. — This instrument was invented
by M. Bouguer in 1747, and is a kind of telescope, con-
sisting of two object-glasses of equal focal distance, placed
by the side of each other, so that the same eye-Mass
serves for both. The tube of this instrument is of a co-
nical form, larger at the upper end, which receives the.
two object-glasses, than at the lower, which is furnished
with an eye-glass and micrometer. By the construction
4 N

MEM

[ 642 ]

HER

of this instrument, two distinct images of an object are
formed in the focus of the eye-glass, whose distance, de-
pending on that of the two object-glasses from each other,
may be measured with great accuracy. Mem. Acad. Sci.
1748. Mr. Servington Savery discovered a similar me-
thod of improving the micrometer, which was communi-
cated to the Royal Society in 1743.

HELIOSCOPE, a kind of telescope peculiarly adapted
for viewing and observing the sun without hurting the eye.
There are various kinds of this instrument, usually made
by employing coloured glass for the object- or eye-glass, or
both; and sometimes only using an eye-glass blackened
by holding it over the smoke or flame of a lamp or candle,
which is Huygens's way. See Dr. Hooke's treatise on
Helioscopes.

HELIOSTATA, an instrument invented by Dr. Grave-
sande, and so called from its property of fixing the sun-
beam in one position, viz, in a horizontal direction across
the dark chamber while it is used. See Gravesande's Phy-
sices Element. Mathematica, torn. 2, pa. 715, ed.3tia,
1742, for an account of the principles, construction and
use of this instrument.

HEL1SPHER1CAL Line, is the rhumb-line in naviga-
tion ; being so called, because on the globe it winds round
the pole heliacally or spirally, approaching always nearer
and nearer to it.

HELIX, a spiral line. See Spiral.

HEMICYCLE, in Architecture, a term applied to se-
micircular vaults.

HEMISPHERE, the half of a sphere or globe, when
divided in two by a plane passing through its centre.

Hemisphere, in Astronomy, is particularly used for
one half of the mundane sphere. The equator divides
the sphere into two equal parts, called the northern and
southern hemispheres. The horizon also divides the sphere
into two equal parts, called the upper and lower hemi-
spheres.

Hemisphere is also used for a map or projection of half
the terrestrial globe, or of half the celestial sphere, on a
plane; being more frequently called a planisphere. The
centre of gravity of a hemisphere, is five-eighths of the
radius distant from the vertex. A hemisphere unites
the parallel rays at the distance of four-thirds of a dia-
meter from the pole of the glass.

Hemispheres of Mugdeburgh, two brass concave he-
mispheres, one of which is furnished with a cock, by which
it may be adjusted to the air-pump. The other has a ring
at the middle of its convexity, by means of which it may
be suspended. When these two hemispheres are placed
together, the air is drawn from within through the me-
dium of the cock by the mechanism of the air-pump ; and
then, the exhaustion being carried far enough, and the
cock turned to prevent the admission of air from without,
a considerable force is requisite to separate the hemi-
spheres ; the force varying with their great circle. This fur-
nishes an easy and obvious method ol proving the pressure
of the air. Otto Gueriche, the inventor of the air-pump, to
whom also is due this invention, made hemispheres of about
a foot radius, and found that the atmospheric pressure on
them was equivalent to about 5400 pounds,

HENDECAGON, a figure of eleven sides, or theE
decagon ; which see.

HENIOCHAS, or Heniochus, a northern constel-
lation, the same as Auriga, which see.

HEPTAGON, in Geometry, a figure of seven sides and
seven angles.— When those sides and angles are all equal,
the heptagon is said to be regular, otherwise it is irregu-
lar.— In a regular heptagon, the angle ^-^
c at the centre is = 5l°|, the angle x<^
dab of the polygon is = 128°£, and
its half cab = 6'4°f-. Also the area
is 3= the square of the side ab! x
3-6'339124 or = ab2 x Zt, where t isjj\
the tangent of the angle cab of 64°|-
to the radius 1 ; or t is the root of
the equation ; -A- -^
ttz - 26tto + 143<3 - 245*6 + 143/4 -26*z + 1 = 0 ; or
_ /* +. « _ i + ^/d- y2) y

the value of x and y are the roots of the equations

x6 — ! x4 -+- i.r'2 — ' = 0,

j,s - Z

*r

if-

-54

:0.

See my Mensuration, pa. 86, 4th edition.

Heptagon, in Fortification, a place fortified or strength-
ened with seven bastions for its defence.

Heptagon al Numbers, are a kind of polygonal num-
bers in which the difference of the terms of the corre-
sponding arithmetical progression is 5. Thus,

Arithmeticals, 1, 6, 11, lfi, 21, 26", &c.

Heptagonals, 1, 7, IS, 34, 55, 81, &c ;
where the heptagonals are formed by adding continually
the terms of the arithmeticals, above them, whose common
difference is 5. — One property, among many others, of
these heptagonal numbers is, that if any one of them be
multiplied by 40, and to the product be added 9, the sum
will be a square number.

Thus 1 x 40 + 9 = 49 = 7';*

and 7 x 40 -+- 9 = 289 = 172;

and 18 x 40 ■+■ 9 = 729 = 27";

and 34 x 40 -+- 9 = 13o"9 = 372; &c.
Where it is remarkable that the series of squares so formed
is 72> 172, 272, 372, &c, the common difference ofwhose
roots is 10, the double of the common difference of the
arithmetical series, from which the heptagonals are formed.

— See POLYGONALS.

HEPTANGULAIl Figure, in Geometry, is one that
has seven angles ; and therefore also seven sides.

HERCULES, in Astronomy, a constellation of the
northern hemisphere, and one of the 48 old constellations
mentioned by ancient writers. The stars in this constel-
lation, in Ptolemy's catalogue, are 29; inTycho's, 28; and
in the Britannic catalogue, 113.

HERISSON, in Fortification, a beam armed with iron
spikes, having their points turned outward. It is sup-
ported in the middle by a stake, having a pivot on which
it turns; and serves'asa barrier to block up a passage. —
Herissons arc frequently placed before gates, especially
the posterns of a town or fortress, to secure those passages
which must of necessity be often opened,

HERMANN (Jami'.s), a learned mathematician of the

HEMISPHEROIDAL, in Geometry, something that ' Academy of Berlin, and member of the Academy of Sci-

approaches to the figure of a hemisphere, but is not ences at Paris, was born at Basil in 16"78. tie was agneat

justly so. traveller ; and for 6" years was professor of mathematics at

HEMITONE, in Music, a half note, or semi-tone. Padua. He afterwards went to Russia, being invited thi-

HER

c

]

HER

ther by the Czar in 172.4, 'as well as his compatriot Daniel
Bernoulli. On his return to his native country, he was
appointed professor of morality and natural law at Basil ;
where he died in 1733, at 55 years of age. Hermann
wrote several mathematical and philosophical pieces, in
the Memoirs of different Academies, and elsewhere ; but
his principal work, is the Phoronomia, or two Books on
the Forces and Motions of both Solid and Fluid Bodies;
4to, 1716: a very learned work on the new mathematical
physics.

HliRMETIC, or Hermetical Art, a name given to
chemistry, on a supposition that Hermes Trismegistiis was
its inventor.

HERMETICAL Philosophy, is that which undertakes
to solve and explain all the phenomena of nature, from
the three chemical principles, salt^ sulphur, and mercury.
A considerable addition was made to the ancient herme-
tical philosophy, by the modern doctrine of alkali and acid.

Hermetical Seal, or Hermetical Sealing, a manner of
stopping or closing glass vessels, for chemical and other
operations, so very closely, that no substance can possibly
exhale or escape. This is usually done by heating the
neck of the vessel in the flame of a lamp, with a blow-pipe,
till it be read}' to melt, and then with a pair of hot pincers
twisting it close together.

HERO, or Heron, the name of two celebrated mathe-
maticians of the Alexandrian school, under Ptolemy Euer-
cetes 11 ;one called the elder, who wrote a very ingenious
treatise on various pneumatics and hydraulic machines,
translated into Latin, and published in several editions and
forms, since the invention of printing; first a very elegant
edition by Commandine, in 4to, 1583, entitled Heronis
Alexandrini Spiritalium Liber. To this was added Qua-
tuor Thoremata Spiritalia, by Joan. Bapt. Aleotti ; trans-
lated from Italian into Latin ; Amsterdam, 168O, in 4to.
The other, called the younger, under the Emperor Hera-
clius, left a Treatise on Mechanics and Mensuration, pub-
lished in Latin, at Venice, 1572, in 4to, entitled Heronis
Mechanici, liber de Machinis Bellicis, nee non liber de
Geodassia. A Francisco Barocia, patritio Veneto latini-
tate donati, multis mendis expurgati, et figuris, ac scho-
liis illustrati.

HERSCHEL. See Uranus.

Herschel's Telescope, a curious one made by that
astronomer, being a reflecting one, and the largest ever
constructed, 40 feet in length. The figure of it is exhi-
bited in plate xv, and thedescription as follows, abridged
from the Philos. Trans, vol. 85, or from my Abridg. vol.
17, pa. 593.

The telescope is placed in a situation due north and
south, and the plate delineates the whole apparatus as
seen by a person placed at a convenient distance from it
towards the south-west. From this view the structure is
sufficiently understood ; and with very little attention, the
mode of pointing this immense body to any part of the
heavens, will be clearly seen. We shall treat of the chief
parts in their order; and first, of the tube itself.

The tube is made of rolled or sheet iron, joined toge-
ther without rivets, by a seaming similar to that which is
used for iron funnels for stoves ; the thickness of the sheets
is somewhat less than a 36th part of an inch, or it may
be found more accurately by taking a square foot of it at
the weight of 14 pounds. Great care was taken in so
joining together the plates, of which the tube is composed,

that the cylindrical form should be secured, and then the
whole was coated over three < r four times with paint, in-
side and outside, to secure it against the damp. The tube
wus-formed at a short distance from its present place, and
removed with great ease by 24 men, divided into six sets;
so that two men on each side, with a pole of 5 feet long in
their hands, to which was affixed a piece of coarse cloth,
7 feet long, going under the tube, and joined to a pole of
5 feet long, ini the hands of two other men, assisted in
carrying the tube. The length of the tube is 39 feet 4
inches, the diameter 4 feet 10 inches; and upon a mo-
derate computation it is supposed that a wooden tube for
the same purpose would have exceeded this in weight by
at least 3000 pounds. The length of the iron plate form-
ing the tube, and composed of smaller ones 3 feet 10 inches
long, and 23§ inches broad ; is nearly 40 feet, and the
breadth 15 feet 4 inches.

The great mirror which, by proper methods, was brought
to the lower part of the tube, is made of metal, 49| inches
in diameter ; but the concave part, or polished surface, is
only 48 inches in diameter. Its thickness is Z\ inches ;
and when it came from the cast, its weight was 2118
pounds, of which a small quantity must have beett lost in
polishing. An iron ring, 4-91 inches in diameter within,
4 inches broad, and 1^ inch thick, with three strong
handles to it, goes round the mirror, and a flat cover of
tin is made to correspond with this ring, that the mirror
may be preserved from damp ; and, by an easy contri-
vance, it is taken off and fixed on at. pleasure.

At the upper end the tube is open, and directed to the
part of the heavens intended for observation, to which the
observer's back is turned, and he, standing on the foot-
board visible in the plate, looks down the tube, and per-
ceives the object by rays reflected from the great mirror,
through the eye-glass at the opening of the tube. Near
the place of the eye-glass is the end of a tin pipe, into
which a mouth-piece may be placed ; so that, during an
observation, a person may direct his voice into this pipe,
while his eye is at the glass. This pipe is 1# inch in dia-
meter; runs down to the bottom of the tube, where it goes
into a turning joint, thence into a drawing tube, and out
of this into another turning joint, from which it proceeds
by aset of sliding tubes towards the front of the foundation
timber. The use of this tube is to convey the voice of
the observer to his assistants ; for at the last place it di-
vides itself into two branches, one going into the observa-
tory, the other into the workman's room, ascending into
both places through the floor, and being terminated in the
usual shape of speaking-trumpets. Though the voice
passes in this manner through a tube with many inflec-
tions, and not less than 115 feet, it requires very little
exertion to be well understood.

To direct so immense a body to any part of the heavens
at pleasure, much ingenuity, and many mechanical con-
trivances are evidently necessary. The whole apparatus
rests upon rollers, and care was previously taken of the
foundation in the ground. This consists of concentrical
circular brick walls, the outermost 42 feet, the innermost
21 feet in diameter; 2 feet 6 inches deep underground ;
2 feet 3 inches broad at bottom, and 1 foot 2 inches at
the top, capped with paving-stones, about 3 inches thick,
and \2\ inches broad. In the centre is a large post of
oak, framed together with braces underground, and walled
fast with brick-work, to make it steady. Round this
4N 2

HER

[ 644 ]

H E T

centre, the whole frame is moved horizontally, by means
of 20 rollers, 1 2 upon ihe outer, and 8 upon the inner wall.
The vertical motion is given to the telescope by means
of ropes and pulleys as seen in the plate, passing over the
main-beam, supported by the ladders. These ladders are
in length 4-9 feet 2 inches; and there is a moveable gal-
lery with 24 rollers to ease its motion. The small stair-
case, visible in the plate, is intended for persons, who wish
to ascend into the gallery, without being obliged to go up
the ladder. The ease with which the horizontal and ver-
tical motions may be communicated to the tube, will be
best conceived from a remark of Dr. Herschel's, that, in
the year 17S9, he several times observed Saturn two or
three hours before and after its meridional passage, with
one single person to continue, at his directions, the ne-
cessary horizontal and vertical motions.

Upon the platform are visible two rooms, the one called
the Observatory, 8 feet 5 inches by 5 feet 5 inches, the
other called the Working-room, 6 feet 6 inches by 4 feet
5 inches. To persons in these rooms, as has been above
remarked, the observer can give his directions by means
of the speaking-pipes; and in the rooms may be placed
things commonly used in an observatory.

From a view of the plate and the description thus given
of it, the reader will form a competent idea of an instru-
ment, which, with proper eye-glasses, magnifies above 6000
times, and is the largest that has ever been made. Astro-
nomers in different parts of the world may be discouraged
from continuing their observations, when it should seem
that their discoveries must be anticipated by our observer ;
but though he has so much the advantage, much is left to
their labour and industry. It did not require a telescope
of this magnitude to observe the object which was first dis-
covered to be a planet by our astronomer, for it had been
seen and taken for a fixed star by many persons in the two
last centuries. And the double ring'of Saturn, which has
indeed been so beautifully observed through Herschel's
magnifier, had been already described by Cassini in his
Memoirs. Such of our readers as wish for a more parti-
cular account of this instrument will find it in the Trans-
actions of the Royal Society for 17.95, second part; in
which there are 18 plates, and 63 pages of letter-press, to
give an ample detail of every circumstance relating to
joiner's work, carpenter's work, smith's work, &c : which
has attended the formation and erection of this instrument.
It was completed on August the 28th, 1789, on which
day the sixth satellite of Saturn was discovered.

HERSE, in Fortification, a lattice or portcullice, in
the form of a harrow, beset with iron spikes, to block up
a gate-way, &c.

HERS1LLON, or little Herse, in Fortification, is a
plank armed with iron spikes, for the same use as the
herse, and also to impede the march of the infantry or ca-
valry.

HERWART(John George), chancellor of Bavaria
at the beginning of the 17th century, was descended
from a Patrician family of Augsburg. He was author of
several curious books. I .Tabulx Arithmetics Universale^
&c. Monach. Bavar. in large folio, 16"I0. This is per-
haps a work of the most stupendous magnitude and la-
bour in numeral calculations that ever was executed, con-
sisting of a table of numbers closely printed on f)9<) large
folio pages, for readily taking out the products and quo-
tients of all numbers by inspection only ; being in fact an

immense multiplication-table, of all factors or numbers,
both up to 1000 each, or 1 million of products. Many
examples of multiplications and divisions are prefixed,
with applications to the calculations of spherical triangles.
And, but for the more compendious invention of logarithms,
which soon after took place, Herwart's tables would doubt-
less still have been in constant use for similar purposes. —

2. Herwart also wrote an Apology for the Emperor Louis
of Bavaria, against the falsehoods asserted by Bzovius. —

3. Chronologia, nova et vera, l6'l2 and \626, 2 parts,
4to.— 4. Another very singular book, which was published
by his son, John Frederick Herwart, entitled, Admiranda
ethicas Theologia?" Mysteria propolata, de antiquissima
veterum Nationum Superstitione, &c; lG24, 4to.

HESSE (William Prince of), rendered his name im-
mortal by his encouragement of learning, by his studies,
and by his observations, for many years, of the celestial
bodies. For this purpose, he erected an observatory at
Cassel, and furnished it with good instruments, well adapt-
ed to that design; calling also to his assistance two emi-
nent artists, Christopher Rothmannand Juste Byrge. His
observations, which are of a very curious nature, were
published at Leyden, in the year l6'l8, by Willebrord
Snell ; and are in part mentioned by Tycho Brahe, as
well in his epistles as in the 2d volume of his Progymnas-
mata ; a signal example to all princely and heroic minds,
to undertake the promoting the arts of peace, and ad-
vancing this truly noble and celestial science. This prince
died in the year 1597-

HETERODROMUS Vectis, or Lever, in Mechanics,
a lever in which the fulcrum, or point of suspension is be-
tween the weight and the power ; being the same as what
is otherwise called a lever of the first kind.

HETEROGENEOUS, or Heterogeneal, literally
imports things of different natures, or something that con-
sists of parts of different or dissimilar kinds ; in opposition
to Homogeneous. Thus,

Heterogeneous Bodies, are such as have their parts
of unequal density.

Heterogeneous Light, that which consists of parts or
rays of different refrangibility, reflcxibility, and colour.
Thus, the common light of the sun, er clouds, is heteroge-
neous, being a mixture of all sorts of rays.

Heterogeneous Numbers, are mixed numbers, con-
sisting of integers and fractions.

Heterogeneous Particles, are such asare of different
kinds, natures, and qualities; of which generally all bodies
consist.

Heterogeneous Quantities, inMathematics,are those
which cannot have proportion, or be compared together
as to greater and less; being of such different kind and
consideration, as that one of them taken any number of
times, never equals or exceeds the other. As lines, sur-
faces, and solids in geometry.

Heterogeneous Surds, are such as have different ra-
dical signs, as a/h and£/6*J or %/ 10 and ^/20.

IIETEROSCII, in Geography, are such inhabitants of
the earth as have their shadows at noon projected always
the same way with regard to themselves, or always con-
trary ways with respect to each other. Thus, all the in-
habitants without the toil id zone are 1 leteroscii, with re-
gard to themselves, since any one such inhabitant has his
shadow at noon always the same way, viz, always north of
him in north latitude, and always south of him in south

H E V

[ 645 ]

H E V

latitude; or these two situations are Heteroscii to each
other, having such shadows projected contrary ways at all
times of the year.

HEVELIUS (John), a very celebrated astronomer, and
a burgomaster of Dantzick, was born iu that city in
l6"ll. He studied mathematics under Peter Cruger, in
which he made a wonderful progress. He afterwards spent
several years on his travels through Holland, England,
Germany, and France, for his improvement in the sciences.
On his return, he constructed excellent telescopes himself,
and began diligently to observe the heavens, an employ-
ment he closely followed during the course of a long life,
which was terminated only in l6S7, at 76 years of age.
Hevelius was author of several remarkable discoveries in
the heavens. He was the first who observed the phenome-
non called the libration of the moon, and made several
other important observations on the other planets. He
also discovered several fixed stars, which he named the
Firmament of Sobieski, in honour of John the 3d, king of
Poland. He also framed a large catalogue of the stars,
and collected multitudes of the unformed ones into new
constellations. His wife was also well skilled in astro-
nomy, and made a part of the observations that were pub-
lished by her husband. Our author's principal publica-
tions are, his Selenographia, or an exact description of the
moon; in which he has engraved all her phases, and re-
markable parts, distinguished by names, and ascertained
their respective bounds by the help of telescopes ; contain-
ing also a delineation of the several visible spots, with the
various motions, changes, and appearances, discovered by
the telescope, as also in the sun and other planets, 164-7.
— In l654j, two epistles; one to the celebrated astrono-
mer Riccioli, concerning the Libration of the Moon; and
the other to Bulliald, on the Eclipses of both luminaries.
—In 1656, a Dissertation, De Natura Saturni faciei, &c.
— In 1668, his Cometographia, representing the whole
nature of comets, their situation, parallaxes, distances,
diverse appearances, and surprising motions, with a history
of all the comets from the beginning of the world down to
the present time; being enriched with curious sculpture
of his own execution : to which he added a treatise on the
planet Mercury, seen in the sun at Dantzick, May 3, 1 66 1 ;
with the history of a new star appearing in the neck of
Cetus, and another in the beak of Cygnus ; besides an il-
lustration of some astronomical discoveries of the late Mr.
Horrox, in his treatise on Venus seen in the sun, Novem-
ber '34', l0*39 ; with a discourse of some curious Parase-
lena and Parhelia observed at Dantzick. He sent copies
of this work to several members of the Royal Society at
London, and among them to Mr. Hooke, in return for
which, this gentleman sent to Hevelius a description of the
Dioptric Telescope, with an account of the manner of using
it; and recommending it to him, as much preferable to
telescopes with plain sights. This gave rise to a dispute
between them,. viz, " whether distances and altitudes could
be taken with plain sights any nearer than to a minute."
Hooke asserted that they could not ; but that, with an in-
strument of a span radius, by the help of a telescope, they
might be determined to the exactness of a second. He-
velius, on the other hand, insisted that, by the advantage
of a good eye and long practice, he was able with his in-
struments to come up even to that exactness; and, appeal-
ing to experience and facts, sent by way of challenge 8
distances, each between two different stars, to be examined
by Ilooke. Thus the affair rested for some time with out-

ward decency, but not withoutsome inward grudge between
the parties.

In 1673, Hevelius published the first part of his Ma-
china Coelestis, as a specimen of the exactness both of his
instruments and observations; and sent several copies as
presents to his friends in England, but omitting Mr. Hooke.
This, it is supposed, occasioned that gentleman to print,
in l6'74, Animadversions on the First Part of the Machina
Coelestis; in which he treated Hevelius with a very magis-
terial air, and threw out several unhandsome reflexions,
which were greatly resented. This was the cause of a
most violent dispute, which arose afterwards to such a
height, and became so notorious, that iri lt>79 Dr. Hal-
ley went, at the request of the Royal Society, to examine
both the instruments and the observations made with them.
Of both these, Halley gave a favourable account, in a let-
ter to Hevelius ; and Hooke managed the controversy so
ill, that he was universally condemned, though the prefe-
rence has since been given to telescopic sights. However,
Hevelius could not be prevailed on to make use of them :
whether he thought himself too experienced to be inform-
ed by a young astronomer, as he considered Hooke; or
whether, having made so many observations with plain
sights, he was unwilling to alter his method, lest he might
bring their exactness into question ; or whether, being by
long practice accustomed to the use of them, and not tho-
roughly apprehending the use of the other, nor well un-
derstanding the differenqe, is uncertain. Besides Hallev's
letter, Hevelius received many others in his favour, which
he took the opportunity of inserting among the astronomi-
cal observations in his Annus Climactericus, printed in
l685. In a long preface to this work, he speaks with more
confidence and greater indignation than he had done be-
fore, and particularly exclaimed against Hooke's dogma-
tical and magisterial manner of assuming a kind of dictator-
ship over him. This revived the contest, and excited so much
interest as to occasion several other learned men to engage in
it. The book itself being sent to the'Royal Society, at their
request an account of it was given by Dr. Wallis; who,
among other things, look notice, that, " Hevelius's obser-
vations had been misrepresented, since it appeared from
this book, that he could distinguish by plain sights to a
small part of a minute." About the same time Mr. Mo-
lyncux also wrote a letter to the Society, in vindication of
Hevelius, against Hooke's animadversions. To all which,
Hooke drew up a letter in answer, which was read before
the society, containing many qualifying and accommo-
dating expressions, but still at least expressing the supe-
riority of telescopic sights over plain ones, excellent as the
observations were that had been made with these.

Hevelius had published in l679> the second part of his
Machina Ccelestis ; but the same year, while he was at a
seat in the country, he had the misfortune to have his
house at Dantzick burnt down. By this calamity it is said
he sustained several thousand pounds damage; having not
only his observatory and all his valuable instruments and
astronomical apparatus destroyed, but also a great many
copies of his Machina Ccelestis, an accident which has
made this second part very scarce.

In I69O, were published a description of the heavens,
called, Firmamentum Sobiescianum, in honour of John
the 3d, king of Poland, as above mentioned ; and also Pro-
dromus Astronomiae, et Nova; Tabulte Solares, una cum
Catalogo Fixarum ; in which are contained the necessary
preliminaries for taking an exact catalogue of the stars,

H I E

[ 646 ]

H I G

Both these works were posthumous; for Hevelius died the
28th of January l687, exactly 76 years of age, as before ob-
served, and universally admired and respected; abundant
evidence of which appears in a collection of letters be-
tween him and many other persons, that was printed at
Dantzick in l6S3.

HEURAET (Vas), a respectable Dutch mathemati-
cian. He contributed some pieces on the nature of equa-
tions, in the commentaries published by Schoeten, on
the geometry of Descartes ; and, by pursuing the analyses
of that author, recommended himself to notice by his in-
vention for rectifying the curve of the semicubical para-
bola. In this, however, he was anticipated by the En-
glish mathematician Neil, who had some years before dis-
covered the rectification of that curve. But it is very
probable that Van Heuraet knew nothing of Neil's disco-
very ; and besides, the two methods are very different from
each other.

HEXACHORD, a certain interval or musical concord,
usually called a sixth.

HEXAEDRON, or Hexahedron, one of the five re-
gular or Platonic bodies ; being indeed the same as the
cube; and is so called from its having 6 faces. — The
square of the side or edge of a hexahedron, is one-third of
the square of the diameter of the circumscribing sphere :
and hence the diameter of a sphere is to the side of its in-
scribed hexahedron, as »/3 to 1.

In general, if a, b, and c be put to denote, respectively
the linear side, the surface, and the solidity of a hexahe-
dron or cube, alsor the radius of the inscribed sphere, and
ii the radius of the circumscribed one ; then we have these
general equations or relations :

1. a = 1r = |kV3 = \/is = Vc-

2. B = 24r4 = 8R2, = 6a2 = 6^/c2.

3. c = 8r3 = |rV3 = a3 = ^By^E.

4: R = r.y/3 =^^3 = Wi& = W$ x Vc-
5. r = § r.v/3 = |a = 1v/|b = J^/c.
HEXAGON, in Geometry, a figure of six sides, and
consequently of as many angles. When these are equal,
it is a regular hexagon. — The angles of a hexagon are each
equal to 120°, and its sides arc each equal to the radius
of its circumscribing circle. Hence a regular hexagon is
inscribed in a circle, by applying the radius 6 times suc-
cessively to the periphery, and joining with right lines the
several divisions. And hence also, to describe a regular
hexagon upon a given line, describe an equilateral triangle
upon it, the vertex of which will be the centre of the
circumscribing circle.

The side of a hexagon being s, its area will be =
2'598076'2s2 = |.s2 x tang. 60° = §*V3.

Hexagon, in Fortification, is a six-lined figuredefended
by six bastions.

HEXASTYLE,in the Ancient Architecture, a building
with 6 columns in front.

HIERO's Crown, in Hydrostatics. The history of this

gold for making a crown, suspected that he had been
cheated, by the workman using a greater alloy of silver
than was necessary in making it ; and he aplied to Archi-
medes to discover the fraud, without defacing the crown.

This celebrated mathematician was led by chance to a
method of detecting the imposture, and of determining
precisely the quantities of gold and silver composing the
crown : for he observed, when bathing in a tub of water,
that the water ran over as his body entered it, and he pre-
sently concluded that the quantity so running over was
equal to the bulk of his body that was immersed. He
was so pleased with the discovery, that it is said he ran
about naked, crying out, EUfTjwa, £upyica, / have found it;
and some affirm that he offered a hecatomb to Jupiter for
having inspired him with the thought.

On this principle he procured a ball or mass of gold,
and another of silver, exactly of the same weight with the
crown ; considering, that, if the crown were of pure gold,
it would be of equal bulk, and expel an equal quantity of
water as the golden ball; and if it were of silver, then it
would be of equal bulk and expel an equal quantity of
water with the ball of silver; but of intermediate quantity,
if it consisted of a mixture of the two, gold and silver.
This, upon trial, he found to be the Case; and hence, by
a comparison of the quantities of water displaced by the
three masses, he determined the exact portions of gold and
silver in the crown.

Now suppose, for example, that each of the three masses
weighed 100 ounces; and that on immersing them seve-
rally in water, there were displaced 5 ounces of water by
the golden ball, 9 ounces by the silver, and 6 ounces
by the compound, or crown; that is, their respective or
comparative bulks are as 5, 9> and. 6, the sum of which
is 20.

Then the method of operation is this :
From 9 6

5

1, whose sum is 4.

100 : : 3 : 75 oz. of gold,

100 : : 1 : 25_oz. of silver.

That is, the crown consisted oT 75 ounces of gold, and 25

ounces of silver.

See Cotes Hydros. Lect. p. 81; or Martin's Philos.
Britan. vol. 1, p. 305, &c. See also Specific, Gravity.

HIGH- Water, that state of the tides when they have
flowed to their greatest height, or have ceased to flow or
rise. At high-water the motion commonly ceases for a
quarter or half an hour, before it begins to ebb again. The
times of high-water of every day of the moon's age, is
usually computed from that which is observed on the day
of the full or change; viz, by taking 4-5ths of the moon's
age on any day of the month, and adding it to the time ot
high-water on the day of the full or change ; then is the
sum nearly equal to the time of high-water on the day of
the month proposed. And as to the times of high-water,

Take 6
rem. 3
Therefore 4
and 4 ;

crown, and of the important hydrostatical proposition 0n the day of the full and change of the moon, at many
which it gave occasion to, is as follows: Hiero, king of different places; they have been observed as they are set
Syracuse, having furnished a workman with a quantity of down in the following table.

H I G

[ 6*7 ]

H I G

Table of the Times of High-tuater on the Days of the New and Fall Moons, at many different Places.

Names of Plates.

Countries,

High

-water.

Names of Places.

Aberdeen

Scotland

0h

45m

Dunbar

Aldborough

England

0

45

Dundee v

Alderney I. .

England

12

0

Dungasvan .

Amazons River

South America .

6

0

Dungeness .

Amsterdam .

Holland .

S

0

Dunkirk

Amsterdam I. of .

South Seas

8

30

Dunnose

Andrew's St. .

Scotland

2

15

Dusky Bay .

Antyolt I.

Denmark ,

0

0

Easter Isle .

Antwerp . .

Flanders

6

0

Edystone

Archangel

Russia .

6

0 •

Elbe R.

Arran I.

Ireland

11

0

Embden

Ashley Riv. .

Carolina

0

45

Estaples

Augustine St. .

Florida

7

30

Falmouth

Bajador Ca.

Negroland .

0

0

Flamborough H.

Baltimore

Ireland

4

30

C. Florida .

Barfleur Ca. .

France

7

30

Flushing

Bayonne

France

3

30

N. Foreland .

Beachy-head

England

0

0

Foulness

N. and S. Bear .

Labradore .

12

0

Fowey

Belfast

Ireland

10

0

Fayal Isl.

Bellisle

France

3

30

Garonne R.

Bermudas I.

Bahama I. '

7

0

Gibraltar

Berwick

England . ,

2

30

C. Good Hope

Blackney

England . • .

6

0

Goree (Isle)

Blanco Cap.

Negroland

9

45

Granville

Blavet

France

3

0

Gravelines

Bourdeaux .

France

3

0

Gravesend

Boulogne

France

10

30

Groin (

Bremen

Germany .

6

0

Guernsej^I. .

Brest

France

3

45

Hague .

Bridlington B.

England .

3

45

Halifax

Brill

Holland .

1

30

Hamburgh

Bristol

England

6

45

Hare Isle

Buchaness

Scotland

3

0

Harlem

Button's Isles

New Brit.

6

50

Hartlepool

Cadiz

Spain

4

30

Harwich

Caen

France

9

0

Havre de Grace

Calais

France

11

30

Holy Head .

Canaria I. • .

Canaries

3

0

Honfleur

C. Cantin

Barbary

0

0

Hull

Cape Town

Caffers

2

30

Humber R. .

Caskets

Guernsey .

8

15

St. John's

Cathness Po.

Scotland

9

0

St. Julian (Port)

Charles Town

Carolina

3

0

Kentishnock

Q. Charlotte's S. .

New Zealand

9

0

Kinsale

Cherbourg .

France

7

30

Land's End .

Churchill R.

Hudson's Bay .

7

20

Leith

Ca. Clear

Ireland

4

30

Leostoff

Concarneau .

France

3

0

Lisbon

Conquet

France

2

15

Liverpool

Coquet Isle .

England .

3

0

Lizard

Cork

Ireland

6'

30

Loire (Riv.) .

C. Corse

Guinea

3

30

London

Cromer

England

7

0

Lundy (Isle)

Dartmouth ■

England

6

30

Madeira

St. David's H.

Wales

6

0

St. Maloes .

Dieppe

France

10

30

Isle of Man .

Dort

Holland .

3

0

Margate

Dover

England

11

30

St. Mary's (Isle)

Downs

England

1

15

Milford

Dublin

Ireland . ,

9

15

Mount's Bay

Countries.

High- water.

Scotland

2*

30m

Scotland

2

15

Ireland .

4

30

England

9

45

France

0

0

I. of Wight
N. Zealand

9
10

45
57

Chili

2

0

English Channel

5

30

Germany .

0

0

Germany .
France

0
11

0
0

England
England -.

5

4

30
0

Florida .

7

30

Holland .

0

45

England
England
England

9
6
5

45
45
15

Azores

2

20

France

3

0

Spain
Caffers

0
3

0
0

Atlantic Ocean .

1

30

France

7

O

Flanders

0

0

England .

Spain

English Channel

1

3
1

30

3

30

Holland .

8

15

Nova Scotia

7

30

Germany .
Canada

6
3

0

30

Holland .

9

0

England
England
France

3

11
9

0
15
0

Wales

1

30

France

9

0

England .

6

0

England

5

13

Newfoundland .

6

0

Patagonia .

4

4.5

English coast

0

0

Ireland

5

15

England .
Scotland

7
4

30
30

England

9

45

Portugal

2

15

England

11

15

England

7

30

France

3

0

England . .

3

0

England
Atl. Ocean

5

12

15
4

France . .

6

0

England . .

9

0

England .

Scilly Isles

11
3

15
45

Wales

5

15

England .

4

30

H

I P

C 648 ]

HI P

Names of Places.

Countries.

High-

water.

Names of Places.

Countries.

High-water.

Nantes

France

s'1

0m

Severn (Mouth.) .

England

6h 0°?

Naze

Norway

11

15

Sheerness

England

0 0

Needles

England

10

15

Sierra Leone

Guinea

S 15

Newcastle

England

3

15

Shetland I. .

Scotland

3 0

Nieuport

Flanders

12

0

Isle of Sky .

Scotland

5 30

Nore

England

0

0

Spurn

England

5 15

North Cape .

Lapland

3

0

Start Point .

England

6 45

Orfordness .

England

9

45

Stockton

England

5 is

Orkneys

Scotland

3

0

Sunderland .

England

3 30

Ostend

Flanders

12

0

Tanua

Pacific Ocean

3 0

Placentia

Newfoundland .

9

0

TenerifF

Canaries

3 0

Plymouth

England

0

0

Texel (Isle) .

Holland .

7 30

Portland

England

8

15

Thames' Mouth

England . .

1 30

Porto Pray a .

Cape Verdes

11

0

Tinmouth

England

3 0

Portsmouth .

England

11

15

Torbay

England

5 15

Quebec

Canada

7

30

St. Valery .

France

10 30

Rhee (Isle) .

France .

3

0

Vannes

France

3 45

Resolution (Bay) .

Ohitahoo . .*

2

30

Ushant

France

4 30

Robin Hood's B. .

England

3

0

Waterford

Ireland

.6 30

Rochefort

France . .v

4

15

Wells

England

6 0

Rochelle

France

3

45

Weymouth .

England

7 20

Rochester

England

0

45

Whitby

England

3 0

Rotterdam

Holland .

3

0

Isle of Wight

England

0 0

Rouen

France

1

15

Winchelsea .

England

0 45

Rye . .

England

11

15

Wintertoness

England

9 0

Sandwich

England

11

30

Yarmouth

England

9 45

Scarborough H. .

England

3

45

New York

America

3 0

Scilly Isles .

England .

3

45

Ysughall

Ireland

4 30

Senegal

Negroland .

10

30

Zuric Sea

Holland .

3 0

HIPS, in Architecture, are those pieces of timber
placed at the corners of a roof. These are much longer
than the rafters, because of their oblique position.

Hip means also the angle formed by two parts of the
roof, when it rises outwards.

Hiv-Roof, called also Italian Roof, is one in which
two parts of the roof meet in an angle, rising outwards :
the same angle being called a valley when it sinks inwards.

HIPPARCHUS, a celebrated ancient astronomer, was
born at Nice in Bithynia,and flourished between the 154-th
and the l63d olympiads ; that is, between 160 and 135
years before Christ ; for in this interval of time it is that
his observations are dated. He is accounted the first,
who from vague and scattered observations reduced astro-
nomy into a science, and prosecuted the study of it
systematically. Pliny often mentions him, and always
with great commendation. He was the first, he tells us,
who attempted to count the number of the fixed stars;
and his catalogue is preserved in Ptolemy's Almagest,
where they are all noted according to their longitudes and
apparent magnitudes. The same author places him among
those men of a sublime genius, who, by foretelling the
eclipses, taught mankind, that they ought not to be
alarmed at these phenomena. Thalcs was the first among
the Greeks, who could discover when there was to be an
eclipse. Sulpitius Gallus among the Romans began to
succeed in this kind of prediction ; and he gave an essay
of his skill very opportunely, the day before a battle was
fought. After these, Hipparchus improved the science of
eclipses very considerably : making ephemerides, or cata-
logues of eclipses, for 600 years. Pliny admires him for
making a review of all theslars, acquainting us with their
situations and magnitudes; for by these means, observes
he, posterity will bfc able to discover, not only whether

they are born and die, but also whether they change their
places, and whether they increase or decrease. He men-
tions also a new star which was produced in his days ; and
that by its motion, when it first appeared, he began to
doubt whether this did not frequently happen, and whether
those stars, which we call fixed, do not likewise move.
Hipparchus is also memorable for being the first who
discovered the precession of the equinoxes, or a very slow
apparent motion of the fixed stars from east to west, by
which in a great number of years they will appear to have
performed a complete revolution. He endeavoured also
to reduce to rule the many discoveries he made, and in-
vented new instruments, by which he marked their mag-
nitudes and places in the heavens : so that by means of
them it might be easily observed, not only whether they
appear and disappear, but also whether they pass each
other, or move, and whether they increase or decrease.

The first observations made by Hipparchus, were in the
isle of Rhodes ; whence he got the name Rhodius; but
afterwards he cultivated this science in Bithyuia and Alex-
andria only. One of his works is still extant, viz, his
Commentary on Aratus's Phenomena. He composed se-
veral other works ; and on the whole it is agreed, that
astronomy is greatly indebted to him, For laying that ra-
tional and solid foundation, on which all succeeding astro-
nomers have built their superstructure.

HIPPOCRATES, of Chios, a respectable Greek geo-
metrician, but originally it seems a merchant, who, from
his honest and unsuspecting simplicity of nature, being
greatly imposed on, and half ruined by the knavery of the
farmers of the revenue at Byzantium (now Constantino-
ple), suspended his Commerce, and repaired to Athens
to retrieve his affairs, where he first became acquainted
with geometry. Curiosity, or amusement, having one day

H I R

[ 649 ]

HOB

led him to visit one of the schools of philosophy, he so
much relished the lectures on geometry, which he there
heard, that he resolved to devote himself to the study of
that science. He soon became a distinguished geometri-
cian. The quadrature of the lunes, which he discovered,
emboldened him to attempt also that of the circle. He
also composed Elements of Geometry, which have not
however reached our times.

H1RCUS, in Astronomy, a fixed star of the first mag-
nitude, the same with Capella.

H incus is also used by some writers for a comet, en-
compassed as it were with a 'mane, seemingly rough and
hairy.

HIRE (Philip de la), an eminent French mathema-
tician and astronomer, was born at Paris in 1640. His
father, who was painter to the king, intending him for the
same occupation, taught him drawing and such branches
of mathematics as relate, to it : but died when the son was
only 17 years of age. Three years after this, he travelled
into Italy for improvement in that art, where he spent 4
years. He applied himself also to mathematics, which
at last engrossed all his attention. ' On his return to Paris,
he continued his mathematical studies with great eager-
ness, and afterwards published some works, which gained
him so much reputation, that in i67S he was named a
member of the Academy of Sciences.

Colbert the minister having formed a design for a bet-
ter chart or map of France than any which had then been
conceived, Lahire was appointed, with Picard, to make
the necessary observations for that purpose. This occu-
pied him some years in several of the provinces ; and was
the main object of his peregrinations. But he was not,
however, unmindful of other branches of knowledge, for
he philosophized on every thing that occurred, and parti-
cularly on the phenomena of the variations of the mag-
netic needle, and refractions, as also on the height of
mountains, as determined by the barometer.

In 16S3, Lahire was employed in continuing the meri-
dian line, which Picard had begun in 1669. He con-
tinued it from Paris northward, while Cassini carried it
on to the south ; but Colbert dying the same year, the
work was relinquished before it was finished. Lahire was
next employed, with other members of the Academy, in
taking the necessary levels for the grand aqueducts which
Louis the 14th was. about to make.

The great number of works published by our author,
together with his continual employments, as professor of
the Royal College and of the Academy of Architecture,
give us some idea of the great labours he underwent. His
days were always spent in study : his nights very often in
astronomical observations ; seldom seeking any other re-
lief from his labours, than a change of one for another.
In his manner, he had the exterior politeness, circum-
spection, and prudence of Italy ; on which account he
appeared too reserved in the eyes of his countrymen;
though he was always esteemed as a very honest disin-
terested man. He died in 1718, at 78 years of age.

Of the numerous works which ho published, tlie prin-
cipalare, 1. Traite deMechanique ; 1665. — 2. Nouvelle
Methode en Geometric pour les Sections des Superficies
Coniques et Cylindriques ; 1673, 4to. — 3. De Cycloide;
1"77, 12mo. — 4. Nouveaux Elemens des Scctiones Co-
niques : les Lieux Geometriqucs : la Construction, ou Ef-
fection des Equations ; 1678, 12mo. — 5. La Gnomonique,
&c; lo'S2, i2rno. — 6. Traite du Nivellement de M.Pi-
Vol. 1.

card, avec des additions ; l6"84.— 7. Sectiones Conies in
novem libros distributae; l6'S5, folio. This was consi-
dered as an original work, and gained- the author great
reputation all over Europe. — 8. Traite du Mouvement des
Eaux, &c; 1686. — 9. Tabula? Astronomies ; 16'87 and
1702, 4to. — 10. Ecole des Arpenteurs ; lo'S.9. — 11. Ve-
terum Mathematicorum Opera, Grasce et Latine, plera-
que nunc primum edita; 1693, folio. This edition had
been begun by Thevenot; who dying, the care of finishing
it was committed to Lahire. It shows that our author's
strong application to mathematical and astronomical
studies had not hindered him from acquiring a very com-
petent knowledge of the Greek tongue. Besides these,
and other smaller works, a vast number of his pieces are
diffused in Journals, and particularly in the Memoirs of
the Academy of Sciences, viz, from 1666 till the year
1718.

HOBBES (Thomas), a famous writer and philosopher,
was bom at Malmsbury in Wiltshire, in 1588, being the
son of a clergyman of that place. He completed his studies
at Oxford, and was afterwards governor to the eldest so-n
of William Cavendish earl of Devonshire, with whom he
travelled through Fiance and Italy, applying himself
closely to the study of polite literature. In 1626 his pa-
tron the earl of Devonshire died ; and in 1628 his son
also ; the same year Mr. Hobbcs published his translation
of Thucydides in English. He soon after went abroad a
second time as governor to the son of Sir Gervase Clifton;
but shortly after returned, to resume his concern for the
hopes of the Devonshire family, to whom he had so early
attached himself; the countess-dowager having desired to
put the young earl under his care, then about 13 years
of age. This charge was very agreeable to Mr. Hobbes's
inclinations, and he discharged the trust with great dili-
gence and fidelity. In 1634 he accompanied his young
pupil to Paris, where he employed his own vacant hours
in the study of natural philosophy, frequently conversing
with Father Mersenne, a man of great and deserved re-
nown, and who corresponded with almost all the learned
in Europe. From Paris he attended his pupil into Italy,
where he became acquainted with the celebrated Galileo,
who freely communicated his notions to him ; and from
hence he returned with his ward to England. But after-
wards, foreseeing the civil wars, he went to seek a retreat
at Paris; where he soon became acquainted with Des-
cartes, with whom he afterwards held a correspondence
on, several mathematical subjects, as appears from the
letters of Mr. Hobbes published in the works of Descartes.
But when this philosopher printed afterwards his Medi-
tations, wherein he attempted to establish points of the
highest consequence from innate ideas, Mr. Hobbes toojc
the liberty of dissenting from him, as did also the French
king's mathematical professor, the illustrious Peter Gas-
sendi, with whom Mr. Hobbes contracted a very close
friendship, which was not interrupted till the death of the
former.

In 1642, Mr. Hobbes printed his famous book De
Cive, which raised him many adversaries, who charged
him with instilling principles of a dangerous tendency.
Among many illustrious persons who, from the troubles
in England, retired to France for safety, was Sir Charles
Cavendish, brother to the duke of Newcastle: and this
gentleman, being well skilled in the mathematics, proved
a constant friend and patron to our author; who, by em-
barking in 1645 in a controversy about squaring the circle,
40

HOD

[ 650 ]

H O L

was become so famous by it, that in 16*47 he was recom-
mended to instruct Charles prince of Wales, afterwards
King Charles the 2d, in mathematical learning. During
this he employed his vacant time in composing his Le-
viathan, which was published in England in 1651,. After
the publication of this work, he returned to England, and
passed the remainder of his long life in a very retired and
studious manner, in the house of the earl of Devonshire,
mostly at his seat in Derbyshire, but accompanying the
earl always to London, fearing to be left out of his imme-
diate protection, lest he should be seized by officers from
the parliament or government, on account of the freedom
of his opinions in politics and religion. He received great
marks of respect from King Charles the 2d at the restora-
tion in l660, with a pension of 100/. a year. From that
time, till his death, he applied himself to his studies, and
in opposing the attacks of his adversaries, who were very
numerous : in mathematical subjects, disputes rose to a
great height between him and Dr. Wallis, on account of
his pretended Quadrature of the Circle, Cubature of the
Sphere, and Duplication of the Cube, which he obstinately
defended without ever acknowledging his error.

His long life was that of a perfectly honest man ; a
lover of his country, a good friend, charitable and obliging.
He accustomed himself much more to thinking than read-
ing ; and was fond of a well-selected, rather than a large
library. He had a hatred to the clergy, having been per-
secuted by them on account of the freedom of his doc-
trine, and having a very indifferent opinion of their know-
ledge and their principles. In his last sickness he was very
anxious to know whether his disease was curable ; and
when intimations were given, that he might have ease, but
no remedy, he said, ' I shall be 'glad to find a hole to creep
out of the world at.' He died the 4th of Dec. 1679, at
91 years of age.

His chief publications were,

1. An English translation of Thucydides's History of
the Grecian War. — 2. De MirabilibusPecci, and Memoirs
of his own Life, both in Latin verse. — 3. Elements of Phi-
losophy.— 4. Answer to Sir William Davenant's Epistle,
or Preface to Gondibert. — 5. Human Nature, orthe Fun-
damental Elements of Policy. — 6. Elements of Law. —
7. Leviathan; or the Matter, Form, and Power of a Com-
monwealth.— 8. A Compendium of Aristotle's Rhetoric.
— 9. A Letter on Liberty and Necessity. — 10. The Ques-
tions, concerning Necessity and Chance, stated. — 11. Six
Lessons to the Professors of Mathematics, of the Institu-
tion of Sir Henry Saville. — 12. The Marks of Absurd
Geometry, &c. — 13. Dialogues of Natural Philosophy. —
Besides many other pieces on polity, theology, mathema-
tics, and other miscellaneoussubjects, to the numberof41.
HOB1TS, in Gunnery. See Howitz.
HODGSON (James), f. it. s. was a respectable ma-
thematician in the early part of the 18th century, and
some lime master of the Royal Mathematical School in
Christ's Hospital, London. Besides a multitude of papers
in the Philos. Trans, from vol.24 (anno 1704), to vol. 49
inclusive, he was 'also author of several mathematical
works: as, 1. Treatise on Navigation, in 4to, 1706 ;
2. System of Mathematics, in 2 vols. 4to, 1723; 3. The
Theory of Jupiter's Satellites, 4to, 1750 ; 4. The Doctrine
of Fluxions, 4to, 1758: 5. The Valuation of Annuities
upon Lives, 4to ; 6'. An Introduction to Chronology.
And perhaps some others.— Mr. Hodgson was born 1672;
elected f. r. s. 1703 ; and died June 25, 1755.

HOGSHEAD, a measure, or vessel, of wine or oil;
containing the 4th part of a tun, the half of a pipe, or 63
gallons.

HOLDER (William), a learned and philosophical
Englishman, was born in Nottinghamshire, educated at
Cambridge, and in 1042 became rector of Blechingdon
in Oxfordshire. In 1660 fe proceeded d. d., and became
afterwards canon of Ely, fellow of the Royal Society,
canon of St. Paul's, sub-dean of the royal chapel, and sub-
almoner to the king. He was a general scholar, a very
accomplished person, and a great virtuoso. — Dr. Holder
greatly distinguished himself, by giving speech to a young
gentleman of the name of Popham, who was born deaf.
This was effected at his own house at Blechingdon in 1659;
but the young man losing what he had been taught by
Holder after he was called home to his friends, he was
sent to Dr. Waliis, who brought him to his speech again.
Holder published a book, entitled The Elements of
Speech ; an essay or inquiry into the natural Production
of Letters : with an appendix, concerning. persons that are
deaf and dumb, 1669, 8vo. In the appendix he relates
how soon, and by what methods, he brought young Pop-
ham to speak. In the Philos. Trans., for July 16'70, was
inserted a letter from Dr.'Wallis, in which he claims to
himself the honour of bringing that gentleman to speak.
By way of answer to which, in 1678, Dr. Holder pub-
lished, in 4to, A Supplement to the Philos. Trans, of
July 1670, with some reflections on Dr. Wallis's letter
there inserted. Upon which the latter soon after pub-
lished A Defence of the Royal Society, and the Philos.
Trans, particularly those of July 16"70, in answer to the
cavils of Dr. William Holder, 1678, 4to.

Dr. Holder's accomplishments were very general. He
was skilled in the theory and practice of music, and wrote
A Treatise of the Natural Grounds and Principles of
Harmony, l6"94, 8vo. He wrote also A Treatise con-
cerning Time, with applications of the Natural Day,
Lunar Month, and Solar Year, &c, 1694, 8vo. He
died at Amen Corner in London, Jan. 24, 16'97> and was
buried in St. Paul's.

HOLLOW, in Architecture, a concave moulding, about
a quarter of a circle, by some called a casement, by others
an abacus.

HoLLOw-Tower, in Fortification, is a rounding made
of the remainder of two brisures, to join the curtin to the
crillon, where the small shot are played, that they may
not be so much exposed to the view of the enemy.

HOLY Thursday, otherwise called Ascension-day, being
the 39th day after Easter Sunday,, and kept in comme-
moration of Christ's ascension into heaven.

Holy Rood, or Holy Cross, a festival kept on the 14th
of September, in memory of the exaltation of the holy
cross.

Holy Week, is the last week of Lent, called also Pas-
sion week.

IIOLYWOOD (John), or Halifax, or Sacrobosco,
an eminent mathematician of the 12th century, whose
birthplace is variously ascribed. According to Leland,
Bale, and Pitts, he was born at Halifax in Yorkshire: ac-
cording to Stanihurst, at Holywood near Dublin; and
according to Dempster and Mackenzie, in Nithsdale in
Scotland : though it is possible there might have been
more than one of the name. Mackenzie, however, informs
us, that having finished his studies, he entered intoorders,
and became a canon regular of the order of St. Augustin

H O M

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H O O

in the famous monastery of Holywood in Nitbsdale. The
English biographers, on the contrary, inform us that he
was educated at Oxford. They all agree however in as-
serting, that he spent most of his life at Paris; where,
.observes Mackenzie, he was admitted a member of the
university, June 5, 1221, under the syndics of the Scotch
nation ; and soon after was elected professor of mathema-
tics, which he taught with applause for many years. Ac-
cording to the same author, he died in 1256", as appears
from the inscription on his monument in the cloisters of
the convent of St. Maturine at Paris.

Holywood was contemporary with Roger Bacon, but
probably older by about 20 years. He wrote, 1. De
Sphtera Mundi ; a work often reprinted, and illustrated
by various commentators. — 2. De Anni Ratione, seu de
Computo Ecclesiastico. — 3. De Algorismo, printed with
Comm. Petri Cirvilli Hisp : Paris, 14£)8. The Algorismus
was probably the treatise, in technical verses, which he
composed on the Arabian arithmetic.

HOMOCENTRIC, the same as Concentric.
HOMODROMUS Vectis, or Lever, in Mechanics, is
a lever in which the weight and power are both on the
same side of the fulcrum, as in the lever of the 2d and 3d
kind ; being so called because here the weight and power
move both in the same direction, whereas in the Hetero-
dromus they move in opposite directions.

HOMOGENEAL, or Homogeneous, consisting of
similar parts, or of the same kind and nature, in contra-
distinction from Heterogeneous, where the parts are of
different kinds. — Natural bodies are usually composed of
homogeneous parts, as a diamond, a metal, &c. But arti-
ficial bodies, on the contrary, are assemblages of hetero-
geneous parts, or parts of different kinds ; as a building,
of stone, wood, &c.

Homogeneal Light, is that whose rays are all of one
and the same colour, refrangibility, &c.

Homogeneal Numbers, are those of the same kind
and nature.

Homogeneal Surds, are such as have one common
radical sign; as */27 and \/30, or */a and */b, or 2^/c
"and \/d.

HOMOGENEOUS Equations, in the doctrine of
Fluxions, particularly treated by John Bernoulli, for in-
tegrating such forms of two fluxions of the first order, in
which the sum of the dimensions of x and y, which affect
x and ji, rise to the same degree in all the terms, then the
indeterminates may be separated by a simple substitution,
as ofyc for* : such for example as thefluxional equations

(axz ■+■ xu)x = (by; + x2) i, or -7- — — = -77! —.

yJ V •* ; J' ^/(xx + yy) -/(xy-yy)

Then substituting, in the former of these, yz for x, and the

fluxion of this for x, by reduction the equation becomes

y __ + z. wjiere tnc inclcterminates are separated, and

y b-az'

the fluents may be found.

In like manner the application is made to such homo-
geneous equations as the following:
(ax ■+- by)x + (ex -+- dy)y = 0,
(ax1 ■+• bxy ■+■ cy2)x + (dxz -+- exy + fyz)y = 0,
(ax3 + bx2y ■+■ cxyz + df) x ■+■ (ex3 -t-fx2y + gxy'1 + hy3)y = 0
&c. &c.

HOMOGENEUM Adfectionis, a name given by Vieta
to the second term of a compound or affected equation,
being that which makes it adfected.

Homogeneum Comparationis, in Algebra, a name

given by Vieta to the absolute known number or term in
a compound or affected equation. This he places on the
right-hand side of the equation, and all the other terms
on the left.

HOMOLOGOUS, in Geometry, is applied to the cor-
responding sides of similar figures, or those that are op-
posite to equal or corresponding angles, and are so called
because they are proportional to each other. For all
similar figures have their like sides homologous, or pro-
portional to one another, also their areas or surfaces are
homologous, or proportional to the squares of the like
sides, and their solid contents homologous, or proportional
to the cubes of the same.

HOOF, or Ungula, in Geometry, is a part cut off a
cylinder, or cone, &c, 'by a plane passing both through
the base and part of the curve surface. It has obtained
its name from its resemblance to the hoof (ungula) of a
horse. For the contents and surfaces of such hoofs, see
my Mensuration, pa. lfjl — 183, 4th edit.

With respect to the surfaces of conic ungulas formed
by planes perpendicular to the base, Father GuidoGrandi
first remarked, that if a polygon be inscribed in the base
of a cone, and if on each side of this polygon a plane be
raised perpendicular to the base, the portion of the co-
nical surface cut off towards the axis, is equal to a recti-
lineal space. The portions also of the cone cut off by the
above planes, towards the base, are in the same ratio as
the segments of the base on which they stand. Indeed,
whatever figure be inscribed in the base, if we conceive a
right prismatic surface raised from the perimeter of the
figure, it will cut off from the conical surface a portion,
which will be to it in the same ratio, viz, that of the radius
of the base to the slant height of the cone.

HOOKE (Robert), a very celebrated mathematician
and philosopher, was born, l6'35, at Freshwater in the
Isle of Wight, where his father was minister. He was
intended for the church ; but being of a weakly constitu-
tion, and very subject to the liead-ach, all thoughts of
that nature were laid aside. Thus left to himself, the boy
followed the bent of his genius, which was turned to me-
chanics; and employed his time in making little toys,
which he executed with wonderful art and dexterity. He
had also a good turn for drawing ; for which reason, after
his father's death, which happened in l6~4S, he was placed
with Sir Peter Lely ; but the smell of the oil-colours in-
creasing his head-ach, he quitted painting in a very short
time. He was afterwards kindly taken by Dr. Busby into
his house, and supported there, while he attended West-
minster-school; where he not only acquired a competent
share of Greek and Latin, together with an insight into
Hebrew and some other Oriental languages, but also made
himself master of a good part of Euclid's Elements ; and,
as Wood asserts, invented 30 different ways of flyin<r.

About the year 1653 he went to Christ-church, Ox-
ford; and in \655 was introduced to the Philosophical
Society there; where, discovering his mechanic genius, he
was first employed to assist Dr. Willis in his chemical
operations, and was afterwards recommended to Mr.
Robert Boyle, whom he served several years in the same
capacity. He. was also instructed in astronomy about this
time by Dr. Seth Ward, Savilian professor of that science;
and henceforward distinguished himself by many noble in-
ventions and improvements of the mechanical kind. He
also invented several astronomical instruments, for making
observations both at sea and land, and was particularly
402

H O O

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H O O

serviceable to Mr. Boyle in completing the invention of
the air-pump. In 1662 he was appointed Curator of Ex-
periments to the Rojral Society; and when that body was
established by royal charter, he was in the list of those
who were first named by the council in May 20, 1663;
and was admitted accordingly June 3, with a peculiar ex-
emption from all payments. Sept. 28 of the same year,
he was named by Lord Clarendon, chancellor of Oxford,
for the degree of m. a.; and Oct. 1 9 it was ordered that
the repository of the Royal Society should be committed
to his care; the white gallery in Gresham-college being
appropriated to that use. In May 1664, he began to read
the astronomy lecture at Gresham-college, for the pro-
fessor Dr. Pope, then in Italy; and the same year he was
made professor of mechanics to the Royal Society by Sir'
John Cutler, with a salary of 50/. per annum, which that
gentleman, the founder, settled upon him for life. Jan. 11,
l665, that society granted a salary also of 30/. a year, for
his office of curator of experiments for life'; and the month
of March the same year he was elected professor of geo-
metry in Gresham-college.

In 1665 he published in folio, his Micrographia, or
some Philosophical Descriptions of Minute Bodies, made
by Magnifying Glasses, with Observations and Inquiries
on them. And the same year, during the recess of the
Royal Society on account of the plague, he attended Dr.
Wilkins and other ingenious gentlemen into Surrey, where
they made several experiments. In 1666 he produced to
the Royal Society a model for rebuilding the city of Lon-
don, then destroyed by the great fire, with which the so-
ciety was well pleased; and the lord mayor and aldermen
preferred it to that of the city surveyor, though it hap-
pened not to be carried into execution. The rebuilding
of the city, according to the act of parliament, requiring
able persons to set out the ground for the proprietors, Mr.
Hooke was appointed one of the surveyors ; an employ-
ment in which he acquired most part of his estate, as ap-
peared from a large iron chest of money found after his
death, locked down with a key in it, and a date of the
time, which showed it to have been so shut up above 30
years. From 166S he was engaged for niany years in a
warm contest with Hevelius, concerning the difference in
accuracy between observing with astronomical instruments
with plain and telescopic sights; in which dispute many
learned men afterwards engaged, and in which Hooke ma-
naged so ill, as to be universally condemned, though it
• has since been agreed that he had the better side of the
question. In 1 67 1 he attacked Newton's New Theory of
Light and Colours; where, though he was obliged to
submit in respect to the argument, it is said he came off
with more credit. The Royal Society having commenced
their meetings at Gresham-college, November l6'74, the
committee in December allowed him 40/. to erect a turret
over part of his lodgings, for trying his instruments, and
making astronomical observations: and the year follow-
ing he published A Description of Telescopes, and some
other instruments made by R. II. with a Postscript, com-
plaining of some injustice done him by their secretary Mr.
Oldenburg, who published the Philosophical Transactions,
in regard to his invention of pendulum watches. This
charge engaged him in a dispute with that gentleman,
which ended in a declaration of the Royal Society in their
secretary's favour. Mr. Oldenburg dying in 16'77, Mr.
Hooke was appointed to supply, his place, and began to
take minutes at the meeting in October, but did not

publish the Transactions. Soon after this, he became
more reserved than formerly ; and though he read his
Cutlerian Lectures, often made experiments, and showed
new inventions before the Royal Society, j'et he seldom
left any account of them to be entered in^their registers ;
designing, as he said, to publish them himself, which how-
ever he never performed. In 16S6, when Newton's work
the Principia was published, Hooke laid claim to his dis-
covery concerning the force and action of gravity, which
~was warmly resented by that great philosopher. Hooke,
though a great inventor and discoverer himself, was yet
so envious and ambitious, that he would fain have been
thought the only man who could invent and discover any
thing. This made him often lay claim to the inventions
and discoveries of other persons; on which occasions, how-
ever, as well as in the present case, the thing was gene-
rally carried against him.

In the beginning of the year l6S7, his brother's daugh-
ter, Mrs. Grace Hooke, who had lived with him several
years, died : and he was so affected with grief at her death,
that he never recovered it, but was observed from that
time to become less active, more melancholy, and more
cynical than ever. At the same time, a chancery suit, in
which he was concerned with Sir John Cutler, on account
of his salary for reading the Cutlerian Lectures, made him
uneasy, and increased his disorder. In 1691, he was em-
ployed in forming the plan of the hospital near Hoxton,
founded by Robert Ask, alderman of London, who ap-
pointed Archbishop Tillotson one of his executors; and in
December the same year, Hooke was created m.d. by a
warrant from that prelate. In July 1696, the chancery
suit with Sir John Cutler was determined in his favour, to
his inexpressible satisfaction. His joy on that occasion
was found in his diary thus expressed; domshlgissa ;
that is, Deo, Optimo, Maximo, sit honor, laus, gloria, in
sxcula scEculorum, Amen. " I was born on this day of July
1635, and God hath given me a new birth : may I never
forget his mercies to me! while he gives me breath may I
praise him!" In the same year 1696, an order was
granted to him for repeating most of his experiments at the
expense of the Royal Society, on a promise of his finish--
ing the accounts, observations, and deductions from them,
and of perfecting the description of all the instruments
contrived by him: but his increasing illness, and general
decay, rendered him unable to perform it. He continued
some years in this wasting condition ; and thus languish-
ing till he was quite emaciated, he died March 3, 1702, in
his 67th year, at his lodgings in Gresham-college, and was
buried in St. Helen's church, Bishopsgate-street ; his corps
being attended by all the members of the Royal Society
then in London.

With regard to Mr. Hooke's character, it is not in all re-
spects one of the most amiable. In his person he made
rather a despicable figure, being but of a short stature,
very crooked, pale, lean, and of a meagre aspect, with
dark-brown hair, very long, and hanging over his face, lank
and uncut. Suitable to his person, his temper was pe-
nurious, melancholy, and mistrustful : and, though pos-
sessed of great philosophical knowledge, and strong inven-
tion, he had so much ambition, that he would be thought
the only man who could invent or discover; and hence he
often laid claim to the inventions and discoveries "f others,
while he boasted of many of his' own which-he never com-
municated. In the religious part of his character, he was
so exemplary, that he always expressed a great veneration

HOR

[ 6S3 ]

HOR

for the Deity; and seldom received any remarkable benefit
in life, or made any considerable discovery in nature, or
invented any useful contrivance, or discovered any difficult
problem, without noting some professions of acknowledg-
ment to God ; as many places in his diary plainly show.
His chief publications are,

1. Lectiones Cutlerianas, or the Cutlerian Lectures. — 2.
Micrographia, or Descriptions of Minute Bodies made by
Magnifying Glasses.- — 3. A Description of Helioscopes. —

4. A Description of some Mechanical Improvements of
Lamps and Water-poises. — 5. Philosophical Collections.
— 6- Posthumous Works, collected from his papers by
Richard Waller, secretary to the Royal Society. Besides
a number of papers in the Philos. Trans, volumes 1, 2, 3,

5, 6, 9, 16, 1/, 22.

HORARY, something relating to Hours. As,
Horary Circles, hour lines, marking the hours, or
drawn at the distance of hours from one another.

Horary Motion, is the motion or space moved in an
hour. Thus, the horary motion of the earth on her axis,
is 15° ; for, completing her revolution of 360°, in 24 hours,
therefore the motion in one hour will be the 24th part of
3fj0°, which is 15 degrees.

higher as at b, the tangents bh and br will reach farther,'
and the visible horizon hor will be larger.

The visible horizon is most accurately observed at sea,
and is sometimes called the horizon of the sea. In observ-
ing this horizon, the visual rays aIi and at will, on account
of the curve surface of the sea, always point a little below
the true sensible horizon ss or ef, and consequenlly below
the rational horizon tt, which is parallel to it.

To find the depression of the horizon of the sea below
the true horizon, which varies with the height of the eye,
and in a small degree with the variation of the refractr

lve
HORIZON, a great circle of the sphere, dividing the power of the atmosphere, see Depression.

world into two parts, or hemispheres ; the one upper, and
visible; the other lower, and hid. The horizon is either
rational or sensible.

Horizon, Rational, True, or Astronomical, called also
simply and absolutely the horizon, is a great circle having
its plane passing through the centre of the earth, and its
poles are the zenith and nadir. Hence all the points of
the horizon, quite around, are at a quadrant distance from
the zenith and nadir. Also the meridian and vertical
circles cut the horizon at right angles, and into two equal
parts.

Horizon, Apparent, Sensible, or Visible, is a lesser circle
of the sphere, parallel to the rational horizon, dividing the
visible part of the sphere from the invisible, and whose
plane touches the spherical surface of the earth. The
sensible horizon is divided into. eastern and western; the
eastern or ortive being that in which the heavenly bodies
rise; and the western, or occidual, being that in which
they set.

Horizon, in Geography, is a circle dividing the visible
part of the earth and heavens from that which is invisible.
This is peculiarly called sensible or apparent horizon, to
distinguish it from therational or true, which passes through
the centre of the earth; as already observed. These two
horizons, though distant from one another by the semidi-
ametcr of the earth, will appear to coincide when continued
to the sphere of the fixed stars ; because the eai th compared
with this sphere is but a point.

By sensible horizon is also often meant a circle which
determines the segment of the surface of the earth, over '
which the eye can reach ; called also the physical horizon.
And in this sense we say, a spacious horizon, a narrow or
scanty horizon, &c; depending chiefly on the height the
eye is elevated above the earth.

For, it is evident that the higher the eye is placed, the
farther is the visible horizon extended. Thus, if the eye
be at a, at the height ad above the earth; draw the two
tangents aIi, Ar ; and let one of these lines a/i, be moved
round the point a, and in its revolution always touch the
surface of the earth ; then the other point h will describe
the visible horizon hor, &c. But if the eye be placed

As to the right-lined distance, or tangent Ah, it may be
found thus; as radius : sin. .£ c : : ca : aIi, or thus;

as radius : tan. Z. c : : ch : a/i; either of which will be
nearly the same as the arc or curved distance vh. Or,
without finding the angle e, thus; the square of Ah is
equal to the difference of the squares of ca and ch, i. e.
a/i2 = ca1 — ch2 = (ca -+- ch) x (ca — ch) = (ca -(- ch)
x ad, and hence a/i = ^/(ca -t- ch x ad), which is also
equal to the curve d/« nearly.

The distance on a perfect globe, if the visual rays came
to the eye in a straight line, would be as above stated :
but by means of the refraction of the atmosphere, distant
objects on the horizon appear higher than they really are,
or appear less depressed below the true horizon ss, and
may be seen at a greater distance, especially on the sea.
M. Legcndre, in his Memoir on Measurements of the
Earth, in the Mem. Acad. Sci. for the year 1787, says
that, from several experiments, he is induced to allow for
refraction a 14th part of the distance of the place ob-
served, expressed in degrees and minutes of a great circle.
Thus, if the distance be 14000 toises, the refraction will
be 1000 toises, equal to the 57th part of a degree, or
l' 3". But the English, and other later measurements,
make the allowance a 12th part; which therefore on the
same distance will be ll6"7 toises, or the 49th part of a
degree, equal to l' 13".

Horizon of the Globe, the broad wooden circle.

HORIZONTAL, something that relates to the horizon,
or that is taken in the horizon, or on a level with or pa-
rallel to it. Thus, we say, a horizontal plane, horizontal
line, horizontal distance, &c.

Horizontal Dial, is one drawn on a plane parallel to
the horizon; having its gnomon or style elevated according
to the altitude of the pole of the place it is designed for.

Horizontal Distance, is that estimated in the direc-
tion of the horizon.

Horizontal Line,\n Perspective, is a right line drawn
through the principal point, parallel to the horizon; or it
is the intersection of the horizontal and perspective
planes.

Horizontal Line, or base of a hill, in Surveying, a

H O R

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HOE

line drawn on the horizontal plane of the hill, or that on
which it stands.

Horizontal Moon. See Apparent Magnitude.

Horizontal Parallax. See Parallax.

Horizontal Plane, is that which is parallel to the
horizon of the place, or not inclined to it.

Horizontal Plane, in Perspective. See Plane.

Horizontal Projection. See Projection, and Map.

Horizontal Range, of a piece of ordnance, is the dis-
tance it throws its shot on a horizontal plane, whatever be
the angle of elevation or direction of the piece. When
the piece is pointed parallel to the horizon, the range is
then called the point-blank or point-blanc range.— The
greatest horizontal range, in the parabolic theory, or in a
vacuum, is that made with the piece elevated to 45 de-
grees, and is equal to double the height from which a
heavy body must freely fall, to acquire the velocity with
which the shot is discharged. Thus, a shot being dis-
charged with the velocity of v feet per second; because
gravity generates the velocity 2g or 32-g feet in the first
second of time, by falling 16-^ or g feet, and because the
spaces descended are as the squares of the velocities, there-
fore as 4,g- : vz : : g : — the space a body must descend to
acquire the velocity v of the shot, or the space due to the
velocity v ; consequently the double of this, or — = — —

is the greatest horizontal range with the velocity v, or at
an elevation of 45 degrees; which is nearly half the square
of a quarter of the velocity.

In other elevations, the horizontal range is as the sine
of double the angle of elevation ; so that, any other eleva-
tion being e, it will be, as radius 1 : sin. 2e : : — : — - x

sin. 2e, the range at the elevation e, with the velocity v.

But in a resisting medium, like the atmosphere, the
actual ranges fall short of those expressed by the above
theorems, in so much that with the great velocities, the
actual or real ranges may be less than the 10th or 20th
part of the potential ranges ; so that some balls, which
actually range but a mile or two, would in vacuo range
20 or 30 miles. And hence also it happens that the ele-
vation of the piece, to shoot farthest in the resisting medium,
is always below 45°, and gradually the more below it, as
the velocity is greater; so that the greater velocities with
which balls are discharged from cannon with gun-powder,
require an elevation of the gun equal to but about 30°, or
little more. And the less the size of the balls is too, the
less must this angle of elevation be, to throw them the far-
thest with a given velocity. See Projectile, and Gun-
nery.

Horizontal Refraction. See Retraction.

Horizontal Speculum, one to find a horizon at sea,
&c, when the atmosphere is hazy near the horizon, by
which the sight of it is prevented. A speculum of this
kind was invented by a Mr. Serson, on the principle of a
top spinning, which always keeps its upright position, not-
withstanding the motion of the substance it spins upon.
This curious instrument, as it has since been improved by
Mr. Smeaton, consists of a well polished metal speculum,
of about 3 inches and a half in diameter, inclosed within
a circular rim of brass ; so fitted that the centre of gravity
of the whole shall fall near the point on which it spins.
This is the' end of a steel axis running through the centre
of the speculum, above which it finishes in a square, for

the conveniency of fitting a roller on it, which sets it in
motion by means of a piece of tape wound about the roller.
Various other contrivances to form artificial horizons have
been invented by different persons, as glass planes floating
on mercury, &c. See Hadley's Quadrant, and seve-
ral inventions of this sort by Elton, Halley, Leigh, &c, in
-the Philos. Trans. vol.37,pa. 273, vol.38, pa. l67,vol.40,
pa. 413,417, &c.

HORN-Work, in Fortification, a kind of out-work,
advancing towards the field, to cover and defend acurtain,
bastion, or other place, suspected to be weaker than the
rest ; as also to possess a height ; carrying in the fore-part,
or head, two demi-bastions, resembling horns: these horns,
epaulments, or shoulderings, being joined by a curtin,
shut up on the side by two wings, parallel to each other,
are terminated at the gorge of the work, and so present
themselves to the enemy.

HOROGRAPHY, the art of making or constructing
dials; called also Dialling, Horologiography, Gnomonica,
Sciatherica, Photosciatherica, &c.

HOROLOGIUM, a common name, among ancient
writers, for any instrument or machine for measuring the
hours. See Clock, Watch, Sun-Dial, Chronometer,
Clepsydra, &c.

HOROLOGY, that branch of mechanical science which
enables us to measure the portions of time. We judge of
the lapse of time by the succession of sensible events ; and
the most convenient and accurate measures of its quan-
tity are derived from motions ; either uniform, or else re-
peated at equal intervals. Of the former kind, the rotation
of the earth on its axis is the most exact, and the situa-
tion of its surface with regard to the fixed stars, or less
simply," with regard to the sun, constitutes the means for
determining the parts of time as they follow each other.
Of the latter kind, the rotation of machinery, consisting
of wheel-work moved by a weight or spring, and regulated
by a pendulum or balance, affords instruments of which
the utility is well known. The ' term horology is at pre-
sent more particularly confined to the principles on which
the art of making clocks and watches is established. A
considerable portion of this extended subject of research is
given under the article Clock. See also Scapement, Watch,
&c.

HOROMETRY, the art of measuring or dividing time
by hours, and keeping the account of time.

HOROPTER, in Optics, is a right line drawn through
the point where the two optic axes meet, parallel to that
which joins the centres of the two eyes, or the two pupils.
As the line ab drawn through c the point of concourse of

the optic axes of the eyes, and parallel to m joining the
centres of the eyes. — This line is called the horopter, be-
cause it is found to be the limit of distinct vision. It has

H O R

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H O R

several properties in Optics, which are described at large
in Aguillonius, Opt. lib. 2, diss. 10.

HOROSCOPE was a mathematical instrument, in man-
ner of a planisphere; but now disused. It was invented
by J. Paduanus, who wrote a special treatise upon it.

HORROR of a Vacuum, an imaginary principle among
the more ancient philosophers, to which they ascribed the
ascent of waterin pumps, and other similar phenomcna,which
are now known to be occasioned by the weight of the air.

HORROX (Jeremiah), an eminent English astrono-
mer, was born at Toxteth in Lancashire, about the year
1619. From a grammar school in the country, he was
sent to Cambridge, where he spent some time in academi-
cal studies. About 16.33 he began to apply himself to the
"study of astronomy: but living at that time with his fa-
ther at Toxteth, in very moderate circumstances, and
being destitute of books and other assistances for such
studies, he could not make any considerable progress in
it. About the year 1636, he formed an acquaintance
with Mr. William Crabtree, of Broughton near Manches-
ter, who was engaged in the same studies, with whom a
mutual correspondence was carried on till his death;
sometimes communicating their improvements to Mr.
Samuel Foster, professor of geometry at Gresham College
in London. Having now obtained a companion in his
studies, Mr. Horrox assumed new vigour, procured other
instruments and books, and was pursuing his studies and
observations with great assiduity, when he was suddenly
cut off by death, the 3d of January l640, in the 22d year
of his age.

What we have of his writings is sufficient to show how
great a loss the world had by his death. He had just
finished his Venus in Sole visa, l639, a little before, as ap-
pears by some of the letters to his friend Mr. Crabtree, by
which also it appears that he made his observations on
that phenomenon at Hool near Liverpool. This tract, of
Venus seen in the Sun, was published at Dantzick in 1668,
by Hcvelius, together with his own Mercurius in Sole
visus May 3, l66i. His other posthumous works, or
rather his imperfect papers, were published by Dr.Wallis,
in 1673, 4to, with some account of his life; in which we
find he first asserts and promotes the Keplerian astronomy
against the hypothesis of Lansberg; which he proves to be
inconsistent wiih itself, and neither agreeing with observa-
tions nor theory. He likewise reasons very justly con-
cerning the celestial bodies and their motions, vindicates
Tycho Brahe from some objections made to his hypothe-
sis, and gives a new theory of the moon : to which are
added the Lunar Numbers of Mr. Flamsteed. There are
also extracts from several letters between him and Mr.
Crabtree, upon various astronomical subjects ; with a ca-
talogue of astronomical observations.

There are two things particularly which will perpetuate
the memory of this very extraordinary young man. The
one is, that he was the first that ever predicted or saw the
planet Venus in the sun ; for we do not find that any per-
sons, besides himself and Mr. Crabtree, ever beheld such
a phenomenon. Though he was not apprised of the great
use that was to be made of this phenomenon, in discover-
ing the parallax and distance of the sun and planets, yet
he made from it many useful observations, corrections,
and improvements in the theory of the motions of Venus.
— Secondly, his New Theory of Lunar Motions, which
Newton himself made the ground-work of all his astro-
nomy, relative to the moon, who always spoke of our au-
thor as a genius of the first rank.

HORSE-Shoe, in Fortification, is a work sometimes of
a round, sometimes of an oval figure, inclosed with a pa-
rapet, raised in the ditch of a marshy place, or in low
grounds; sometimes also to cover agate; or to serve as a
lodgment for soldiers, to prevent surprises, or to relieve
an over-tedious defence.

HORSLEY, (Samuel, ll.d. and f. r. s.), a very
able mathematician and a learned prelate, was born in
October 1732, and died October 4, 1806; consequently
about 74 years of age. — He received the elements of his
education at Westminster-school, whence he was removed
to the university of Cambridge. There he chiefly ap-
plied to the study of mathematics; and not content with
carefully consulting the writings of the acutest of the
moderns, he recurred to the profoundest of the ancients
in that line, and made himself master of their most intri-
cate reasonings. Having taken his degree as master of
arts, he accepted an invitation to accompany the late Earl
of Aylesford, as private tutor, to Oxford. Here, in 1769,
Dr. Horsley printed his edition of the Geometrical Incli-
nations of Apollonius, and here he first conceived the de-
sign of publishing a complete edition of the works of Sir
Isaac Newton, for which he began to prepare the mate-
rials. And from that time, it has been more the fashion
to cultivate the mathematical sciences at that university.
On leaving this place, Dr.Horsley came to London, where
he was elected F. r. s. and enriched the Philos. Trans,
with many valuable essays, from vol. 57, to vol. 66". He
was also chosen one of the secretaries in 1773, on the re-
signation of Dr. Morton ; an office he continued to serve,
with the greatest credit to himself and benefit to science,
till he thought it proper to resign along with the late pre-
sident Sir John Pringle, in 1778. Soon after settling in
the metropolis, Dr. Horsley was noticed by Bishop Louth,
and was appointed his lordship's domestic chaplain, as
well as was presented by that prelate successively to seve-
ral different livings. In 1776, Dr. Horsley published
proposals for a complete edition of the works of the
immortal Newton, with notes, which appeared in L779,
in 5 vols. 4to, with an elegant Latin dedication to the
king. In 177S he preached and published^ a sermon
against the principles of materialism, agitated between the
doctors Price and Priestley &c : and on the publication,
in 1783, of Dr. Priestley's History of the Corruptions of
Christianity, Dr.Horsley again attacked the positions and
principles of that writer and his adherents, with great ap-
plause and success, and there ensued an exchange of seve-
ral pamphlets, &c. on both sides; which chiefly laid the
foundation of Dr. Ilorsley's elevation to the episcopal dig-
nity in 1788, assisted probably, by the very celebrated
speeches he made in the course of the violent debates, that
took place in the Royal Society 2 or 3 years before, on
the occasion of the removal of Dr. Hutton from the office
of foreign secretary, when Dr. Horsley, and several other
members, forsook, as expressed in his own forcible lan-
guage, " that temple where Philosophy once reigned, and
where Newton presided as her officiating minister." In
179*> Dr. Horsley was translated from St. David's to the
see of Rochester, with the deanery of Westminster; and in
1802, to the lucrative see of St. Asaph. In 1801 he pub-
lished the first, and in 1S03 the last, of 3 octavo volumes
of practical mathematics, for the use of students ; consist-
ing of Euclid's Elements and Data, with notes; the pro-
perties and projection of the sphere, and spherical trigo-
nometry ; Archimedes on the mensuration of the circle ;
a tract on the nature and use of logarithms, &c. Besides

HOU

[ 656 ]

HOU

these, numerous and important were the writings and la-
bours of Dr. Horsley in theology, politics, &c, too nume-
rous to be here entered into in detail.

Doctor Horsley went down to Brighthelmstone in the
latter end of September 1806, where he had not been two
weeks, before he was seized with a complaint in the
bowels, then frequent in.the place, which soon turned to a
mortification ; and terminated his valuable existence on
the 4th day of his illness. Though Doctor Horsley had
reached the 74th year of his age, the powers both of his
body and mind were so vigorous, as to promise still a con-
siderable length of years; indeed he was always of a very
remarkably active and energetic cast of mind, and firm
compact constitution of body. Perhaps no man' of the
age, possessed more of what is termed recondite learning; or
was more profoundly versed in classical chronology. As
a senator, he ranked in the first class; there were few
important discussions in the house of lords, in which he
did not participate; especially when the topics related to
the hierarchical establishments of the country, or to the
French revolution, or to the African slave trade, of which
he was a systematic opposcr. As an orator, his voice was
deep, full-toned, and commanding; his enunciation was
distinct and clear; and his delivery in other respects,
highly advantageous ; abating only for a remarkable mix-
ture of certain guttural sounds, such as are almost pecu-
liar to the inhabitants of the. county of Northumberland ;
occasioned, he used to say, by the circumstance of a nurse
from that part of England being "long employed in his fa-
ther's family.

HOSPITAL, or Hopital (William-Francis An-
thony, marquis of), a celebrated French mathematician,
was born of an ancient family in l66l. He was a mathe-
matician almost from his infancy; for being one day at
the duke of Rohan's, where some able mathematicians
were speaking of a problem of Pascal's, which appeared to
them very difficult, he ventured to say, that he believed
he could resolve it. They were surprised at such pre-
sumption in a boy of 15 : however, in a few days he sent
them the solution. M. l'Hospital entered early into the
army, and ^as a captain of horse; but being' very short-
sighted, and on that account exposed to perpetual incon-
veniences and errors, he at length quitted the army, and
applied himself entirely to his favourite amusement. — He
contracted a friendship with Malbranche, and took his
opinion upon all occasions. — In l6S»9 he was received an
honorary member of the Academy of Sciences at Paris. —
He was the first person in France who wrote upon New-
ton's analysis, and on this account was regarded almost as
a prodigy. His work was entitled l'Analyse des Infini-
niens Petits, lo'oG. He engaged afterwards in another
mathematical work, in which he included Les Sections
Coniques, les Lieux Geometriques, la Construction des
liquations, et une Theoric des Courbes Mcchaniques:
but, a little before he had finished it he was seized with a
fever, which carried him off, the 2d of February 1704, at
43 years of age. The work was published after his death,
viz, in 1707. Six of his pieces are inserted in different vo-
lumes of the Memoirs of the Academy.

HOUR, in Chronology, an aliquot part of a natural
day, usually the 24th, but sometimes a 12th part. With
us, it is the 24th part of the earth's diurnal rotation, or
the time from noon to noon, and thcrefore'it answers to 15
degrees of the whole- circle of longitude, or of 36'0L'. The
hour is divided by 6'0ths, viz, first into Co minutes, then'
each minute into O'O seconds, &c. The division of the day

into hours is very ancient ; as is shown by Kircher, Oedip.
yEgypt. torn. 2, par. 2, class 7, cap. S. The most ancient
hour is that of the 12th part of the day. Herodotus,
lib. 2, observes, that the Greeks learnt from the Egyptfans,
among other things, the method of dividing the day into
12 parts. And the astronomers of Cathaya, &c, still re-
tain this division.

The division of the day into 24 hours, was not known
to the Romans before the Punic war. Till that time they
only regulated their days by the rising and setting of the
sun. They divided the 12 hours of their day in four; viz,
Prime, which commenced at 6 o'clock, Third at 9, Sixth
at 12, and None at 3. They also divided the night into
four watches, each containing 3 hours. Various kinds of
hours are used by chronologers, astronomers, dialists, &c.
Sometimes too,

Hours are divided into equal and unequal.
Equal Hours-, are the 24th parts of a day and night
precisely ; that is, the time in which the 15 degrees of the
equator pass the meridian. These are also called equinoc-
tial hours, because measured on the equinoctial ; and astro-
nomical, because used by astronomers.

Astronomical Hours, arc equal hours, reckoned from
noon to noon, in a continued series of 24.

Babylonish Hours, are equal hours, reckoned from sun-
rise in a continued series-of 24.

European Hours, used in civil computation, are equal
hours, reckoned- from midnight; 12 from thence till noon,
and 12 more from noon till midnight.

Ancient or Jewish Hours, are twelfth parts of the artifi-
cial day, or of the night. Hence, as it is only at the time
of the equinoxes, that the artificial day is equal to the
night, it is then only that the hours of the day are equal
to those of the night, or to the 24th part of the natural
day. From the vernal to the autumnal equinox, the
hours of the day exceed those of the night; but during
the interval between the autumnal and vernal equinoxes,
the hours of the night are longer than those of the day.
It is therefore manifest, that when it is said the third hour
was about nine in the morning, and the ninth about three
in the afternoon, this is not to be understood as rigorously
exact. The third hour was the middle time between sun-
rising and noon ; which, if the sun rose a.tfizie, was half an
hour after eight; if at seven was half an hour after nine, Sfc.

The following table exhibits the time of the sun-rising
and setting, and the length of the Jewish hour, both of day
and night; as calculated for about the middle of every
Jewish month, and for the latitude of Jerusalem.

Length of die Hour

Names of ihe Months.

Sun

Sun sets.

of .lay.

of

light.

11.

a.

ll- in-

h. m-

1.

m-

Nisan, or Abib

5

4()

6 14

1 0

0

5S

Ijar, or Zif -

5

20

6' 40

1 7

0

53

Sivan

5

0

7 0

1 10

0

50

Thamuez

4

56

7 4

1 11

0

4.9

Ab

5

10-

6 50

1 s

0

52

Elul -

5

36

6 21

1 4

0

56

Tisri,or Ethanim -

(»'

6

5 54

0 59

1

1

Nearchevvan,or Bui

(i

34

5 2<>

0 54

1

6"

Chisleu

6

55

5 5

0 51

1

9

Tebcth

7

0

5 0

0 50

1

10

Shebat

a

42

5 18

0 53

1

7

Adar

6

1(>

5 44

0 57

1

3

HUE

[ &7 J

HUE

Italian Hours, are equal hours, reckoned from sunset,
in a continued series of 24.

Unequal or Temporary Hours, are 1 2th parts of the ar-
tificial day and night. The obliquity of the sphere ren-
ders these more or less unequal at different times; so that
they only agree with the equal hours at the times of the
equinoxes..

tiov R-Circles, or Horary- Circles, are great circles,
meeting in the poles of the globe or world, and crbssing
the equinoctial or equator at right angles; the same as
meridians. They are supposed to be drawn through every
15th degree of the equinoctial and equator, each answer-
ing to an hour, and dividing them into 24 equal parts;
and on both globes they are supplied by the meridian
hour-circle and index.

HoVR-Glass, a popular kind of chronometer or clepsy-
dra, serving to measure time by the descent or running of
sand, water, &c, out of one glass vessel into another. The
best, it is said, are such as, instead of sand, have egg-
shells, well dried in the oven, then beaten fine and sifted.

HovR-Lines, on a dial, are lines which arise from the
intersections of the plane of the dial, with the several
planes of the hour-circles of the sphere ; and therefore
must be all right lines on a plane dial.

HouR-Scale, a divided line on the edge of Collins's qua-
drant, being only two lines of tangents of 45 degrees each,
set together in the middle. Its use, together with the
lines of latitude, is to draw the hour-lines of dials that have
centres, by means of an equilateral triangle, drawn on the
dial-planes.

HOWITZ, or Howitzer, in artillery, a kind of mor-
tar, or something between a cannon and mortar, partaking
of the nature of both, being either a very short gun or a
long mortar. It is of German invention, and is mounted
on a carriage like a travelling gun-carriage, with its trun-
nions placed nearly in the middle. The howilz is one of
the most useful kinds of ordnance, as it can be employed
occasionally either as a cannon or mortar, "discharging
either shells or grape-shot, as well as balls, and so doing
great execution.- They are also very easily travelled about
from place to place.

iHUDDE (John), a burgomaster of Amsterdam, and
a respectable mathematician, who died at a great age in
1704. Having been as great in politics as in mathema-
tics, he served his country in distinguished situations, and
contributed, by his discoveries, to the advancement of the
sciences. He applied himself particularly to the analysis
of equations. One of his two letters, inserted in Schooten's
Commentary on the Geometry of Descartes, contains a
very ingenious method for discovering whether an equa-
tion of any dimension has equal roots, and for determining
those roots. In the same letters we find Hudde's methods '
of drawing tangents to curves, and of investigating the-1
maxima and minima in mathematics.

HUES (Robert), was born at Little Hereford, in He-
refordshire, about 1552, and studied at Oxford. He be-
came eminent for his knowledge in geography, navigation,
and mathematics ; which acquired him particularly the
favour of that noble Maecenas, Henry carl of Northumber-
land, being one of the earls so called three magi ; from
whom he enjoyed a liberal pension, for the encouragement
and promotion of his studies. He died in 1632, at 79
years of age, and was buried at Christ's Church, Ox-
on. — His publications were chiefly two; viz. 1, Brevia-
rium totius Orbis; printed several times at London, in

Vol. I.

Latin and English, in Svo and 12mo. — 2. De Globis ce-
lest. et terrest. Tract. 2; several times printed in Latin
and English, in 8vo. The first edition was printed at
London 1593, in Svo. It was afterwards improved by
annotations, and illustrated with figures, by Joh. Isaac
Pontanus, professor of philosophy at Harderwicke, in Gel-
derland, p'rinted at Amsterdam, l6l7, l6~24, &c, at Lon-
don, 16'3S>, and at Oxon, l6'63, in Svo and 12mo. See
Wood, vol. 1, pa. 4S9.

Hues was a man of letters, and had acquired a practi-
cal knowledge" of maritime affairs, by voyages at sea.
Among other curious particulars, he gives a good account
of the attempts that had been made at various times to
measure the earth : in the epistle to Sir Walter Raleigh,
he enumerates the many discoveries made by our mari-
ners in different parts of the world. His book was re-
ceived with great applause, and has been indeed a pattern
lor such as afterwards treated on the same subject. Among
other problems, he solves by the globe, the problem of
determining the latitude, from two observations of the
sun's altitude, when the time between the two observa-
tions is given ; and in the last part of his book, he per-
forms the usual questions in navigation, premising a very
sensible discourse on the rhumb-lines, opposing the opi-
nion of Gemma Frisius, who had asserted, that they meet
in the poles. At the conclusion, he highly praises a trea-
tise of Mr. Thomas Hariot, hoping it would soon be pub-
lished, in which that author had treated this subject on
geometrical principles, with great sagacity and judgment.
But unfortunately that manuscript has not been found
among the papers of that author., lately discovered by Dr.
Zach.

It was in this book that Hues proposed a curious nau-
tical problem, which has made a considerable sensation in
the mathematical world. It was in these words, as trans-
lated into English : " The difference of longitude and the
distance being given, how to find the rumbe, and the dif-
ference of latitude." After which he adds these words :
" There is not anything in all this art more difficult and
hard to be found, than the rumbe out of the distance and
difference of longitude given. Neither can it be done
upon the globe without long and tedious practice, and
many repetitions and mensurations."

The next person who appears to have mentioned this
problem, was Willebrord Snell, professor of mathematics
at Leyden, in his Tiphys Batavus, printed in 1624, being
the 34th or last proposition of the first book of that work.
Snell solved this problem in an indirect way ; and it seems
probable that his solution was the first that ever was given
of it.

The problem was afterwards proposed to the learned
world, by the celebrated Dr. Halley, in the year 1696, in
the Philos. Trans, vol. 19, No. 2 It); whence it was repeat-
ed in the 2d vol. of the Miscellanea Curiosa in I7O8, as
he had probably not seen it in either of the foregoing au-
thors. Dr. Halley proposes it in the following words:
" A,,ship sails from a given latitude, and, having run a
certain number of leagues, has altered her longitude by a
given angle; it is required to find the course steered."
And he then adds, " The solution hereof would be very
acceptable, if not to the public, at least to the author of
this tract, being likely to open some further light into the
mysteries of geometry."

Since that time, this problem has been solved in an in-
direct manner, by several writers on navigation, and
4P

H U Y

t 658 ]

HYD

others : — As by M. Bouguer, in his Nouveau Traite de
Navigation ; by Mr. Robertson, in the 2d vol. of liis Ele-
ment of Navigation ; by Mr. Emerson, in his Theory of
Navigation ; by Mr. Israel Lyons, in the Nautical Alma-
nack, 1772; and by M. Bezout, in his Traite de Naviga-
tion ; and lately, Baron Maseres, with the assistance of
Mr. Atwood, has given a direct solution of this problem,
being the first direct solution that has been given of it.
All which solutions may be seen in the 4th vol. of Baron
Maseres's work, entitled Scriptores Logarithmic!.

HUMIDITY, or moisture, the power or quality of wet-
ting or moistening other bodies, and adhering to them.
Fluids are moist to some bodies, and not to others. Thus,
quicksilver is not moist in respect to our hands or clothes,
and other substances, which it will not adhere to; but it
may be called humid in reference to gold, tin-, or lead, to
the surfaces of which it will presently adhere, and render
them soft and moist. Even water itself, which wets almost
every thing, and is the great standard of moisture and hu-
midity, is not capable of wetting all things; for it stands
or runs off in globular drops from any thing greased or
oiled, or the leaves of cabbages, and many other plants;
and it will not wet the feathers of ducks, geese, swans,
and other water-fowl.

HUNDRED, the number of ten times ten, or the
square of 10. The place of hundreds makes the third in
order in Arabic or modern numeration, being denoted
thus, 100. In the Roman notation it is denoted by the
letter c, being the initial of its name, Centum.

Hundred Weight, or the great hundred, contains 112
pounds weight. It is subdivided into 4 quarters, and each
quarter into 28 lbs.

HURTERS, in Fortification, denote pieces of timber,
about 6 inches square, placed at the- lower end of the
platform, next to the parapet, to prevent the wheels of
the gun-carriages from damaging the parapet.

HUYGENS (Christian), a very eminent astronomer
and mathematician, was born at the Hague in Holland, in
1629, being the son of Constantine Huygens, lord of Zuyli-
chem, who had served three successive princes of Orange
in the quality of secretary. He spent his whole life in
cultivating the mathematics; and not in the speculative way
only, but also in making them subservient to the. uses of
life. From his infancy he discovered an extraordinary
fondness for the mathematics; in a short time made a
great progress in them ; and perfected himself in those stu-
dies under professor Schooten, at Leyden. In 1649 he
went to Holstein and Denmark, in the retinue of Henry
count of Nassau ; and was extremely desirous of going to
Sweden, to visit Descartes, who was then in that country
with the queen Christina, but the count's short stay in
Denmark would not permit him. — In 1651 he gave the
world a specimen of his genius for mathematics, in a trea-
tise entitled, Theoremata de Quadratura Hyperbola;, El-
lipsis, et Circuli, ex dato Portionum Gravitatis Centro;
in which he clearly showed what might be expected from
him afterwards. — In 1655 he travelled into France, and
took the degree of ll. d. at Angers. — In 165S he pub-
lished his Horologium Oscillatoriuni, sive de Motu Pcn-
dulorum, &c, at the Hague. He had exhibited in a for-
mer work, entitled Brevis Institutio de Usu Horologiorum
ad inveniendas Longitudines, a model of a new-invented
pendulum ; but as some persons, envious of his reputation,
were labouring to deprive him of the honour of the inven-
tion, he wrote this book to explain the construction of it;

and to show that it was very different from the pendulum
of astronomers invented by Galileo. — In 1659 he published
his Systema Saturninum, &c ; in which he first of any one
explained the ring of Saturn, and discovered also one of
the satellites belonging to that planet, which had hitherto
escaped the eyes of astronomers : new discoveries, made
with glasses of his own forming, which gained him a high
rank among the astronomers of his time.

In 1660, he took a second journey into France, and
the year after passed over into England, where he commu-
nicated his art of polishing glasses for telescopes, and was
made Fellow of the Royal Society.- About this time the
air-pump was invented, which received considerable im-
provements from him. This year also he discovered the
laws of the collision of elastic bodies; as did also about
this time Wallis and Wren, with whom he had a dispute
about the honour of this discovery. On his return to
France, in l663, the minister Colbert, being informed of
his great merit, settled a considerable pension on him, to
engage him to fix at Paris ; to which Mr. Huygens con-
sented, and staid there, from the year 1666 to l6'81, where
he was admitted a member of the Academy of Sciences.
All this time he-spent in mathematical pursuits, wrote se-
veral books, which were published from time to time, and
invented and perfected several useful instruments and ma-
chines : particularly he had a dispute, about the year
160S, with Mr. James Gregory, concerning the quadra-
ture of the circle and hyperbola of the latter, then just
published, in which Huygens it seems had the better side
of the question. But continual application gradually im-
paired his health ; and though he had visited his native
country twice, viz, in 1670 and l675, for the recovery of
it, he was now obliged to betake himself to it altogether.
Accordingly he left Paris in 168I, and retired to his own
country, where he spent the remainder of his life in the
same pursuits and employments. He died at the Hague,
June 8, 1695, in the 6'7th year of his age, while his Cos-
motheoros, or treatise concerning a plurality of worlds,
was printing; so that this work did not appear till 169S.
Mr. Huygens loved a quiet and studious manner of life,
and frequently retired into the country to avoid interrup-
tion, but did not contract that moroseness which is so
commonly the effect of solitude and retirement. Pie was
one of the purest and most ingenious mathematicians of
his age, and indeed of any other ; and made many valuable
discoveries. He was the first who discovered Saturn's
ring, and a third satellite of that planet, as mentioned
above. He invented the means of renrlering clocks exact,
by applying the pendulum, and of rendering all its vibra-
tions equal, by the cycloid. He brought telescopes to
perfection, and made many other useful discoveries.

He was the author of many excellent works. The prin-
cipal of these arc now contained in two collections, of 2
volumes each, printed in 4to, under the care of professor
Gravcsande. The first was at Leyden, in lo"82, under the
title of Opera Varia ; and the second at Amsterdam, in
1728, entitled Opera Reliqua".

HYADES, a cluster of 5 stars in the face of the con-
stellation Taurus, or the Bull.

HYALOIDES, the vitreous humour of the eye, con-
tained between the tunica-retina and the uvea.
HVBERNAL Occident. See Occident.
Hybernal Orient. See Orient.
HYDATOIDES, the watery humour of the eye, con-
tained between the cornea and the uvea.

H Y D [

HYDRA, a southern constellation, consisting of a
number of stars, imagined to represent a water-serpent.
The stars in the constellation Hydra, in Ptolemy's cata-
logue, are 27; in Tycho's, 19; in Hevelius's, 31 ; and in
the Britannic catalogue, 60.

HYDRAULICS, the science of the motion of water
and other fluids, with its application in artificial water-
works of all kinds. — As to what respects merely the equi-
librium of fluids, or their gravitation or action at rest, be-
longs to hydrostatics. On removing or destroying that
equilibrium, motion ensues; and here hydraulics com-
mence. Hydraulics therefore suppose hydrostatics ; and
many writers, from the near relation between them, like
mechanics and statics, join the two together, and treat of
them conjointly as one science. — The laws of hydraulics
are given under the word Fluid. And the art of raising
water, with the several machines employed for that pur-
pose, are described under their several names, Fountain,
Hydrocanistenum Pump, Siphon, Syringe, &c.

The principal writers who have cultivated and improved
hydraulics and hydrostatics, are Archimedes, in his Libris
de Insidentibus Humido ; Hero of Alexandria, in his Liber
Spiritualium ; Marinus Ghelaldus, in his Archimedes
promotus ; Mr. Oughtred ; Jo. Ceva, in his Geometria
Motus ; Jo. Bap. Balianus, De Motu Naturali Gravium,
Solidorum et Liquidoi urn ; Mariotte, in his treatise of the
Motion of Water and other Fluids; Boyle, in his Hydro-
statical Paradoxes ; Fran. Tertius de Lanis, in his Ma-
gisterium Naturae et Artis ; Lamy, in hisTraite de l'Equi-
libre des Liqueurs; Rohault; Dr. Wallis, in his Mecha-
nics; Dechales; Newton, in his Principia ; Guglielmini,
in his Mensura Aquarum Fluentium ; Herman; Wolfius;
Gravesande ; Musschenbroek; Leopold; Schottus, in his
Mechanica Hydraulico-Pneumatica; Geo. Andr. Bockler,
in his Architectura Curiosa Germanica ; August. Ram-
milleis; Lucas Antonius Portius; Slurmy, in his treatise
on the Construction of Mills; Switzer's Hydrostatics;
Varignon, in the Mem. Acad. Sci. ; Jurin; Belidor ;
Bernoulli; Desaguliers ; Clare; Emerson; Ferguson;
Ximenes ; Bossu ; D'Alembert; Buat ; &c, &c. — See
also Pumps, Steam Engines, &c.

HYDRAULICO-Pneumatical, a term applied by
some authors to such engines as raise water by means of
the weight or spring of the air.

HYDRAULIC Ram, the name of a machine by Mont-
golfier, for raising water.

The ingenious idea of raising water by the momentum
of water itself, was first suggested by Mr. Whitehurst in
the Philosophical Transactions for 1775. The same
principle, in an improved form, has lately been revived in
France, and has excited considerable attention both on
the continent and in this country. Whatever credit there-
fore is due to the inventor of the Hydraulic Ram, pro-
perly belongs to our countryman Mr. Whitehurst, and
Montgolfier can lay claim to nothing more than the merit
of an improver.

Mr. Whitehursl's machine, which is represented in fig. 1.
pi. l6, was actually erected at Oulton in Cheshire, and
completely answered the expectation of its inventor, am
is the original reservoir, whose surface is on a level with
B, the bottom of the reservoir bn. The main pipe ae,
is If inches diameter, and nearly two hundred yards long,
and the branch pipe ep is of such a size that the cork f
is about 16 feet below the surface m of the reservoir. D
is a valve box, with its valve a, and c is an air vessel,

659 J H Y D

i into which are inserted the extremities m, n. of the main

pipe, bent downwards to prevent the air from being driven
out when the water is, forced into it.

Now, since the difference of level between the cock f
and the top of the reservoir am is lC feet, on opening the
cock f the water will rush out with a velocity of nearly
30 feet per second. A column of water therefore 200
yards long, is thus put in motion, and though the aperture
ot the cock f be small, it must have a very considerable
momentum. Let the cock f be now suddenly stopped,
"the water must evidently rush through the valve a into
the air vessel c, and condense the included air. This
condensation must take place every time the cock is opened
and shut, and the included air being highly compressed,
will press upon the water in the air vessel, and raise it into
the reservoir bs.

From this brief description of Whitehurst's machine,
the reader will easily perceive its resemblance to Mont-
golfier's hydraulic ram, a section of which is represented
by fig. 2. r is the reservoir, us the height of the fall, and
st the horizontal tube which conducts the water to the
engine abiitc. e and d are two valves, and fg a pipe
reaching within a very little of the bottom ce. Now let
water descend from the reservoir, it will rush out at the
aperture ?nn till its velocity becomes so great as to force,
up the valve e. The water being thus suddenly checked,
and unable to find a passage at inn, will move forwards
with great violence and force towards H, and raise the
valve d. A portion of water being admitted into the
vessel abc, the impulse of the column of fluid is expended,
the valves d and e fall, and the water rushes outat in n as
before, when its motion is again stopped, and the s;ime
operation repeated which has now been described. Every
time therefore that the valve e closes, a portion of water
will force its way into the vessel abc, and condense the air
which it contains ; for the included air has no communi-
cation with the atmosphere after the water is higher than
the bottom of the pipe fg. This condensed air will con-
sequently exert great force on the surface op of the water,
and raise it in the tube fg, to a height proportioned to
the elasticity of the imprisoned air. The external ap-
pearance of the engine, copied from one in the possession
of Professor Leslie, is exhibited in fig: 3 ; where abc is
the air vessel, f the valve box, g the extremity of the
valve, and m n screws for fixing the horizontal tube to the
machine. A piece of brass, a, with a small aperture, is
screwed on the top, when the engine is employed, to form
a jet of water.

From this description, it will easily be perceived that
the only difference between the engines of Montgolfier and
Whitehurst is, that the one requires a person to turn the
cock, while the other has the advantage of acting sponta-
neously. Montgolfier assures us (Journal des Mines, vol.
13, No. 73) that the honour of this invention does not
belong to England, but that he is the sole inventor, and
did not receive a hint from any person whatever. We
leave it to the reader to determine to what degree of credit
these assertions are entitled.

It would appear, from some experiments of Montgol-
fier, that the effect of the water-ram is equal to between
-| and i of the power expended ; which renders it supe-
rior to most hydraulic engines. Appendix to Brewster's
Ferguson's Lectures, pa. 4 19-

The principles on which the ram of Montgolfier is con-
structed, are susceptible of a very extensive application :
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thus, it wou-ld be easy to raise other water besidis that
which runs through the-ram; as for example, with a
Fall of water we might by a slight modification of this
machine raise water from the bottom of wells, or even of
mines.

Other machines analogous to Montgolfier's engine have
been invented by Messrs. Sarjeant and Boulton ; and the
celebrated hydraulic engine in Chemnitz in Hungary is
closely allied to the ram in principle. M. Montgolficr
suggests the utility of his invention in the draining of
mines ; and in that case, the hydraulic ram and the Hun-
garian machine would become very nearly if not altoge-
ther identical as to principle.

HYDRODYNAMICS, that branch of mechanical
science which relates to the motion of non-elastic fluids,
and the forces with which they act upon bodies opposed
to them.

There is no part of mechanics the true theory of which
we are so little acquainted with as that which respects the
motion of fluids. Nor is it strange that the knowledge
we do possess of this science should be so dubious and'
confined, when we consider the many powerful obstacles
which continually present themselves in the cultivation of
it. The mass, the figure, and the number of particles of
a fluid in motion, are particulars with which we are too
superficially acquainted to determine the laws of its mo-
tion, with any satisfactory degree of accuracy ; and even
if we were in possession of the data by which those laws
might be determined, it is doubtful whether we should be
much farther advanced, as it might be extremely difficult
to deduce any convenient and practical results from the
intricate and complex expressions which might be ob-
tained from the investigation. The greatest mathemati-
cians have been forced to yield to the numerous difficulties
of these researches, and to confess that the methods so
successfully followed in the mechanics of solid bodies, do
not afford any conclusions with respect to fluids, but such
as are too general and uncertain for the greater part of
particular cases. Whatever we have of this science is
almost exclusively due to the labours of modern analysts
and geometers ; for the only work which has reached us
from the ancients is that of Archimedes, in two hooks,
entitled De Insidentibus humido, in which the only en-
quiries respect the sinking and floating of bodies in fluids,
their relative gravities, levities, situations and positions,
when in equilibrio. It is indeed true that some hints, as
well as rules, on the motion of fluids, are contained in a
treatise attributed to Sextus Julius Frontinus, inspector of
public fountains at Rome, under the emperors Nerva,
Cocccius, and Trajan, entitled, De Aqua ductibus urbis
Roma' commentarius; but they are not of sufficient im-
portance to deserve much attention by a student in this
science. Benedict Castelli was the first who opened the
way to a true measure of the flux of waters, in his Trea-
tise Delia Mesura dell' Aquae Current], which measure
he found to depend on the area of the section and the ve-
locity of the water conjointly. The most valuable and
important discoveries and theorems in this department of
science, are given in Sir Isaac Newton's Principia, lib. 2,
pr. 36, with the comment : Dan. Bernoulli's Hydrody-
namique ; D'Alembert's Traite des Fluides; D.George
Juan's Examen marjtimo Tlieorico Practico ; M. Bos-
sut's Hydrodynamique; Buat's Principes d' Hydraulique,
and Mr. Eytelwein's Ilandbuch der Mechanick und der
Jiydraulique. To which may be added an ingenious paper

on the Motion and Resistance of Fluids, by Mr. Yince,
in the Philos. Transact, for 1795 ; and those by the late
Dr. Matthew Young in the Irish Trans. — See Hydrau-
lics.

HYDROGEN, the basis of what has been generally
called inflammable air, and is one of the components of
water.

HYDROGEN Gas, an invisible elastic aeriform fluid
uncondensible by any known cold, and is the only form
in which pure hydrogen has ever been exhibited. The
specific gravity of this gas varies considerably, according
to the manner in which it has been procured. In general,
when pure, it is from 6 to 10 or even, to 12 times lighter
than common air, so that S times may be considered as a
general average. It is one of the most inflammable sub-
stances we are acquainted with ; that is to say, it com-
bines with more oxygen than any other body, and occa-
sions more heat by its combustion; it being a well esta-
blished fact, that hydrogen and oxygen gases, whenmixed,
produce the most intense heat yet known.

One of the most striking properties of this gas, is that
of reducing metallic oxyds to the reguline or nearly regu-
line state. This is effected by the union of the hydrogen
with the oxygen of the metallic oxyd, in consequence of
which the hydrogen loses, its gaseous form, and with the
oxygen produces water. The metallic reduction by hy-
drogen gas appears to have been first noticed by Dr. Priest-
ley, but the consequent production of water in these cases
was suggested by the most important researches of Mr.
Cavendish, on the union of hydrogen and oxygen.

From tiie great levity of hydrogen gas, it has generally
been used for filling air-balloons: indeed this appears the
only use to which it has at present been applied. The
following is the best way of procuring it : Put a quantity
of filings of zinc into a vessel which has a glass tube pre-
fixed ; then pour upon them sulphuric acid, diluted with
6 or 8 times its quantity of water, and an effervescence
will immediately take place; the water will be decom-
posed, and the oxygen of it will become united to the
metal, and the hydrogen gas will be disengaged, and may
be conveyed into the body pf the balloon. This gas can
be procured pure only from water, which in all cases must
undergo a decomposition. For the usual properties of
this gas, the reader may consult with much interest
Aikins's Chemical Diet.

HYDROGRAPHICAL Charts or Maps, more usually
called sea-charts, are projections of some part of the sea,
or coast, for the use of navigation. In these are laid down
all the rhumbs or points of the compass, the meridians,
parallels, &c,with the coasts, capes, islands, rocks, shoals,
shallows, &c, in their proper places, and proportions.

For the construction and use of the several kinds of hy-
drographical maps, see Chart, and Sailing.

HYDROLOGY', is that part of natural history which,
examines and explains the nature and properties of water
in general.

HYDROMETER, an instrument for measuring the
properties and effects of water, as its density, gravity, force,
velocity, &c. That with which the specific gravity of
water is determined, is often called an aerometer, or
water-poise.

The general principle on which the construction anil
use of tiie hydrometer depends, has been illustrated under
the article Specific Gravity ; where it is shown that a
body specifically lighter than several fluids, will serve to.

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find outtheir specific gravities ; because it will sink deepest
in the fluids whose specific gravity is the least. In fig. 5,
pi. 14, if a b represent a small even glass tube, hermeti-
cally sealed, having a scale of equal divisions marked upon
it, with a hollow ball of about an inch in diameter at bot-
tom, and a smaller ball c under it, communicating with
the first ; and into the little ball be put mercury or small
shot, before the tube is sealed, so that it may sink in water
below the ball, and float or stand upright, the divisions on
the stem showing how far it sinks. — If this instrument be
dipped in common water, and sink to D,'it will sink only
to some lower point e in "salt water ; but in port wine it
will sink to some higher point f, and in brandy perhaps
to b. It is evident that an hydrometer of this kind will
only show that one liquid is specifically heavier than an-
other; but the true specific weight of any liquid cannot
be determined without a calculation for this particular in-
strument, the tube of which should be truly cylindrical.
Besides, these instruments will not serve for fluids whose
densities are much different.

Mr. Clarke constructed a new hydrometer, showing
whether any spirits be proof, or above or below proof, and
in what degree. This instrument was made of a ball of
copper (because ivory imbibes spirituous liquors, and
glass is apt to break), to which is soldered a brass wire
about a quarter of an inch thick; upon this wire is marked
the point to which it exactly sinks in proof spirits ; as also
•two other marks, one above and one below the former,
exactly answering to one-tenth above and one-tenth below
proof. There are also a number of small weights made
to add to it, so as to answer to the other degrees of
strength besides those above, and for determining the spe-
cific gravities of different fluids. Philos. Trans. Abr. vol.
vii, pa. 392.

Dr. Desaguliers contrived an hydrometer for determin-
ing the specific gravities of different waters, to such a de-
gree of nicety, that it would show when one kind of water
was but the 40,000th part heavier than another. It con-
sists of a hollow glass ball of about 3 inches in diameter,
charged with shot to a proper degree, and having fixed in
it a long and very slender wire, of only the 40th part of an
inch in diameter, and divided into tenths of inches, each
tenth answering to the 40,000th part, as above. See his
Exper. Philos. vol.2, pa. 234.

Mr. Quin and other persons have also constructed hy-
drometers, with other and various contrivances, and with
different degrees of accuracy; but all nearly on the same
general principles.

But there is one circumstance which deserves particular
attention in the construction and graduation of hydrome- '
ters, for determining the precise strength of different
brandies, and other spirituous liquors. Mr. Reaumur
discovered, in making his spirit-thermometers, that when
rectified spirit and water, or phlegm, the other constituent
part of brandy, are mixed together, there appears to be a
mutual penetration of the two liquors, and not merely
juxtaposition of parts ; so that a part of the one fluid seems
to be received into the interstices of the other; by which
it happens, that if a pint of rectified spirit be added to a
pint of water, the mixture will be sensibly less than a
quart. The variations hence produced in the bulk of the
mixed fluid render the hydrometer, when graduated in the
usual way by equal divisions, an erroneous measure of its
strength ; because the specific gravity of the compound is
found not to correspond to the mean gravity of the two

ingredients. M. Montigny constructed- a scale for this
instrument, in the manner before suggested by Dr. Lewis,
on actual observation of the sinking or rising of the hydro-
meter in various mixtures of alcohol and water, made in
certain known proportions. Hist, de l'Acad. Roy. des
Sci. 1768; also Neumann's Chem. by Lewis, pa. 450,
note r.

M. De Luc has lately published a scheme for the con-
struction of a comparable hydrometer, so that a work-
man, after having constructed one upon his principles,
may make all others similar to each other, and capable of
indicating the same degree on the scale, when immersed
in the same liquor of the same temperature. This instru-
ment is proposed to be constructed of a ball of flint glass,
communicating with a small hollow cylinder, containing
such a quantity of quicksilver for a ballast, that the instru-
ment may sink nearly to its top, in the most spirituous
liquor, when heated as much as possible; to which is also
attached a thin silvered tube, for a scale, &c. The whole
description may be seen at large in the Philos. Trans.
vol. 68, pa. .500. M. Le Roi also published a proposal for
constructing comparable hydrometers. See Hist, de
l'Acad. des Scien. for 1770, Mem. 7.

Mr. Nicholson has made an improvement, by- which
the hydrometer is adapted to the general purpose of find-
ing the specific gravity, .both of solids and fluids (fig. 4,
pi. 16). a is a hollow ball of copper, b a dish affixed
to the ball by a short slender stem, d ; c is another dish
affixed to the opposite side of the ball by a kind of stirrup.
In the instrument actually made, the stem d is of hardened
steel, -^of an inch in diameter, and the dish c is so heavy
as in all cases to keep the stem vertical when the instru-
ment is made to float in any liquid. The parts are so ad-
justed that the addition of 1000 grains in the upper dish
b will just sink it in distilled water (at the temperature of
6"0° of Fahrenheit's thermometer), so far that the surface
shall intersect the middle of the stem d.

Let it now be required to find the specific gravity of any
fluid. Immerse the instrument in it, and by placing
weights in the dish b cause it to float, so that the middle
of its stem d shall be cut by the surface of the fluid.
Then, as the known weight of the instrument, added to
1000 grains, is to the same known weight added to the
weights used in producing the last equilibrium, so is the
weight of a quantity of distilled water displaced by the
floating instrument, to the weight of an equal bulk of the
fluid under examination.

Again, let it be required to find the specific gravity of
a solid body, whose weight is less than 1000 grains.
Place the instrument in distilled water, and put the body
in the dish b. Make the adjustment of sinking the instru-
ment to the middle of the stem, by adding weights in the
same dish. Subtract those weights from 1000 grains, ~
and the remainder will be the weight of the body. Place
now the body in the lower dish c, and add more weight in
the upper dish b, till the adjustment is again obtained.
The weight last added will be the loss the solid sustains
by immersion, and is the weight of an equal bulk of water.
Consequently the specific gravity of the solid is to that
of water, as the weight of the body to the loss occasioned
by the immersion. Mr. Nicholson observes, " This in-
strument was found to be sufficiently accurate to give
weights true to less than ^th of a grain." Nicholson's-
Philosophy, vol.2, p. 16.

HYDROMETR1A, Hydrometry, the mensuration,

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of water and other fluid bodies, their gravity, force, velo-
city, quantity, &c ; including both hydrostatics and hy-
draulics.

HYDROSCOPE, an instrument anciently used for the
measure of time. It was a kind of clepsydra or water-
clock, consisting of a cylindrical tube, conical at bottom :
the cylinder was graduated with -divisions, to which the
top of the water becoming successively contiguous, as it
trickled.out of the vertex ot the cone, pointed out the hour.
HYDROSTATICAL Balance, a balance contrived for
the exact and easy finding the specific gravities of bodies,
both solid and fluid, and thereby of estimating the degree
of purity of bodies of all kinds, with the quality and rich-
ness of metals, ores, minerals, &c, and the proportions in
any mixture, adulteration, or the like. This is effected
by weighing the body both in water, or other fluid, and
out of it ; and for this purpose, one of the scales has
usually a hook at the bottom, for suspending the body by
some very fine thread. And the use of the instrument is.
founded on this theorem of Archimedes, that any body
weighed in water, loses as much of its weight as is equal
to the weight of the same bulk of the water. Thus then
is known the proportion of the specific gravities of the so-
lid and fluid, or the proportion of their weights under the
same bulk, viz, the proportion of the weight of the body
weighed out of water, to the difference between the same
and its weight in water. Hence also, by doing the same
thing for several different solids, with the same fluid, or
different fluids with the same solid, all their specific gra-
vities become known.

This instrument re-
quires but little descrip-
tion, ab is a nice ba-
lance beam, with its
scales c and d, turning
with the small part of a
grain, the one of them,
d, having a hook in the
bottom, to receive the
loop of a horse hair &c,
E, by which the body f
is suspended, gh is a.
jar of water, in which

the body is immersed when weighing. The pieces in the
scale c denote the weight of the body out of water; then,
upon immerging it, put weights in the scale d to restore
the balance again, and their quotient will show the specific
gravity of the body. 4

There have been various kinds of the hydrostatica! ba-
lance, and improvements made on it, by different persons.
Thus, Dr. Desaguliers set three screws in the foot of the
sland, to move any side higher or lower, till the stem be
quite upright, which is known by aplummet hanging over
a fixed point in the pedestal. Desag. Exp. Philos. vol. 2,
pa. 1.96'. And for sundry other constructions of this in-
strument, designed for greater. accuracy than the common
sort, see Martin's Phil. Britan. or Gravesande's Physices
- Elem. Math. torn. 1, lib. 3, cap. 3, &c.

The specific gravities of small weights may be deter-
mined by suspending them in loops of horse hair, or fine
silken threads, to the hook at the bottom of the scale.
Thus, if a guinea suspended in air weigh 129 grains, and
on being immersed in water require 7\ grains to be put in
the scale over it, to restore the equilibrium ; we thus find
that a quantity of water of equal bulk with the guinea,

weighs 7y grains, or 7'2 ; therefore dividing the 129 by
the 7"2, the quotient 17"8S shows that the guinea is so
many times heavier than its'bulk of water. Whence, if
any piece of gold be tried, by weighing it first in air, then
in water, and if, upon dividing the weight in air by the
loss in water, the quotient be 17*88, the gold is good ; if
the quotient be 18 or more, the gold is more fine ; but if
it be less than 17'88, the gold is too much alloyed with
other metal. If silver be tried in the same manner, and
found to be 1 1 times heavier than water, it is very tine;
if it be 1QI times heavier, it is standard ; but if less, it is
mixed with some lighter metal, such as tin.

When the body, whose specific- gravity is sought, is
lighter than water, so that it will not quite 'sink; annex
to it a piece of another body heavier than water, so that
the mass compounded of the two may sink together.
Weigh the denser body, and the compound mass, sepa-
rately, both in water and out of it, thereby finding how
much each loses in water ; and subtract the less of these,
two losses from the greater; then say,
As the remainder

is to the weight of the lighter body in air,
so is the specific gravity of water
to the specific gravity of the lighter body.
HYDROSTATICAL Bellows, a machine for showing
the upward pressure of fluids and the hydro- £v#:v
statical paradox. It consists of two thick
boards, a, d, each about 1-6 or 18 inches
diameter, more or less, covered or connect-
ed firmly with leather round the edges, to
open and shut like a common bellows, but
without valves ; only a pipe, b, about 3
feet high is fixed into the bellows at e.
If water be poured into the pipe at c, it
will run into the bellows, gradually sepa-
rating the boards, by raising the. upper one.
And if several weights, as three hundred ■
weights, be laid upon the upper board, by
pouring the water in at the pipe till it be full, it will sus-
tain them all, though the water in the pipe should not
weigh a quarter of a pound ; for the pipe or tube may be
as small as we please, provided it"be but long enough, the
whole effect depending upon the height, and notat all on
the width of the pipe : for the proportion is always this,
As the area of the orifice of the pipe
is t.) the area of the bellows-board,
so is the weight of water in the pipe
' to the. weight it will sustain on the board.
Hence if a man stand upon the upper board, and blow
into the pipe b, he will raise himself on the board; and
the smaller the pipe, the easier he will be able to raise
himself; then by putting his finger upon the top ot the
pipe, he can support himself as long as he pleases, pro-
vided the bellows be air-tight.

Mr. Ferguson has described another machine, which
may be substituted instead of this common hydrostatical
bellows. It is however on the same principle of the hy-
drostatical paradox ; and may be seen in the Supplement
to his Lectures, pa. 19.

HYDROSTATICAL Paradox, is a principle in hydro-
statics, so called because it has a paradoxical appearance
at first view : it is this ; that any quantity of water, or
other fluid, how small soever, may be made to balance
and support any quantity, or any weight, how great so-
ever. This is partly illustrated in the last article, on the

I

.

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L

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W:

hydrostatical bellows, where k appears that any weigh't
whatever may be blown up and supported by the breath
from a person's mouth. And the principle may be ex-
plained as follows : it is well known that water in a pipe
or canal, open at both ends, always rises to the same
height at both ends, whether those ends be wide or narrow,
equal or unequal. Thus, the «& ^„
small pipe gh being close P^-^l j'l
joined to another open vessel i!|iEij;| "
a i, of any size whatever ; by *™
pouring water into one of
these, it will rise up in the
other, and stand at the same
height, or horizontal line df
in both of them, and that j:;| | |>C £#'
whether they be upright, or
inclined in any position. So
that all the water that is in
the large vessel from a to I,
is supported by that which is
in the small vessel from d to i|
i only. And as there is no ^-^Sg^f
limit to this latter one, but £

that it may be made as fine even as a hair, it hence evi-
dently appears that any quantity of water may be thus
supported by any other the smallest quantity.

Since then the pressure of fluids is directly as their per-
pendicular heights, without any regard to their quantities,
it appears that whatever the figure or size of the vessels
may be, if they ate but of equal heights, and the areas of
their bottoms equal, the pressures of equal heights of
water are equal upon the bottoms of these vessels; even
though the one should contain a thousand or ten thousand
times as much as the other.

Mr. Ferguson confirms and illustrates this paradox by
the following experiment.

Let two vessels be prepared of equal heights, but very
unequal contents,- such as ab and CD ; each vessel being
open at both ends, and their bottoms e and f of equal
widths. Let a brass bottom G and h be exactly fitted to
each vessel, not to go into it, but for it to stand upon ;
and let a piece of wet leather be put between each vessel
and its brass bottom, to render it as close and tight as
possible. Join each bottom to its vessel by a hinge d, so
that it may open like the lid of a box ; and let each bottom
be kept up to its vessel by equal weights w, hung to lines
which go over the pulleys p, whose blocks are fixed to

the sides of the vessel at /, and the lines tied to hooks at
d, fixed in the brass bottoms opposite to the hinges d. In
this adjustment of parts, hold one vessel upright in the
hands over a bason on a table, and cause water to be
poured slowly into it, till the pressure of the water bears
down its bottom at the side d, and raises the weight e;
and then part of the water will run out at d. Maik the
height at which the surface h of the water stood in the
vessel, when the bottom began to give way at d; after
which, holding up the other vessel in the same manner,
cause water to be poured into it ; and it will be seen that
when the water rises in this vessel just as high as it did in
the former, its bottom will also give way at d, and it will
also lose part of the water. «

The reason of this surprising phenomenon is, that since
all parts of a fluid at equal depths below the surface, are
equally pressed in all manner of directions, the water im-
mediately below the fixed part B/will be pressed as much
upward against its lower surface within the vessel, by the
action of the column a^, as it would be by a column of
the same height, and of any diameter whatever ; and there-
fore, since action and reaction are equal and contrary to
each other, the water immediately below the surface b/
will be pressed as much downward by it, as if it were im-
mediately touched and pressed by a coltimn of the height
Ag, and of the diameter b/; and therefore the -water in
the cavity BDrffwill be pressed as much downward upon
its bottom g, as the bottom of the other vessel is pressed
by all the water above it. Lectures, pa. 105.

HYDROSTATICS, is that science which treats of the
nature, gravity, pressure, and equilibrium of fluids ; and of
the weighing of solids in them. That part of the science
of fluids which considers theirmotions,beingincluded under
the article Hydrodynamics. Hydrostatics and hydrody-
namics together constitute a branch of philosophy that is
justly considered as one of the most curious, ingenious,
and useful of any ; affording theorems and phenomena
not only of the first use and importance, but also surpris-
ingly amusing and pleasant; as appears in the numberless
writings on the subject ; the principal points of which may-
be found under the several particular articles of this work,"
as Pressure, Gravity, &c.

The chief writers who have cultivated and improved this
science are, Archimedes, in his Libris de Insidentibus
Humido; Hero of Alexandria, in his Liber Spiritualium ;
Marinus Ghctaldus, in his Archimedes promotus ; Ought"
red; Jo. Ceva, in his Geometria Motus; Jo. Bap. Ba-
lianus De Motu Naturali Gravium, Solidorum etLiquido-
rum ; Marioke, in his treatise of the Motion of Water and
other Fluids ; Boy le in his Hydrostatical Paradoxes ; Boyle
in Tcrtius de Lanis, in his Magisterium Natura et Artis;
Lamy, in his Traite de l'Equilibro des Liqueurs ; Rohault;
Dr. Wallis, in his Mechanics; Dechales; Newton, in his
Principia ; Gulielmini, in his Mensura Aquarum Fluen-
tium ; Herman; Wolfius; Gravesandc ; Muschenbroek ;
Leopold ; Schottus, in his Mechanica Hydraulico-Pneu-
matica; Geo. Andr. Bockler, in his Architectura Curiosa
Germanica ; August Rammilhis ; Lucas Antonius Portius ;
Switzer; Va'rignon, in the Mem. Acad. Sci. ; Jurin;
Belidor; Bernoulli ; Desaguliers ; Clare ; Emerson ; Fer-
guson ; Ximenes ; Bossu ; D'Alembert; Buat ; Vince ;
Olinthus Gregory, in his Mechanics, '&c, &c.

HYDRUS, or Water Serpent, one of the new southern
constellations, including only ten stars.

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H YEMAL Solstice, the same with Winter Solstice. See
Solstice.

' HYGROMETER, or Hygroscope, or Notiometer,
an instrument for measuring the degrees of moisture in the
air. There are various kinds of this instrument ; for what-
ever body either swells by moisture, or shrinks by dry-
ness, is capable of being formed into an hygrometer. Such
are woods of most kinds, particularly deal, ash, poplar,
&c. Sucli also is catgut, the beard of a wild oat, and a
Jwisted cord, &c. The best and most usual contrivances
for this purpose are as follow.

1. Stretch a common cord, or a fiddle-string, abd along
a wall, passing it over a pulley b ; fixing it at one end a,
and to the other end hanging a weight e, carrying a style
or index p. Against the same wall fit a plate of metal
hi, graduated, or divided into any number of equal parts;
and the hygrometer &c, is complete. For it is matter of
constant observation, that moisture sensibly shortens cords
and strings ; and that, as the moisture evaporates, they
return to their former length again- The like may be said
of a fiddle-string : and hence it happens that such strings
are apt to break in damp weather, if they be not slackened
by the screws of the violin. Hence it follows, that the
weight E will ascend when the air is more moist, and de-
scend again when it becomes drier. By which means the
index f will be carried up and down, and, by pointing to
the several divisions on the scale, will show the degrees
of moisture or dryness.

2. Or thus, for a more sensible and accurate hygrome-

V

ter : strain a whipcord, or catgut, over several pulleys B>
c, d, E, F ; and proceed as before for the rest of the con-
struction. It matters not whether the several parts of the
cord be parallel to the horizon, as expressed in the an-
nexed figure, or perpendicular to the same, or in any
other position ; the advantage of this, over the former
method, being merely the having a greater length of cord
in the same compass; for the longer the cord, the greater
is the contraction and dilatation, and consequently of the
degrees of variation of the index over the scale, for any
given change of moisture in the air.

3. Or thus : Fasten a twisted cord,
or catgut, ab, by one end at a, sus-
taining^ weight at b, carrying an in-
dex c round a circular scale de de-
scribed on a horizontal board or table.
For a cord or catgut twists itself as it
moistens, and untwists again as it dries.
Hence, on an increase or decrease of
ahe humidity of the air, the index will
show the quantity of twisting or un-
twisting, and consequently the increase
or decrease of moisture or dryness.

4. Those Dutch toys, called weather houses, where a
small image of a man, and another of a woman, are fixed
upon the ends of an index, are constructed on this prin-
ciple. For the index, being sustained by a cord, or
twisted catgut, turns backwards and forwards, sending out
the man in wet weather, and the woman in dry.

5. Or thus: Fasten one end of a
cord, or catgut, ab, to a hook at a;
and to the other end a ball d of about
one pound weight; upon which draw
two concentric circles, and divide
them into any number of equal parts,
for a scale ; then fit a style or index
EC into a proper support at e, so as the
extremity c may almost touch the di-
visions of the ball. — Here the cord
twisting or untwisting, as in the former
case, will indicate the change of
moisture, by the successive application
of the divisions of the circular scale, as
the ball turns round, to the index c.

6. Or an hygrometer may be made
of the thin boards of ash or fir, by their
swelling or contracting. But this, and
all the other kinds of this instrument,
above described, become in time sen-
sibly less and less accurate ; till at last
they lose their effect entirely, and suffer
no alteration from the weather; in
consequence of which, several others of another kind have
been invented, much more durable, serving for many years
with tolerable accuracy. They are as follow :

7. Take the mano-
scope, described under
that article, and instead
of the exhausted ball E,
substitute a sponge, or
other body, that easily
imbibes moisture. To
prepare the sponge, it
may' be proper first to
wash it in water very

clean ; and, when dry again, in water or vinegar in which
there has been dissolved sal ammoniac, or salt of tar-
tar; after which let it dry again. — Now, if the air be-
come moist, the sponge will imbibe it and grow heavier,
consequently will preponderate, and turn the index to-
wards c ; on the contrary, when the air becomes drier, the
sponge is lighter, and the index turns towards a ; and thus
showing the state of the air.

8. In the last-mentioned hygrometer, Mr. Gould, in the
Philos. Trans, instead of a sponge, recommends oil of

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vitriol, which grows sensibly lighter or heavier from the being scarcely any variation in the night. 3d, That there

degrees of moisture in the air; so that being saturated in is a motion even in dry weather, the wood swelling in the

the moistest weather, it afterwards retains or loses its ac- morning, and shrinking in the afternoon. 4th, That the

quired weight, as the air proves more or less dry. The wood, by night as well as by day, usually contracts when

alteration in this liquor is so great, that in the space of 57 the wind is in the north, north-east, and east, both in

da}'s it has been known to change its weight from 3 summer and winter. 5th, That by constant observation

drachms to 9; and has shifted a tongue or index of a ba- of the motion and rest of the wood, with the help of a ther-

lance 30 degrees. So that in this way a pair of scales may mometer, the direction of the wind maybe told nearly

atford a very nice hygrometer. The same author suggests, without a weather-cock. He adds also, that even the time

that oil of sulphur or campanam, or oil of tartar per deli- of the year may be known by it; for in spring it moves

quiurn, or the liquor of fixed nitre, might be used iustead
of the oil of vitriol.

9- This balance may be contrived in two ways ; by
either having the pin in the middle of the beam, with a
slender tongue a foot and a half long, pointing to the di

more and quicker than in winter; in summer it is more
contracted than in spring ; and has less motion in autumn
than in summer.

See an account of a method of constructing these and
other hygrometers, in various places of the Philos. Trans.

visions on an arched plate, as represented in the last as vol. 11, pa. 6'47 and 715; vol. 15, pa. 1032; vol. 43,
figure. Or the scale with the liquor may be hung to the pa. 6; vol.44, pa. 95, 1 69, and 184; vol.54, pa.259;

point of the beam near the pin, and the other extremity

made so long, as to describe a large arch on a board placed
for the purpose ; as in the figure here annexed.

10. Mr. Arderon has proposed some improvement in
the sponge hygrometer. He directs the sponge a to be so

cut, as to contain as large a superficies as possible, and to
hang by a fine thread of silk upon the beam of a balance b
and exactly balanced on the other side by another thread
of silk at d, strung with the smallest lead shot, at equal
distances, so adjusted as to cause an index e to point at
c, the middle of a graduated archrGH, when the air is in
a middle state between the greatest moisture and the
greatest dryness. Under this silk so strung with shot, is
placed a small table or shelf 1, for that part of the silk or
shot to rest upon which is not suspended. When the
moisture imbibed by the sponge increases its weight, it will
raise the index, with part of the shot, from the table, and
vice versa when the air is dry. Phil. Trans, vol.44, pa. 96.

11. From a series of hygroscopical observations, made
with an apparatus of dealwood, described in the Philos.
Trans. No. 480, Mr. Coniers concludes, 1st, That the
wood contracts most in summer, and swells most in win-
ter, but is most liable to change in the spring and fall. 2d,
That this motion happens chiefly in the day time, there

Vol. I.

ol. 6T,pa. 19S; vol.63,pa. 404, &c. In myAbridg. also
the General Index at the end points out the places of re-
ference.

12. Dr. Hooke's hygrometer was made of the beard of a
wild oat, set in a small box, with a dial plate and an index.
See his Micrographia, pa. 150.

13. The doctors Hales and Desaguliers both contrived
another form of sponge hygrometer, on this principle.
They made an horizontal axis, having a small part of its

mmm*-.

length cylindrical, and the remainder tapering conicallv
with a spiral thread cut in it, after the manner ofthefuzee
of a watch. The sponge is suspended by a fine silk thread
to the cylindrical part of the axis, upon which it winds.
This is balanced by a small weight w, suspended also by
a thread, which winds upon the spiral fuzee. Here, when
the sponge grows heavier, in moist weather, it descends
and turns the axis, and so draws up the weight, which
coming to a thicker part of the axis it becomes a balance
to the sponge, and its motion is shown by an attached scale.
And vice versa when the air becomes drier. — Salt of tartar,
or any other salt, or pot-ashes, may be put into the scale
of a balance, and used instead of the sponge. Desag.
Exper. Philos. vol. 2, pa. 300.

14. Mr. Ferguson made an hygrometer of a thin deal
pannel ; and to enlarge the scale, and so render its vari-
ations more sensible, he employed a wheel and axle,
making one cord pass over the axle, which turned a
wheel ten times as large, over which passed a line with a
weight at the end of it, whose motion was therefore ten
times as much as that of the deal pannel. The board
should be changed in 3 or 4 years. See Philos. Trans,
vol. 54, art. 47.

15. Mr. Smeaton gave also an ingenious and elaborate
construction of an hygrometer ; which may be seen in the
Philos. Trans, vol. 61, art. 24.

16. Mr. Deluc's contrivance for an hygrometer is very
ingenious, and on this principle. Finding that even ivory

4 Q

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swells with moisture, and contracts with dryness, he made
a small and very thin hollow cylinder of ivory, open only
at the upper end, into which is fitted the under or open
end of a very fine long glass tube, like that of a thermo-
meter, ln'o those is introduced some quicksilver, filling
the ivory cylinder, and a small part of the length Up the
glass tube. The consequence is this: when moisture swells
the ivory cylinder, its bore Or capacity grows larger, and
consequently the mercury sinks in the fine glass tube ;
and vice versa, when the air is drier, the ivory contracts,
and forces the mercury higher up the glass tube. It is
evident that an instrument thus constructed is in fact also
a thermometer, and must necessarily be affected by the
vicissitudes of heat and cold, as well as by those of dry-
ness and moisture ; or that it must act as a thermometer
as well as an hygrometer. The ingenious contrivances in
the structure and mounting of this instrument may be
seen in the Philos. Trans, vol. 63, art. 38 ; where it may
also be seen how the above imperfection is corrected by
some simple and ingenious expedients, employed in the
original construction and subsequent use of the instru-
ment; in consequence of which, the variations in the tem-
perature of the air, though they produce their full effects
on the instrument, as a thermometer, do not interfere with
or embarrass its indications as an hygrometer.

17. In the Philos. Trans, for 179*, Mr. Deluc has
given a second paper on hygrometry. This has been
chiefly occasioned by a memoir of M. de Saussure on the
same subject, entitled Essais sur l'Hygrometrie, in 4to,
17S3. In this work M.de S. describes a new hygrometer
of his construction, on the following principle. — It is a
known fact that a hair will stretch when it is moistened,
and contract when dried: and M. de Saussure found, by
repeated experiments, that the difference between the
greatest extension and contraction, when the hair is pro-
perly prepared, and has a weight of about 3 grains suspend-
ed by it, is nearly one-40th of its whole length, or one inch
in40. This circumstance suggested the idea of a new hygro-
meter. To render these small variations of the length of the
hair perceptible, an apparatus was contrived, in which one
of the extremities of the hair is fixed, and the other, bear-
ing the counterpoise abovementioned, surrounds the cir-
cumference of a cylinder, which turns upon an axis to
which a hand is adapted, marking upon a dial in large
divisions the almost insensible motion of this axis. About
12 inches high is recommended as the most convenient and
useful: and torenderthem portable,a contrivance is added,
by which the hand and the counterpoise can be occasion-
ally fixed.

But, for all the reasonings of these ingenious philoso-
phers on this interesting subject, and complete information,
seethe Philos. Trans, an. 1791, pa. 1, and 3S9; and De-
luc'sMeteorologie, vol. 1, an. 1786; as also the Monthly
Review, vol. 51, pa. 224, vol. 71, pa. 213, vol. 76, pa.
3 1 6, vol. 78, pa. 236, and vol. 6', of the new series for the
year 1791, !>«• 133.

HYGROMETRY, the science of the measurement of
the moisture of the atmosphere. The chief writings on
this science are those of M. Deluc and M.de Saussure,
for which see the last article.

HYGROSCOPIC, is commonly used in the same sense
with Hygrometer. Wolfius, however, regarding the ety-
mology of the word, makes some difference. According
to him, the hygroscope only shows that there arc altera-
tions of the air in respect of humidity and dryness, but

the hygrometer measures them. A hygroscope therefore
is only an indefinite or less accurate hygrometer.

HYPATIA, a very learned and beautiful lady, was
born at Alexandria, about the end of the 4th century, as
she flourished about the year of Christ 430. She was the
daughter of Theon, a celebrated philosopher and mathe-
matician, and president of the famous Alexandrian school.
Her father, encouraged by her extraordinary genius, had
her not only educated in all the ordinary accomplish-
ments of her sex, but instructed in the most abstruse
sciences. She made such great progress in philosophy,
geometry, astronomy, and the mathematics in general,
that she passed for the most learned person of her time.
She published commentaries on Apollonius's Conies, on
Diophantus's Arithmetic, and other works. At length
she was thought worthy to succeed her father in that dis-
tinguished and important employment, the government of
the school of Alexandria; and to deliver instructions out
of that chair where Ammonius, Hierocles, and many other
great men, had taught before ; and this at a time when
men of great learning abounded both at Alexandria and
in many other parts of the Roman empire. Her fame
being so extensive, and her wonh so universally acknow-
ledged, it was no wonder that she had a crowded audi-
tory. " She explained to her hearers (says Socrates, an
ecclesiastical historian of the 5th century, born at Con-
stantinople) the several sciences that go under the general
name of philosophy ; for which reason there was a con-
fluence to her, from all parts, of those who made philoso-
phy their delight and study."

The pupils of this lovely and surprising female, were
not less eminent than they were numerous. One of them
was the celebrated Synesius, who was afterwards bishop of
Ptolemais. This ancient Christian Platonist always ex-
presses the strongest, as well as the most grateful testimony
of the virtue of his tutoress; never mentioning her with-
out the most profound respect, and sometimes in terms of
affection but, little short of adoration. But it was not
Synesius only/, and the disciples of the Alexandrian school,
who admired Hypatia for her virtue and learning: never
was woman more caressed by the public, and yet never
had woman a more Unspotted character. She was held as
an oracle for her wisdom, for which she was consulted by
the magistrates in all important cases; a circumstance
which often drew her among the greatest concourse of men,
without the least censure of her manners. In short, when
Nicephorus intended to pass the highest compliment on
the princess Eudocia, he thought he could not do it better
than by calling her another Hypatia.

While Hypatia thus reigned the brightest ornament of
Alexandria, Orestes was governor of that place for the
emperor Thcodosius, and Cyril was bishop or patriarch.
Orestes, having had a liberal education, could not but ad-
mire Hypatia; and as a wise governor often consulted
her. This, together with an aversion which Cyril had
against Orestes, proved fatal to the lady. About 500
monks assembling, attacked the governor one day, and
would have killed him, had he not been rescued by the
townsmen ; and the respect which Orestes had for Hypatia
causing her to be traduced among the Christian multitude,
they dragged her from her chair, tore her in pieces, and
burnt her limbs. This horrible catastrophe was perpe-
trated during the lent of the year 4l6.

Cyril is strongly suspected of having fomented this tra-
gedy. Cave indeed endeavours to remove the imputation

HYP

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HYP

of so horrid an action from ihe patriach; and places it on
- the Alexandrian mob in general, whom he calls " a very
trifling inconstant people." But though Cyril should be
allowed neither to have been the perpetrator, nor even the
contriver of it, yet it is much to be suspected that he did
not discountenance it in the manner he ought to have
done: a suspicion which must needs be greatly confirmed
by reflecting, that he was so far from censuring the vio-
lence committed by the monks upon Orestes, that he after-
wards received the dead body of Ammonius, one of the
most conspicuous in that brutality, who had grievously
wounded the governor, and who was justly punished with
death. Upon this riotous ruffian, Cyril made a panegyric
in the church where he was laid, in which he extolled his
courage and constancy, as one that had contended for the
truth ; and changing his name to Thaumasius, or the Ad-
mirable, ordered him to be considered as a martyr. " How-
ever (continues Socrates), the wisest part of Christians did
not approve the zeal which Cyril showed on this man's
behalf, being convinced that Ammonius had justly suffered
for his desperate attempt."

HYPERBOLA, one of the conic sections, being that
which is made by a plane cutting a cone, so that, entering
one side of the cone, and not being parallel to the oppo-
site side, it may cut the circular base when the opposite
side is ever so far produced below the vertex, or shall cut
the opposite side of the cone produced above the vertex,
or shall make a greater angle with the base than the oppo-
site side of the cone makes; all these three circumstances
amounting to the same thing, but in other words.

1. Thus, the figure dae is an hyperbola, made by a
plane entering the side vq of a cone pvq at a, and either
cutting the base peq when the plane is not parallel to vp,
and this is ever so far produced ; or
when the angle arq is greater than
the angle vpq; or when the plane
cuts the opposite side in b above the
vertex.

2. By the hyperbola is some-
times meant the whole plane of the
section, and sometimes only the
curve line of the section.

3. Hence, the cutting plane meets
the opposite cone in b, and there
forms another hyperbola d b e, equal
to the former one, and having the
same transverse axis ab; and the same vertices a and b.
Also the two are called Opposite Hyperbolas.

4. The centre c is the middle of the transverse axis.

5. The semi-conjugate axis is cl, a mean proportional
between ci and ck, the distances to the sides of the oppo-
site cone, when ci is drawn parallel to the diameter pq of
the base of the cone. Or the whole conjugate axis is a
mean proportional between af and bh, which are drawn
parallel to the base of the cone.

6. If dae and fbg be two opposite hyperbolas, having
the same transverse and conjugate axes ab and ab, per-
pendicularly bisecting each other; and if dae and fh«
be two other opposite hyperbolas, having the same axes
with the two former, but in the contrary order, viz, hav-
ing ab for their first or transverse axis, and ab for their
second or conjugate axis: then any two adjacent curves
are called Conjugate Hyperbolas," and the whole figure
formed by all the four curves, the figure of the Conjugate
Hyperbolas. And if the rectangle hikl be inscribed
within the four conjugate hyperbolas, touching the ver-
tices a, b, a, b, and having their sides parallel and equal
to the two axes; and if then the two diagonals hck, icl,<
ol the parallelogram be drawn, these diagonals are the
asymptotes of the curves, being lines that continually ap-
proach nearer and nearer to the curves, without meeting
them, except at an infinite distance, where each asymp-
tote and the two adjacent sides of the two conjugate hy-
perbolas may be supposed all to meet; the asymptote
being there a common tangent to them -both, viz, at that
infinite distance.

7. Hence the four hyperbolas, meeting and running into
each other at the infinite distance, may be considered as
the four parts of one entire curve, having the same axes,
tangents, and other properties.

8. A diameter in general, is any line, as mk, drawn
through the centre c, and meeting, or terminated by the
opposite legs of the opposite hyperbolas. And if parallel
to this diameter there be drawn two tangents, at m and n,
to the opposite legs of the other two opposite hyperbolas,
the line men joining the points of contact, is the conjugate
diameter to mn, and the two mutually conjugates to each
other. Or, if to the points m or n there be drawn, a tan-
gent, and through the centre c the line mn parallel to it,
that line will be the conjugate to mn. The points where
each of these meets the curves, as m, n, m, n, are the ver-
tices of the diameters; and the tangents to the curves at
the two vertices of any diameter, are parallel to each
other, and also to the other or conjugate diameter.

<cK\

9. Moreover, if those tangents to the four hyperbolas,
at the vertices of two conjugate diameters, be produced
till they meet, they will form a parallelogram opqr ; and
the diagonals oq and pa of the parallelogram will be the
asymptotes of the curves; which therefore pass through
the opposite angles of all the parallelograms so inscribed
between the curves. Also it is a property of these paral-
lelograms, that they are all equal to each other, and there-
fore equal to the rectangle of the two axes ; as will be
further noticed below. Further, if these diagonals or
asymptotes make a right angle between them, or if the in-
scribed parallelogram be a square, or if the two axes be
equal to each other, then the hyperbola is called a right-
angled or an equilateral one.

4Q2

HYP

[

]

HYP

10. An ordinate to any diameter, is a line drawn pa-
rallel to its conjugate, or to the tangent at its vertex, and
terminated by the diameter produced and the curve. So
MS and tn are ordinates to the axis ab; also ad and bg
are ordinates to the diameter jin (last fig. but one).
Hence the ordinates to the axis are perpendicular to it;
but ordinates to the other diameters are oblique to them.

11. An absciss is a part of any diameter, contained be-
tween its vertex and an ordinate to it; and every ordinate
has two abscisses: as at and bt, or mv and sv,

12. The parameter of any diameter, is a third propor-
tional to the diameter and its conjugate. — The parameter
of the axis is also equal to the line AG or sg (fig. 1), if
fg be drawn to make the angle af'g = the angle bav, or
the line Hg to make the angle Biig= the angle abv.

13. The focus is the point in the axis where the ordi-
nate is equal to half the parameter of the axis; as s and t
(fig. 2) if ms and tn be half the parameter, or the 3d pro-
portional to ca and ca. Hence there are two foci, one
on each side the vertex, or one for each of the opposite
hyperbolas. These two points in the axis are called foci,
or burning points, because it is found by opticians that
rays of light issuing from one of them, and falling upon
the curve of the hyperbola, are reflected into lines that
verge towards the other point or focus.

To describe an Hyperbola, in various ways.

14. (1st Way by points.) — In the transverse axis af
produced, take the foci f and /, by making cf and
c/= a« or sa, assume any point i: Then with the

16*. (3d Way by points, to describe the curve through a
given point.) — cg and ch being the asymptotes, and P the
given point of the curve; through the point p draw any
line gph between the asymptotes, upon which take
gi = ph, so shall i be another point of the curve. And
in this manner may any number of points be found, draw-
ing as many lines through the given point p.

17. (4th Way by a continued Motion.) — If one end of a

radii ai, bi, and centres f,/, describe arcs intersecting in
e, which will give two points in each of the opposite
curves eae, ede. In like manner, assuming other points
I, as many other points will be found in the curve. Then,
with a steady hand, draw the curve line through all the
points of intersection e. — In the same manner are to be
constructed the other pair of opposite hyperbolas, using
the axis.aft instead of ab.

15. (2d Way by points, for a Right-angled Hyperbola
only.) — On the axis produced if necessary, take any

long ruler /mo be fastened at the point/, by a pin on a
plane, so as to turn freely about that point as a centre. Then
take a thread mo, shorter than the ruler, and fix one end
of it in f, and the other to the end o of the ruler. Then if
the ruler /mo be turned about the fixed point / at the
same time keeping the thread omf always tight, and its
part mo close to the side of the ruler, by means of the pin
m; the curve line ax described by the motion of the pin
m is one part of an hyperbola. And if the ruler be turned,
and move on the other side of the fixed point F, the other
part az of the same hyperbola may be described after the
same manner. — But if the end of the ruler be fixed in f,
and that of the thread in /, the opposite hyperbola xaz
may be described. ,

18. (5th Way, by a continued Motion.) — Letc and f be
the two foci, and E and k the two vertices of the hyper-
bola. (See the last fig. above.) Take three rulers CD,
dg, gf, so that cd=gf = ek, and dg = cf; the rulers
CD and gf being of an indefinite length beyond c and g,
and having slits in them for a pin to move in ; and the
rulers having holes in them at c and f, to fasten them to
the foci c and f by means of pins, and at the points d and
G they are to be joined by the ruler dg. Then, if a pin
be put in the slits, viz, the common intersection of the
rulers CD and gf, and moved along, causing the two rulers
gf, CD, to turn about the foci c and f, that pin will de-
scribe the portion F.e of an hyperbola. — The foregoing are
a few among various wajs given by several authors.
Some of the chief Pi operties of the Hyperbola.

19. (1st) The squares of the ordinates, of any diameter,
point 1, through which draw a perpendicular line, upon arc to each other, as the rectangles of their abscisses; i.e.
which set off im and in, equal to the distance la or lb, de2 : GH* : : ad . BD : AG. bg.

from 1 to the extremities of the otheraxis; and M and n 20. As the square of any diameter, is to the square of
will be points in the curve. its conjugate; so is the rectangle of two abscisses, to the

HYP

[ 669 ]

HYP

square of their ordinate. That is, ab* : aP : : ad , bd :

DE8.

Or, because the rectangle ad . bd is = the difference
of the squares cds — cbz, the same property is,
As ab!: ab2 : : cd'2 — cb1 : de%

r\ a!>~ „ i o

Or ab : — : : CD' — cb': de-,
ae

That is ab : p : : cd5 — cb2: de2, where p is the pa-
rameter of the diameter ab, or the 3d proportional to ab

and ab, or — .

AB

And hence is deduced the common equation of the hy-
perbola, by which its general nature is expressed. Thus,
putting d = the semidiameter ca or cb,
c = its semiconjugate ca or cb,

p = its parameter or — ,

x == the absciss bd from the vertex,

y =: the ordinate DE,and

•a = the absciss cd from the centre:

that

Then is d2 : c~ :

: (2d -+- x)x : y'

or dz : c1 :

: (d + v)x :y*

or d1 : c2 :

: v2 - d1 : /

or d : p :

: vz - d1 : y-

„ idx -t- a
V = = —

3 a _ °a - d°

—t--p> any

of which equations or proportions express the nature of
the curve. And hence arises the name hyperbola, signi-
fying to exceed, because the ratio of d2 to c2, or of d top,
exceeds that of Idx toy2; that ratio being equal in the
parabola, and defective in the ellipse, from which circum-
stances these also take their names.

21. The distance between the centre and. the focus, is
equal to the distance between the extremities of the trans-
verse and conjugate axes. That is, cf = Act or Ab, where
r is the focus.

22. The conjugate semi-axis is a mean proportional be-
tween the distances of the focus from both vertices of the
transverse. That is, ca is a mean between af and bf, or
af : ca:: ca : bf, or af . bf = ca2.

23. The difference of two lines drawn from the foci, to
meet ill any point of the curve, is equal to the transverse
axis. That is,/E — fe = ab, where f and /are the two
foci.

24. All the parallelograms inscribed between the four
conjugate hyperbolas are equal to one another, and there-
fore each equal to the rectangle of the two axes. That is,
the parallelogram opqr = ab . tib (fig. to art. 9).

25. The difference of the squares of every pair of con-
jugate diameters, is equal to the same constant quantity,
viz, the difference of the squares of the two axes. That is,
jiN8 — mn2 = ab8- ab2, (fig. to art. 6) where mn and
mn are any two conjugate diameters.

26". The rectangles of the parts of two parallel lines,
terminated by the curve, are to each other, as the rectangles
of the parts of any other two parallel lines, any where
cutting the farmer. Or the rectangles of the parts of two
intersecting lines, are as the squares of their parallel dia-
meters, or squares of their parallel tangents.

27- All the rectangles are equal which are made of the
segments of any parallel lines, cut by the curve.and limited
by the asymptotes, and each equal to the square of their
parallel diameter. That is, he . ek or lie . ck = cq2 or

28. All the parallelograms are equal, which are formed
between the asymptotes and curve, by lines parallel to the
asymptotes. That is, the paral. cgek = cpbq. — Hence
is obtained another method of expressing the nature of the
curve by an equation, involving the absciss taken on one
asymptote, and ordinate parallel to the other asymptote.
Thus, if x = CK,y = ke, a = cq, and b = bq the ordi-
nate at the vertex b of the curve; then, by the property
in this article, ab = xy, or a: x::y : b; that is, the rect-
angle of the absciss and ordinate isevery where of the same
magnitude, or any ordinate is reciprocally as its absciss.

29. If the abscisses cq, ck, cb, &c, taken on the one
asymptote, be in geometrical progression increasing ; then
shal.1 the ordinates qb, ke, lm, &c, parallel to the other
asymptote, be a like geometrical progression in the same
ratio, but decreasing; and all the rectangles are equal,
under every absciss and its ordinate, viz, cq . qb = ck .
ke = cl. lm, &c.

30. The abscisses cq, ck, cl, &c, being taken in geo-
metrical progression; thespaces or asymptotic areas bqke,
eklm, &c, will be all equal ; or, the spaces bqke, bqlm,
&c, will be in arithmetical progression ; and therefore these
spaces are the hyperbolic logarithms of those abscisses.

These, and many other curious properties of the hyper-
bola, may be seen demonstrated in my Treatise on Conic
Sections, and several others. See also Conic Sections.

Acute Hyperbola, one whose asymptotes make an
acute angle.

Ambigenal Hyperbola, is that which has one of its in-
finite legs falling within an angle formed by the asymptotes,
and the other falling without that angle. This is one of
Newton's triple hyperbolas of the 2d order. See his Enu-
meratio Lin. tert. Ord. See also Ambigenal.

Common, or Conic Hyperbola, is that which arises
from the section of a cone by a plane ; called also the
Apollonian hyperbola, being that which was treated on by
the first and chief author Apollonius.

Conjugate Hyperbolas, are those formed or lying to-
gether, and having the same axes, but in a contrary order,
viz, the transverse of each equal the conjugate of theother;
as the two Conjugate Hyperbolas pee and eee in the last
figure but one.

Equilateral, or Rectangular Hyperbola, is that whose
two axes are equal to each other, or whose asymptotes
make a right angle. — Hence, the property or equation of
the equilateral hyperbola, is;yx = ax ■+- x2, where a is the
axis, x the absciss, and y its ordinate; which is similar to
the equation of the circle, v\z,y2 = ax — x2, differing only
in the sign of the second term, and where a is the diameter
of the circle.

Infinite Hyperbolas, or Hyperbolas of the higlter
kinds, arc expressed or denned by general equations similar

HYP

C 670 ]

HYP

to that of the conic or common hyperbola, but having ge-
neral exponents, instead of the particular numeral ones,
but so as that the sum of those on one side of the equation,
is equal to the sum of those on the other side. Such as,
aym + n = bxm {d -+- x)n, where x and y are the absciss
and ordinate to the axis or diameter of the curve; or
xmyn = am + ", where the absciss x is taken on one asym-
ptote, and the ordinate^ parallel to the other.

As the hyperbola of the first kind, or order, viz the
conic hyperbola, has two asymptotes ; that of the 2d kind
or order has three; that of the 3d kind, four; and so on.

Obtuse Hyperbola, is that whose asymptotes form an
obtuse angle.

Opposite Hyperbolas, are those, pee, Qee, having the
same transverse and conjugate axes.

Rectangular Hyperbola, the same as equilateral hy-
perbola.

Hyperbolic Arc, is the arc of an hyperbola.
Put a = ca the semitransverse axe,
c = ca the semiconjugate, y = an or-
dinate pq to the axe drawn from the
end q of the arc aq, beginning at the

. . an + cc
vertex a : then putting q = — ,

a = the hyp. log. 01

= 2*302585093 x common log. of
y + ^/(cc + yy) i/y/fcc + yy) — ccA

Generally, if xmyn = am + n be an equation expressing
an hyperbola of any order ; then its asymptotic area will

be xy ; which space therefore is always quadrable, in

all the orders of hyperbolas, except the first or common hy-
perbola only, in which m and n being each 1, the deno-
minator n — m becomes 0 or nothing.

Hyperbolic Conoid, a solid formed by the revolution
of an hyperbola about its axis, otherwise called an- hyper-
boloid.

To find the Solid Content of an Hyperboloid.

Let AC be the semitransverse of the generating hyper-
bola, and ah the height of the solid ; then as 2ac ■+• ah
is to 3ac -1- ah, so is the cone of the same base and alti-
tude, to the content of the conoid.

t c-

y*y/(cc + yy) — SccB

, &c ; then is the length of the arc aq

expressed by c x (a

1.3.50* 0 , lSc^t -\- (21c' + '\9t*)x ,

— - — E&c; or by — = 3 -, — x y, nearly;

2.4.6.8 ■> 15^4 + (21^ -I- 91') X J1 J

where t is the whole transverse axe 2ca, c = 2ca the con-
jugate, x = ap the absciss, and 3/ = pq the ordinate.

These and other rules may be seen demonstrated in my
Mensuration, pa. 310 and 313, 4th edit.

Hyperbolic Area, that included by the hyperbolic
curve and other lines.

Putting o=ca the semitransverse, c = ca the semicon-
jugate,^ = pq the ordinate, and v = cp its distance from
the centre; then is the

t r 1 1 c ay + ra
area apq = \vy — \ac x hyp. log. ot ■ — ;

sector caq = \ac x hyp. log. of
area apq =%xy x (f- jj_- ^y- -

where x = a V, and z = ;

ay + cv

&c),

•2f.T
15(2

apq = — x [</<2ux ■+■ 4V(2a* + £*•*)] nearly.

Let ct and ce be the two asymptotes, and the ordi-
nates da, ef parallel to the other asymptote ct; then the
asymptotic space adef or sector caf is

CF.

= cn x da x hyp. log. of — , or

= CD x da x hyp. log. — ,or

r>E de' de* de1 „ j

= cn x da x t h- ; — 3 &c; and

CD 2CD' i(CBa 4 CD4

this last series was first given by Mercator in his Logarith-
motechnia.

Sec my Mensuration, pa. 315, &c, 4th edit.

To find the Curve Surface of an Hyperboloid,
Let ac be the semitransverse, and ab perpendicular to
it, and equal to the semiconjugate of ade the generatinf
hyperbola, or section through the axis of the solid. Join
cb ; makecF = ca, and on ca let fall the perpendicular
FG ; then with the semitransverse cg, and semiconjugate
gh = ab, describe the hyperbola gik ; then as the di-
ameter of a circle is to its circumference, so is the hy-
perbolic frustum ilamk to the curve surface of the co-
noid generated by dae. See my Mensur. pa. 328, &c,
4th edit.

Hyperbolic Cylindroid, a solid formed by the revolu-
tion of an hyperbola about its conjugate axis, or line
through the centre perpendicular to the transverse axis.
This solid is treated of in the Phil.Trans. by SirChristopher
Wren, who shows some of its properlies, and applies it to
the grinding of hyperbolical glasses ; affirming that they
must be formed this way, or not at all. See Philos. Trans,
vol. 4, pa. 96I.

Hyperbolic Leg, of a curve, is that having an asym-
ptote, or tangent at an infinite distance. Newton reduces
all curves, both of the first and higher kinds, into hyper-
bolic and parabolic legs, i.e. such as have asymptotes, and
such as have not, or such as have tangents at an infinite
distance, and such as have not.

Hyperbolic Line, is used by some authors for what is
more commonly called the hyperbola itself, being the curve
line of that figure; in which sense the surface terminated
by it is called the hyperbola.

Hy.perbolic Logarithm, a logarithm so called as
being similar to the asymptotic spaces of the hyperbola.
The hyperbolic logarithm of a number, is to the common
logarithm, as 2-3025850929.O40457 to 1, or as 1 to
•4342944S19032518. The first invented logarithms, by
Napier, are of the hyperbolic kind ; and so are Kepler's.
See Logarithm.

Hyperbolic Mirror, is one ground into that shape.

Hyperbolic Space, the same as Hyperbolic Area.

HYPERBOLICUM Acutum, a solid made by the re-
volution of the infinite area or space contained between
the curve of the hyperbola, and its asymptote. This pro-
duces a solid, which though infinitely long and gene-

HYP

C 671 ]

HYP

rated by an infinite area, is nevertheless equal to a finite
solid body; as is denominated by Torricelli, who gave it
this name.

HYPERBOLIFORM Figures, are such curves as ap-
proach, in their properties, to the nature of the hyperbola;
called also hyperboloides.

HYPERBOLOIDS, are hyperbolas of the higher kind,
whose nature is expressed by this equation, aym + n =
bxm(a -+- i)n. See Hyperbola. It also means the hy-
perbolic conoid. See that article.

HYPERBOREANS, the most northern nations, or
regions, as dwelling beyond or about the wind Boreas : as
the Siberians, Samoieds, &c.

HYPERTHYRON, in Architecture, a sort of table,
usually placed over gates or doors of the Doric order, above
the chambranle, in form of a frize.

HYPETHRE, in Ancient Architecture, two rows of
pillars surrounding, and ten at each face of a temple, &c,
with a peristyle within of six columns.

HYPOGEUM,in the Ancient Architecture, a name com-
mon to all the parts of a building that are under ground ;
as the cellars, butteries, &c.

HYPOMOCHLION, the fulcrum or prop of a lever;
or the point which sustains its pressure^ when employed
either in raising or lowering bodies. The hypomochlion is
frequently a roller set under the lever ; or under stones or
pieces of timber, &c, that they may be the more easily lifted
up, or removed.

HYPOTENUSE, or Hypothen use, in a right-angled
triangle, is the side which subtends, or is opposite to the
right angle, being always the longest of the three sides; as
the side ac, opposite to the right angle b.

It is a celebrated theo-
rem in plane geometry,
being the 47th prop, of the
1st book of Euclid, that
in every right-angled tri-
angle abc, the square
formed on the hypo-
tenuse ac, is equal to
both the two squares
formed upon the other
two sides ab and ec ; or
that ac2 = ab1 + bc2.
This is particularly called
the Pythagorean theorem, from its reputed inventor, Py-
thagoras, who it is said sacrificed a whole hecatomb to
the muses, in gratitude for the discovery. But the same
thing is true of circles or any other similar figures, viz, that
any figure described on the hypotenuse, is equal to the
sum of the two similar figures described on both the other
two sides.

HYPOTHENUSE. See Hypotenuse.
HYPOTHESIS, in Geometry, or Mathematics, means
much the same thing with supposition, being an assump-
tion of something as a condition, upon which to raise a de-
monstration, or from which to draw an inference.

Hypotheses, or postulatums, are considered by Dr.
Barrow as propositions assuming or affirming some evi-

dently possible mode, action, or motion of a thing, and
that there is the same affinity between hypotheses and pro-
blems, as between axioms and theorems: a problem show-
ing the manner, and demonstrating the possibility of some
structure, and an hypothesis assuming some construction
which is manifestly possible.

In disputation, false hypotheses are often employed in
order to draw the antagonists into absurdities; and even
in geometry, truths are often to be deduced from such
false hypotheses.

Every conditional, or hypothetical proposition, maybe
distinguished into hypothesis and thesis : the first asserts
theconditions underwhich any thing is affirmed or denied ;
and the latter is the thing itself affirmed or denied. Thus,
in the proposition, a triangle is half of a parallelogram, if
the bases and altitudes of the two be equal; the latter
part, namely, " if the bases," &c, is the hypothesis; and
the former part, " a triangle is half a parallelogram," the
thesis.

In strict logic, we are never to pass from the hypothesis
to the thesis: that is, the principle supposed must be
proved to be true, before we require the consequence to be
made.

Hypothesis, in Philosophy, denotes a kind of system
laid down from our own imagination, by which to ac-
count for some phenomenon or appearance of nature.
Thus, there are hypotheses to account for the tides, for
gravity, for magnetism, for the deluge, &c. The real and
scientific causes of natural things generally lie very deep
in obscurity : observation and experiment, the proper
means of arriving at them, are in most cases extremely
slow; and the human mind in consequence gets verv im-
patient: hence we are often induced to feign or invent
something that may seem like the cause, and which is cal-
culated to answer the several phenomena, so that it may
possibly be the true cause. Philosophers are divided as to
the use of such fictions or hypotheses, which are much
less current now than they were formerly. The latest and
best writers are for excluding hypotheses, and standing
entirely on observation and experiment. Whatever is
not deduced from phenomena, says Newton, is an hypo-
thesis ; and hypotheses, whether metaphysical, or physi-
cal, or mechanical, or of occult qualities, have no place
in experimental philosophy. Phil. Nat. Prin. Math, in
Calce.

Hypothesis is more particularly applied, in astro-
nomy, to the several systems of the heavens ; or the divers
manners in which different astronomers have supposed the
heavenly bodies to be ranged, or moved. The principal
hypotheses are the Ptolemaic, theTychonic, and the Co-
pernican. This last is now so generally received, and so
well established and warranted by observation, that it is
thought derogatory to it to call it an hypothesis.

HYPOTRACHELION, in Architecture, is used for a
little frize in the Tuscan and Doric capital, between the
astragal and annulets; called also the colerin and gor-
gerin. The word is applied by some authors in a more
genera] sense, to (he neck of any column, or that part of its
capital berow the astragal.

J A C

[ 672 J

ICE

I. J.

JACK, in Mechanics, is an instrument in common use
for raising heavy timber, or very great weights of any
kind ; being a certain powerful combination of teeth and
pinions; the whole inclosed in a strong wooden stock or
frame bc, and moved by a winch or handle HP; the out-
side appearing as in the figure an-
nexed.

In fig. G, pi. 14, the wheel or
rack-work is shown, being the view
of the inside when the stock is re-
moved. Though it is not drawn in
the just proportions and dimensions,
for the rack ab must be supposed
at least four times as long in propor-
tion to the wheel Q,as the figure re-
presents it; and the teeth, which
will be then four times more in num-
ber to have about 3 in the inch.
Now if the handle up be 7 inches
long, the circumference of this radius
will be 44 inches, which is the di-
stance or space the power moves
through in one revolution of the
handle: but as the pinion of the handle has but 4 leaves,
and the wheel Q suppose 20 teeth, or 5 times the number,
therefore to make one revolution of the wheel Q, it re-
quires 5 turns of the handle, in which case it passes
through 5 times 44 or 220 inches : but the wheel having
a pinion R of 3 leaves, these will raise the rack 3 teeth, or
one inch, in the same space. Hence then, the handle or
power moving 220 times as fast as the weight, will raise
or balance a weight of 220 times its own power. And if
this be the hand of a man, who can sustain 100 pounds
weight, he will, by help of this jack, be able to raise, or
sustain a weight or force of 22000lb,'or about 10 tons
weight. This machine is sometimes open behind from the
bottom almost up to the wheel q, to let the lower claw,
which in that case is turned up as at is, raise any weight.
When the weight is drawn or pushed sufficiently high, it
is kept from going back by hanging the end of the hook s,
fixed to a staple, over the curved part of the handle at h.
Jack is also the name of a well-known engine in the
kitchen, used for turning a spit. Here the weightis the
power applied, acting by a set of pulleys ; the friction of
the parts, and the weight with which the spit is charged,
make the force to be overcome; and a steady uniform mo-
tion is maintained by means of the fly. See the fig. of this
machine, pi. 14, fig. 7.

Smoke Jack, is an engine used for the same purpose
with the common jack, and is so called from its being
moved by means of the smoke, or rarefied air, ascending
the chimney, and striking against the sails of the horizontal
wheel ah (plate 14, fig. 8), which being inclined to the
horizon, is moved about the axis of the wheel, together
with the pinion C, which carries the wheels i> and e; and
B carries the chain r, which turns the spit. The wheel
A is should !»• placed in the narrow part of the chimney,
where the molion of the smoke is swiftest, and where also

the greatest part of it must strike on the sails. The
force, of this machine depends upon the draught of the
chimney, and the strength of the fire.

jACK-^4rc/(, is an arch of one brick thickness.
J ACK-Head, a part sometimes annexed to the forcing
pump.

J ACOR's-Staf, a mathematical instrument for taking
heights and distances; the same with the Cross-staff; which
see.

JAMBS, or Jaums, in Architecture, are the upright
sides of chimneys, from the hearth to the mantle-tree.
Also door posts, or the upright posts at the ends of the
window frames.

St. JAMES's Day, a festival in the calendar, observed
on the 25th of July, in honour of St. James the apostle.

JANUARY, the first month of the year, according to
the computation now used in the West, and containing 31
days; so called by the Romans from Janus, one of their
divinities, to whom they gave two faces ; because on the
one side, the first day of this month looked towards the
new year, and on the other towards the old one. The
name may also be derived from Janua, a gate; this month,
being the first of the year, may be considered as the gate
or entrance of it. January and February were introduced
into the year, by Nuuia Pompilius ; Romulus's year be-
ginning with the month of March.
JAUMS. See Jambs.

ICE, a brittle transparent body, formed of some fluid,
frozen or fixed by cold. The specific gravity of ice
to water, is various, according to the nature and circum-
stances of the water, degree of cold, &c. Dr. Irving
(Pliipps's Voyage towards the North Pole) found the
densest ice he could meet with about a 14-th part lighter
than water. M. de Mairan found it, at different trials,
1-14-th, 18th, or 19th lighter than water ; and when the
water was previously purged of air, only a 22d part.

The rarefaction of ice has been supposed owing to the
air-bubbles produced in ice while freezing; these, being
considerably large in proportion to the water frozen, ren-
der the ice so much specifically lighter. It is well known
that a considerable quantity of air is lodged in the inter-
stices of water, though it has there little or no elastic pro-
perty, on account of the disunion of its particles ; but on
these particles coming closer together, and uniting as the
water freezes, light, expansive, and elastic air-bubbles are
thus generated, and increase in bulk as the cold grows
stronger, and by their elastic force bursts to pieces any
vessel in which the water is closely contained.' Rut snow-
water, or any water long boiled over the (ire, affords an
ice more solid, and with fewer bubbles. Pure water long
kept in vacuo and frozen afterwards there, freezes much
sooner, on being exposed to the same degree of cold, than
water unpurged of its air and set in the open atmosphere.
And the ice made of water thus divested of its air, is much
harder, more solid and transparent, and heavier than com-
mon ice.

But M. de Mairan, in a dissertation on ice, attributes
the increase of the bulk of the water under this form,

I c o

r 673 j

JET

chiefly to a different arrangement of its parts : the icy
skin on water being composed of filaments which arc found
to be joined constantly and regularly atan angle of 60°, and
which, by this disposition, occupy a greater volume than
if they were parallel. Besides, after ice is formed, he
found it continue to expand by cold; a piece of ice, which
was at first only a 14th part specifically lighter than water,
on being exposed some days to the frost, became a 12th
part lighter ; and thus he accounts for the bursting of ice
in ponds.

It appears from an experiment of Dr. Hooke, in \663,
that ice refracts the light less than water ; whence he in-
fers, that the lightness of ice, which causes it to swim in
water, is not produced merely by the small bubbles which
are visible in it, but that it arises from the uniform con-
stitution or general texture of the whole mass : a fact
which was afterwards confirmed by M. Lahire. See
Hooke's Exper. by Derham, pa. 26, Acad. Per. 1693, Mem.
pa. 25.

Sir Robert Barker thus describes the process of making
ice in the East Indies, in a country where he never saw
any natural ice; On a large plain is dug three_ or four
pits, each about 30 feet square, and 2 feet deep ; the bot-
toms of which arecovered, about 8 or 12 inches thick, with
sugar-cane, or the stems of the large Indian corn, dried.
On this bed are placed in rows a number of small shal-
low unglazed earthen pans, formed of a very porous earth,
a quarter of an inch thick, and about an inch and a quarter
deep ; which, at the dusk of the evening, they fill with
soft water that has been boiled. In the morning before
sunrise the ice-makers attend at the pits, and collect what
has been frozen in baskets, which they convey to the place
of preservation. This is usually prepared in some high
and dry situation, by sinking a pit It or 15 feet deep,
which they line first with straw, and th. n with a coarse
kind of blanketing. The ice is deposited in this pit, and
beaten down with rammers, till at length its own accu-
mulated cold again freezes it, and it forms one solid mass.
The mouth of the pit is well secured from the exterior air
with straw and blankets, and a thatched roof is thrown
over the whole. Philos. Trans, vol. 65, pa. 252.

For the force of ice and other very interesting particu-
lars relating to it, see the articles Freezing, Cold, and
Congelation.

1CHNOGRAPHY, in Architecture, is a transverse or
horizontal section of a building, exhibiting the plot of the
whole edifice, and of the several rooms and apartments in
any story ; together with the thickness of the walls and
partitions; the dimensions of the doors, windows, and
chimneys ; the prefectures of the columns and piers, with
every thing visible in such a section.

Ichnographt, in Fortification, is the plan or repre-
sentation of the length and breadth of a fortress; the dis-
tinct parts of which are marked out, either on the ground
itself, or upon paper.

Ichnography, in Perspective, the view of any thing
cut off by a plane parallel to the horizon, just by the base
or bottom of it ; being the same with what is otherwise
called tl?e plan, geometrical plan, or ground-plot, of any
thing, and is opposed to Orthography or Elevation.

ICOSAEDRON, or Icosahedron, one of the five
regular bodies or solids, terminated by twenty equilateral
and equal triangles. It may be considered as consisting
of 20 equal and similar triangular pyramids, whose ver-
tices meet in the centre of a sphere conceived to circum-
Vol. I.

scribe it, and therefore having all their heights and bases
equal: hence the solidity of one of those pyramids mul-
tiplied by 20, the number of them, gives the solid con-
tent of the icosaedron.

To form or make the Icosaedron. — Describe upon a card
paper, or some other such like substance, 20 equilateral
triangles, as in the figure at the article Regular Body.
Cut it out by the extreme edges, and cut all the other
lines half through, then fold the sides up by these ed^es,
and the solid will be formed.

The linear edge or side of the icosaedron being a, then
will the surface be 5 ax^/3 = S-6602540a% and the so-
lidity = ^7'-^^ 2-1816950A3.

More general equations, and relations, may be seen in
my Mensuration, pa. 190, 4th edit.

IDES, in the Roman Calendar, a name given to a series
of S days in each month ; which, in the full months,
March, May, July, and October, commenced on the loth
day; and in the other months, on the 13th day; from
thence reckoned backward, so as in those four months to
terminate on the 8th day, and in the rest on the 6th.
These came between the calends and the nones. And this
way of counting is still used in the Roman Chancery, and
also in the Calendar of the Breviary.

JET d'eau, a French word, signifying a fountain that
throws up water to some height in the air. A jet of water
is thrown up by the weight of the column of water above
its ajutage, or orifice, up to its source or reservoir; and
therefore it would rise to the same height precisely as the
head or reservoir, if certain causes were removed which
prevent it from rising quite so high. For first, the velo-
city of the lower particles of the jet being greater than that
of the upper, the lower water strikes that which is next
above it ; and as fluids press every way, by its impulse it
widens,, and consequently shortens the column. Secondly,
the water at the top of the jet does not immediately fall
off, but forms a kind of ball or head, the weight of which
depresses the jet ; but if the jet be a little inclined, or
not quite upright, it will play higher, though it will not
in such case be quite so beautiful. Thirdly, the friction
against the sides of the pipe and hole of the ajutage, pre-
vents the jet from rising quite so high, and a small one
will be more impeded than a large one. And the fourth
cause is the resistance of the air, which is proportional to
the square of the velocity of the water nearly; and there-
fore the defect in the height is nearly in the same propor-
tion, which is also the same as the proportion of the heights
of the reservoirs above the ajutage. Hence, and from ex-
perience, it is found that a jet, properly constructed, rises
to different heights according to the height of the reser-
voir, as in the following table, Of the heights of reservoirs
and the heights of their corresponding jets; the former in
feet, and the latter in feet, and tenths of afoot.

Heig)

ts of Reservoirs at

d their Jets.

Res.

Jet.

Res.

Jet.

Res.

Jet.

Res.

Jet.

5

4'9

12

11-6

19

17-9

26

24-1

6

5-9

13

12-5

20

18-8

27

24-9

7

&8

14

13-4

21

197

28

25-8

8

7-8

15

14-3

22

20-6

29

266

9

87

16

152

23

21-5

30

27-5

10

97

17

l6l

24

22-3

31

28-3

11

10-6

1 8

17-0

25

23-2

32

292

4R

I G N

[ 674 3

IMA

Res. |

Jet. 1

Res. |

Jet.

Res.

Jet.

Res.

Jet.

33

30-0

52 j

45-2

71

593

90

72-5

34

30-S

53

46-0

72

60-0

91

73-2

35

31-6

54

467

73

607

-92

73-8

36

32-5

55

47"5

74

61-4

93

74-5

37

33-3

56

48-2

75

62- 1

94

75-2

3S

34-1

57

490

76

62-8

95

75-8

39

34-g

58

497

77

63-5

96

76-5

40

357

59

50-5

78

64-2

97

77-2

41

36-6

60

51-2

79

64-9

. 98

77-S

42

37-4

O'l

52-0

80

65-6

99

78-5

43

3S-1

62

527

81

66-3

100

79- 1

44

38-9

63

53-5

82

67-0

110

85-6

45

397

64

54-2

S3

677

120

91-9

46

40-5

65

54-9

84

68-4

130.

9S-0

47

41-3

66

557

S5

69-1

140

104

48

42-1

67

56-4

86

69-8

150

no

■ 4£l

42-9

68

57-1

87

70-5

160

116

50

437

69

57-8

88

7ri

51

44-4

70

58-6

89

71-8

Height
Reservoir.
0 feet

Diam. of the

Ajutage,
i to | inch

10
15

.

"

1 to £ -

20

-

-

T

25

-

-

■%-—..-

30
40

-

.

*to£

3

50
60

T

"■

3^
1

SO

-

-

'1 -

100

-

-

If to I*. ■

By various experiments that have been made by Ma-
riotte, Desagulicrs, and others, it has been found, that if
the reservoir be 5 feet high, a conduct-pipe If inch dia-
meter will admit a hole in the ajutage from i to -| of an
inch ; and so on as in the following table:

Diam. of the
Conduct-Pipe.

- If inch

- 2

- 2J

- 24

- a*

- H

- '44 >

- 5| or 6

- 6i or 7

- 7 or 8

But the size of-the pipe is more or less, with the distance.

If it be required to keep any number of jets of given di-
mensions playing, by one common conduct-pipe, the dia-
meter of an ajutage must be found that shall be equal to
all the small ones that are given, and from this its proper
conduct-pipe. Thus, if there be 4 ajutages, each f of an
inch diameter; then the square off is ■£%, which multi-
plied by 4, the number of them, makes if, the square
root of which is -| or 1-|, the diameter of an ajutage equal '
to all the other four ; to which in the table answers a
pipe of 8 inches diameter. In general, the diameter of
the conduct-pipe should be about 6 times that of the aju-
tage.— See Mariotte's Mouvement des Eau.x; Desaguliers's
Exper. Philos. vol. 2, pa. 127, &c ; Clare's Motion of
Fluids, pa. 109; &c.

JETTE, the border made round the stilts under a pier,
in certain old bridges, being the same with starling, con-
sisting of a strong framing of timber filled with stones,
chalk, &e; to preserve the foundations of the piers from
injury.

IGNIS Eatuus, a common meteor, chiefly seen in
dark nights about meadows, marshes, and other moist
places, as also in bufying-grounds, and near dung-hills.
It is known among the people by the appellations, Will
with a Wisp, and Jack wftba Lantern. Dr. Shaw describes
a remarkable ignis fatuus, which he saw in the Holy Land,

that was sometimes globular, or in the form of the flame
of a candle; and presently afterwards it spread itself so
much as to involve the whole company in a pale inoffen-
sive light, and then contract itself again, and suddenly
disappear. But in less than a minute it would become
visible as before ; or, running along from one place to
another, with a swift progressive motion, would expand
itself at certain intervals over more than 2 or 3 acres of
the adjacent mountains. The atmosphere had been thick
and hazy, and the dew on the horses' bridles was uncom-
monly clammy and unctuous. In the same weather he
observed those luminous appearances, which skip about
the masts and yards of ships at sea, and which the sailors
call corpusanse, by a conuption of the Spanish cuerpo-
santo. Shaw's Travels, pa. 363.

Newton calls it a vapour shining without heat; and
supposed it to be of the same nature with the light issuing
from putrescent substances. Willoughby and B.ay were
of opinion that it is occasioned by shining insects: but all
the appearances of it observed by Derham, Beccaria, and
others, sufficiently evince that it must be an ignited va-
pour. Inflammable air has been found to be the most
common of all the factitious airs in nature; and that it is
the usual product of the putrefaction and decomposition
of vegetable substances in water. Signor Volta writes to
Dr. Priestley, that he fires inflammable air by the electric
spark, even when thr electricity is very moderate; and
he supposes that this experiment explains theinflammation
of the ignes fatui, provided they consist of inflammable
air, issuing from marshy ground by help of the electricity
of fogs, and by falling stars, which have probably an
electrical origin. See Priestley's Obs. on Air, vol. 3, pa.
382 ; the Philos. Trans. Abr. vol. 7, pa. 374, &c.

IGNITION, properly signifies the setting fire to any
substance; but the sense is commonly restrained to that
kind of burning which is not accompanied with flame,
such as of charcoal, cinders, metals, stones, and other
solid substances, The effects of ignition are first to dis-
sipate what is called the phlogiston of the ignited body,
after which it is reduced to ashes. Vitrification next fol-
lows ; and lastly, the substance is totally dissipated in
vapour. All these effects, however, depend on the presence
of the air,- for in vacuo the phlogiston of any substanee
cannot be dissipated. Neither can a body which is to-
tally destitute of phlogiston be ignited in such a manner
as those which contain it; for as long as the phlogiston
remains, the heat is kept up in the body by the action of
the external air upon it : but when the phlogiston is en-
tirely evaporated, the air always destroys, instead of aug-
menting the heat. Philosophers have therefore been
greatly embarrassed in explaining the phenomena of igni-
tion. See Phlogiston.

ILLUMINATION, the act or effect of a luminous
body, or a body that emits light ; sometimes also the state
of another body that receives it.

Circle of Illumination. , See Circle.

ILLUMINATIVE Lunar Month, the space of time that
the moon is visible, between one conjunction and another.

IMAGE, in Optics, is the spectre or appearance of an
object, made cither by reflection or refraction. In all
plane mirrors, the image is of the same magnitude as the
object ; and it appears es far behind the mirror as the ob-
ject is before it. In convex mirrors, the image appears
less than the object ; and farther distant from the centre
of the convexity, than from the point of reflection. Mr.

I M A

[ 675 ]

I M A

Molyneux gives the following rule for finding the diame-
ter of an image, projected in the distinct base of a convex
mirror : viz. As the distance of the object from the mirror
is to the distance of the image from the glass, so is the
diameter of the object to the diameter of the image. See
Lens, Mirror, Reflection", and Refraction.

IMAGINARY Quantities, or Impossible Quantities, in
Algebra, are the even roots of negative quantities; which
expressions are imaginary, or impossible, or opposed to
real quantities ; as ^/ — aa, or \/ — a*, &c. For, as every
even power of any quantity whatever, whether positive or
negative, is necessarily positive, or having the sign -+-, be-
cause ■+- by H-, or — by — give equally -1- ; hence it fol-
lows that every even power, as the square for instance,
which is negative, or having the sign — , has no possible
root; and therefore the even rootsofsuch powers or quan-
tities are said to be impossible or imaginary. The mixt
expressions arising from imaginary quantities joined to real
ones, are also imaginary ; as a — */ — aa, or b -+- y7 — aa.
The rootsof negative quantities were, perhaps, first treated
of in Cardan's Algebra. As to the uneven roots of such
quantities, he shows that they are negative, and at the
same time assigns them : but the even roots of them he
rejects, observing that they are nothing as to common use,
being neither one thing or other ; that is, they are merely
imaginary or impossible. And since his time, it has gra-
dually become a part of algebra to treat of the roots of
negative quantities. Albert Girard, in his Invention Nou-
velle en l'Algebre, pa. 42, gives names to the three kinds
of roots of equations, calling them, greater than nothing,
less than nothing, and envelopee, as ^/- 3: but this was
soon after called imaginary or impossible, as appears by
Wallis's Algebra, pa. 264 &c ; where he observes that the
square root of a negative quantity, is a mean proportional
between a positive and a negative quantity ; as */ — be is
the mean proportional between -+- b and — c, or between
— b and ■+- c; and this he exemplifies by geometrical con-
structions. See also pa. 313. «

The arithmetic of these imaginary quantities has not yet
been generally agreed on ; viz, as to the operations of mul-
tiplication, division, and involution ; some authors giving
the results with -1-, and others on the contrary with the
negative sign — . Thus, Euler, in his Algebra, pa. 106
&c, makes the square of \/ — 3 to be — 3, of </ — 1 to
be — 1, &c; and yet he makes the product of two impos-
sibles, when they are unequal, to be possible and real : as
y7 - 2 x y' - 3 = y'fj ; and ,/ - 1 x </ - 4 = ^4 or
2. But how can the equality or inequality of the factors
cause any difference in the signs of the products ? If
1/— 2 x \/ — 3 be = ■v/-,-rJ> how can y7 — 3 x „/ — 3,
which is the square of y^ — 3, be — 3 ? Again, he makes
y'_ 3 x y' -t- 5 = .y/— 15. Also in division, he makes
y'- 4_i.^/_ 1 tobe=A/+ 4 or 2; and y'+S-^y' — 3

= v'— 1; also that 1 or y'+l -i-\/— 1 =V/ZT1 =
ij — 1 ; consequently, multiplying the quotient rooty' — 1
by the divisor y' — 1 , must give the dividend </ ■+■ 1 ; and
yet, by squaring, he makes the square of y7— I, or the
product y7— 1 x y7— 1, equal to — 1.

But Emerson makes the product of imaginaries to be
imaginary ; and for this reason, that " otherwise a real
product would be raised from impossible factors, which is
absurd. Thus, y* ' — a x y' — 6 = y' — ab, and y' — a
x — y' — b= — y' — ab, &c. Also y7 — a x y' — a
= — a, and y' — a x — y' — a = ■+■ a, &c." And thus

most of the writers on this part of algebra are pretty
equally divided, some making the product of impossibles
real, and others imaginary. — In the Phil: Trans, for 1778,
pa. 31S &c, Mr. Playfair has given an ingenious disser-
tation on the Arithmetic of Impossible Quantities. But
this relates chiefly to their applications and uses, and not
to the algorithm of theni,or .rules for their products, quo-
tients, squares, &c. From some operations however here
performed, we learn that he makes the product of y'— 1
by i/~ 1, or the square of y'— 1, to be— 1 ; and yet
in another place he makes the product of y'— 1 and
\/(l — z1) to be \/( — 1 -+- z2). This author concludes,
" that imaginary expressions are never of use in investiga-
tions but when the subject is a property common to the
measures both of ratios and of angles ; that they never lead
to any consequence which might not be drawn from the
affinity between those measures ; and that they are indeed
no more than a particular method of tracing that affinity.
The deductions into which they enter are thus reduced
to an argument from analogy, but the force of them is
not diminished on that account. The laws to which this
analogy is subject; the cases in which it is perfect, in
which it suffers certain alterations, and in which it is
wholly interrupted, are capable of being precisely ascer-
tained. Supported on so sure a foundation, the arith-
metic of impossible quantities will always remain an use-
ful instrument in the discovery of truth, and may be of
service when a more rigid analysis can hardly be applied.
For this reason, many researches concerning it, which in
themselves might be deemed absurd, are nevertheless not
destitute of utility. M. Bernoulli has found, for exam-
ple, that if r be the radius of a circle, the circumference

is = — ?: ,., Considered as a quadrature of the

y- 1 ^

circle, this imaginary theorem is wholly insignificant, and
would deservedly pass for an abuse of calculation ; at the
same time we learn from it, that if in any equation the

lojj. >/ — 1 , . '
quantity — should occur, it may be made to dis-
appear, by the substitution of a circular arch, and a pro-
perty, common to both the circle and hyperbola, may be
obtained. The same is to be observed of the rules which
have been invented for the transformation and reduction
of impossible quantities * ; they facilitate the operations of
this imaginary arithmetic, and thereby lead to the know-
ledge of the mostbeautiful and extensive analogy which
the doctrine of quantity has yet exhibited.

* The rules chiefly referred to, are those for reducing
the impossible roots of an equation to the form a + By/' — 1.

Imaginary Roots, of an equation, are those roots or
values of the unknown quantity in an equation, which con-
tain some imaginary quantity. So the roots of the equa-
tion xx -t- aa = 0, are the two imaginary quantities
■+- y7 — aa and — y* — , aa, or -+- ny' — 1 and — at/ — 1 ;
also the two roots of the equation xx -+■ ax ■+■ aa = 0,
are the imaginary quantities ± ia — fay' — 3; and the
three roots of the equation i3 — 1 == 0, or £3 = 1, are 1,

— — , and — — — : the first real, and the

2 2

two latter imaginary. Sometimes the real root of an

equation may be expressed by imaginary quantities; as in

the irreducible case of cubic equations, when the root is

expressed by Cardan's rule; and that happens whenever

the equation has no imaginary roots at all ; but when it

has two imaginary roots, then the only real root is ex-

4R2

IMA

[ 676 ]

I M P

pressed by that rule in an imaginary form. See my paper
on Cubic Equations, in my Tracts, vol. 2, pa. 78. Philos.
Trans, for 1780, pa. 406 &c.

Albert Girard first treated expressly on the impossible
or imaginary roots of equations, and stated that every
equation has as many roots, either real or imaginary, as the
index of the highest power denotes. Thus, the roots of the
biquadratic equation xi = 4x — 3, are four, which he
shows to be two real and two imaginary, viz, 1, 1, — 1 ■ +
■v/— 2, and — 1 — ^/ — 2 ; and he renders the relation
general, between all the roots and the coefficients of the
terms of the equation. See his Invention Nouvelle en
l'Algebre, anno 1629, theor. 2, pa. 40 &c.

M. Dalembert demonstrated, that every imaginary
root of any equation can always be reduced to the form
e+f^/— 1, where e and / are real quantities. And
hence it was also shown, that if

one root of an equation be - - e -*-/v — 1,

another root of it will always be - - e —f\/ — 1 :

and hence it appears that the number of the imaginary
roots in any equation is always even, if any ; i. e. either
none, or else two, or four, or six, &c. Memoirs of the
Academy of Berlin, 1746.

To discover how many impossible roots are contained
in any proposed equation, Newton gave this rule, in his
Algebra, viz, Constitute a series of fractions, whose deno-
minators are the series of natural numbers 1, 2, 3, 4, 5,
&c, continued to the number showing the index or expo-
nent of the highest term of the equations, and their nume-
rators the same series of numbers in the contrary order :
and divide each of these fractions by that next before it,
and place the resulting quotients over the immediate terms
of the equation ; then under each of the intermediate
terms, if its square multiplied by the fraction over it, be
greater than the product of the terms on each side of it,
place the sign -t- ; but if not, the sign — ; and under the
first and last term place the sign -+- . Then will the equa-
tion commonly have as many imaginary roots as there are
changes of the underwritten signs, from ■+- to — , and from

— to H-. So for the equation x3 — 4a-1 ■+■ 4* —6 = 0,
the series of fractions is 3, ■§, | ; then the second divided
by the first gives £ or •§•, and the third divided by the se-
cond gives | also ; hence these quotients placed over the
intermediate terms, the whole will stand

thus, *3 — 4x- + 4x — 6.
■+■ ■+■ — ■+■
Now because the square of the 2d term multiplied by its
superscribed fraction, is '-/a4, which is greater than 4a4
the product of the two adjacent terms, therefore the sign
■+■ is set below the 2d term ; and because the square of
the 3d term multiplied by its overwritten fraction, is '-/a'2,
which is less than 24*' the product of the terms on each
side of it, therefore the sign — is placed under that term ;
also the sign -+- is set under the first and last terms.
Hence the two changes of the underwritten signs -t- -+-

— rh, the one from ■+■ to — , and the other from — to 4-,
show that the given equation has two impossible roots.
When two or more terms are wanting together, under the
place of the 1st of the deficient terms write the sign — ,
under the 2d the sign +, under the 3d — , and so on, al-
ways varying the signs, except that under the last of the
deficient terms must always be set the sign -+- , when the
adjacent terms on both sides of the deficient terms have
contrary signs. As in the equation

+ «s = 0,

which has four imaginary roots ; there being four changes
in the signs as written. The author remarks, that this
rule will sometimes fail of discovering all the impossible
roots of an equation, for some equations may have more
of such roots than can be found by this rule, though this
seldom happens.

Mr. Maclaurin has given some account of this rule of
Newton's, together with one of his own, which he says
will never fail; and the same has also been done by Mr.
Campbell; but none of them obtain universally. See
Philos. Trans, vol.34, pa. 104, and vol.35, pa. 5 15. More
accurate rules have' been since given by Dr. Waring.

The real and imaginary roots of equations may be found
from the method of fluxions, applied to the doctrine of
maxima and minima, that is, to find such a value of x in
an equation, expressing the nature of a curve, made equal
to y, an abscissa which corresponds to the greatest and
least ordinate. But when the equation is above 3 dimen-
sions, the computation is very laborious. See Stirling's
treatise on the lines of the 3d order, Schol.pr.8, pa. 59 &c.
IMBIBE, is commonly used in the same sense as ab-
sorb, viz, where a dry porous body takes up another that
is moist.

IMMENSE, that whose amplitude or extension cannot
be equalled by any measure whatever, or how often soever
repeated.

IMMERSION, the act of plunging into water, or some
other fluid.

Immersion, in Astronomy, is when a star, or planet
comes so near the sun, that it cannot be seen; being as
it were enveloped, and hid in the rays of that luminary.

Immersion also denotes the beginning of an eclipse,
or of an occultation, when the body, or any part of it just
begins to disappear, either behind the edge of another
body, or in its shadow. As, in an eclipse of the moon,
when she begins to be darkened by enteringinto the shadow
of the earth : or the beginning of an eclipse of the sun,
when the moon's disc just begins to cover him : or the
beginning of the eclipses of any of the satellites, as those
of Jupiter, by entering into his shadow : or, lastly, the
beginning of an occultation of any star or planet, by
passing behind the body of the moon or another planet.
Ill all these cases, the darkened body is said to immerge,
or to be immerged, or begin to be hid, by entering into
the shade. In like manner, when the darkened body
begins to appear again, it is said to emerge, or come out
of darkness again.

IMPACT, the simple or single action of one body upon
another to put it in motion. Point of impact, is the
place or point where a body acts. — Impact is either direct
or oblique : if the body struck moves in the direction of
the stroke, the impact is said to be direct ; but if not, it
is called an oblique impact.

IMPENETRABILITY, the faculty which a body has
to exclude every other body from the place that it occu-
pies: in such manner that two bodies, placed in contact,
can never occupy less 'space than that which they filled
when they were separate.

Impenetrability, as it respects solid bodies, does not
require to be proved, it is at once obvious; but fluids
having their moleculie moveable in all directions, and
yielding to every impression exerted on them, their im-
penetrability does not manifest itself so perceptibly as that

I M P

[ 677 ]

I N C

of solid bodies. Taking the air for an example : it is evi-
dent that so long as this fluid is not enclosed, its facility
of motion causes it to admit an easy and free passage to
all bodies that are urged through it; ^nd in this case ft
cannot be said to be penetrated, but displaced ; for, if the
air be included within the sides of a vessel, and another
body be then presented to take its place, without suffering
-it to escape, it will exercise its impenetrability in the same
manner as solid bodies. This may be sufficiently evinced
by the following very easy and simple experiment : Let a
vessel be vertically detruded into another vessel filled
with water to a'certain height, with its orifice downwards :
the surface of the water corresponding with the orifice of
the first vessel, is depressed as this vessel itself descends ;
and this depression may be rendered more sensible by
means of a little plate or slip of- cork, placed so as to float
on the surface of the water: nevertheless, this water is
not excluded by the air which occupies the immersed ves-
sel ; it is always raised within it by a certain quantity
which increases as the vessel is immersed to a greater
depth, similar to what is observed in the diving-bell ; but
it is evident that this ascension of the fluid is occasioned
by the circumstance that the air is a compressible fluid,
and not from any penetration of the water into the bulk
of air in the vessel ; and therefore its volume is only con-
tracted into a smaller space by a superior upward pressure
of the surrounding water in virtue of its weight.

We must here notice a difficulty which appears to re-
sult from this, that when certain bodies are mingled to-
gether, the volume of the mixture is less than the sum of
the volumes taken separately. This happens, for example,
in the mixture of equal parts of alcohol and water; the
same also obtains when copper by fusion is mingled with
zinc, in order to form the compound metal called brass :
it is then observed that the density of the mixture is aug-
mented by about its tenth part. This apparent penetra-
tion is owing to the circumstance, that the moleculse of
the two bodies, in consequence of their respective forma-
tion, have a stronger attraction than in the two bodies
taken separately ; there hence results in the figure of the
pores sucli a change as diminishes the space equal to the
sum of these pores. On the contrary, in the' alloy of
silver with copper, a kind of rarefraction is produced,
such that the volume of the mixture is somewhat greater
than the sum of the capacities of the two bodies previous
to fusion.

IMPENETRABLE, that cannot be penetrated.

IMPERFECT Number, is that whose aliquot parts,
taken all together, do not make a sum that is equal to the
number itself, but either exceed it, or fall short of it ;
being an abundant number in the former case, and a de-
fective number in the latter. Thus, 12 is an abundant
imperfect number, because the sum of all its aliquot parts,
1, 2, 3, 4, 6, is 16, which exceeds the number 12. And
10 is a defective imperfect number, because, its aliquot
parts, ], 2, 5, taken all together, make only 8, which is
less than the number 10 itself.

IMPERIAL Table, is an instrument made of brass, with

a box and needle, and staff, &c, used for measuring of angle abc

Impetus, in Gunnery, is the altitude due to the first
force of projection, or the altitude through which a body
by falling acquires that velocity. So that, if the velocity
be v, the impetus or altitude will be a = »*-r- 64j feet.

IMPOSSIBLE Quantity, or Root, the same as

Imaginary ones ; which' see.

Impossible Binomial; see Binomial.

IMPOST, in Architecture, a capital or plinth, to a
pillar, or pilaster, or pier, that supports an arch, &c.

IMPROPER Fraction, is a fraction whose numerator
is either equal to, or greater than, its denominator. As -f-
or I or *■£ . An improper fraction is reduced to a whole
or mixt number, by dividing the numerator by the deno-
minator; the quotient is the integer, and the remainder
set over the divisor makes the fractional part, of the value
of the original improper fraction. Thus \ = 1, and
f = 1-|, and '-g9 = 3-g-. So that, when the numerator is
just equal to thedenorhinator, the improper fraction is ex-
actly equal to unity or 1 ; buF when the numerator is the
greater, the fraction is greater than 1.

IMPULSE, the single or momentary action or force
by which a body is impelled ; in contradistinction to con-
tinued forces ; like the blow of a hammer.

IMPULSIVE, a term applied to actions by impulse.

INACCESSIBLE Height or Distance, is that which
cannot be approached, or measured by actual measure-
ment, by reason of some impediment ; as water, &c. — See
Heights and Distances.

INCEPTIVE, of Magnitude, a term used by Dr. Wal-
lis, to express such moments, or first principles, as, though
of no magnitude themselves, are yet capable of producing
such as are. See Infinite, and Indivisible. Thus,
a point has no magnitude, but is inceptive of a line, which
it produces by its motion. Also a line, though it has not
breadth, is yet inceptive of breadth ; that is, it is capable,
by its motiom, of producing a surface, which has breadth.

INCH, a common English measure, being the 12th part
of a foot, or 3 barley-corns in length.

INCIDENCE, or Line of Incidence, in Mechanics,
implies the direction or inclination in which one body
strikes or acts on another. — In the incursions of two mo-
ving bodies, their incidence is said to be direct or oblique,
as the directions of their motion make a straight line, or
an angle at the point of impact.

Angle of Incidence, by some writers, denotes the
angle comprehended be-
tween the line of incidence,
and a perpendicular to the
body acted on at the point
of incidence. Thus, sup-
pose ab an incident line,
and bf a perpendicular to

the plane cb at the incident point b ; then abf is the
angle of incidence, or of inclination.

But, according to Dr. Barrow, and some other writers,
the angle of incidence is the complement of the former,
or the angle made between the incident line, and the plane
acted on, or a tangent at the point of incidence : as the

land.

IMPERVIOUS, not to be pervaded or entered either
because of the closeness of the pores, or the particular
configuration of its parts.

IMPETUS, in Mechanics, force, momentum, motion,
&c.

\t is demonstrated by optical writers, 1st, That the
angle of incidence of the rays of light, is always equal to
the angle of reflection; and that they lie in the same plane.
And the same is proved by the writers on mechanics, con-
cerning the reflection of clastic bodies. That is, the
Z.abf = the Z.fbd, or the £ABC = the Zdbe.—

INC

[ 678 ]

INC

2d, That the sines of the angles of incidence and refraction
are to each other, either accurately, or very nearly, in a
given or constant ratio. — 3dly, That from air into glass,
the sine of the angle of incidence is to the sine of the angle
of refraction, as 300 to 1.93, or nearly as 14 to 9 : and, on
the other hand, that out of glass into air, the sine of the
angle of incidence, is to the sine of the angle of refraction,
as 193 to 300, or as 9 to 14 nearly.

Incidence of Eclipse. See Eclipse, and Immersion.
Axis of Incidence, is the line fb perpendicular to the
reflecting plane at the point of incidence b.

Cathetusof Is civesce. See Cathetus, and Reflection.

Line of Incidence, in Catoptrics, denotes a right line,

as ab, in which light is propagated from a radiant point a ,

to a point b, in the surface of a speculum. The same

line is also called an incident ray.

Line of Incidence, in Dioptrics, is a right line, as ab,
in which light is propagated unrefracted, in the same me-
dium, from the radiant point to the surface of the re-
fracting body, cbe.

Point o/Incidence, is the point b on the surface of
the reflecting or refracting medium, on which the incident
ray falls.

Scruples of Incidence. See Scruples.
INCIDENT Ray, is the line or ray ab, falling on the
surface of any body, at b.

INCLINATION, in Geometry, Mechanics, or Physics,
denotes the mutual tendency of two lines, planes, or bo-
dies, towards each other ; so that their directions make
at the point of concourse some certain angle.

Inclination of the Axis of the Earth, is the angle it
makes with the plane of the ecliptic ; or the angle between
the planes of the equator and ecliptic.

Inclination of a Line
to a Plane, is the acute
angle, as cde, which the
line cd makes with another
line de drawn in the plane
through the incident point
d and the foot of a perpen-
dicular -e, from any point
of the line, upon the plane.

Inclination of an Incident Ray, is the angle of inclina-
tion, or angle of incidence.

Inclination of the Magnetical Needle. See Dipping
Needle.

Inclination of Meridians, in Dialling, is the angle
that the hour-line on the globe, which is perpendicular to
the dial-plane, makes with the meridian.

Inclination of the Orbit of a Planet, is the angle formed

by the planes of the ecliptic and of the orbit of the planet.

The quantity of this inclination for the several planets, is

as follows, viz.

Mercury

Venus

Earth

Moon

Mars

Jupiter

Saturn

Herschel

Inclination of a Plane, in Dialling, is the arch of a
vertical circle, perpendicular both to the plane and the
horizon, and intercepted between them.

Inclination of a Planet, is the arch or angle com-

6°

54'

0'

3

23

35

0

0

0

5

18

0

1

51

0

1

18

56

2

29

50

0

46

20

prehended between the ecliptic and the place of the pla-
net in its orbit. The greatest inclination, or declination,
is the same as the inclination of the orbit ; which see above.

Inclination of a Reflected Ray, is the angle which a
ray after reflection makes with the axis of inclination; as
the angle fed, in the last fig. but one.'

Inclination of Two Planes, is the angle made by two
lines drawn in those planes perpendicular to their common
intersection, and meeting in any point of that intersection.

Angle of Inclination, is the same as what is other-
wise called the angle of incidence.

Argument of Inclination. See Argument.

INCLINATIONS {Problem of), is one of the last works
of Apollonius, described by Pappus, as consisting of two
books. Of these, restorations have been attempted by se-
veral of the moderns, with various success. 1st, By Ghe-
taldus, in two books entitled Apollonius Redivivus, 4to,
Venice, 1607 & 1613. — 2d, Alex. Anderson, Supplem.
Apol. Rediv. Inclin. Paris, 1612, & l6l5, 4to.— 3d,
Horsley, Apol. Inclin. lib. 2, Oxon, 17/0, 4to. — 4th, Reu-
ben Burrow published an English edition in 1779, *to.

INCLINED Plane, in Mechanics, is a plane inclined to
the horizon, or making an angle with it. It is one of the
simple mechanic powers, and the double inclined plane
makes the wedge.

1. The power gained'by the inclined plane, is in pro-
portion as the length of the plane is to its height, or as
radius to the sine of its inclination ; that is, a given weight
hanging freely, will balance upon the plane another weight,
that shall be greater in that proportion. So, when the
greater weight w on the plane, is ba- c
lanced by the less weight w hanging
perpendicularly, then is vj : w : : bc :
AC : : sin. Z. a : radius. Or, in other
words, the relative gravity of a body
upon the plane, or its force in de-
scending down the plane, is to its' ab-
solute gravity or weight, in the same proportion of the height
of the plane to its length, or of the sine of inclination to radius.

2. Hence therefore the relative gravities of the same
body on different inclined planes, or their forces, to de-
scend down the planes, are to each other, as the sines of
the angles of inclination, to radius 1, or directly as the
heights of the planes, and inversely as their lengths.

3. Hence, if the planes have the
sameheight,and the absolute weights
of the bodies be directly proportional
to the lengths of the planes, then the
forces to descend will be equal. Con-
sequently, if the bodies be then con-
nected by a string acting parallel to the planes, they will
exactly balance each other; as in the annexed figure.

4. The relative force of gravity on the plane being in a
constant ratio to the absolute weight of the body, viz, as
sine of inclination to radius; therefore all the laws rela-
ting to the perpendicular free descents of bodies by gra-
vity, hold equally true for the descents on inclined planes;
such as, that the motion is a uniformly accelerated one ;
that the velocities are directly as the times, and the spaces
as the square of either of them ; using only the relative
force upon the plane for the absolute weight of the body,
or instead of 32A feet, the velocity generated by gravity in
the first second of time, using 32-is, where s is the sine of
the inclination to t lie radius 1.

5. The velocity acquired by a body in descending

INC

C 679 ]

I N C

down an inclined plane ac, when the
body arrives at a, is the same as
the velocity acquired by descending
freely down the perpendicular alti-
tude bc, when it arrives at b. But
the times are very different; for the
time of descending down the inclined
plane, is greater than down the per-
pendicular, in the same proportion
es the length of the plane ac, is to the height cb : and so
the time of descending from any point c to a horizontal
line or plane abg &c, down any oblique line, or inclined
plane, is directly proportional to the length of that plane,
Ca, or cd, or ce, or cb, or cf, &c.

6. Hence, if there be drawn ah perpendicular to ac,
meeting cb produced in h; then the time of descending
down any plane ca, is equal to the time "of descending
down the perpendicular en. So that, if upon ch, as a
diameter, a circle be described, the times of descent wilL
be exactly equal, down every chord in the circle, begin-
ning at c, and terminating any where in the circumfe-
rence, as ci, ca, ck, ch, &c, or beginning any where in
the circumference, and ending at the lowest point of the
circle, as ch, ih, ah, kh, &c. ■

7. When bodies ascend inclined planes, their motion is
uniformly retarded ; and all the former laws for descents, "
or the generation of motion, hold equally true for ascents,
or the destruction of the same.

Inclined Towers, are towers inclined, or leaning from
the perpendicular, as those of Pisa or Bologna in Italy.
See Towers.

INCLINERS, are inclined dials. See Dial.

INCOMMENSURABLE, Lines, or Numbers,orQuan-
tities in general, are such as have no common measure, or
no line, number, or quantity of the same kind, that will
measure or divide them both without a remainder. Thus,
the numbers 15 and \6 are incommensurable, because,
though 15 can be measured by 3 and 5, and \6 by 2, 4,
and 8, there is yet no single number that will divide or
measure them both. — Euclid demonstrates (prop. 1 17,
lib. 10) that the side of a square and its diagonal are in-
commensurable to each other. And Pappus, prop. 17,
lib. 4, speaks of incommensurable angles.

Incommensurable in Power, is said of quantities
whose 2d powers, or squares, are incommensurable. As
^/2 and t/'d, whose squares are 2 and 3, which are incom-
mensurable. It is commonly supposed that the diameter
and circumference of a circle are incommensurable to each
other; at least their commensurability has never been
proved. And Dr. Barrow surmises even that they are in-
finitely incommensurable, or that all possible powers of
them arc incommensurable.

INCOMPOS1TE Numbers, are the same with those
called by Euclid prime numbers, being such as are not
composed by the multiplication together of other numbers.
As 3, 5, 7, 11, &c.

INCOMPRESSIBLE, incapable of being squeezed into
less room.

INCOMPRESSIBILITY, incapacity to be compressed
into less room. — During a long period, water was consi-
dered as a fluid perfectly unelastic, that is, incompressible
or unyielding: and this opinion was corroborated by an
experiment of the Academy del Cimento in Italy. By
that experiment, the members of the academy endeavour-
ed to ascertain whether water was capable of being com-

pressed in any degree. For this purpose, they filled a
hollow metallic globe with that fluid, and closed the ori-
fice very accurately. This ball was then forcibly pressed
in a proper machine; but no contraction could be ob-
served, neither indeed was the apparatus capable of show-
ing small degrees of compression: hence however they
concluded that water was not capable of compression.
This opinion prevailed till the year 176.1, when Mr. Can-
ton discovered the compressibility of water, and of other
liquids. He took a glass tube, having a hollow ball at one
end, but open at top; he filled the ball and part of the
tube with water, which he had deprived of air as much as
possible; he then placed the glass thus filled under the
receiver of an air-pump; and on exhausting the receiver,
which removed the pressure of the atmosphere from over
the water and the glass vessel, the consequence was, that
the water rose a little way in the tube, that is, it expand-
ed itself. He then placed the apparatus under the re-
ceiver of a condensing engine, and on forcing the air into
it, which increased the pressure on the water, a diminu-
tion of bulk visibly took place, the water descending a
little, in the tube. " In this manner," says Mr. Canton,
" I have found by repeated trials, when the heat of the
air has been about 509, and the mercury at a mean height
in the barometer, that the water will expand and rise in
the tube, by removing the weight of the atmosphere, one
part in 21710, and will be as much compressed under the
weight of an additional atmosphere. Therefore the com-
pression of water by twice the weight of the atmosphere, is
one part in 10S70. Water has also the remarkable pro-
perty of being more compressible in winter than in sum-
'mer, which is contrary to what I have observed both in
spirits of wine and oil of olives." By the same means,
and in the same circumstances, Mr. Canton ascertained
compressibility in some other fluids, the results of which
are as below : viz,

Compression of Millionth parts.

Spirit of wine - - 66

Oil of olives - - 48

Rain-water - - 46

Sea-water - - 40

Mercury - - 3

INCREMENT, is the small increase of a variable
quantity. Newton, in his Treatise on .Fluxions, calls
these- by the name Moments, and observes that they are
proportional to the velocity or rate of increase of the
flowing or variable quantities, in an indefinitely small
time; he denotes them by subjoining a cipher 0, to the
varying quantity whose moment or increment it is ; thus
xO is the moment of x. In the doctrine of increments, by
Dr. Brooke Taylor and Mr. Emerson, they are denoted
by points below the variable quantities; as f Some have
also represented them by accents underneath the letter,
as x; but it is now more usual to express them by ac-
cents over the same letter ; as x'.

INCREMENTS, Method of, a branch of Analytics, in
which a calculus is founded on the properties of the suc-
cessive values of variable quantities, and their differences,
on increments. The inventor of this method was the
learned Dr. Taylor, who, in the year 1715, published a
treatise upon it; and afterwards gave some further ac-
count and explication of it in the Philos. Trans, as applied
to the finding the sums of series. Another ingenious and
easy treatise on the same, was published by Mr. Emerson,
in the year 1763. The method is nearly allied to New-

INC [ 6S0

ton's doctrine of fluxions, and arises put of it. Also the
differential method of Mr. Stirling, which he applies to
the summation and interpolation of series, is of the same
nature as the method of increments, but not so general
and extensive.

From the method of increments, Mr. Emerson observes,
" the principal foundation of the method of fluxions
may be easily derived. For, as in the method of incre-
ments, the increment may be of any magnitude, so in the
method of fluxions, it must be supposed infinitely small ;
whence all preceding and successive values of the variable
quantity will be equal, from which equality the rules for
performing the principal operations of fluxions are imme-
diately deduced. That I may give the reader," continues
he, "a more perfect idea of the nature of this method : sup-
pose the abscissa of a curve to be divided into any num-
ber of equal parts, each part of which is called the incre-
ment of the abscissa; and imagine so many parallelo-
grams to be "erected thereon; either circumscribing the
curvilineal figure, or inscribed in it ; then the finding the

]

INC

gral. — 2. The increase of any quantity from its present
value, to the next succeeding value, is called an incre-
ment: or, if it decreases, a decrement. — 3. The increase
of any increment, is the second increment ; and the in-
crease of the 2d increment, is the 3d increment; and so on.
— 4. Succeeding values, are the several values of the inte-
gral, succeeding each other in regular order, from the
present value ; and preceding values, are such as arise
before the present value. All these are called by the ge-
neral term Factors. — 5. A perfect quantity is such as
contains any number of successive values without inter-
mission ; and a defective quantity, is that which wants
some of the successive values. Thus imisa perfect

3 3 4 5

quantity ; and x x x, an imperfect or defective one.

2 4 5

Notation. This, according to Mr. Emerson's method, is
as follows ; 1. Simple integral quantities are denoted by
any letters whatever, as z, y, x, u, &c. — 2. The several
values of a simple integral, are denoted by the same letter
with small figures under them : so if z be an integral,

of all these parallelograms belongs to the method of then 2, z, 2, z, &c, are the present values, and the 1st
increments. But if the parts of the abscissa be taken infi-
nitely small, then these parallelograms degenerate into the
curve; and then it becomes the method of fluxions, to find
the sum of all, or thearea of the curve. So that the me-
thod of increments finds the sum of any number of finite
quantities; and the method of fluxions the sum of any
infinite number of infinitely small ones: and this is

the essential difference between these two methods." Again,
" There is such a near relation between the method of
fluxions, and that of increments, that many of the rules
for the one, with little variation, serve also for the other.
And here, as in the method of fluxions, some questions
may be solved, and the integrals found, in finite terms;
while in others we are forced to have recourse to infinite
series for a solution. And the like difficulties will occur
in the method of increments, as usually happen in fluxions.
For while some fluxionary quantities have no fluents, but
what are expressed by series ; so also some increments
have no integrals, but what infinite series afford ; which
will often, as in fluxions, diverge and become useless."

By means of the method of increments, many curious
and useful problems are easily resolved, which scarcely
admit of a solution in any other way. As, suppose seve-
ral series of quantities be given, whose terms are all formed
according to some certain law, which is also given; the
method of increments will determine a general series,
which comprehends all particular cases, and from, which
all of that kind may be found. The method of increments
is also of great use in finding any term of a series proposed ;
having the law given by which the terms are formed ;
which will either be expressed in finite quantities, or by an
infinite series. Another application of the method of in-
crements, is to find the summation of series, which it will
often effect in finite terms. And when the sum of a series
cannot be had in finite terms, we must have recourse to in-
finite scries ; for the integral being expressed by such a
series, the sum of a competent number of its terms will
give the sum of the series required. This is equivalent to
transforming one series into another, converging quicker:
and sometimes a very few terms of this series will give the
sum of the series sought.

Definitions in the Method of Increments.
1. When a quantity is considered as increasing, or de-
creasing, by certain steps or degrees, it is called an Inte-

2d, 3d, &c, successive values of it; and the preceding
values are denoted by figures with negative signs, thus
z, 2, 2, 2, are the 1st, 2d, 3d, 4th preceding values; and

-1 -2 -3 --4

the figure denoting any value, is the characteristic.

3. The increments are denoted with the same letters,
and points under them : thus, x is the increment of x, and
•z is the increment of 2. Also x is the increment of x; and

x of jt, &c.

4. The 2d, 3d, and other increments, are denoted with
two, three, or more points : so z is the 2d increment of z,
and * is the 3d increment of s, and so on. And these are
denominated increments of such an order, according to
the number of points.

5. If x be any increment, then [x] is the integral of
it; also 2[x] denotes the integral of [x], or the 2d in-
tegral of x ; and 3[x] is the 3d integral of x, or an in-
tegral of the 3d order, &c.

6. Quantities written thus,

x ... x mean the same asm x x, or signify that the

1 5 1234s

quantities are continued from the first to the last, without

break or interruption.

To find the Increment of any integral, or variable quantity.

Rule\. If the proposed quantity be not fractional, and
be a perfect integral, consisting of the successive values of
the variable quantity, which increases uniformly : Multi-
ply the proposed integral by the number of factors, and
change the lowest factor for an increment. So the incre-
ment of a — 3x + 6z is — 3x ■+■ 6*?; for the increment
of the constant quantity a is 0 or nothing. So likewise,

The increment of c x x x x, is 4c* x x x.

123 '123

The increment of ax x x, is 3a x x x.

The increment of x . . . x, is (m -*- n -+- 1) . x x . . . x.

— m n 1—nn- 1 n

Rule 2. In fractional quantities, where the denominator
is perfect, and the variable quantity increases uniformly.
Multiply the proposed integral by the number of factors,
and by the constant increment with a negative sign, and
take the next succeeding value into the denominator. Thus,
, a . — Sax
The increment ol , is — •

INC

]

INC

The increment of

Rule 3. The increment o( any power, as xn is
(.r -i- *)" — xn; that is, the difference between the present
value x" and the next succeeding value (x •+- f)R. And
generally, the increment of any quantity whatever, is
found by subtracting the present value, or ihe given quan-
tity, from its next succeeding value." Also by expanding
the compound quantity in a series, and subtracting xn
from it, the increment will be either
= ni"~' x -+- — x"""2 x- ■+- - — : —x"~3 x3 Sec, or

2 . 3
n.n — i.i

^&C.

So the increment of x\ is (x -f- x )_' — x* —
4,x3x -+- 6x"x2 •+- 4XX3 -+- x\

The increment of — or x~3,

is • — = (x -t- x) -3 — I"3

Ix + xf x3 N '

= - 3x'*x -+- 6x-s x- - Wx-6 X3 &c

AX 6x" 1 OX3

x°

&c.

The increment of ax, a being constant, is
a1 — aK — a " — a* = ax. (a" — 1).
The increment of — is —

a*— a „x _ a* + *■ l—o-

aal '~~^ax + x ax + x

The increment of xz is rz — xz =

(x -+- a?) . (z + z) — xz = zx ■+■ xz -t- x%.

And so on for any form of integral whatever, subtract-
ing the given quantity from its next succeeding value. So,

The increment of the log. of x is log. x — log. x

= log. x -t- x — log. x = log. , which, by the nature

of logarithms, is — r ■+■ -=-r — -=-— &c.

a; 2jt 3.ar 4.c

Schol. Hence may be deduced the principles and rules
of fluxions; for the method of fluxions is only a particu-
lar case of the method of increments, fluxions being infi-
nitely small increments; therefore, if in any form of incre-
ments} the increment be taken infinitely small, the form
or expression will be changed into a fluxional one. Thus,
in zx -t- x? -i- xz, which is the increment of the rect-
angle xz, if x and z be changed for * and z, the expression
will become zx ■+■ xz -+- xz for the fluxion of xz, or only
zx -)_ rz, because xz js infinitely less than the rest.
So likewise, if * be changed for * in this
_ , v.n — 1 „ „ , n.n — \.n. — 2 ,. „ ,

71Xn"' X -f. x" -x -t- X n dxJ

■2 2.3

&:c, which is the increment of x", it becomes

,,_• • n.n — l „__o •, n.n— l.n — 2 _ „ ...

.n-1 xi

&c, or nxn~ ' •*, for the fluxion of the power x", as all the
terms after the first will be nothing, because x* and x3 &c
are infinitely less lhan x. And thus may all the other
forms of fluxions oe derived from the corresponding incre-
ments. And in like manner, the finding of the integrals,
is only a more general way of finding fluents, as appears
in what follows.
Vol. I.

To find the Integral of any given Increment.

Rule 1. When the variable quantity increases uniform-
ly, and the proposed integral consists of the successive
values of it multiplied together, or is a perfect increment
not fractional : Multiply the given increment by the next
preceding value of the variable quantity, then divide by
the new number of factors, and by the constant increment.

Example. Thus, the integral of '4c* xxx is exxxx.

The integral of 3ax x x is ax x x.

Rule 2. In a fractional expression, where the variable
quantity increases uniformly, and the denominator is per-
fect, containing the successive values of the variable quan-
tity : Expunge the greatest value of the variable letter,
then divide by the new number of factors, and by the con-
stant increment with a negative sign. So,

The integral of "• is ~ a .
x ... x bx . .x

5 4

The integral of — is — '— .

X . .. .X X . . . ,x
-3 6 -3 5

Rule 3. Various other particular rules are given, but
these and the two foregoing are all best included in the fol-
lowing general table of the most useful forms of incre-
ments and integrals, to be used in the same way as the si-
milar table of fluxions and fluents, to which these cor-
respond.

A Table of Increments and their Integrals.

Forms

Increments.

Integrals.

1

x when constant
not constant

x, or x, or x, or x &c.

1 1 3

x only.

2

* constant

m -+- n

ax

— a

3

x constant

(m -+- n — 2) . x .... x

4

Zx -+- xz

xz

zx — xz

X

a

zz

Z

6

ax

x given

ax

c$:-i

Integrals, when found from given increments, are cor-
rected in the very same way as fluents, when found from
given fluxions, viz, instead of each several variable quan-
tity in the integral, substituting such a determinate Value
of them as the)7 are known to have in some particular
case; and then subtracting each side of the resulting
equation from the corresponding side of the integral the
remaining equation will be the correct state of the inte-
grals.

For an example of the use of the method of incre-
ments, suppose it were required to find the sum of any
number of terms of the series 1.2 -+-2.3 4-3. 4 &c
Let x be the number of the terms,
and 2 the sum of them.

4 S

1 N D

[ C8? ]

LND

Then, by the progression of the series, the last, or the x
term, is xx, and the next term after that will be xx, that

is % = xx, where x = I. Hence the integral is

2 = f xxx =\x .x -+- 1.1 + 2, which is the sura of x

terms of the given series. So if the number of terms x be
JO, this becomes \ . 10. 11 . 12 = 440, which is the sura
of 10 terms of the given series 1.2h-2.3h-3.4&c.
Or, when x = 100, the sum of 100 terms of the same
series is {. 100. 101 . 102= 100. 101.34 = 343400.
Again, to find the sum z of n terms of the series

1 -+■ — — &c.

1.3,5 8.5.7 5.7-9

Here the nth term is — — — - — — — —.

in— 1 .2)1+ 1 .in + 3

Put x = 2n — 1 ; then is x = In = 2, and the nth term
is — : = — ; and the n -t- lth term or * is

X . X -f- 2 . X + 4 XXX

— ; the general integral of which is z = = .

XIX " ° 12XX 4 n

But this wants a correction ; for when n = 0, then
x = — 1, and the sum z = 0, and the integral becomes

— i — l , . i . . .

z or 0 = = ■ ; that is is the correction,

4.1.3 12 12

which being subtracted, the correct state of the integrals
, ill

becomes z = — — — = —

. which

ix iw i x 4 . Ill + 1 . 2H -+- 3

is the sum of n terms of the proposed series. And when
n is infinite, the latter fraction is nothing, and the sum of
the infinite series, or the infinite" number of the terms, is

accurately — .

When n = 100, the sum of 100 terms of the series be-
i i ~ i i

comes z = ■ = — — — .

12 4.201.203 12 163212

For more ample information and application on this
science, see Emerson's Increments, Taylor's Methodus ln-
crementorum, and Stirling's Summatio et Interpolatio
Serierum.

INCURVATION of the Rays of Light. See Light, and
Refraction.

INDEFINITE, Indeterminate, that which has no cer-
tain bounds, or to which the human mind cannot affix
any. Descartes uses the word, in his Philosophy, instead
of infinite, both in numbers and quantities, to signify an
inconceivable number, or number so great, that an unit
cannot be added to it; and a quantity so great, as not to
be capable of any addition. Thus, he says, the stars, visi-
ble and invisible, are in number indefinite, and not, as the
ancients held, infinite ; and that quantity may be divided
into an indefinite number of parts, not an infinite num-
ber.— Indefinite is now commonly used for indeterminate,
number or quantity, that is, a number or quantity in ge-
neral, in contradistinction from some particular known and
given one.

IN UETERMINED,or Indeterminate, in Geometry,
is understood of a quantity, which has no certain or de-
finite bounds.

Indeterminate Analysis, that particular branch of
analysis which treats of the solution of indeterminate
problems.

In detkiimi Bf ate Problem, is that which admits of in-
numerable different solutions, and sometimes perhaps only

of a great many different answers; otherwise called aii
unlimited problem. In problems of this kind the number
of unknown quantities concerned, is greater than the num-
ber of the conditions or equations by which they are to
be found; from which it happens, that generally some
other conditions or quantities are assumed, to supply the
defect, which being taken at pleasure, give the same
number of answers as varieties in those assumptions. As,
if it were required to find two square numbers whose dif-
ference shall be a given quantity d. Here, if x2 and y~ de-
note the two squares, then will x" — y1 = d, by the ques-
tion, which is only one equation, for finding two quanti-
ties. Now by assuming a third quantity z, so that s = x -+■ „

the sum of the two roots; then is x = , and y —

2z J

»2 — d . . , ..,.,;„

, which are the two roots, having the difference ol

their squares equal to the given quantity d, and are ex-
pressed by means of an assumed quantity z ; so that there
will be as many answers to the question, as there can be
taken values of the indeterminate quantity z, that is, innu-
merable.

Again, If it were required to divide 25 into two such
parts, that one of them may be divisible by 2, and the
other by 3. Then, putting one of the parts sought = 2x,
and the other = 3y, we shall have 2x -+- 3y = 25 ; one
equation as before with two unknown quantities. By rc-

25 3j;

duction, x = - ; from which we may conclude in

the first place, that 3y must be less than 25 ; and conse-
quently^ is less than 8. By division, x = 12 —y -t -;

from which it is evident that 1 — y, or rather y — l,must
be divisible by 2. Let us therefore assume a third quan-
tity z, such that y — 1 = 2z; and we get y = Qz + 1 ; so
that x = 12 — 2z — 1 — z = 11 — 3z. Now since y
cannot be greater than S, we must not substitute any num-
bers for z which would make 2; -+- I greater than 8. Con-
sequently s must be less than -I-; that is to say, z cannot be
. taken greater than 3 ; for which reasons, are obtained the
following answers:

1 If z = 0; theny = 1, and x = 11.

2 If s = 1 j then y =3, and X = 8.

3 I f z = 2 ; then y = 5, and x = 5.

4 If z = 3 ; then y = 7, and x = 2.

Hence, and from the equation it appears, that the two
parts sought are I. 22 -+- 3, II. 10' ■+- 9, ill. 10 ■+• 15,
iv. 4 -i- 21, the several sums of which are evidently the
number given 25.

Diophantus was the first writer on indeterminate pro-
blems, viz, in his Arithmetic or Algebra, which was fust
published in 1575 by Xilander,and afterwards in 1 6*2 1 by
Bachet, with a large commentary, and many additions to
it. His book is wholly upon this subject ; whence it has
happened, that such kind of questions have been called
by the name of Diophantine problems. Ferimit, Des-
cartes, Frenicle in 'France, and Wallis and others in Eng-
land, particularly cultivated this branch of algebra, on
which they held a correspondence, proposing difficult
questions to each other; an instance of which are those
two curious ones, proposed by M. Fermat, as a challenge
to all the mathematicians of Europe, viz, 1st, To find a
cube number which added to all its aliquot parts shall
make a square number ; and 2d, To find a square num-
ber which added to a 1 its aliquot parts shall make

1 N D

[ 683 ]

r n d

«. cubic number; which problems were answered after
several ways by Dr. Wallis, as well as some others
ef a different nature. See the letters that passed be-
tween Dr. Wallis, the lord Brounker, Sir Kenelm Digby,
&c, in the doctor's Works ; and the W orks of Fermat,
which were collected and published by his son. Most
authors on algebra have also treated more or less on this
part of it, but more especially Kersey, Prestet, Ozanam,
Kirkby, &c. But afterwards, mathematicians seemed to
have forgotten such questions, if they did not even despise
them as useless, when Euler drew their attention by some
excellent compositions, demonstrating some general theo-
rems, which had only been known by induction. M. La-
grange has also taken up the subject, having resolved very-
difficult problems in a general way, and discovered more
direct methods than heretofore. The 2d volume of the
French translation of Euler's Algebra contains an ele-
mentary treatise on this branch of analysis, and, with La-
grange's additions, an excellent theory of it ; treating very
generally of indeterminate problems, of the first and second
degree, of solutions in whole numbers, of the method of in-
determinate coefficients, &c. Also, much in the same way-
were such indeterminate problems treated by the Indian
algebraists ; as may be seen by the account of their al-
gebra given in my Tracts, vol. 2, pa. 151, &c.

Finally, Mr. John Leslie has given, in the 2d volume
of the Edinburgh Philos. Transactions, an ingenious paper
on the resolution of indeterminate problems, resolving
them by a new and general principle. " The doctrine of
indeterminate equations," says Mr. Leslie, " has been sel-
dom treated in a form equally systematic with the other
parts of algebra. The solutions commonly given are devoid
of uniformity, and often require a variety of assumptions.
The object of this paper is to resolve the complicated expres-
sions which we obtain in the solution of indeterminate
problems, into simple equations, and to do so, without
framing a number of assumptions, by help of a single prin-
ciple, which, though extremely simple, admits of a very
extensive application."

" Let a x n be any compound quantity equal to an-
other, c x d, and let m be ary rational number assumed
at pleasure; it is manifest that, taking equimultiples,
a x wjb = c x mT). If therefore we suppose that

a = mn, it must follow that mj = c, or B = — . Thus

two equations of a lower dimension are obtained. If these
be capable of farther decomposition, we may assume the
multiples?! and p, and form four equations still more sim-
ple. By the repeated application of this principle, a higher
equation, admitting of divisors, will be resolved into those
of the first order, the number of which will be one greater
than that of the multiples assumed.

For example, in the problem, to find two rational num-
bers, the difference of the squares of which shall be a given
number. Let the given number be the product of a and
b; then by the hypothesis, x" — y2 =s ab j but these com-
pound quantities admit of an easy resolution, for (r ■+- y)
a (x — y) = a x b. If therefore we suppose r x y = ma,

we shall obtain r — y c= — ; where m is arbitrary, and
J m J

if rational, x and y must also be rational. Hence the re-
solution of these two equations gives the values of x and y,
the numbers sought, in terms of m ; viz,

_m"a + b , __ m'a -b

INDEX, in Arithmetic, is the same with what is other-
wise called the characteristic or the exponent of a loga-
rithm; being that which showsof how many places theabso-
lute or natural number belonging to the logarithm consists,
and of what nature it is, whether an integer or a fraction ;
the index being less by 1 than the number of integer
figures in the natural number, and is positive for integer
or whole numbers, but negative in fractions, or in the de-
nominator of a fraction; and in decimals, the negative in-
dex is 1 more than the number of ciphers in the decimal,
after the point, and before the- first significant figure; or,
still more generally, the index shows how far the first
figure of the natural number is distant from the place of
units, either towards the left hand, as in whole numbers, or
towards the right, as in decimals; these opposite cases
being marked by the correspondent signs -+- and — , of
opposite affections, the sign — being set over the index,
and not before it, because it is this index only which is un-
derstood as negative, and not the decimal part of the lo-
garithm. Thus, in this logarithm 2'4234097, the figures
of whose natural number are 2651, the 2 is the index, and
being positive, it shows that the first figure of the number
must be two places removed from the units place, or that
there will be three places of integers, the number of these
places being always 1 more than the index; so that the na-
tural number will be 265- 1. But if the same index be
negative, thus 2-4234097, it shows that the natural num-
ber is a decimal, and that the
first significant figure of it is
in the 2d place from units, or
that there is one cipher at the
beginning of the decimal, being
1 less than the negative index;
and consequently that the na-
tural number of the logarithm
in this case is -02651. Hence,
by varying the natural num-
ber, with respect to the decimal
places in it, as in the former of the two columns here an-
nexed, the index of their logarithm will vary as in the 2d
column.

Mr. Townley introduced a peculiar way of noting these
indices, when they become negative, or express decimal
figures, which is now much in use, especially in the log.
sfnes and tangents, &c, viz, by taking, instead of the true
index, its arithmetical complement to 10; so that, in this
way, the logarithm 2-4234097 is written 8-4.234097. For
the addition and subtraction of indices, see Logarithm.
Index of a Globe, is a little stile fitted on to the north-
pole, and turning round with it, pointing out the divisions
of the hour-circle.

Index of a Quantity, in Arithmetic and Algebra, other-
wise called the exponent, is the number that shows to
what power it is understood to be raised: as in lO^oro3,
the figure 3 is the index or exponent of the power, signify-
ing that the root or quantity, 10 or a, is raised to the 3d
power. See this fully treated under Expoy ent.

INDICTION, or Roman Indiction, a kind of epoch,
or manner of counting time, among the Romans ; contain-
ing a cycle or revolution of 15 years. The popes have
dated their acts by the year of the indiction, which was
fixed to the 1st of January anno domini 813, ever since
Charlemagne made them sovereign ; before that time, they
dated them by the vears of the emperors.— At the time of
4S2

Number.

Logarithm.

265 1 -

- 3-4234097

265-1 -

- 2-4234097

26-51 -

- 1-4234097

2-651 -

- 0-4234097

•2651 -

- 1-4234097

•02651 -

- 2-4234097

•002651

- 3-4234097

I N D

[ 684 ]

INF

reforming the calendar, the year 1582 was reckoned the
10th year of the indiction ; so that beginning to reckon
from hence, and dividing the number of years elapsed be-
tween that time and this, by 15, the remainder, with the
addition uf 10, rejecting 15 if the sum be more, will be
the year of the indiction.

But the indiction will be easier found thus: Add 3 to
the given year of Christ; divide the sum by 15, and the
remainder after the division, will be the year of the indic-
tion : if there be no remainder, the indiction is 15. In
either of these ways, the indiction for the year 1795 is 13.

INDIVISIBLES, are those indefinitely small elements,
or principles, into which any body or figure may ulti-
mately be divided. — A line is said. to consist of points, a
surface of parallel lines, and a solid of parallel surfaces :
and because each of these elements is supposed indivisible,
if in any figure a line be drawn perpendicularly through all
the elements, the number of points in that line, wdll be the
same as the number of the elements.

Whence it appears, that a parallelogram, or a prism, or
il cylinder, is resolvable into elements, as indivisibles, all
equal to each other, parallel, and like or similar to the
base; for which reason, one of these elements multiplied
by the number of them, that is the base of the figure mul-
tiplied by its height, gives the area or content. And a
triangle is resolvable into lines parallel to the base, but
decreasing in arithmetical progression; so also do the
circles, which constitute the parabolic conoid, as well as
those which constitute the plane of a circle, or the surface
of a cone. In all which cases, as the last or least term of
the arithmetic progression is 0,and the length of the figure
the same thing as the number of the terms, therefore the
greatest term, or base, being multiplied by the length of the
figure, half the product is the sum of the whole, or the con-
tent of the figure.

And in any other figure or solid, if the law of the de-
crease of the elements be known, and thence the relation of
the sura to the greatest term, which is the base, the whole

sponding radius of the sphere, and ik that of the cone; and-
the circular sections of these bodies, are as the squares of
their radii; therefore the section of the cylinder is every
where equal to the sum of the sections of the hemisphere
and cone; and, as the number of all those sections, which
is the common height of the figures, is the same, therefore
all the sections, or elements, of the cylinder, will be equal
to the sum of all those of the hemisphere and cone taken
together; that is, the cylinder is equal to both the hemi-
sphere and cone: but as the cone itself is equal to one-third
part of the cylinder ; therefore the hemisphere is equal to
the other two-thirds of it.

The method of indivisibles was introduced byCavalerius,
in l6'35, in his Geometria' Indivisibilium. The same was
also pursued by Torricelii in his Works, printed l6'44 : and
again by Cavalerius himself in another treatise, published
in I6'47-

INEQUALITY Optic. See Optic Inequality.

INERTIA of Matter, in Philosophy, is defined by New-
ton to be a power implanted in all matter, by which it re-
sists any change endeavoured to be made in its actual
state; that is, by which it becomes difficult to alter its
state, either of rest or motion. This power then, agrees
with the vis resistendi, or power of resisting, by which
every body endeavours, as much as it can, to persevere in
its own state, whether of rest or uniform rectilinear mo-
tion ; which power is still proportional to the body, or to the
quantity of matter in it, the same as the weight, or gra-
vity of the body; and yet it is quite different from, and
even independent of the force of gravity, and would be
and act just the same if the body were devoid of gravity.
Thus, a body by this force resists the same in all direc-
tions, upwards, or downwards, or obliquely ; whereas
gravity acts only downwards.

Bodies only exert this power in changes brought in their
state by some vis impressa, or force impressed on them.
And the exercise of this power is, in different respects,
both resistance and impetus ; resistance, so far as the body

number of them being the altitude of the figure, then the * opposes a force impressed on it to change its state; and

said sum of the elements is always the content. A cy-
linder may also be resolved into cylindrical curve surfaces,
having all the same height, and continually decreasing
inwards, as the circles of the base decrease, on which they
insist.

This way of considering magnitudes, is called the method
of indivisibles, which is only the. ancient method of ex-
haustions, a little disguised and contracted. And it is
found of good use, both in computing the contents of
figures in a very short and easy way, as above instanced,
and in shortening other demonstrations in mathematics ;
an instance of which may here be given in that celebrated
proposition of Archimedes, that a sphere is two-thirds of
its circumscribed cylinder; Tims, suppose a cylinder, a
hemisphere, and an inverted cone, having all the same
base and altitude, and cut by an infinite number of planes
nil parallel to the base, of which efg ii is one ; it is evi-
dent that the square of El, the radius of the cylinder, is
every where equal to the square of
sf, the radius of the sphere; and
also that the square of EI, or ol'sr,
is equal to the sum of the squares
ot ik and is, or of if and ik, be-
cause i k = is; that, is, IE1 == if1 -+■
i k ', in every position ; but ie i> the
radius ofllie cylinder, IF thecorre-

impetus, as the same body endeavours to change the state
of the resisting obstacle. Phil. Nat. Princ. Math lib. 1.

Inertia, the same great author elsewhere observes, is a
passive principle, by which bodies persist in their motion
or rest, and receive motion, in proportion to the force
impressing it, and resist as much as they are resisted. See
Resistance.

INKRTLE Vis. See Vis Inertia.

INFINITE, is applied to quantities which are cither
greater or less than any assignable ones. In which sense
it differs but little bom the terms Indefinite and Indeter-
minate. Thus, an

Infinite, or Infinitely great line, denotes only an inde-
finite or indeterminate line. ; or a line to which no certain
bounds or, limits are prescribed.

Infinite Quantities, Though the idea of magnitude
infinitely great, or such as exceeds any. assignable quan-
tity, does include a negation of limits, yet such magni-
tudes are not all equal among themselves ; but besides in-
finite length, and infinite area, there are no less than three
several sorts of infinite s didiiy : all of which are quantities
Stti generis ; and those of each species are in given propor-
tions.— Infinite length, or a line infinitely long, may be
considered, either as beginning at a point, and so infinitely
extended one way : or else both ways from the same point.
— As to infinite surface or area, any right line infinitely

I N F

[ 085 ]

I N F

extended both ways on a plane infinitely extended every
way, divides that plane into two equal parts, one on each
side of the line. But if from any point in such a plane,
two right lines be infinitely extended, making an angle
between them; the infinite area, intercepted between these
infinite right lines, is to the whole infinite plane, as that
angle is to 4 right angles. And if two infinite and parallel
lines be drawn at a given distance on such an infinite
plane, the area intercepted between them will be likewise
infinite; but yet it will be infinitely less than she whole
plane ; and even infinitely less than the angular or sec-
toral space, intercepted between two infinite lines, that
are inclined, though at never so small an angle ; because
in the onp case, the given finite distance of the parallel
lines diminishes the infinity in one of the dimensions ;
whereas in a sector, there is infinity in both dimensions.
And thus there are two species of infinity in surfaces, the
one infinitely greater than the other.

In like manner there are species of infinites in solids,
according as only one, or two, or as all their three dimen-
sions, are infinite ; which, though they be all infinitely
greater than a finiie solid, yet are they in succession in-
finitely greater than each other. — Some farther properties
of infinite quantities are as follow : the ratio between a fi-
nite and infinite quantity, is an infinite ratio. — If a finite
quantity be multiplied by an infinitely small one, the pro-
duct will be an infinitely small one; but if the former be
divided by the latter, the quotient will be infinitely great.
— On the contrary, a finite quantity being multiplied by
an infinite!', great one, the product is infinitely great ; but
the former divided by the latter, the quotient will be in-
finitely little. — The product or quotient of an infinitely
great or an infinitely small quantity, by a finite one, is
respectively infinitely great, or infinitely little.— An infi-
nitely great multiplied by an infinitely little, is a finite
quantity ; but the former divided by the latter, the quo-
tientis infinitely infinite. — The mean proportional between
infinitely great, and infinitel little, is finite.

Arithmetic of Infinities. See Arithmetic. Also
Wallis's treatise on this subject; and another by Ltnerson,
at the beginning of his Conic Sections; also Bulliald's trea-
tise Antlimetica Infinitorum.

Infinite Decimals, such as do not terminate, but go
on without end; as •333&c = =§=, or '142857 &c = -f .
See Repetend.

Infinitely Infinite Fractions, or all the powers of the
fractions whose numerator is 1 ; which are all together
equal to unity, as is demonstrated byDr.Wood,in Hook's
Philos. Coll. No 3, pa. 45; where some curious proper-
ties are deduced from the same.

Infinite Series, a series considered as infinitely conti-
nued as to the number of its terms. See Srries.

INFINITESIMALS, are certain infinitely or indefi-
nitely small parts; as also the method of computing by
them. — In the method of infinitesimals, the element by
which any quantity increases or decreases, is supposed to
be infinitely small, and is generally expressed by two or
more terms, some ot which are infinitely less than the rest,
which being neglected as of no importance, the remain-
ing terms form what is called the Difference of the pro-
posed quantity. Tin; terms that are neglected in this
manner, as infinitely less than the other terms of the ele-
ment, are the very same which arise in consequence of the
acceleration, or retardation, of the generating motion,
(luring the infinitely small time in which the clement is

generated; so that the remaining terms express the ele-
ment that would have been produced in that time, if the
generating motion had continued uniform. Therefore,
those differences are accurately in the same ratio to each
other, as the generating motions or fluxions. And hence,
though in this method, infinitesimal parts of the elements
are neglected, the conclusions are accurately true with-
out even an infinitely small error, and agree precisely with
those that are deduced by the method of fluxions.

But however safe and convenient this method may be,
some will always scruple to admit infinitely little quanti-
ties, and infinite orders and infinitesimals, into a science
that boasts of the most evident and accurate principles, as
well as of the most rigid demonstrations. In order to avuid
such suppusitions, Newton considers the simultaneous in-
crements of the flowing quantities as finite, and then inves-
tigates the ratio which is the limit of the various propor-
tions which those increments bear to each other, while he
supposes them to decrease together till they vanish; which
ratio is the same with the ratio of the fluxions. See Mac-
laurin's Fluxions, in the Introduc. pa. 3Q &c, also art
435 to 502.

INFLAMMABILITY,that property of bodies by which
they kindle, or catch fire.

INFLECTION, in Optics, called also Diffraction, and
Deflection of the rays of light, is a property of them, by
reason of which, wtien they come within a certain dis-
tance of any body, they are either bent from the body, or
towards it; being a kind of imperfect reflection or refrac-
tion. Some writers ascribe the first discovery of this pro-
perty to Grimaldi, who published an account of it, in his
treatise De Luminc, Coloribus, c-t Iride, printed in 1666.
But Dr. Hooke also claims the discovery of it, and com-
municated his observations on this subject to the Royal
Society, in l6'72. He shows that this property differs
both from reflection and refraction ; and that it seems to
depend on the unequal density of the constituent parts of
the ray, by which the light is dispersed from the place of
condensation, and rarefied or gradually diverged into a
quadrant; and this deflection, he observes, is made to-
wards the superficies of the opaque body perpendicularly.
Newton discovered, by experiments, this inflection of the
rays of light; which may be seen in his Optics.

M. Lahire observed, that when we look at a candle, or
any luminous body, with our eyes nearly shut, rays of light
are extended from it, in several directions, to a consider-
able distance, like the tails of comets. The true cause of
this phenomenon, which has exercised the sagacity of
Descartes, Rohault, and others, seems to be, that thV
light passing among the eyelashes, in this situation of ihe
eye, is inflected by its near approach to them, and there-
fore enters the eye in a great variety of directions. He
also observes, that he found that the beams of the stars
being observed, in a deep valley, to pass near the brow
of a hill, arfc always more refracted than if there were no
such hill, or the observation was made on. the top of it;
as if the rays of light were bent down into a curve, by
passing near the surface of the mountain.

Point of Inflection, or of contrary flexure, in a curve,
is the point or place in the curve where it begins to bend
or turn a contrary way; or which separates the concave
part from the convex part, and lying between the two; or
where the curve changes from-coucavc to convex, or from
convex to concave, on the same side of the curve ; such
as the point e in the annexed figures; where the former of

I N F

[ 686 ]

I N S

the two is concave towards the axis ad from a to e, and
convex from e to f ; but, on the contrary, the latter fi-
gure is convex from a to e, and concave from e to r.

Hence also, from the original equatioB,

— = ■?-; = i-a, the ordinate to the point of jn-

T *<Z * ' r

There are various ways of finding the point of inflexion ;
but the following, which is new, seems to be the simplest
and easiest of all. From the nature of curvature it is
evident that, while a curve is concave towards an axis,
the fluxion of the ordinate decreases, or is in a decreasing
ratio, with regard to the fluxion of the absciss ; but, on
the contrary, that the,said fluxion increases, or is in an
increasing ratio to the fluxion of the absciss, where the
curve is convex towards the axis ; hence it follows that
those two fluxions are in a constant ratio at the point of
inflection, where the curve is neither concave nor convex,
That is, if i = ad the absciss, and y = de the ordinate,
then xis toi in a constant ratio, or-= or ^ is a constant
quantity. But constant quantities have no fluxion, or

» of + x'

flection sought.

When the curve has but one point of inflection, it will
be determined by a simple equation, as above; but when
there arc several points of inflection, by the curve bending
several times from the one side to the other, the resulting
equation will be of a degree corresponding to them, and
its roots will determine the abscisses or ordinates to the
same. Other methods of determining the points of inflec-
tion in curves, may be seen in most books on the doctrine
of fluxions.

To know whether a curve be concave or convex to-
wards any point assigned in the axis; find the value of y
at that point; then if this value be positive, the curve
will be convex towards the axis, but if it be negative, it
will be concave.

INFORMED Stars, or Informes Stella-, are such stars
as have not been reduced into any constellation ; other-
wise called Sporades. — A great many of this kind were
left by the ancient astronomers; but Hevelius and some
others of the moderns have provided for the greater part
of them, by making new constellations. And I think tiiat
the later celestial globes made in England, have the con-
stellations contrived to include the old unformed stars.
INGINEER. See Engineer.

INGRESS, in Astronomy, the sun's entrance into one
of the signs, especially Aries.

INNOCENTS Day, a feast celebrated on the 2Sth day

their fluxion is equal to nothing; so that in this case the
fluxion of * or of ■? is equal to nothing. And hence \ve

y x

have this general rule: viz,

Put the given equation of the curve into fluxions ; from of December, in commemoration of the infants said to be

which equation of the fluxions find ? or - ; then take the
fluxion of this ratio or fraction, and put it equal to 0 or
nothing and from this last equation find also the value
of the same % or 4": then put this latter value equal to the
former, which will be an equation from whence, and the
first given equation of the curve, x and y will be deter-
mined, being the absciss or ordinate answering to the point
of inflection in the curve.

Or, putting the fluxion of - equal to 0, that is,

^~ "$ = 0, or xj - xy = 0, or xy = xy, or x :j : : xfy,
that is, the 2d fluxions have the same ratio as the 1st flux-
ions, which is a constant ratio ; and therefore if x be
constant, or x = 0, then shall y be = 0 also ; which gives
another rule, viz; Take both the 1st and 2d fluxions of
the given equation of the curve, in which make both i and
y = 0, and the resulting equations will determine the va-
lues of x &ndy, or absciss and ordinate to the point of in-
flection.

For example, if it be required to find the point of inflec-
tion in the curve whose equation is ax'1 = a?y ■+- x\.
Now the fluxion of this is 2axx = a1} ■+■ 2xyx ■+- x%

which gives - = — — . Then the fluxion of this again

b y Sax — ?.ry "

made = 0,gives2.ri (ax — xy) = (a* -+■ x*) . (ax —yx — xy);

and this gives again - = -s — '-^ * . Lastly, this

° y a — 0. a — y "

value, of -> put = the former, gives
if + r

1 - x', or 3x*

and hence 2j" =

a1, anil x = «\/t> l'ie absciss.

murdered by Herod.

INORDINATE Proportion, is where the order of the
terms compared, is disturbed or irregular. As, for exam-
ple, in two ranks of numbers, three in each rank, viz, in
one rank, - - - - 2, 3, 9,

and in the other rank - - 8, 24, 36",

which are proportional, the former to the latter, but in
a different order, viz, - - 2 : 3 : : 24 : 36,

and - - 3 : o : : s : 24,

then, casting out the mean terms in each rank, it is con-
cluded that - - - 2 : 9 : : S : 36",
that is, the first is to the 3d in the first rank,
as the first is to the 3d in the 2d rank.
INSCRIBED Figure, is one that has all its angular
points touching the sides of another figure in which the
former is said to be inscribed.

Inscribed Hyperbola, is one that lies wholly within
the angle of its asymptotes; as the common or conical
hyperbola doth.

INSTANT, otherwise called a moment, an infinitely
small part of duration, or in which we perceive no suc-
cession, or which takes up the time of only one idea in our
mind. It is a maxim in mechanics, that no natural elfect
can be produced in an instant, or without some definite
time ; also that the greater the time, the greater the effect.
And hence may appear the reason, why a burthen seems
lighter to a person, the faster he carries it ; and why, the
faster a person slides or skates on the ice, the less liable
it is to break, or bend.

INSTITUTE is a name which has lately been substi-
tuted for academy or school. When royalty was lately
abolished iu France, the several royal academies were
abolished., and were all condensed into o«>e general aca-
demy, under the name of the

I N T

[ 687 ]

I N T

National Institute, or Nciv Academy of Arts and
Sciences. This academy was founded on a decree of the
new government constitution in the year 1795. It con-
sists of 268 members, half of whom reside in Paris, the
other half in the departments; and to them are added 24
foreigners, as honorary members. The academy is di-
vided into 3 classes, each class into sections, each section
containing 12 members.

1st Class. Mathematics and natural philosophy. This
class is divided into 10 sections. 1. Mathematics. 2. Me-
chanical arts. 3. Astronomy. 4. Experimental philosophy.
5. Chemistry. 6. Natural history. 7. Botany. 8. Ana-
tomy and animal history. 9- Medicine and surgery.
10. Animal economy, and the veterinary science.

2d Class. Morality and politics. This class consists of
6 sections. 1. Analysis of sensations and ideas. 2. Mo-
rals. 3. Legislature. 4. Political economy. 5. History.
6". Geography.

3d Class. Literature and the fine arts, consisting of 8
sections. I. Universal grammar. 2. Ancient languages.
3. Poetry. 4. Antiquities. 5. Painting. G. Sculpture.
7- Architecture. 8. Music.

For each class a particular room in the Louvre is ap-
propriated. No person can be a member of two classes
at the same time : but a member of one class may be
present at the meetings of any other. Each class prints,
yearly, an account of its transactions.

Four times a-year public meetings are held, when the three
classes meet together. At the end of each year, they
give a circumstantial account to the legislative body, of
the progress made in that year in the arts and sciences.
The prizes given annually by each class, are published at
certain times. The sums requisite for the support of the
institution, are decreed yearly by the legislative body, on
a requisition made by the executive directory.

The first 4S members were chosen by the executive
directory, to whom the choice of the remaining members
was confided. To the members resident in Paris is re-
served the choice, both of the department and foreign
members. On a vacancy in any class, three candidates are
named by the class, for the choice of the body at large.
Each class is to have, at its place of meeting, a collection
of the products, both of nature and art, and a library, ac-
cording to its particular wants. The regulations of the
institution, with respect to the times of meeting, and its
employments, are to be drawn up by the body at large,
and laid before the legislative assembly.

INSULATE, or Insulated, a term applied to a co-
lumn or other edifice, which stands alone, or free and de-
tached from any adjacent wall, &c, like an island in the
sea.

Insulated, in Electricity, is a term applied to bodies
that rue supported by electrics, or non-conductors; so
that their communication with the earth, by conducting
substances, is interrupted.

INSURANCE, the same as Assurance ; which sec.

INTAC'iVE, are right lines to which curves do conti-
nually approach, and yet can never meet them, more
usually railed Asymptotes.

INTEGERS, ate whole numbers, as contra-distin-
guished from fractions. Integers may be considered as
numbers which refer to unity, as a whole to a part.

INTEGRAL Number, an integer; not a fraction.

Integral Calculus, in the New Analysis, is the reverse
of i lie, differential calculus, and is the finding the integral

from a given differential ; being similar to the inverse me-
thod of fluxions, or the finding the fluent to a given fluxion.
The rules for the integration of differentials, likethose for
the extraction of roots, are established by inspection of the
direct process by which, from given functions, differentials
are derived : thus, the differential of xm or d(xm), is
mxm~ dx; therefore, from mx dx, the integral may be
found from this process: increase the index in — 1, or
power of the vatiable quantity, by unity or 1, and divide
by the index so increased, and by the differential of the
variable quantity, for the integral required. Let d~ , be
the sign of the operation by which the integral is obtain-
ed, or the reverse of that operation which d indicates ;
then will d~ (mx" " dx) = xm; or, d~ d(xm) = xm. Also,

d" ' (x n dx) —

Again the integrals

of axdx, exdx ; are ax, eK ; and are obtained on the very
same grounds of inspection as the above.

The differentials of a ■+■ bx, a -+- b.r, & -+- bx, &c, or
d(a ■+■ bx), d(d -*- bx), d(a -+- bx), &c. are dbx,dbx, dbx,
&c. Hence inversely, the integrals are d~* (dbx), d~* (dbx)
d (dbx) &c: where it is to be noted, that the constant
quantities a, a, a", &c, are the corrections of the inte-
grals ; and are determined in the same manner as those in
the inverse method of fluxions, from the particular nature
of the problem under discussion.

It may be here observed, that as in the analysis of
finites, any quantity may be raised to any degree or power ;
but vice versa, the root cannot be extracted out of any
number required : so in the analysis of infinites, any va-
riable or flowing quantity may be differenced, but vice
versa, any differential cannot be integrated. And as, in
the analysis of finites, we are not yet arrived at a method
of extracting the roots of all equations, so neither has the
integral calculus arrived at its perfection : and as in the
former we are obliged to have recourse, to approximation,
so in the latter we have recourse to infinite series, where
we cannot attain to a perfect integration. For the several
authors on this calculus, see Fluxions.

INTEGRANT \P«rAs, in Philosophy, are the similar
parts of a body, or parts of the same nature with the
whole; as filings of iron are the integrant parts of iron,
having the same nature and properties with the bar or
mass they were filed oft' from.

INTENSITY, or Intension, in Physics, is the degree

. or rate of the power or energy of any quality ; as heat,

cold, &c. The intensity of qualities, as gravity, light,

heat, &c, vary in the reciprocal ratio of the squares of

the distances from the centre of the radiating quality.

INTERCALARY Day, denotes the odd day inserted in
the leap-year. See Bissextile.

INTERCEPTED Axis, in curves, the same with what
is otherwise called the absciss or abscissa.

1NTERCOLUMNATION, or Intercolumniation,
is the space between column and column, which is always
to be proportional to the height and bulk of the columns.
From a medium, some authors have laid down, the follow-
ing proportions: in the Tuscan order, the intercolumna-
tion to be 4 diameters of the body of the column below;
in the Doric 3, in the Ionic 2f, in the Corinthian 2|, and
in the Composite 1-'-.

INTEREST, is a sum reckoned for the loan or for-

1 N T

[

1

INT

bearance of another sum, or principal, lent for, or due at,
a certain time, according to some certain rate or propor-
tion; being estimated usually at so much per cent, or by
the 100. This forms a particular rule in arithmetic. The
highest legal interest now allowed inEngiand, is after the rate
of 5 per cent, per annum, or the 20th part of the princi-
pal for the space of a year, and so in proportion for other
times, either greater or less : except in the case of pawn-
brokers, to whom it has lately been made legal to take a
much higher interest, for one of the worst and most de-
structive purposes that can be suffered in any state.

Jntercst is either Simple or Compound.

Simple Interest, is that which is counted and allowed
upon the principal only, for the whole time of forbear-
ance.— The sum of the principal and interest is called the
amount.

As the interest of any sum, for any time, is directly
proportional to the principal sum and time ; therefore
the interest of 1 pound for one year being multiplied by
any proposed principal sum, and by the time of its for-
bearance, in years and parts, will be its interest for that
time. That is, if

r = the rate of interest of 1/. per annum,
p=any principal sum lent,
t = the time it is lent for, and
a = the amount, or sum of principal and interest;
then is prt = the interest of the sum p, for the time t, at
the rater; and consequently p -*-prt=:p x (1 H-rt)=a,
is the amount of the same for that time. And from this
general formula, other theorems can easily be deduced for
finding any of the quantities above-mentioned ; which col-
lected all together, will be as follow :
1st, a=p -+- prt the amount,

the principal,
the rate,

4th, t == '' the time.

For Example, let it be required to find, in what time any
principal sum will double itself, at any rate of simple in-
terest. In this case we may use the 1st theorem a = p
■+■ prt, in which the amount a must be = 1p, or double

the principal, i. e. p + prt == 2p; and hence t = -;

where r being the interest of 1/. for one year, it follows
that the time of doubling at simple interest, is equal to the
quotient of any sum divided by its interest for one year.
So that, if the rate of interest be 5 percent, then 100-f-5
= 20, is the time of doubling. — Or the 4th theorem im-

a — p «2p — p '2 — 1 1

1 C i ° ft pr r r'

For some compendious tables and rules for computing
the interest on money, see my small book of Arithmetic.

Compound Interest, called also laterest-upon-lnterest,
is that which is counted, not only on the principal sum
lent, but also for its interest, as it becomes due, at the end
of each stated time of payment.

Though it be not lawful to lend money at compound in-
terest, yet in purchasing anrfni ties, pensions, &c, and leases
in reversion, it is usual to allow compound interest to the
purchaser tor his ready money; and therefore it is neces-
sary to understand this subject.

Besides the quantities concerned in simple interest, viz,
the principal p, the rate or interest of 1/. for 1 year r, the

amount a, and the time t, there is another quantity em-
ployed in compound interest, viz, the ratio of the rate of
interest, which is the amount of 1/. for 1 time of payment,
and which here let be denoted by r, viz, r = 1 -t- r.
Then, the particular amounts for the several times may
be thus computed, viz, As 1. pound, is to its amount for
any time, so is any proposed principal sum, to its amount
for the same time; i. e.

11. : r : : p : pR the 1st year's amount,
1/. : R : : j3R : p&2 the 2d year's amount,
\l. : r : : pu2 : pa.3 the 3d year's amount,
and so on.
Therefore in general, pa.1 = a, is the amount for the /year,
or t time of payment. Whence the following general theo-
rems are deduced:

1st, a = /m' the amount,

2d, p = — the principal,

3d, R = \/
4th, t =

P

log. of a

the ratio,
los. of p

the time.

t =■

lug. Of R

From which any one of the quantities may be found, when
the rest are given.

For example, suppose it were required to find, in how
many years any principal sum will double itself, at any
proposed rate of interest. In this case must be employed
the 4th theorem, where a will be = 2j9, and then it is
l.a—l.p 1.2p— ,1. p log. 2

log. R log. R ~~ log. R '

So, if the rate of interest be 5 per cent, per annum; then
lt.= 1 -t- '05 = 1 '05, and hence

lojr. 2 -3010300 , , „„,-_ , ,,

t =, = = 14'20cv nearly; that is, anv

lpg. 1-05 -0211893 ■" ' J

sum doubles in 14^- years nearly, at the rate of 5 percent,
per annum compound interest.

Hence, and from the like question in simple interest,
above given, arc deduced the times in which any sum
doubles itself, at several rates of interest, both simple and
compound : viz,

1

At

1

2

per cent, per an.

interest, 1/. or
other sum
double in

any
will

"At Simple Interest.

At Comp. Interest.

Veuis.

Years.

100

6>6'6'03

50

35'0028

40

2S'0701

33j

23:4408

2S£

20M4S8

25

17-6730

22;-

15-7473

20

14-2007

16}

11-8957

14f

I0-244S

12i

9*0065

Ml-

8-0433

L 10

7-2725

3
3-_

4|

5

6

7

8

9
10 .

The following Table will very much facilitate the cal-
culation of the compound interest for any sum, for any
number of years, ai various rates ol interest.

The use of this table, which contains all the powers tt1,
to the 20lh power, or the amounts of I/, is chielly to cal-
culate the interest, or the amount, of an\ principal sum,
for any time, not more than 20 years. For example, re-
quired to find to how much 523/. will amount in 15 years,
at the rale of 51. per cent, per annum compound interest.

INT

[ 689"]

INT

In the table, on the line 15, and in the column 5 per

cent, is the amount of 1l. viz.
this multiplied by the principal
gives the amount

2-078.9,

52,3,
1087-2647

or 10S7/. 5s. ;

and therefore the interest is - 56-J-/. 5s. I

The Amount of 1 1, in any Number of Years.

Yrs.

3

3|

4

4§ 1 5

6

1

10300

1-0350

1 -0400

1 -0450

1-0500

1 -0600

2

1-0009

1 0712

1 -08 1 6

1-0920

1-1025

1-1236

3

1 -0927

1-1087

1-1249

1-1412

1-1576

11910

4

1-1255

1-1475

1-1699

1-1925

1-2155

1 -2625

5

1-1593

1-1877

1-2167

1-2462

1-2763

1-3382

6

1-1941

1-2293

1-2653

1-3023

1-3401

1-4185

7

1-2299

1-2723

1-3159

1 3609

1-4071

1-5036

8

1-2668

1-3168

1-3686

1-4221

1-4775

1-5939

9

1-3048

1-3629

1-4233

1-4S61

I -55 13

1-6895

10

1-3439

1-4106

1 -4802

1-5530

1-6289

1-7909

11

1-3842

1 -4600

1-5395

1-6229

1-7103

1-8983

12

1-4258

1-5111

1-6010

1-6959

1-7959

2-0122

13

1-4685

1-5640

1-6651

1-7722

1-8356

2-1329

14

1-5126

1-6187

1-7317

1-8519

1-9799

2-2609

15

1-5580

1-6753

1-8009

1-9353

2-0789

2-3966

16

1-6047

1-7340

1-8730

20224

21829

2-5404

17

1-6528

1-7947

1-9479

21 134

2-2920

2-6928

18

1-7024

1-8575

2-0258

2-2085

2-4066

2-8543

19

J-7535

1-9225

2-1068

2-3079

2-5270

20256

20

1-8061

1-9898

2-1911

2-4117

2-6533

2-207 1

See Annuities; Discount; Reversion; Smart's
Tables of Interest ; the Philos. Trans, vol. 6,-pa. 508 ; and
most books on Arithmetic.

I NTERIOR Figure, Angle of. See Angle.

Interior Polygon. See Polygon.

Interior Talus. Sec Talus.

Internal Angles, are the angles made within any
figure, by the sides of it. In a triangle abc, the two angles

a and c are peculiarly called internal and opposite, in re-
spect of the external angle cbd, which is equal to them
both together.

Intern al Angle is also applied to the two angles formed
between two parallels, by aline intersecting those parallels,
on each side of the intersecting line. Such are the angles
a, b, c, d, formed between the parallels ef and on, on each
side of the intersecting line. — The two adjacent internal
angles a and b, or c and d, are together equal to two right
angles.

Internal and Opposite Angles, is also applied to the
two angles a and b, which are respectively equal to the two
n and m, called the external and opposite angles. Also
the alternate internal angles are equal to each other ; viz,
a = d, and b = c.

INTERPOLATION, in the modern Algebra, is used
for finding an intermediate term of a series, its place in the
series being given. The method of interpolation was first
invented by Mr. Briggs, and by him applied to the calcu-
lation of logarithms, &c, in his ArithmeticaLogarithmica,

Vol. I.

and his Trigonometria Brilaivnica ; where he explains,
and fully applies the method of interpolation by differences.
His principles were followed by Reginal and Mouton in
France, and by Cotes and others in England. Wallis em-
ployed the method of interpolation in several parts of his
works; as his Arithmetic of infinites, and his Algebra, for
quadratures, &c. The same was also happily applied by
Newton in various ways : by it he investigated his bino-
mial theorem, and quadratures of the circle, ellipse, and
hyperbola: see Wallis's Algebra, chap. 85, &c. Newton
also, in lemma 5, lib. 3 Princip. gave a most elegant solu-
tion of the problem for drawing a curve line through the
extremities of any number of given ordinates; and in the
subsequent proposition, he applied the solution of this
problem to that of finding, from certain observed places
of a comet, its place at any given intermediate time. And
Dr. Waring, who adds, that a solution still more elegant,
on some accounts, has been since discovered by Mess.
Nichol and Stirling, has also resolved the same problem,
and rendered it more general, without having recourse to
finding the successive differences. Philos. Trans, vol. 69,
part I, art. 7' Mr. Stirling indeed pursued this branch
as a distinct science, in a separate treatise, viz, Tractatus
de Summatione et Interpolatione Scrierum Infinitarum,
in the year 1730.

When the 1st, 2d, or other successive differences of the
terms of a series become at last equal, the interpolation of
any term of such a series, maybe found by Newton's Dif-
ferential Method. When the algebraic equation of a»
series is given, the term required, whether it be a primary
or intermediate one, may be found by the resolution of af-
fected equations ; but when that equation is not given, as
it often happens, the value of the term sought must be ex-
hibited by a converging series, or by the quadrature of
curves. See Stirling, ut supra, pa. 86. Meyer, in Act.
Petr. torn. 2, pa. 180.

A general theorem for interpolating any term is as fol-
lows : Let a denote any term of an equidistant series of
terms, and a, b, c, &c, the first of the 1st, 2d, 3d, &c, or-
ders of differences ; then the term z, whose distance from
a is expressed by x, will be this, viz, Theorem I,
x — 1 , x — \ x — 2 '

J=A+IS+I. — ■ *+I. — . — C &C

Hence, if any of the orders of differences become equal
to one another, or = 0, this series for the interpolated
term will break off, and terminate, otherwise it will run
out in an infinite series.

Example. To find the 20th term of the series of cubes
1, S, 27, 64, 125, &c, or l'3, 23, 33, 43, 53, &'c.

Set doun the series in a
column, and take their Conti-
nual differences as here an-
nexed, where the 4th differ-
ences, and all after it be-
come = 0, also a = 1, a = 7 ,
b = 12,c = 6, and x — ]9;
therefore the 20th term sought
is barely

z=I+lSx7+19x^

= 1 -+- 133 + 2052 -4- 5814 = 8000.

T/ieor. 2. In any series of equidistant terms, a, b, c, d,

&c, whose first differences are small ; to find any term

wanting in that scries, having any number of terms given.

Take the equation which stands against the number of

A T

A
1

a

7

b

c

8

1.9

12

6

d

27

37

18

6

•0

6'4

61

24

25

12

H-

iy .

:„- • -r • 6

INT

1

1 N T

given terms, in the following table ; and by reducing the
equation, that term will be found.

No.

Equations.

1

a - b = 0

2

a — 26 + c = 0

3

a — 36-+- 3c — d = 0

4

a — 46 ■+- 6c — 4rf + e = 0

5

a — 56 -+- 10c - lOd + 5e — /= 0

6

a _ 66 -f- 15c — 20<Z + 15e - 6/ ■+■ g = 0

&.C.

&c.

- nb + n.^-^c-n.-^ .-T-d,&cc = 0:

where it is evident that the coefficients in any equation, are
the unciae of a binomial 1 + 1 raised to the power denoted
by the number of the equation.

- Example. Given the logarithms of 101, 102, 104, and
105; to find the log. of 103.

Here are 4 quantities given ; therefore we must take the
4th equation a-46-*-6c-4d-he = 0, in which it is
the middle quantity or term c that is to be found, because
103 is in the middle among the numbers 101, 102, 104,
105; then that .equation gives the value of c as follows,

4 . ft + d) — (a + e)
VIZ c = g •

Now the logs, of the given numbers will be thus :

°2-0043214 = a

2-0086002 = 6

2-0170333 = d

2-02118.93 = e

4-0256335 = 6

4

d

16-1025340 = 4. (6
subtr. 4-0255107 == «

d)

6) 12-0770233

2-0 I2S372 the log. of 103.
Theor. 3. When the terms a, 6, c, d, &c, are at un-
equal distances from each other; to find any intermediate
one of these terms, the rest being given.

Letjj, q, r, s, &c, be the several distances of those terms
from each other ; then let

J — a El- B Cl — C D 1 — D

B = ; c = ; D = ; E = —

p i> + 9 p + q +r p +

c-b

Bl= ;cl = — — ;d! = -

9
e — d

V

+ 1

Til

— Bl

'/

-+- r

nn

— B-2

q + r + t

b3 =

&c &c &c. Then the term 2, whose

distance from the beginning is x, will be
z = a ■+- bz ■+- ex . (x — p) + nx . (x — p) . {x — p — g)
-+- tlx.(x — p). (x — p — q).{x —p — q — r) + &c;
to be continued to as many terms as there are terms in the
given series.

By this series may be found the place of a comet, or the
sun, or any other object at a given time ; by knowing the
places of the same for several other given times. Other
methods of interpolation may be found in the Philos.
Trans, number 362; or Stirling's Summation and Inter-
polation of Series.

INTERSCENDENT, in Algebra, is applied to quanti-
ties, when the exponents of their powers are radical quan-
tities. Thus x* , x^a, &cc, are interscendent quantities.
See Function.

INTERSECTION, the cutting of one line, or plane,
by another ; or the point or line in which two lines or two
planes cut each other. — The mutual intersection of two
planes is a right line. The centre of a circle, or conic
section, &c, is in the intersection of two diameters ; and
the central point of a quadrangle, is the intersection of
two diagonals.

INTERSTELLAR, intervening between the stars; a
word used by some authors, to express those parts of the
universe, that are without and beyond the limits of our
solar system. — In the interstellar regions, it is supposed
there are several other systems of planets, moving round
the fixed stars, as the centres of their respective motions.
And if it be true, as it is not improbable, that each fixed
star is thus a sun to some habitable orbs, or earths, that
move round it, the interstellar world will be infinitely the
greatest part of the universe.

INTERTIES, or Interduces, in Architecture, those
small pieces of timber which lie horizontally between the
summers, or between them and the cell or rising plate.

INTERVAL, in Music, the difference between two
sounds, in respect of acute and grave. Authors distin-
guish several divisions of an interval, as first into Simple
and Compound. The

Simple Interval is that without parts, or division;
such are the octave, and all that are within it ; as the 2d,
3d, 4th, 5th, 6th, and 7th, with their varieties.

Compound Interval consists of several lesser intervals;
as the 9th, 10th, 11th, &c, with their varieties.

The simple interval was by the ancients calleda diastem,
and the compound they called a system.

An interval is also divided into just or true, and into
false. The

Just or True Intervals, are such as all those above
mentioned, with their varieties, whether major or minor.

False Intervals, are the diminutive or superfluous
ones.

An interval is also divided into the consonance and dis-
sonance ; which see.

INTESTINE Motion of the parts of fluids, that which
is among its corpuscles or component parts. When the
attracting corpuscles of any fluid are elastic, they must
necessarily produce an intestine motion; and that, greater
or less, according to the degrees of their elasticity and at-
tractive force. For, two elastic particles, after meeting,
will fly from each other, with the same degree of velocity
with which they met; abstracting from the resistance of
the medium. But when, in leaping back from each other,
they approach other particles, their velocity will be in-
creased.

INTONATION, in Music, the action of sounding the
notes in the scale with the voice, or any other given order
of musical tones. — Intonation may be either true or false ;
either too high or too low, either too sharp or too flat ; and
then this word intonation attended with an epithet must
be understood concerning the manner of performing the
notes.

INTRADOS, the interior and lower side, or curve, of
an arch. In contradistinction from the extrados, or ex-
terior curve, or line on the upper side of the arch.. See
my Tracts, vol. 1.

INTRENCHMENT, in Fortification, any work that
shelters a post against the attacks of an enemy. It is ge-
nerally taken for a ditch, or trench with a parapet. In-
trenchments arc sometimes made of fascines with earth

i N V

t to ]

J O I

thrown over them, of gabions, hogsheads, or bags filled
with earth, to cover the men from the enemy's fire.

INVERSE, is applied to a manner of working the rule
of three, or proportion, which seems to go backward, i.e.
reverse or contrary to the order of the common and direct
rule: so that, whereas, in the direct rule, more requires
more, or less requires less ; in the inverse rule, on the con-
trary, more requires less, or less requires more. — For in-
stance, in the direct rule it is said, If 3 yards of cloth cost
20 shillings, how much will 6 yards cost ? the answer is
40 shillings; where more yards require more money, and
less yards require less money. But in the inverse rule it
is said, If 20 men perform a piece of work in 6 days, in
how many days will 40 men perform as much ? where the
answer is 3 days ; and here the more, men require the less
time, and the fewer men the more time.

Inverse Method of Fluxions, is the method of finding
fluents, from the fluxions being given; and is similar to
what the foreign mathematicians call the calculus inte-
gralis. See Fluents.

Inverse Method of Tangents, is the method of finding
the curve belonging to a given tangent ; as opposed to the.
direct method, or the finding trie tangent to a given curve.
— As, to find a curve whose subtangent is a third propor-
tional to r — y andj/, or whose subtangent is equal to the
semiordinate, or whose subnormal is a constant quantity.
The solution of this problem depends chiefly on the in-
verse method of fluxions. See Tangent. See also
Montucla, v. 3, pa. 16"4.

Inverse Proportion, or Inverse Ratio, is that in
which more requires less, or less requires more. As for
instance, in the case of light, or heat from a luminous
object, the light received is less at a greater distance, and
greater at a less distance ; so that here more, as to di-
stance, gives less, as to light, and less distance gives more
light. This is usually expressed by the term inversely, or
reciprocally; as in the case above, where the light is in-
versely, or reciprocally as the square of the distance; or
in the inverse or reciprocal duplicate ratio of the distance.

INVERSION, Invertendo, or by Inversion, by the 14th
def. of Euclid, lib. 5, is inverting the terms of a propor-
tion, by changing the antecedents into consequents, and
the consequents into antecedents. As in these, a :b::c:d,
then by inversion b : a ; : d ; c.

INVESTIGATION, the searching or finding any thing
out, by means of certain steps, traces, or ways.

INVOLUTE Figure or Curve, is that which is traced
out by the outer extremity of a string as it is folded or
wrapped upon another figure, or as it is unwound from off
it. The involute of a cycloid, is also a cycloid equal to
the former, which was first discovered by Huygens, and
by means of which he contrived to make a pendulum vi-
brate in the curve of a cycloid, and so, theoretically at
least, vibrate always in equal times, whether the arch of
vibration were great or small ; which is a property of that
curve. See Evolute.

INVOLUTION, in Arithmetic and Algebra, is the
raising of powers from a given root ; as opposed to Evo-
lution, which is the extracting, or developing of roots
from given powers. So the. involution of the number 3,
or its powers, are thus raised :

3 - or 3' or 3 is the 1st power, or root,

3x3 or 3 x or 9 is the 2d power, or square,

3 x 3 x 3 or 3 3 or 27 is the 3d power, or cube,
and so on. — And hence, to find any power of a given root,

or quantity, let the root be multiplied by itself a number
of times which is one less than the number of the index ;
i. e. once multiplied for the 2d root, twice for the 3d root,
thrice for the 4th root, &c.

So also, in algebra, to involve the binomial a -+- /;,
or raise its powers.

a -h b - - - - 1st power, or root

a h- b

2d power

a'1 ■+■ ab

■+■ ab ■+■

b2

a'2 -+- 2ab ■+■
a ■+■ b

b1

a3 ■+- 2a2b ■+■
-f- a?b -(-

ab2 + bz

3a2b ■+■ 3ab2

3d power

And in like manner for any other quantities, whatever
the number of their terms may be. But compound alge-
braic quantities are best involved by the Binomial Theo-
rem; which see.

Simple quantities are involved, by raising the numeral
coefficients to the given power, and the literal quantities
are raised by multiplying their indices by that of the root;
that is, the raising of powers is performed by the multi-
plication of indices, the same as the multiplication of lo-
garithms. Thus,

The 2d power of a is a2.

The 2d power of 2az is 2VX 2 or 4«4.
The 3d power of 3azb3 is 27a°b9.

_I 2 3

The 3d power of a2iT is a^b*.

Thereth power of aracP is amnc?n or (amcP)\

INWARD Flanking Angle, in Fortification, is that made
by the curtin and the razant flanking line of defence.

JOINTS, in Architecture, are the separations between
the stones or bricks ; which may be filled with mortar,
plaster, or cement.

Joint, in Carpentry, &c, is applied to several manners
of assembling, setting, or fixing pieces of wood together.
As by a mortise joint, a dove-tail joint, &c.

Universal Joint, in Mechanics, an excellent invention
of Dr. Hooke, adapted to all kinds of motions and flexures;
of which he has given a large account in his Cutlerian
Lectures, printed in l6"78. This seems to have given oc-
casion to the gimbols used in suspending the sea compass;
the mechanism of which is the same with that of Desagu-
liers's rolling lamp.

JoiNT-Liws, are such as continue during the same time,
or that exist together. See Life- Annuities.

JOISTS, or Joysts, those pieces of timber framed
into the girders and summers, and on which the board-
ing of floors is laid. They are of various kinds: as, 1.
Common Joists, being those which are framed level or flush
with the upper surface of the girders, but are seldom of
equal depth. 2. Trimming Joists, those which are framed
into two other joists, that other joists may be framed into -
them, where the opening is made for a stair-case, or a
chimney hearth. 3. Binding Joists, those which are laid
across from girder to girder, and about 3 or 4 inches lower
than their upper surface, in order that other joists, called
4. Bridging Joists, may be laid over them parallel to the
girders. 5. Ceiling Joists, which are turned into the bind-
ing joists, and are generally made slender, having but
little weight to support. — The chief or flooring joists,
should be abuut 8 inches deep, the common breadth of a
4T2

JON [692

deal board ; their breadth more or less, according to their
length, viz, from 2 to 4 inches broad, or different lengths
from 6 to 12 feet.

" JONES (William), f. r. s. a very eminent mathe-
matician, was born at the foot of Bodavon mountain
[Mynydd Bodafon] in the parish of Lianfihangel tre'r
Bard, in the Isle of Anglesy, North Wales, in the year
1675. His 'father's name was John George, his surname
being the proper name of his father. For itis'a custom
in several parts of Wales for the proper name of the father
to become the surname of his children. John George the
father was commonly called Sion Siors of Llanbabo, to
which place he moved, and where his children were
brought up. Accordingly our author, whose proper name
was William, took the surname of Jones from the proper
name of his father, who was a farmer, and of a good fa-
mily, being descended from Hwfa ap Cynddelw, one of
the 15 tribes of North Wales. He gave his two sons the
common school education of the country, reading, writing,
and accounts, in English, and the Latin Grammar. Harry
his second son took to the farming business; but William
the eldest, having an extraordinary turn for mathematical
studies, determined to try his fortune abroad from a place
where the same was but of little service to him. He ac-
cordingly came to London, accompanied by a young man,
Rowland Williams, afterwards an eminent perfumer in
Wych-street. The report in the country is, that Mr. Jones
soon got into a merchant's counting house, and so gained
the esteem of his master, that he gave him the command
of a ship for a West India voyage ; and that upon his re-
turn he set up a mathematical school, and published his
book of Navigation ; and that upon the death of the mer-
chant he married his widow : that, lord Macclesfield's son
being his pupil, he was made secretary to the chancellor,
and one of the deputy tellers of the exchequer: — and they
have a story of an Italian wedding, which caused great
disturbance in lord Macclesfield's family, but was com-
promised by Mr. Jones ; which gave rise to a saying,
' that Macclesfield was the making of Jones, and Jones
the making of Macclesfield.' " The foregoing account of
Mr. Jones, 1 found among the papers of the late Mr. John
Robertson, librarian and clerk to the Royal Society, who
had been well known to Mr. Jones, and possessed many of
his papers.

Mr. Jones having by his industry acquired a competent
fortune, lived upon it as a private gentleman for many
years, in the latter part of his life, in habits of intimacy
with Sir Isaac Newton and others the most eminent mathe-
maticians and philosophers of his time; and died July 3,
1749, at 74 years of age, being one of the vice-presidents
of the Royal Society ; leaving at his death one daughter,
and a son very young, who was the late Sir William Jones,
one of the judges in India, and highly esteemed for his
great abilities, extensive learning, &c. — Mr. Jones's publi-
cations were,

1. A new Compendium of the whole Art of Navigation,
&c; in small 8vo, London, 1702. This is a neat little
piece, and dedicated to the Rev. Mr. John Harris, the au-
thor of the Lexicon Technicum, or Universal Dictionary
of Arts and Sciences, in whose house Mr. Jones says he
composed his book.

2. Synopsis Palinariorum Matheseos,: or a New Intro-
duction to the Mathematics, &c ; Svo, London, 1706" ;
being a very neat and useful compendium of all the ma-
thematical sciences, in about 000 pages.

]

J O U

His papers in thePhilos. Trans, are the following:

3. A Compendious Disposition of Equations for exhi-
biting the relations of Goniometrical Lines ; vol. 44,
pa. 500.

4. A Tract on Logarithms; vol. pi, pa. 455.

5. Properties of the Conic Sections, deduced by a com-
pendious method; Vol.63, pa. 340.

In all these works of Mr. Jones, a remarkable neatness,
brevity, and accuracy, every where prevail. He seemed
to delight in a very short and comprehensive mode of ex-
pression and arrangement; in so much that, sometimes
what he has contrived to express in two or three pages,
would occupy a little volume in the ordinary style of
writing.

Mr. Jones, it is said, possessed the best mathematical
library in England; scarcely any book of that kind but
what was there to be found. He had collected also a
great quantity of manuscript papers and letters of former
mathematicians, which have often proved useful to writers
of their lives, &c. After his death, these were dispersed,
and fell into different persons' hands; many of them, as
well as of Mr. Jones's own papers, were possessed by the
late Mr. John Robertson, before mentioned, at whose death
I purchased a considerable quantity of them. From such
collections as these it was that Mr. Jones was enabled to
give that first and elegant edition, in 4to, 1711, of several
of Newton's papers, that might otherwise have been lost,
entitled, Analysis per quantitatum Series, Fluxioncs, ac
Differentias: cum Enumeratione Lincarum Tertii Ordinis.

IONIC Column, or Order, the 3d of the five orders, or
columns, of architecture. The first idea of this order was
given by the people of Ionia; who, according to Vitru-
vius, formed it on the model of a young woman, dressed
in her hair, and of an easy elegant shape, as the Doric had
been formed on the model of a strong lobust man. This
column is a medium between the massive and the more
delicate orders, the simple and the rich. It is distinguish-
ed from the composite, by having none of the leaves of
acanthus in its capital; and from the Tuscan, Doric, and
Corinthian, by the volutes, or rams horns, which adorn its
capital ; and from the Tuscan and Doric, by the chan-
nels, or fluting, in its shaft. The height of this column
is IS modules, or 9 diameters of the column taken at the
bottom: indeed at first its height was but 16" modules ;
but, to render it more beautiful than the Doric, its height
was augmented by adding a base to it, which was un-
known in the Doric. Si. Leclerc makes its entabla-
ture to be. 4 modules and 10 minutes, and its pedestal 6
modules; so that the whole order makes 28 modules 10
minutes

JOURNAL, in Merchants' Accounts, is a book into
which every particular article is posted out of the waste-
book, according to the order of time, specifying the debtor
and creditor in each account and transaction.

Jou rnal, in Maritime Affairs, is a register kept by the
pilot, and others, noticing every thing that happens to the
ship, from clay to day, and from hour to hour, with regard
to the winds, the rhumbs or courses, the knots or rate of
running, the rake, soundings, astronomical observations,
for the latitudes and longitudes, &c; to enable them to
adjust the reckoning, and determine the. place where the
ship is. In all sea journals, the day, or what is called the
24 hours, is divided into twice I 2 hours; those before noon
marked a.m. for ante meridiem, and those from noon to
midnight markod p.m. post meridiem, or afternoon.

J o u

[ 693 ]

JOU

There are various ways of keeping a sea journal, ac- and Conferences of Arts and Sciences, by M. Dennis,

cording to the different notions of mariners concerning the during the years 1672, 16/3, and l674; New Discoveries

articles to be entered. Some writers direct the keeping in all the parts of Physic, by M. de Blegny; the journal of

such a kind of journal as is only an abstract of each day's Physic, begun in 1684, and some others, discontinued al-

transactions, specifying the weather, what ships or lands most as soon as begun. Rozier's Journal de Physique, be-

were seen, accidents on board, the latitude, longitude, gun in July 1771, and continued till, iu the year 1780, there

meridional distance, course, and run:' these particulars were 19 vols, quarto.

are to be drawn from the ship's log-book, or from that kept The Nouvelles de la Republique des Lettres, News

by the person himself. Other authors recommend the from the Republic of Lettres, were begun by M. Bayle in

keeping only of one account, including the log-book, I6'S4, and carried on by him till the year 1687, when M.

and all the work of each day, with the deductions drawn Bayle being disabled by sickness, his friends, M. Bernard

from it. and M. de Laroque, took them up, and continued them

Journal is also used for the title of several books till 1699. After an interruption of nine years, M. Ber-

published at stated times; giving accounts and abstracts nard resumed the work, and continued it till the year

of the new books that are published, with the new im- 1710. The History of the Works of the Learned, by M.

provements daily made in arts and sciences. The first Basnage, was begun in the year l6"SS, and ended in 1710.

journal of this kind was, the Journal des Scavans, printed The Universal Historical Library, by M. Leclerc, was

at Paris: the design was set on foot for the ease of such as continued to the year 1693, and contained twenty-five

are too busy, or too indolent, to read the entire books volumes. The Bibliotheque Choisie of the same author,

themselves. It seems a ready way of satisfying a man's began in 1703. The Mercury of France is one of the

curiosity, and becoming learned on easy terms: and so use- most ancient journals of that country, and is continued by

ful has it been found, that it has been executed in most other different persons : the Memoirs of a History of Sciences,

countries, though under a great variety of titles. and Arts, usually called Memoires des Trevoux, from the

Of this kind are the Acta Eruditorum of Leipsic ; the place where they are printed, began in 1701. « The Essays

Nouvelles de la Republique des Lettres ofM. Bayle, &c; of Literature reached but to a 12th volume in 1702, 1703,

the Bibliotheque Universelle, Choisie, et Ancienne et Mo- and 1704; these only take notice of ancient authors,

derne, of M. Leclerc ; the Memoirs de Trevoux, &c. In The Journal Literaire, by Father Hugo, began and ended

1692, Juncker printed in Latin, An Historical Treatise of in 1705. At Hamburgh they have made two attempts

the Journals of the Learned, published in the several parts for a French journal, but the design failed : an Epheme-

of Europe; and VVolfius, Struvius, Morhoff, Fabricius, &c, rides Scavantes has also been undertaken, but that soon

have done something of the same kind. disappeared. A Journal des Scavans, by M. Dartis, ap-

The Philosophical Transactions of London, and those peared in 1694, but was discontinued the year following,

of Edinburgh; the Memoirs of the Royal Academy of That of M. Chauvin, begun at Berlin in 1696, held out

Sciences; those of the Academy of Belles Lettres; the three years; and an essay of the same kind was made at

Miscellanea Natura: Curiosorum: the Experiments of the Geneva. To these may be added, the Journal Literaire

Academy del Cimento, the Acta Philo-exoticorum Na- begun at the Hague 1715, and that of Verdun, and the

turre et Artis, which appeared from March 1686 to April Memoires Literaires de la Grande Bretagne by M. de La-

1687, and which are a history of the Academy of Bresse ; roche ; the Bibliotheque Angloise, and Journal Britan-

the Miscellanea Berolinensia, or Memoirs of the Aca- nique, which are confined to English books alone. The

demy of Berlin; the Commentaries of the Academy of Italian journals arc, that of Abbot Nazari, which conti-

Petersburg; the Memoirs of the Institute at Bologna; the nued from 1668 to l6Sl,andwas printed at Rome. That

Acta Literaria Suecire; the Memoirs of the Royal Aca- of Venice began in J 671, and ended at the same time

demy of Stockholm, begun in 1740; the Commentarii with the other: the authors were Peter Moretti and

Societatis Regia; Gottingensis, begun in 1750, &c, &c, are Francis Miletti. The Journal of Parma, by Roberti and

not so properly journals, though they are frequently ranked Father Bacchini.was declined in I69O, and resumed again

in the number. in I692. The Journal of Ferrara, by the abbe de Latorre,

Juncker and Wolfius give the honour of the first inven- began and ended in I69L La Galerio di Minerva, begun

tion of journals to Photius. His Bibliotheca, however, is in 1 696, is the work of a society of men of letters. Seignior

not altogether of the same nature with the modern jour-
nals ; nor was his design the same. It consists of abridg-
ments and extracts of books which he had read during
his embassy in Persia. M. Salo first began the Journal
des Scavans at Paris, in 1665, under the name of the Sicur

Apostolo Zeno, secretary to that society, began another
Journal in 1710, under the protection of the grand duke:
it is printed at Venice, and several persons of distinction
are concerned in it.

The Fasti Euriditi della Bibliotheca Volantc, were pub-

de Hedonville; but his death soon after interrupted the lished at Parma. There has appeared since, in Italy, the

work* The abbe Gallois then took it up, and he, in the Giornale dei Letterati.

year 1 674, gave way to the abbe de la Roque, who conti- The principal among the Latin journals, is that of Leip-

nued it nine years, and was succeeded by M. Cousin, who sic, under the title of Acta Eruditorum, begun in 1682:

carried it on till the year 1702, when the abbe Bignon P.P. Manzani began another at Parma. The Nova Litera-

instituted a new society, and committed the care of con- ria Maris Balthici lasted from 1698 to 1708. The Nova Li-

tinuing the journal to them, who improved and published teraria Germania?, collected at Hamburgh, began in 1703.

it under a new form. This society is still continued, and The Acta Literaria ex Manuscriptis, and the Bibliotheca

M. de Loyer has had the inspection of the journal; which Curiosa, begun in 1705, and ended in 1707, are the works

is no longer the work of any single author, but of a great of Struvius. Messrs. Kustcr and Sike, in 1697, began a

number. The other French journals are the Memoirs Bibliotheca Novorum Librorum, and continued it for two

I R I

[ 694 ] I R R

colours; whence the eye is called grey, or black, &c. In
its middle is a perforation, through which appears a small
black speck, called the sight, pupil, or apple of the eye,
round which the iris forms a ring.

Iris is also applied to those changeable colours, which
sometimes appear in the glasses of telescopes, microscopes,
&c ; so called from their similitude to a rainbow. — The
same appellation is also given to that coloured spectrum,
which a triangular prismatic glass will project on a wall,
when placed at a proper angle in the sun-beams.

litis Marina, the Sea-Rainbow. This elegant appear-
ance is generally seen after a violent storm, in which the
sea water has been in vast emotions. The celestial rain-
bow however has great advantage over the marine one, in
the brightness and variety of the colours, and in their
distinctness one from the other; for in the sea-rainbow,
there are scarce any other colours than a dusky yellow
on the part next the sun, and a pale green on the op-
posite side. The other colours are not so bright, or so

years. Since that time, there have been many Latin
journals; such, besides others, is the Commentarii de
Rebus in Scientia Naturali et Medicina gestis, by M.
Ludwig. The Swiss journal, called Nova Literaria Hel-
vetia;, was begun in 1702, by M. Scheuchzer; and the
Acta Medica Hafnensia, published by T. Bartholin, make
five volumes from the year 1671 to 1679- There are
two Low-Dutch journals; the one under the title of
Boockzal van Europe, which was begun at Rotterdam in
1692, by Peter Rabbus; and continued from 1702 to
170S, by Sewel and Gavern : the other was conducted by
a physician, called Ruitcr, who began it in 1710. The
German journals of best note are, the Monathlichen Un-
terredungen, which continued from l6S9 to 169S. The
Bibliotheca Curiosa, began in 1704, and ended in 1707,
both by M. Tenzel. The Magazin d'Hambourg, com-
menced in 174S : the Physicalische Belustiguuzen, or Phi-
losophical Amusements, begun at Berlin in 1751. The
Journal of Hanover began in 1700, and continued for two
vears bv 61 Eccard, under the direction of M. Leibnitz, distinct, as to be well determined; but the sea-rainbows

J _'J . . , , mi rm,L .1 - ■ .1 c .__J , .1 <.! tl ;t

and afterwards carried on by others. The Theological
Journal, published by M. Loescher, under the title of Altes
und Neues, that is, Old and New. A third at Leipsic and
Francfort, the authors Mess. Walterck, Krause, and Gro-
schuffius; and a fourth at Hall, by M. Turk.

The English journals are, The History of the Works of
the Learned, begun at London in 1699. Censura Tem-
porum, in 1708. About the same time there appeared two
new ones, the one under the title of Memoirs of Litera-
ture, containing little more than an English translation of
some articles in the foreign journals, by M. De Laroche ;
the other a collection of loose tracts, entitled, Bibliotheca
Curiosa, or a Miscellany. These, however, with some
others, are now no more, but are succeeded by the An-
nual Register, which began in 1758; the New Annual
Register, begun in 1780; the Monthly Review, which
began in the year 1749, and gives a character of all Eng-
lish literary publications, with the most considerable of the
foreign ones: the Critical Review, which began in 1756",
and is nearly on the same plan: as also the London Re-
view, by Dr. Kenrick, from 1775 to 1780; Maty's Re-
view, from Feb. 1782, to Aug. 17S6; the English Review
begun in Jan. 1783; and the Analytical Review begun in
May 1788, both now extinct. The Edinburgh Review
begun in 1802, the Quarterly Review in 1S08, and the
New Quarterly in 1813, all these published four times a
year; also Tilloch's Philosophical Magazine, begun 179S;
Nicholson's Philosophical Journal in 1797, but ended with
1813; and Thomson's Annals of Literature; all published
monthly. Besides these, we have several monthly
pamphlets, called Magazines, which, together with a chro-
nological series of occurrences, contain letters from cor-
respondents, communicating extraordinary discoveries in
nature and art, with controversial pieces on all subjects.
Of these, the principal are those called, the Gentleman's
Magazine, which began with the year 1731 ; the London
Magazine, which began a few months after, aud has lately
been discontinued; the Universal Magazine, which is
nearly of as old a date ; and best of all the Monthly Maga-
zine began in 179"-

IRIS, another name for the Rainbow; which see.
Iris also. denotes the striped variegated circle round
the pupil of the eye, formed of a duplicature of the uvea.
In different subjects, the iris is of several very different

are more frequent and more numerous than the others: it
is not uncommon to see 20 or 30 of them at a time at noon-
day.

IRRADIATION, the emission of subtile effluvia, in all
directions, like the rays of the sun.

IRRATIONAL Numbers, or Quantities, are the same as
surds, orsuch routs as cannot be accurately extracted, being
incommensurable to unity. See Suiids.

IRREDUCIBLE Case, in Algebra, is used for that case
of cubic equations where the root, according to Cardan's
rule, appears under an impossible or imaginary form, and
yet is real; viz, in the form x3 — ax = ±_c, having ^a3
greater than ic2, or 4a3 greater than 27c1. Thus, in the
equation r3 — 15i = 4, the root, according to Cardan's
rule, is x = V (2 + ^ - 121) +'i/ (2- y/ - 12l),which
is in the form of an impossible expression, and yet it is
equal to the- quantity 4 :

for ?/(2 + ^ - 121) =2 +V-1.
and^/(2-v'- 121) = 2-^-1,
therefore their sum is x = 4.

The other two roots of the equation are also real.

Algebraists, for almost three centuries, havi in vain en-
deavoured to resolve this case, and to bring it under a real
form; and the problem is not less celebrated among them,
than the squaring of the circle is among geometricians. R
is to be observed, that, as in some other cases of cubic
equations, the value of the root, though rational, is found
under an irrational or surd form ; because the root in this
case is compounded of two equal surds with contrary
signs, which destroy each other; asif ,r=2-t- \/3 ■+■ '2 — a/3,
then x =4. In like manner, in the irreducible ca^e, where
the root is rational, there are two equal imaginary quan-
tities, with contrary signs, joined to real quantities; so
that the imaginary quantities destroy each other: as in the
case above of the root of the equation as3— 15x = 4, which
was found to be 2 -t- *J - 1 -+■ 2 - </ - 1 = 4. It is re-
markable that this case always happens, viz one root, by
Cardan's rule, in an impossible form, whenever the equa-
tion has three real roots, and no impossible ones, but at
no time else.

If we were possessed of a general rule for accurately
extracting the cube root of a binomial radical quantity, it
is evident wc might resolve the irreducible case generally,
which consists of two of such cubic binomial roots. But

I S L [ 6gs

the labours of the algebraists, from Cardan's down to the
present time, have not been able to remove this difficulty.
Dr. Wallis thought that he had discovered such a rule;
but, like most others, it is merely tentative, and can only
succeed in certain particular circumstances.

It has been sometimes asserted that, in the irreducible
case of cubic equation, if the coefficients of all the terms
be rational, the equation will have at least one rational
root: which may be demonstrated in this manner.

The given equation being x3 — ax = c, where the co-
efficients a and c are rational.

Let i 2m denote one of the roots ;

then + 7ft -t- ?i 7 j u »u »i_

■ -J. _ > may denote the other two roots.

Hence a or m~ — n2 — 4>m° is rational :

but 4m1 is rational, therefore m~ — n'is rational.

Again, c — 2m. {nr — rr) is rational:

but m" — n" is rational, therefore 2m is rational,
which is one of the roots.

One of the most convenient methods of resolving such
cubic equations, is by means of tables of sines, &C See
the article Cubic Equations.

Mr. Maseres, cursitor baron of the exchequer, has
lately deduced, by a long train of algebraical reasoning,
from Newton's celebrated binomial theorem, an infinite
series, which will resolve this case, without any mention
of either impossible or negative quantities. And 1 have also
discovered several other series which will do the same thing,
in all cases whatever; both inserted in the Phil. Trans.
See Cardan's Algebra; the articles Algebra, Cubic Equa-
tions; Wallis's Algebra, chap. 48; De Moivre in the Ap-
pendix to Sanderson's Algebra, pa. 744; Philos. Trans,
vol. 68, part 1, art. 42, and vol. 70, pa. 387 ; and my New
Tracts, vol. 2, pa. 98, &c.

IRREGULAR, something that deviates from the com-
mon forms or rules. Thus, we say an irregular fortifica-
tion, an irregular building, &c.

.Irregular Figure, in Geometry, whether plane or
solid, is that whose sides, as well as angles, are not all equal
and similar among themselves.

IRREGULARITIES in the Moon's motion. See Moon.

ISAGONE, in Geometry, is sometimes used for a figure
consisting of equal angles.

ISLAND, or Isle, a tract of dry land encompassed by
water; whether by the sea, a river, or lake, &c. In which
sense Island stands contradistinguished from Continent, or
terra firma; like Great Britain, Ireland, Jersey, Sicily,
Minorca, &c.

Some naturalists imagine that islands were formed at the
deluge : others think they have been rent and separated
from the. continent by violent storms, inundations, and
earthquakes; while .others are thrown up by volcanoes,
or otherwise grow or emerge from the sea.

Varenius thinks most of these opinions true in some in-
stances, and believes that there have been islands produced
each of these ways. St. Helena, Ascension, and other
steep rocky islands, he supposes have become so, by the
sea's overflowing their neighbouring champaigns. By the
heaping up huge quantities of sand, &c, he thinks the
islands of Zealand, Japan, &c, were formed : Sumatra and
Ceylon, and most of the East-India islands, he rather
thinks were rent off from the main land, as England pro-
bably was from France. It is also certain that some have
emerged from the bottom of the sea; as Santorini formerly,
and three other isles near it lately; the last in 1707, which

3

I s o

rose from the bottom of the sea, after an earthquake : the
ancients had a tradition that De.los rose from the bottom
of the sea; and Seneca observes that the island Therasia
rose out of the iEgean sea in his time, of which the mari-
ners were eye witnesses ; as they have been within these
10 years, in the sea between Norway and Iceland, as also
in the Atlantic ocean, where an island has just cmerscd,
near St. Michael's. The late circumnavigators too have
made it probable, that many of the South sea islands have
bad their foundations, of coral rod, gradually increasing,
and growing out of the sea.

ISLES, or rather Ailes, in Architecture, the wings or
sides of a building.

ISOCHRONAL, or Isochronous, is applied to such
vibrations of a pendulum as are performed in equal times.
Of which kind are all the vibrations of the same pendulum
in a cycloidal curve, and in a circle nearly, whether the
arcs it describes be longer or shorter ; for when it describes
a shorter arc,' it moves so much the slower ; and when a
long one, proportionably faster.

Isochronal Line, is that in which a heavy body is
supposed to descend with a uniform velocity, or without
any acceleration. Leibnitz, in the Act. Erud. Lips, for
April 16SQ, has a discourse on the Linea Isochrona, in
which he shows, that a heavy body, with the velocity ac-
quired by its descent from any height, may descend from
the same point by an infinite number of isochronal curves,
which are all of the same species, differing from one an-
other only in the magnitude of their parameters (such as
are all the quadratocubical paraboloids), and consequently
similar to one another. He shows also, how to find a
line, in which a heavy body descending, shall recede
uniformly from a given point, or approach uniformly
to it.

ISOMERIA, in Algebra, a term of Vieta, deno-
ting the freeing an equation from fractions; which is
done by reducing all the fractions to one common deno-
minator, and then multiplying each member of the equa-
tion by that common denominator, that is rejecting it out
of them all.

ISOPERIMETRICALKgt/ra, are such as have equal
perimeters, or circumferences. It is demonstrated in geo-
metry, that among isoperi metrical figures, that is always
the greatest, which contains the most sides or angles.
Whence it follows, that the circle is the most capacious of
all figures which have the same perimeter with it. — That
of two isoperimetrical triangles, which have the same base,
and one of them two sides equal, and the other unequal ;
that is the greater whose sides are equal. That of isoperi-
metrical figures, whose sides are equal in number, that is
the greatest which is equilateral, and equiangular. Hence
arises the solution of that popular problem, To make the
hedging or walling, which will fence in a certain given
quantity of land, also to fence in any other greater quan-
tity of the same. For let x be one side of the rectan-
gle that will contain the quantity aa of acres ; then will

~ be its other side, and double their sum, viz, 2x n ,

will be the perimeter of the rectangle : let also bb be any
greater number of acres, in the form of a square, then is b
one side of it, and 46 its perimeter, which must be equal

to that of the rectangle; and hence the equation 2x h

= 4&, or x1 -t- a? = 2bx, in which quadratic equation the
two roots are x = b ± y/(b2 — a2), which are the lengths

ISO [696

of the two dimensions of the rectangle, viz, whose area b*,
is in any proportion less than the square ax, of the same
perimeter. As, for example, if one side of a square be 10,
and one side of a rectangle be 19, but the other only 1 ;
such square and parallelogram will be isoperimetrical, viz,
each perimeter 40; yet the area of the square is 100,
and of the parallelogram only 19-

Isoperimetrical lines and figures have greatly engaged
the attention of mathematicians at all times. The 5th
book of Pappus's Collections is chiefly on this subject ;
where a great variety of curious and important properties
are demonstrated, both of planes and solids, some of which
were then old in his time, and many new ones of his own.
Indeed it seems he has here brought together, into this
book, all the properties relating to isoperimetrical figures
then known, and their different degrees of capacity.

The analysis of the general problem concerning figures
that, among all those of the same perimeter, produce
maxima and minima, was given by Mr. James Bernoulli,
from computations that involve 2d and 3d fluxions. And
several enquiries of this nature have been since prosecuted
in like manner, but not always with equal success. Mr.
Maclaurin, to vindicate the doctrine of fluxions from the
imputation of uncertainty or obscurity, has illustrated this
subject, which is considered as one of the most abstruse
parts of this doctrine, by giving the resolution and compo-
sition of these problems by first fluxions only; and in a
manner that suggests a synthetic demonstration, serving
to verify the solution. See Maclaurin's Fluxions, pa. 486;
Analysis Magni Problematis Isoperimetrici Act. Erud.
Lips. 1701, pa. 213; Mem. Acad. Scienc. 1705, 1706",
1718 ; and the works of John Bernoulli, torn. 1, pa. 202,
20S, 424, and torn. 2, pa. 235 ; where is contained what
he and his brother James published on this problem. Mr.
John Bernoulli, in his first paper, considered only two
small successive sides of the curve; whereas the true me-
thod of resolving this problem in general, requires the con-
sidering three such small sides, as may be perceived by
examining the two solutions.

M. Euler has also published, on this subject, many pro-
found researches, in the Petersburg commentaries ; and
there was printed at Lausanne, in 1744, a pretty large
work upon it, entitled, Method us inveniendi lineascurvas,
maximi mininiive proprietatc gaudentes: sive Solutio pro-
blematis isoperimetrici in kuissimo sensu accepti. — M.
Cramer too, in the Berlin Memoirs for 1752, has given a
paper, in which he proposes to demonstrate in general,
what can be demonstrated only of regular figures in the
elements of geometry, viz, that the circle is the greatest of
all isopelimetrical figures, regular or irregular — M. Lhuil-
lier has some elementary problems on isoperimeters, in
his Polygonomctry ; and a very considerable tract on this
subject is given in my Course of Mathematics, vol. 3, pa.
31—53.

On this head, see also Simpson's Tracts, pa. 98 ; and
tin' Philos. Trans, vol. 49 and 50.

lSOSCIiXF.S Triangle, is a triangle ',

that has two sides equal. In the 5th
prop, of Euclid's' I si book, which prop. /' \

is usually called the pons asinorum, / j \

or asses bridge, it is demonstrated, that \

the angles, a and b, at the base of the
isosceles triangle, are equal to each
other; and that if the equal sides be
produced, the two angles, c and d,

]

JUL

L

below the base, will also be equal. It is also inferred, that
every equilateral triangle is also equiangular.

Other properties of this figure are, that the perpendicu-
lar ap, from the vertex to the base, bisects the base, the
vertical angle, and also the whole triangle. And that if a
line be drawn from the vertex to any point in the base, the
square of this line, together with the rectangle of the seg-
ments of the base, will be equal to the square of one of
the equal sides of the triangle. Also, if the vertical angles
of two isosceles triangles be equal, the two triangles will
be equiangular.

ISTHMUS, in Geography, a narrow neck or slip of
land, that joins iwo other large tracts together,- and sepa-
rating two seas, or two parts of the same sea. The most
remarkable isthmuses are, that of Panama or Straits of
Darien, connecting north and south America; that of
Suez, which joins Asia and Africa; that of Corinth, or
Peloponnesus, in the Morea ; that of Crim Tartaiy, other-
wise called Taurica Chersonesus; that of the Peninsula
Romania and Erisso, or the isthmus of the Thracian Cher-
sonesus, 12 furlongs broad, and which Xerxes undertook
to cut through. The ancients had several designs of cut-
ting the isthmus of Corinth, which is a rocky hillock,
about 10 miles over; but without effect, the invention of
sluices being not lhen known. There have also been at-
tempts for cutting the isthmus of Suez, to make a com-
munication between the Mediterranean and the Red-sea.

JULIAN Calendar, is that depending on,and connected
with the Julian Year and account of time; so called from
Julius Ctesar, by whom it was established. See Calen-
dar.

Julian Epoch, is that of the institution of the Julian
reformation of the calendar, which began the 46th year
before Christ.

Julian Period, is a cycle of 7980 consecutive years,
invented by Julius Scaliger, from whom it was named ;
though some say his name w<is Joseph Scaliger, and that
it was called the Julian period, because he made use of
Julian years. This period is foimed by multiplying con-
tinually together the three following cycles, \iz, that of
the sunof2S years, that of the moon of 19 years, and
that of the indiction of 15 years; so that this epoch,
though but artificial or feigned, is yet of good use; in
that every year within the period is distinguishable by a
certain peculiar character ; for the year of the sun, moon,
and indiction, will not be the same again till the whole
7980 years have revolved. Scaliger fixed the beginning
of this period 764 years before the creation, or rather the
period naturally reduces to that year, taking the numbeis
of the three given cycles as he then found them; and ac-
counting 3950 years from the creation to the birth of
Christ, this makes the 1st year of the Christian era answer
to the 4714th year of the Julian period ; therefore, to find
the year of this period, answering to any proposed year of
Christ, to (he constant number 4713, add the given year
of Christ, and the sum will be the year of the Julian pe-
riod : thus, to 4713 adding J7£H. 'he sum 6504 is the
year of this period for the year of Christ 179'- Hence
the first revolution of the Julian period will not be com-
pleted till the year of Christ 3267, after which a new re-
volution of this period will commence.

. But the year of the Julian period may be found for any
time, from the numbers of the three cycles that compose
it, without making use of the given year of Christ ; thus,
multiply the

J U N

r 48451

J 4200 V .
L6916) thc year of the

C 697 ]

J U P

J indiction :

then add the three products together, and divide the sum
by 79SO; so shall the remainder, after division., be the year
of the Julian period corresponding to the given years of
the other three cycles. Thus, for the year 1791, the years
of the solar, lunar, and indiction cycles, arc 8, 6", and 9 ;
therefore multiplying by these, &c, according to the rule,
thus

4845
S

4200

6

25200

3S76'0

62244

6916
9

38760

62244

79SO)

126204 ( 15
7980
46404
39900

remains 6504 the year of the Julian period.

Julian Year, is the old account ofthe year, established
by Julius Cajsar, and consisted of 36*5f days. This year
continued in use in all Europe, till it was superseded in
most parts by the new or Gregorian account, in the year
1582. In England however it continued to be used till
the year 1752, when it was abolished by act of parliament,
and eleven days added to the account, to bring it up to
the new style. In Russia, the old or Julian year and style
are still in use.

JULY, the 7th month of the year, consisting of 31 days;
about the 21st of which the sun usually enters the sign SI
leo. It was so named by Mark Antony, from Julius Cassar,
who was born in this month.

JUNGIUS (Joachim), a learned German, who was
born at Lubeck. He studied at Giessen, where he applied
to mathematics and metaphysics, and wrote Latin trage-
dies. He was some time professor of mathematics, but
quitting that station, he studied physic at Augsburg, and
took his doctor's degree at Padua. He was esteemed a
very learned man, being author of several valuable works,
and died at Hamburg, in l657, where his Geometria Em-
pyrica was published, in 1669, 4to.

Dr. Birch (Hist. Royal Soc. vol. 2, pa. 342) says, Jun-
gius was rector ofthe Gymnasium at Hamburg. In the
original draught of a letter of Mr. (afterwards Dr.) John
Pell, to Sir Charles Cavendyshe, dated at Amsterdam,
1644, that eminent mathematician recommends Dr. Jun-
gius and John Adolphus Tassius, corrector ofthe same
Gymnasium, to the acquaintance of Sir Charles, who was
then at Hamburg, as both mathematicians, of whose abi-
lities and worth he had a great opinion. He remarked
that Jungius had many things in print. As Apodictica,
and Apodidactica. That Tassius had, two years before,
reprinted Jungius's Geometria Empyrica in 5 sheets of
paper, but without diagrams : that Tassius was a very
courteous, affable, open man ; Jungius a little more re-
served. He added, " The most barbarous nations had
something to worship ; and there are few men that have
not some idol, some man or woman, whom they esteem
and admire above all the rest of mankind; and Jungius
is mine. For I esteem of men more or less, as I find them
more or less rational ; and therefore havingseen something
of Jungius's writing, wherein he seemed to me to make a
true and better use of his reason, and to manage that di»

Vol. I.

vine instrument of instruments, with more dexterity and
skill, than any other son of Adam, all other writers must
pardon me, if I profess to expect more solidity in Jungius's
writings, than in any other man now living. If you lind
but the one half of what I imagine to be in Jungius, you
will never be able to relish any other modern philoso-
pher." Sir Charles Cavendyshe accordingly, in a letter to
Mr. Pell, without a date, but received by the latter at
-Amsterdam, Sept. if, 1644, mentions his having seen
Jungius, whom he found very free, and who, he observed,
preferred the analytics of the antients before Vieta's by
letters, as being more subject to errors and mistakes,
though more facile and quick of dispatch.

Mr. Oldenburg, secretary of the Royal Society, read to
the Society a Latin letter to him, from Dr. Martin Foge-
lius of Hamburg, dated there Jan. l66|, containing an ac-
count of some manuscripts.left by Jungius, concerning
motion, and Apollonius de Locis Planis, in 2 books; of
the former of which subjects were transmitted the chief
heads, and the method observed in the latter: the writer
of which letter expressed his inclination to publish those
writings, if encouragement were given to him. It does not
appear that any thing was done in consequence of it, but
the latter may be seen in the letter-book of the Royal So-
ciety, vol. 3, pa. 5.

JUNO, a name given to one of the small newly-disco-
vered planets, viz, that found by M. Harding in the year
1804. See Astronomy, Harding, Excentricity, &c.

JUPITER, If., one of the superior planets, remarkable
for its splendour, being the brightest of all, except some-
times the planet Venus, and is much the largest of all the
planets. Jupiter is situated between Mars and Saturn,
being the 9th in order ofthe primary planets from the sun.
His diameter is more than 10 times the diameter of the
earth, and therefore his magnitude more than 1000 times.
His real diameter is 89170 miles. His annual, revolution
about the sun is almost 12 years, viz, 4350 days 14 hours
39 minutes 2 seconds, going at the rate of more than 25
thousand miles per hour, at the mean distance of 4y0
millions of miles; and he revolves about his own axis in
the short space of 9 hours 56' minutes, by which his equa-
torial parts are carried round at the amazing rate of 26
thousand miles per hour, which is about 25 times faster
than the like parts of our earth revolve.

Jupiter is surrounded by faint substances, called zones
or belts, in which so many changes appear, that they are
generally ascribed to clouds : for some of them have been
first interrupted and broken, and then have vanished en-
tirely. They have sometimes been observed of different
breadths, and afterwards have all become nearly of the
same breadth. Large spots have been seen in these belts ;
and when a belt vanishes, the contiguous spots disappear
with it. The broken ends of some belts have often been
observed to revolve in the same time with the spots ; only
those nearer the equator in somewhat less time than those
nearer the poles ; perhaps on account of the sun's greater.
heat near the equator, which is parallel to the belts and
course of the spots. Several large spots, which appear
round at one time, grow oblong by degrees, and then di-
vide into two or three round spots. The periodical time
of the spots near the equator is 9 hours 50 minutes, but
of those near the poles 9 hours 56 minutes. See Dr. Smith's
Optics, §1105 and IIO9.

The axis of Jupiter is so nearly perpendicular to his
orbit, that he has no sensible change of seasons ; which is
4U

J U P

[

3

J U R

a great advantage, and considered as wisely ordered by
the Author of Nature. For, if the axis of this planet were
inclined any considerable number of degrees, justso many
decrees round each pole would ill their turn be almost 6
years together in darkness. And, as each degree of a
great circle on Jupiter contains about 706 miles, it is
easy to judge what vast tracts of land would be rendered
uninhabitable by any considerable inclination of his axis.

The difference between the equatorial and polar dia-
meters of Jupiter, is upwards of 6000 miles ; the former
boing to the latter as 13 to 12 : so that his poles are more
than 3000 miles nearer his centre than the equator is.
This happens from his quick motion round his axis; for
the fluids, together with the light particles, which -they
can carry or wash away with them, recede from the poles
which are at rest, towards the equator where the motion
is quickest, until there be a sufficient number accumu-
lated to make up the deficiency of gravity lost by the
centrifugal force, which always arises from a quick mo-
tion round an axis : and when the deficiency of weight or
oravity of the particles is made up by a sufficient accu-
mulation, there is then an equilibrium, and the equatorial
parts rise no higher.

Jupiter's orbit is 1° 18' 56" inclined to the ecliptic.
The place of his aphelion 11° S' of £±, the place of his
ascending node 7° 29' of qs, and that of his south or
descendinguode7°29'of Vf. The excentricity of his orbit
is | of his mean distance from the sun.

The sun appears to Jupiter but the 48th part so large
as to us ; and his light and heat are in the same small pro-
portion, but compensated by the quick returns of them,
and by 4 moons, some of them larger than our earth,
which revolve about him ; so that there is scarce any part
of this huge planet but what is, during the whole night,
enlightened by one or more of these moons, except his
poles, whence only the farthest moons can be seen, and
where theirlight is not wanted, because the sun constantly
circulates in or near the horizon, and is very probably
kept in view of both poles by the refraction of Jupiter's
atmosphere, which, if it be like ours, has certainly re-
fractive power enough for that purpose. This planet,
seen from its nearest moon, appears 1000 times as large as
otir moon does to us ; increasing and waning in all her
monthly shapes, every 42£ hours. The periods, di-
stances, in semidiameters of Jupiter, and angles of the
orbits of these moons, seen from the earth, are as follow :

No.

1

2

3

4

The three nearest moons of Jupiter fall into his shadow,
and are eclipsed in every revolution : but the orbit of the
+th satellite is so much inclined, that it passeth by its op-
position to Jupiter, without falling into his shadow, two
years in every six. By these eclipses, astronomers have
not only discovered that the sun's light takes up 8 minutes
of time in coming to us; but have also by them deter-
mined the longitudes of places on this earth, with greater
certainty and facility, than by any other method yet
known. The outermost of these! satellites will appear
nearly as large as the moon does to us. See M. Laplace's
Theory of Jupiter's Satellites, in the Memoircs do l'Acad.
and in the Connoissance des Temps for 1792, pa. 273.

Periods round
Jupiter.
1<< 18h36m

Distances.

5|

Angles of
oiVits.
3' 55"

3 13 15

9

6 14

7 3 59
16 18 30

14$

25^

9 58
17 30

The English astronomer Mr. Pound long since ob-
served, that the irregularities of the three interior satel-
lites of Jupiter were repeated in a period of 437 days ;
and this observation is found Lo be just to the present
time. For the following curious circumstance has now
been established, viz. that

247 revolutions of the 1st occupy 437d 3h 44m

123 - - 2d - 43Y 3 42

6l - - 3d - 437 3 36

26 - - 4th - 435 14 16

This naturally led mathematicians to examine their

motions, and see in what manner their relative positions,

or configurations, as they are called, corresponded to this

period ; and it is found, that the mean longitude of the

1st satellite,

minus 3 times that of the 2d,

plus 2 times that of the 3d,

always amounted to ISO degrees.

This requires that the mean motion of the first, added'
to twice that of the 3d, shall be equal to 3 times the mean
motion of the 2d. — This correspondence of the mean mo-
tions is of itself a singular thing, and the odds or chance
against its probability seems infinitely great; and when
we add to this the particular positions of the satellites in
any one moment, which is necessary for the above con-
stant relation of their longitudes, the improbability of the
coincidence, as a thing quite fortuitous, becomes infinite-
ly greater. Doubts were at first entertained of the coin-
cidence, because it was not indeed accurate to a second.
The result of the investigation is curious. When the con-
sequences of mutual gravitation are followed out, we find,
that though neither the primitive motions of projection,
nor the points of the orbit from which the satellites were
projected, were precisely such as suited these observed
relations of their revolutions, and their contemporaneous
longitudes ; yet, if they differed from them only by very
minute quantities, the mutual gravitations of the satellites
would in time bring them into those positions, and those
states of mean motion, that would induce the observed
relations; and when they are once reduced, they will be
continued for ever.

Though there be S primary planets below Jupiter, yet
an eye placed on his surface would never perceive any of
them; unless perhaps as spots passing over the sun's disc,
when they happen to come between the eye and the sun.
The parallax of the sun, viewed from Jupiter, will scarce
be sensible, being not much above 20 seconds; and the
sun's apparent diameter in Jupiter, but about 6 minutes.
Dr. Gregory adds, that an astronomer in Jupiter would
easily distinguish two kinds of planets, four nearer him,
viz his satellites, and two more remote, viz the sun and
Saturn; the former however will fall vastly short of the
sun in brightness, notwithstanding the great disproportion
in the distances and apparent magnitude.

JUR1N (Dr. James), a very distinguished person in
several walks of literature, particularly medicine, mathe-
matics, and philosophy, which he cultivated with equal
success. He was secretary of the Royal Society in Lon-
don, as well as president of the College of Physicians
there, at the time of his death, which happened March 22,
1750. Doctor Jurin was author of several ingenious
compositions; particularly An Kssay upon Distinct and
Indistinct Vision ; printed at the end of the 2d volume of
Dr. Smith's System of Optics: also several controversial
papers ; ngninst Michellotti, on the momentum of running

JUR

[

J

JUR

waters; against Robins, on distinct vision ; against Bishop
Berkeley, on the doctrine of fluxions; and against the
partisans of Leibnitz, on the forces of moving bodies ; &c.
His papers inserted in the Philos. Trans, are the following :
1. On the Suspension of Water in Capillary Tubes:
vol. 30, pa. '739. — 2. Observations on the Motion of
Running Water: pa. 748. — 3. On an old Roman Inscrip-
tion : pa. 813. — 4. A Discourse on the Power of the
Heart : pa. 863 and 929.-5. On the Specific Gravity
of Human Blood, pa. 1000. — 6. Defence of his Doctrine
of the Power of the Heart against the Objections of Dr.
Keill : pa. 1039. — 7- On the Action of Glass Tubes upon
Water and Quicksilver: pa.-1083. — 8. On the Specific

Gravity of Solids when weighed in Water: vol.31, pa. 223-
— 9- On the Motion of Running Water, against Michel-
lotti: vol.32, pa. 179- — 10. Remarkable Instance of the
Small-pox: vol.32, pa. 191. — 11. Inoculated and Na-
tural Small-pox compared: vol.32, pa. 213. — 12. On
Meteorological Diaries: vol.32, pa. 422. — 13. On the
Measure and Motion of Running Water: vol.41, pa. 5
and 65. — 14. Meteorological Observations in Charles-
Town : vol. 42, pa. 491. — 15. On the Action of Springs :
vol. 43, pa. 46.— lS. On the Force of Bodies in Motion :
pa. 423. — 17. Dynamic Principles, or Meteorological
Principles of Mechanics: vol.66, pa. 103.

K.

KAN

K E I

"|7"iESTNER ( ), a celebrated German ma-

-*-*■ thematician; died about the end of the year 1800,in
the 80th year of his age, having filled the chair of pro-
fessor of mathematics and natural philosophy in the uni-
versity of Gottingen from the year 1756, being as it were
the Patriarch of European mathematicians. From his
pen we have several elementary works on different branches
of the mathematics, which have all met with great suc-
cess ; his Elements of Arithmetic, of Plane and Spherical
Trigonometry, and of Perspective, having passed through
five editions between the years 1758 and 1794. He was
one of those who have the most materially contributed to
introduce a sound method into the study of mathematics.
The catalogue of his different works fills above nine pages
in the last edition of Meusel's German Literature.
Among the number are the translations of several im-
portant works from the French, the English, and the Low-
Dutch. His History of the Mathematics, in 4 volumes,
may be considered as a descriptive catalogue of his own
library : for he possessed a precious collection of all the
most rare and valuable works in the mathematical de-
partments.

Ksestner's mode of life was somewhat singular : during
the latter years of his life he never went abroad except on
Sundays, when he regularly attended the sermons at his
parish-church, and on the days when the Royal Literary
Society at Gottingen held their sittings. A few months
before his death, he was afflicted with a paralytic stroke
in his righthand; but so assiduous and indefatigable was
he in the prosecution of his studies, that he began to write
with his left; and so continued till the time of his decease.

KANG HI, one of the greatest princes of China,, who
reigned in the 17th century. He, wishing to have a chart
of his empire and its environs, employed for that purpose
several European missionaries, to whose labour we are in-
debted for the grand chart of China, first published in
1735, in France, by M. Duhalde ; and since with some
corrections by Danville. He also caused to be translated
into the Chinese and Tartar languages, several treatises
on astronomy, and some astronomical tables ; as also the
Elements of Euclid, which he very much admired. Ver-
biost calculated for him, in 1683, a table of all the eclipses
of the sun and moon that would happen in the course of

2000 years. See Astron. Europasa by Verbiost. The
invention of logarithms, and the tables of sines, delighted
him beyond all expression: he soon caused a small edi-
tion of them to be made, that he always carried about
with him, and which after his death were sent as a curiosity
to the library of the Jesuits at Lyons. He also very much
admired the European art of writing ; but his ears being
accustomed to the grave music and monotony of his coun-
try, he could never relish the lively and animating music
of the Europeans ; what we call harmony seemed to him
only a confused noise. He had several sons, who were
instructed in most of the European sciences, at their own
request, thus imitating the example given them by their
father.

KEILL (Dr. John), an eminent mathematician and
philosopher, was born at Edinburgh in l671, and studied
in the university of that city. His genius leading him to
the mathematics, he made a great progress under David
Gregory the professor there, who was one of the first that
had embraced and publicly taught the Newtonian philo-
sophy. In 1694 he followed his tutor to Oxford, where,
being admitted of Baliol-college, he obtained one of the
Scotch exhibitions in that college. It is said he was the
first who taught Newton's principles by the experiments
on which they are founded: and this it seems he did by
an apparatus of instruments of his own providing ; by
which means he acquired a great reputation in the uni-
versity. The first public specimen he gave of his skill in
mathematical and philosophical knowledge, was his Exa-
mination of Dr. Burnet's Theory of the Earth : with Re-
marks on Mr. Whiston's New Theory ; which appeared in
1698. These theories were defended by their respective
authors ; which drew from him, in 1699; An Examination
of the Reflections on the Theory of the Earth, together
with A Defence of the Remarks on Mr. Whiston's New
Theory. Dr. Burnet was a man of great humanity, mo-
deration, and candour ; and it was therefore supposed that
Keill had treated him too roughly, considering the great
disparity of years between them. Keill however left the
doctor in possession of that which has since been thought
the great characteristic and excellence of his work ; and
though he disclaimed him as a philosopher, yet allowed
him to be a man of a fine imagination.
4 U 2

K E I

[ 700 ]

KEP

The year following, Dr. Millington, Sedleian professor
of natural philosophy in Oxford, who had been appointed
physician to King William, substituted Keill as his deputy,
to read the lectures in the public school. This office he
discharged with great reputation ; and, the term of enjoy-
ing the°Scotch exhibition at Baliol-college now expiring,
he accepted an invitation from Dr. Aldrich, dean of Christ-
church, to reside there.

In 1701 he published his celebrated treatise, entitled
Introductio ad Veram Physicam, which is supposed to be
the best and most useful of all his performances. The
first edition of this book contained only fourteen lectures;
but to the second, in 1705, he added two more. This
work was deservedly esteemed, both at home and abroad,
as the best introduction to the Principia, or the new me-
chanical philosophy, and was reprinted in different places ;
also a new edition in English was printed at London in
1736, at the instance of M. Maupertuis, who was then in
England.

Being made Fellow of the Royal Society, he published,
in the Philos. Trans. 17PS, a paper on the Laws of At-
traction, and its physical principles: and being offended
at a passage in the Acta Eruditorum of Leipsic, where
Newton's claim to the first invention of the method of
fluxions was called in question, he warmly vindicated that
claim against Leibnitz. In 1709 he went to New-England
as treasurer of the Palatines ; and soon after his return in
1710, he was chosen Savillian professor of astronomy at
Oxford. In 1711, being attacked by Leibnitz, he en-
tered the lists with that mathematician, in the dispute
concerning the invention of fluxions. Leibnitz wrote a
letter to Dr. Hans Sloane, then secretary to the Royal
Society, requiring Keill, in effect, to make him satisfaction
for the injury he had done him in his paper relating to the
passage in the Acta Eruditorum ; he protested, that he
was far from assuming to himself Newton's method of
fluxions; and therefore desired that Keill might be obliged
to retract his false assertion. On the other hand, Keill
desired that he might be permitted to justify what he had
asserted ; and he made his defence to the approbation of
Newton, and other members of the Society. A copy of
this was sent to Leibnitz ; who, in a second letter, re-
monstrated still more against Keill's want of candour and
sincerity ; adding, that it was not fit for one of his age
and experience to engage in a dispute with an upstart,
who acted without any authority from Newton, and de-
siring that the Royal Society would enjoin him to silence.
On this, a special commitiee was appointed ; who, after
examining the facts, concluded their report with " de-
claring Newton to be the inventor of fluxions: and that
Mr. Keill, in asserting the same, had been in no respect
injurious to M. Leibnitz." The whole proceedings on
this matter may be seen in Collins's Commercium Episto-
licum, with many valuable papers of Newton, Leibnitz,
Gregory, and other mathematicians. In the mean time
Keill behaved himself with great firmness and spirit;
which he also showed afterwards in a Latin epistle, written
in 1720, to John Bernoulli, mathematical professor at
Basil, on account of the same usage shown to Newton :
in the title-page of which he put the arms of Scotland, viz,
a thistle, with this motto, Nemo me impunc lacessit.

About the year 1711, several objections being urged
against Newton's philosophy, in' support of Descartes's
notionsof aplenum, Keill published a paper in the Philos.
Trans, on the Rarity of Matter , and the Tenuity of its

Composition. While he was engaged in this dispute,
Queen Anne was pleased to appoint him her decipherer ;
and he "continued in that place under King George the
1st till the year 171(5- The university of Oxford con-
ferred on him the degree of m. d: in 1713; and two years
after, he published an edition of Commandine's Euclid,
with additions of his own. In 17 IS he published his In-
troductio ad Veram Astronomiam ; which was afterwards,
at the request of the duchess of Chandos, translated by
himself into English ; and, with several emendations, pub-
lished in 1721, under the title of An Introduction to
True Astronomy, Sec. This was his last gift to the pub-
lic ; for being that summer seized with a violent fever, it
terminated his life Sept. 1, in the 50th year of his age. —
His papers in the Philos. Trans, above alluded to, are con-
tained in volumes 26 and 29.

Keill (Dr. James), an eminent physician and philoso-
pher, and younger brother of Dr. John Keill above men-
tioned, was also born in Scotland, in l673. Having tra-
velled abroad, on his return he read lectures on anatomy
with great applause in the universities of Oxford and
Cambridge, by the latter of which he had the degree of
m. d. conferred upon him. In 1703 hesettled at North-
ampton as a physician, where he died of a cancer in the
mouth in 1719- His publications are

1. An English translation of Lemery's Chemistry.

2. On Animal Secretion, the quantity of Blood in the
Human Body, and on Muscular Motion.

3. A treatise on Anatomy.

4. Some pieces in the Philos. Trans, vols. 25 and 30.

KEPLER (John), a very eminent astronomer and ma-
thematician, was born at Wiel, in the duchy of Wirtem-
berg, in 1571. He was the disciple of Ma;stlinus, a
learned mathematician and astronomer, of whom he
learned those sciences, and became afterwards professor
of them to three successive emperors, viz, Matthias, Ru-
dolphus, and Ferdinand the 2d.

To this sagacious philosopher we owe the first discovery
of the great laws of the planetary motions, viz, that the
planets describe areas that are always proportional to the
times; that they move in elliptical orbits, having the sun
in one focus ; and that the squares of their periodic times,
are proportional to the cubes of their mean distances;
which are now generally known by the name of Kepler's
Laws. As this great man stands as it were at the head of
the modern astronomers, we shall enter at some length
into his history, and chiefly from the words of that cele-
brated mathematician Mr. Maclaurin.

Kepler had a particular passion for finding analogies
and harmonies in nature, after the manner of the Pytha-
goreans and Platonists; and to this disposition we owe
such valuable discoveries, as are more than sufficient to
excuse his conceits. Three things, he tells us, he anxiously
sought to find out the reason of, from Jiis early youlh ;
viz, Why the planets were 6 in number? Why the dimen-
sions of their orbits were such as Copernicus had described
from observations? And what was the analogy or law of
their revolutions? He sought for the reasons of the two
first of these, in the properties of numbers and plane
figures, without success: but at length reflecting, that
while the plane regular figures may be infinite in number,
the regular solids are only five, as Euclid had long ago
demonstrated; he imagined, that certain mysteries in na-
ture might correspond with this remarkable limitation in-
herent in the essencesof things; and the rather, as hi found

K E P

[ 701 ]

K E P

that the Pythagoreans had made great use of those five
regular solids in their philosophy. He therefore endea-
voured to find some relation between the dimensions of
these solids and the intervals of the planetarj' spheres ;
thus, imagining that a cube, inscribed in the sphere of Sa-
turn, would touch by its six planes the sphere of Jupiter;
and that the other four regular solids in like manner
fitted the intervals that are between the spheres of the other
planets : he at length became persuaded that this was the
true reason why the primary planets were precisely six in
number,and that the Author of the World had determined
their distances from the sun, the centre of the system,
from, a regard to this analogy. Being thus possessed, as
he thought, of the grand secret of the Pythagoreans, and
greatly pleased with his discovery, he published it in
1596, under the title of Mysterium Cosmographicum :
and was for some time so charmed with it, that lie said he
would not give up the honour of having invented what
was contained in that book, for the electorate of Saxony.

Kepler sent a copy of this book to Tycho Brahe, who
did not approve of those abstract speculations concerning
the system of the world, but wrote to Kepler, first to lay a
solid foundation in observations, and then, by ascending
from them, to endeavour to come at the causes of things.
Tycho, however, pleased with his genius, was very desirous
of having Kepler with him to assist him in his labours:
and having settled, under the protection of the emperor,
in Bohemia, where he passed the last years of his life, af-
ter having left his native country on account- of some ill
usage, he prevailed upon Kepler to leave the university of
Gratz, and remove into Bohemia, with his family and li-
brary, in the year 1600. But Tycho dying the next year,
the task of arranging the observations devolved upon Ke-
pler, and from that time he had the title of Mathematician
to the Emperor all his life, and gained continually more
and more reputation by his works. The emperor Rudolph
ordered him to finish the tables of Tycho Brahe, which
were to be called the Rudolphine Tables, &c. Kepler ap-
plied diligently to the work : but unhappy are those
learned men who depend on the good-humour of the in-
tendants of the finances; the treasurers were so ill-affect-
ed towards our author, that he could not publish these
tables till l6'27- He died at Ratisbon, in 1(530, where
he was soliciting the payment of the arrears of his
pension.

Kepler made many important discoveries from Tycho's
observations, as well as from his own. He found, that
astronomers had erred, from the first rise of the science, in
ascribing always circular orbits and uniform motions to
the planets; that, on the contrary, each of them moves in
an ellipsis which has one of its foci in the sun: that the
motion of each is really unequable, and varies so, that a
ray supposed to be always drawn from the planet to the
sun describes equal areas in equal times.

It was some years later before he discovered the ana-1
logy existing between the distances of the several planets
from the sun, and the periods in which they complete their
revolutions. He easily saw, that the higher planets not
only moved in greater circuits, but also slower than those
near the sun ; so that, on a double account, their perio-
dic times were greater. Saturn, for example, revolves at
the distance from the sun 9| times greater than the earth's
distance from it ; and the orbit of Saturn is therefore in
the same proportion : but as the earth revolves in one
year, so, if their velocities were equal, Saturn ought to re-

volve in 9 years and a half; whereas the periodic time of
Saturn is about 29 years. The periodic times of the
planets increase, therefore, in a greater proportion than
their distances from the sun: but yet not in so great a
proportion as the squares of those distances; for if that
were the law of the motions, (the square of 9| being 90|,)
the periodic time of Saturn ought to be above QO years.
A mean proportion between that of the distances of the
planets, and that of the squares of those distances, is the
true proportion of the periodic times; as the mean be-
tween 9f and its square 90|, gives the periodic time of
Saturn in years. Kepler, after having committed seve-
ral mistakes in determining this analogy, hit upon it at
last, May 15. l6l8; for he is so particular as to mention
the precise day when he found that " The squares of
the periodic times were always in the same proportion
as the cubes of their mean distances from the sun."

When Kepler saw, according to better observations,
that his disposition of the five regular solids among the
planetary spheres, was not agreeable to the intervals be-
tween their orbits, he endeavoured to discover other
schemes of harmony; for which purpose, he compared the
motions of the same planet at its greatest and least di-
stances, and of the different planets in their several orbits,
as they would appear viewed from the sun ; and here he
fancied that he found a similitude to the divisions of the
octave in music. These were the dreams of this ingenious
man, which he was so fond of, that, hearing of the disco-
very of four new planets (the satellites of Jupiter) by Ga-
lileo, he owns that his first reflections were from a con-
cern how he could save his favourite scheme, which was
threatened by this addition to the number of the planets.
The same attachment led him into a wrong judgment
concerning the sphere of the fixed stars: for being obliged,
by his doctrine, to allow a vast superiority to the sun in
the universe, he restrains the fixed stars within very nar-
row limits : nor did he consider them as suns, placed in
the centres of their several systems, having planets revol-
ving around them; as the other followers of Copernicus
had allowed them to be, from their having light in them-
selves, from their immense distances, and from the ana-
logy of nature. Not contented with these harmonies,
which he had learned from the observations of Tycho, he
gave himself the liberty to imagine several other analogies,
that have no foundation in nature, and are overthrown by
the best observations. Thus from the opinions of Kepler,
though most justly admired, we are taught the danger of
espousing principles, or hypotheses, borrowed from abs-
tract sciences, and of applying them, with such freedom,
to natural enquiries.

A more recent instance of this fondness for discover-
ing analogies between matters of abstract speculation,
and the constitution of nature, we find in Huygens, one
of the greatest geometricians and astronomers any age has
produced: when he had discovered that satellite of Sa-
turn, which from him is still called the Huygenian satel-
lite, this, with our moon, and the four satellites of Jupiter,
completed the number of six secondary planets then dis-
covered in the system ; and because the number of pri-
mary planets was also six, and this number is called by
mathematicians a perfect number (being equal to the sum
of its aliquot parts, 1, 2, 3,) Huygens was hence induced
to believe that the number of the planets was complete,
and that it was in vain to look for any more. This is not
mentioned to lessen the credit of this great man, who

K E P

[ 702 ]

KEP

never perhaps reasoned in such a manner on any other the repelling force, which he supposed to begin at the pe-
occasion ; but only to show, by another instance, how ill rihelion, would cause it to ascend in a figure convex to-
grounded reasonings of this kind have always proved.
For, not long after, the celebrated Cassini discovered four
more satellites about Saturn, not to mention the three
others that have been discovered belonging to that planet
by Dr. Herschel, with another new primary planet and
its 6 satellites, besides the other new planets lately disco-
vered. The same Cassini having found that the analogy,
discovered by Kepler, between the periodic times and the
distances from the centre, takes place in the lesser systems
of Jupiter and Saturn, as well as in the great solar system ;
his observations overturned that groundless analogy which
had been imagined between the number of the planets,
both primary and secondary, and the number six : but
established, at the same time, that harmony in their mo-
tions, which will afterwards appear to flow from one real
principle extended throughout the universe.-

But to return to Kepler ; his great sagacity, and conti-
nual meditations on the planetary motions, suggested to
him some views of the true principles from which these
motions flow. In his preface to the commentaries con-
cerning the planet Mars, he speaks of gravity as of a
power that was mutual between bodies, and tells us, that
the earth and moon tend towards each other, and would
meet in a point, so many times nearer to the earth than to
the moon, as the earth is greater than the moon, if their
motions did not prevent it. He adds also that the tides
arise from the gravity of the waters towards the moon.
But not having notions sufficiently just of the laws of mo-
tion, it seems he was not able to make the best use of
these thoughts; nor does it appear that he adhered to
them steadily, since in his Epitome of Astronomy, pub-,
lished many years after, he proposes a physical account
of the planetary motions, derived from different principles.
He supposes, in that treatise, that the motion of the
sun on his axis, is preserved by some inherent vital prin-
ciple; that a certain virtue, or immaterial image of the
sun, is diffused with his rays into the ambient spaces, and,
revolving with the body of the sun on his axis, takes hold
of the planets, and carries them along with it in the same
direction ; like as a loadstone turned round near a magne-
tic needle, makes it turn round at the same time. The
planet, according to him, by its inertia, endeavours to con-
tinue ih its place, and the action of the sun's image and
this inertia are in a perpetual struggle. He adds, that
this action of the sun, like his light, decreases as the di-
stance increases ; and therefore moves the same planet
with greater celerity when nearer the sun, than at a greater
distance. To account for the planet's approaching to-
wards the sun as it descends from the aphelion to the pe-
rihelion, and receding from the sun while it ascends to
the aphelion again, he supposes that the sun attracts one
part of each planet, and repels the opposite part; and that
the part attracted is turned towards the sun in the descent,
and the other towards the sun in the ascent. By suppo-
sitions of this kind, he. endeavoured to account for all the
other varieties of the celestial motions.

But, now that the laws of motion are better known than
they were in Kepler's time, it is easy to show the fallacy
of every part of this account of the planetary motions.
The planet does not endeavour to stop in consequence of
its inertia, but to persevere in its motion in a right line.
An attractive force makes it descend from the aphelion to
the perihelion in a curve concave towards the sun : but

wards the sun. There will be occasion to show afterwards
from Sir Isaac Newton, how an attraction or gravitation
towards the sun, alone produces the effects, which, accord-
ing to Kepler, required both an attractive and repelling
force; and that the virtuewhich he ascribed to the sun's
image, propagated in to the planetary regions, is unnecessary,
as it could be of no use for this effect, though it were ad-
mitted. For now his own prophecy, with which he con-
cludes his book, is verified; where he tells us, that, " the
discovery of such things was reserved for the succeeding
ages, when the Author of Nature would be-pleased to reveal
these mysteries." — The works of this celebrated author
are many and valuable; as,

1. His Cosmographical Mystery, 1596.

2. Optical Astronomy, in l6(H.

3. Account of a New Star in Sagittarius, 1605.

4. New Astronomy ; or, Celestial Physics, in Commen-
taries on the planet Mars.

5. Dissertations ; with the Nuncius Siderius of Galileo,

1610.

6. New Gauging of Wine Casks, iff 15. Said to be
written on occasion of an erroneous measurement of the
wine at his marriage by the revenue officer.

7. New Ephemerides, from l6l7 to l6'20.

8. Copernican System, three first books of the, l6l8.
g. Harmony of the World ; and 3 books of Comets, 1619.

10. Cosmographical Mystery, 2d edit, with Notes, 1621.

11. Copernican Astronomy; the last 3 books, 1622.

12. Logarithms, 1624; and the Supplement, in 1625.

13. His Astronomical Tables, called the Rudolphine
Tables, in honour of the emperor Rudolphus, his great
and learned patron, in 1627.

14. Epitome of the Copernican Astronomy, 1635.
Besides these, he wrote several pieces on various other

branches, as Chronology, Geometry of Solids, Trigono-
metry, and an excellent treatise of Dioptrics, for that
time. His numerous manuscripts were purchased from
his representatives, by Hevelius, and were probably con-
sumed by a fire at the observatory of the latter, in the
year 1679, which destroyed many of his valuable books
and instruments.

Kepler's Laws, are those laws of the planetary mo-
tions discovered by Kepler. These discoveries in the
mundane system, are commonly accounted two, viz. 1st,
That the planets describe about the sun, areas that are
proportional to the times in which they are described,
namely, by a line connecting the sun and planet; and 2d,
That the squares of the times of revolution, are as the
cubes of the mean distances of the planets from the sun.
Kepler discovered also that the orbits of the planets are
elliptical.

These discoveries of Kepler, however, were only found
out by many trials, in searching among a great number of
astronomical observations and revolutions, what rules and
laws were found to obtain. On the other hand, Newton
has demonstrated, a priori, all these laws, showing that
they must obtain in the mundane system, from the laws of
gravitation and centripetal force; viz, the first of these
laws resulting from a centripetal force urging the planets
towards the sun, and the 2d, from the centripetal force
being in an inverse ratio of the square of the distance. And
the elliptic form of the orbits, from a projectile force re-
gulated by a centripetal one.

KEY

[ 703 ]

K I R

Kepler's Problem, is the determining the true from the
mean anomaly of a planet, or the determining its place,
in its elliptic orbit, answering to any given time; and so
named from the celebrated astronomer Kepler, who first
proposed it. See Anomaly.

The general state of the problem is this: To find the
position of a right line, which, passing through one of the
foci of an ellipsis, shall cut off an area which shall be in-
any given proportion to the whole area of the ellipsis;
which results from this property, that such a line sweeps
areas that are proportional to the times. Many solutions
have been given of this problem, some direct and geome-
trical, others not: viz, by Kepler, Bulliald, Ward, New-
ton, Keill, Machin, &c. See Newton's Princip. lib. 1.
prop. 31, Keill's Astron. Lect, 23, Philos. Trans. Abr.
vol. 8, pa. 177, &c.

In the last of these places, Mr. Machin observes, that
many attempts have been made at different times, but
with no great success, towards the solution of the problem
proposed by Kepler: To divide the area of a semicircle
into given parts, by a line drawn from a given point in the
diameter, in order to find a universal rule for the motion
of a body in an elliptic orbit. For, among the several
methods offered, some are only true in speculation, but
are really of no service; others are not different from his
own, which he judged improper. And as to the rest, they
are all so limited and confined to particular conditions and
circumstances, as still to leave the problem in general un-
touched. To be more particular; it is evident, that all
constructions by mechanical curves are seeming solutions
only, being in reality unapplicable; that the roots of in-
finite series are, on account of their known limitations in
all respects, so far from being sufficient rules, that they
serve for little more than exercises in a method of calcu-
lation. And then, as to the universal method, which pro-
ceeds by a continued correction of the errors of a false
position, it is no method of solution at all in itself; because,
unless there be some antecedent rule or hypothesis to be-
gin the operation (as suppose that of an uniform motion
about the upper focus, for the orbit of a planet; or that
of a motion in a parabola for the perihelion part of the
orbit of a comet, or something of a similar nature,) it
would be impossible to proceed one step in it. But as no
general rule has ever yet been laid down, to assist this me-
thod, so as to make it always operate, it is the same in
effect as if there were no method at all. And accordingly
in experience it is found, that there is no rule now subsist-
ing but what is absolutely useless in the elliptic orbits of
comets ; for in such cases there is no other way to pro-
ceed but that which was used by Kepler: to compute a
table for some part of the orbit, and in it examine if the
time to which the place is required, will fall out any where
in that part. So that, upon the whole, it appears evident,
that this problem, contrary to the received opinion, has
neveryet been advanced one step towards its true solution.

Mr. Machin then proceeds to give his own solution of
this problem, which is particularly necessary in orbits of
a great excentricity ; and he illustrates his method by
examples for the orbits of Venus, of Mercury, of the comet
of the year 1682, and of the great comet of the year 1680,
sufficiently showing the universality of the method.

KEY, in Music, is a certain fundamental note, or tone,
to which the whole piece, be it concerto, sonata, cantata,
&c, is accommodated; and with which it usually begins,
and always ends. — See Clef.

Keys denote also, in an organ, or harpsichord, &c,
the pieces of wood or ivory which are struck by the
fingers, in playing upon the instrument.

Keystone, is the middle voussoir, or the arch stone in
the top, or immediately over the centre of an arch. — The
length of the keystone, or thickness of the archivolt at
top, is allowed by the best architects, to be about the
15th or l6th part of the span.

KILDERKIN, a kind of liquid measure, containing
two firkins, or 18 gallons, beer-measure.

KiNG-piece, or KiNG-post, is a piece of timber set up-
right in the middle, between two principal rafters, and
having struts or braces going from it to the middle of each
rafter.

KIRBY (Joshua), was the eldest of the five sons of
Mr. John Kirby ; who was originally a schoolmaster at
Orford ; but afterwards took an actual survey of the
county of Suffolk in the year 1732, &c, and published a
small map of it; as also the Suffolk Traveller, 1735,
1 2mo ; a new and enlarged edition of which was published
by subscription in 1764, in which the name of " Joshua
Kirby, Esq. designer in perspective to his majesty," occurs
for 50 copies. Mr. John Kirby died at Ipswich, Dec. 13,
1753, aged 63.

Our author Joshua was born in 17 16, at Parham, near
Wickham-Market; and settled as a house-painter at
Ipswich about the year 1738. He always had a genius
for drawing, and published 12 prints of castles, ancient
churches and monuments, in Suffolk, with a small
pamphlet, containing further illustrations of them. But
the study which led him to eminence, was that of the art
of perspective, in his improvement of which it may almost
be said that he invented a new art ; so simple was his me-
thod, in comparison "with the systems at that time in use.
He had made a considerable progress in his treatise, when
he met with Dr. Brook Taylor's book, which contributed
to the perfecting of those rules, by which he rendered this
art so easy, that on the publication of his work he was re-
quested by the Society of Artists to read lectures ; for
which he received the thanks of its members, in the pub-
lic papers.

Having obtained the friendship and intimacy of Sir
Joshua Reynolds, Mr. Hogarth, and other eminent artists,
he quitted Ipswich, and removed to London; where he
was patronized by the earl of Bute, who introduced him to
his present majesty, when prince of Wales, and was ap-
pointed clerk of the works at Kew. By his majesty's
patronage he published, in 176 1, his splendid work, The
Perspective of Architecture, in 2 vols, folio; being a mas-
terly performance, and the elegant designs with which it is
illustrated reflect honour on the. artists of our country. —
In 1766, in conjunction with his brother William, then of
Witnisham, in the county of Suffolk, attorney at law (who
died Sept. 25, 1791, aged 72), he published an improved
edition of their father's map of Suffolk, on a larger scale.
Mr. K. was a member both of the Royal and Antiquarian
Societies; and, at the first forming of the Royal Academ,y,
he was president of the Society for Artists, from which that
institution emanated. In the year 1768 he published a
3d edition of his Treatise on Perspective. Mr. K. died
June 20, 1774, aged 58, and was buried in Kew church-
yard, leaving his widow, who died the year after, and a
daughter, the celebrated Mrs. Trimmer, of Old Brentford,
esteemed for numerous works for the instruction of youth.
KIRCH (Christian Frederic), of Berlin, a ceh>

KNO | 7C

bvatecl astronomer, was born at Guben in l6~94. He ac-
quired great reputation in the observatories of Dantzic and
Berlin, Godfrey Kirch his father, and Mary his mother,
also acquired considerable reputation by their astronomi-
cal observations. This family corresponded with all the
learned societies of Europe, and their astronomical works
are in great repute.

KIRCHLR (Athanasius), a noted philosopher and
mathematician, was born at Fulde in 1601. He entered
into the society of the Jesuits in 16 IS, and taught philo-
sophy, mathematics, the Hebrew and Syriac languages, in
the university of Wirtsburg, with great applause, till the
year 1631; when he retired to France, on account of the
ravages committed by the Swedes in Franconia, and lived
some time at Avignon. He was afterwards called to Rome,
where he taught mathematics in the Roman college, col-
lected a rich cabinet of machines and antiquities, and died
in l6SO, in the 80th year of his age.

The quantity of his works is immense, amounting to 22
volumes in folio, 1 1 in quarto, and three in octavo;
enough to employ a man for a great part of his life even to
transcribe them. Most of them are rather curious than
useful; many of them visionary and fanciful; and it is
not to be wondered at if they are not always accompanied
with the greatest exactness and precision. The principal
of them are,

1. Prslusiones Magneticae.

2. Primitias Gnomonicse Catoptricae,

3. Ars magna Lucis et Umbra;.

4. Musurgia Universalis.

5. Obeliscus Pamphilius.

6". Oedipus iEgyptiacus; 4 volumes folio.

7. Itinerarium Extaticum.

8. Obeliscus iEgyptiacus; 4 volumes folio.

9. Mundus Subterraueus.
10. China Illustrata.

KNOT, a tye, or complication of a rope, cord, or string,
or of the ends of two together. There are divers sorts of
knots used for different purposes, which may be explained
by showing the figures of them open, or undrawn, thus.
1. Fig. 1, plate 17, is a Thumb knot. This is the simplest
of all. It is used to tye at the end of a rope, to prevent its
opening out: it is also used by taylors, &c, at the end of
their thread.

Fig. 2, a Loop knot. Used to join pieces of rope, &c,
together. — Fig. 3, a Draw knot, which is the same as the
last ; only one end or both return the same way back, as
abed. By drawing at a, the part bed comes through,
and the knot is loosed. — Fig. 4, a Ring knot. This serves
also to join pieces of cord, &c, together. — Fig. 5 is another
knot for tying cords together. This is used when any
cord is often to be loosed. — Fig.6", a Running knot, to draw
any thing close. By pulling at the end a, the cord is drawn

i I K O E

through the loop b, and the part c d is drawn close about a
beam, &c. — Fig. 7 is another knot, to tye any thing to a
post. And here the end may be put through as often as
you please. — Fig. 8, a Very small knot. A thumb knot is
first made at the end of each piece, and then the end of
the other is passed through it. Thus, the cord a c runs
through the loop d, and b d through c ; and then drawn
close by pulling at a and b. If the ends e and /be drawn,
the knot will be loosed again. — Fig. Q, a Fisher's knot, or
Water knot. This is the same as the 4th, only the ends are
to be put twice through the ring-, which in the former was
but once; and then drawn close. — Fig. 10, a Meshing knot,
for nets ; and is to be drawn close. — Fig. 11, a Barber's
knot, or a knot for cawls of wigs ; and is to be drawn close.
— Fig. 12, a Bowline knot. When this is drawn close, it
makes a loop that will not slip, as fig. 7 ; and serves to
hitch over any thing. — Fig. 13, a Wale knot, which is made
with the three strands of a rope, so that it cannot slip.
When the rope is put through a hole, this knot keeps it
from slipping through. When the three strands are wrought
round once or twice more, after the same manner, it is
called crowning. By this means the knot is made larger
and stronger. A thumb knot, No. 1, may be applied to
the same use as this.

Knots mean also the divisions of the log line, used at
sea. These are usually 7 fathoms, or 42 feet asunder;
but should be 8| fathoms, or 50 feet. And then, as
many knots as the log line runs out in half a minute, so
many miles does the ship sail in an hour; supposing her
to keep going at an equal rate, and allowing for yaws, lee-
way, &c.

KOENIG (Samuel), a learned philosopher and ma-
thematician, was a Swiss by birth, and came early into
eminence by his mathematical abilities. He was professor
of philosophy and natural law at Franeker, and afterwards
at the Hague, where he became also librarian to the
stadtholder, and to the princess of Orange; and where he
died in 1757.

The Academy of Berlin enrolled him among her mem-
bers; but afterwards expelled him on the following occa-
sion. Maupertuis, the president, had inserted in the vo-
lume of the Memoirs for 1746, a discourse on the Laws
of Motion ; which Koenig not only attacked, but also attri-
buted the memoir to Leibnitz. Maupertuis, stung with
the imputation of plagiarism, engaged theAcademy of Ber-
lin to call upon him for his proof; which Koenig failing to
produce, he was struck out of the academy. All Europe
was interested in the quarrel which this occasioned between
Koenig and Maupertuis. The former appealed to the pub-
lic; and his appeal, written with the animation of resent-
ment, procured him many friends. He was author of some
other works, and had the character of being one of the
best mathematicians of the age.

LAC

[ 705 ]

LAC

L.

T ABEL, a long thin brass ruler, with a small sight at
J-i one end, and a central hole at the other; commonly
used with a tangent-line on the edge of a circumftrentor, to
take altitudes, and other angles.

LACAILLE (Nicholas Lewis), an eminent French
mathematician and astronomer, was born at Rumiguy in
the diocese of Rheims, in 1713. His father having quitted
the army, in which he had served, amused himself in his
retirement with studying mathematics and mechanics, in
which he proved the happy author of several inventions of
considerable use to the public. From this example of his
father, our author almost in his infancy took a fancy to
mechanics, and at school discovered very early tokens of
genius. In 1729 he came to Paris, where he studied the
classics, philosophy, and mathematics. He afterwards
studied divinity in the college de Navarre, with the view
of embracing the ecclesiastical life: however he never en-
tered into priest's orders, apprehending that his astrono-
mical studies, to which he had become much devoted,
might too much interfere with his religious duties. His
attachment to astronomy soon connected him with the
celebrated Cassini, who procured him an apartment in
the observatory ; where, assisted by the counsels of this
celebrated man, he soon acquired a name among the as-
tronomers. In 1739 he was engaged with M. Cassini de
Thury, son to M. Cassini, in verifying the meridian through
the whole extent of France: and in the same year he was
named professor of mathematics in the college of Maza-
rine. In 1741 our author was admitted into the Academy
of Sciences as an adjoint member for astronomy ; and
had many excellent papers inserted in their Memoirs; be-
side which he published several useful treatises, viz, Ele-
ments of Geometry, Astronomy, Mechanics, and Optics.
He also carefully computed all the eclipses of the sun and
moon that had happened since the Christian aera, which
were printed in the work entitled l'Art de verifier les
Dates, &c, Paris 1750, in 4to. Healsocompiled a volume
of astronomical ephemerides for the years 1745 to 1755;
another for the years 1755 to 1765; and a third for the
years 1765 to 1775 : as also the most correctsolar tables of
any; and an excellent work en ti tied Astronomiajfundamenta
novissimis solis et stellarum observationibus stabilita.

, Having gone through a series of seven years' astrono-
mical observations in his own observatory in the Mazarine
college, lie formed the project of going to observe the
southern stars at the Cape of Good Hope: being coun-
tenanced by the French court, he. set out upon this expe-
dition in 1750, and in two years he observed the places of
about 10 thousand stars in the southern hemisphere that
are not visible in our latitudes, as well as many other im-
portant elements, viz, the parallaxes of the sun, moon, and
some of the planets, the obliquity of the ecliptic, the re-
fractions, &cc. Having thus executed the purpose of his
voyage, and no present opportunity offering for his return,
lie thought of employing the vacant time in another ar-
duous attempt, namely that of taking the measure of the
earth, as he had already done that of the heavens. This
indeed had been done before by many learned men both in
Europe and America; some determining the quantity of

Vol. I.

a degree at the equator, and others at the arctic circle ;
but it had not yet been decided whether in the southern
parallels of latitude the same dimensions obtained as in the
northern. His labours were rewarded with the satisfac-
tion he wished for; having determined a distance of
41 OS 14 feet from a place called Klip-Fontyn to the Cape,
by means of a base of 3SS02 feet, three times actually
measured : he discovered a new secret of nature, namely,
that the radii of the parallels in south latitude are not of
the same length as those of the corresponding parallels in
north latitude. About the 23d degree of south latitude
he found a degree on the meridian to contain 342222 Paris
feet. Agreeably to the orders from the' court of Ver-
sailles, he fixed the situation of the Isles of France and of
Bourbon. While at the Cape, he observed a wonderful
effect of the atmosphere in some states of it : though the
sky at the Cape be" generally pure and serene, yet when
the south-east wind blows, which is pretty often, it is
attended with some strange and even terrible effects: the
stars look larger, and appear to dance ; the moon has an
undulating tremor; and the planets have asort of beard
like comets.

M. Lacaille returned to France in the autumn of 1754,
after an absence of 4 years ; loaded, not with the spoils of
the east, but with those of the southern heavens, before
then almost unknown to astronomers. On his return, he
first drew up a reply to some strictures which the cele-
brated Euler had published relative to the meridian: after
which he settled the results of the comparison of his ob-
servations for the parallaxes, with those of other astrono-
mers : that of the sun he fixed at 9|" ; of the moon at 56'
56"; of Mars in his opposition, 36"; of Venus 38". He
also settled the laws by which astronomical refractions
are varied by the different density or rarity of the air, by
heat or cold, and by dryness or moisture. And lastly he
showed an easy and practicable method of finding the lon-
gitude at sea by means of the moon. His fame being now
generally established, he was elected a member of most of
the academies and societies of Europe, as London^Bologna,
Petersburg, Berlin, Stockholm, and Gottingen.

In 176O our author was attacked with a severe fit of the
gout; which however did not interrupt the course of his
studies; for he then planned a new and large work, in-
tended as a history of astronomy through all ages, with a
comparison of the ancient and modern observations, and
the construction and use of the instruments employed in
making them. Towards the latter part of 176l, his con-
stitution became greatly reduced ; though his mind con-
tinued unaffected, and he resolutely persisted in his studies
to the last ; death only putting an end to his labours the
21st of March 1762, at 49 years of age; after having com-
mitted his manuscripts to the care and discretion of his
esteemed friend M. Maraldi.

Besides the publications before mentioned, and perhaps
some others also, he had a vast number of papers inserted
in the volumes of the Memoirs of the French Academy of
Sciences, much too numerous indeed, though very im-
portant, to be here all mentioned particularly; suffice it
therefore just to distinguish the years of those volumes in
4 X

LAG

[ 706 ]

LAG

which his pieces are to be found, by the following list of
them, viz, 1741, 1742, 1743, 1744, 1745, 1746, 1747,
1748, 1749, 1750, 1751, 1752, 1753, 1754, 1755, 1756",
1757, 175S, 1759, 1760, I/61, 1763; in all or most
of which years there are two or three or more of his
papers.

LACERTA, Lizard, one of the new constellations of the
northern hemisphere, added by Hevelius to the 48 old
ones, near Cepheus and Cassiopeia. This constellation
contains, in Hevelius's catalogue 10 stars, and in Flam-
steed's 16.

LACUNAR, an arched roof or cieling; more especially
the planking or flooring above the porticos.

LADY-Dny, the 25th of March, being the Annunciation
of the Holy Virgin.

LAGNY (Thomas Fantet de), an eminent French
mathematician, was born at Lyons. Fournier's Euclid,
and Pellt-tier's Algebra, by chance falling in his way, de-
veloped his genius for the mathematics. It was in vain
that his father designed him for the law; he went to Paris
to deliver himself wholly up to the study of his favourite
science; and in 1697, the Abbe Bignon, protector-general
of letters, got him appointed professor-royal of hydrogra-
phy at Rochefort. Soon after, the duke of Orleans, then
regent of France, fixed him at Paris, and made him sub-
director of the general bank, in which he lost the greatest
part of his fortune in the failure of that establishment.
He had been received into the ancient academy in 1696 ;
on the renewal of which he was named associate-geo-
metrician in l699, and pensioner in 1723. After a life
spent in close application, he died, April 12, 1734.

De Lagny particularly excelled in arithmetic, algebra,
and geometry, in which he made many improvements and
discoveries. He, as well as Leibnitz, invented a binary
arithmetic, in which only two figures are concerned.
He rendered much easier the resolution of algebraic equa-
tions, especially the irreducible case in cubic equations;
and the numeral resolution of the higher powers, by
means of short approximating theorenu. — He delivered
the measures of angles in a new science, called Goniome-
try; in which he measured angles by a pair of compasses,
without scales, or tables, to great exactness; and thus
cave a new appearance to trigonometry. — Cyclometry, or
the measure of the circle, was also an object of his atten-
tion ; and he calculated, by means of infinite series, the
ratio of the circumference of a circle to its diameter, to
120 places of figures. — He gave a general theorem for
the tangents of multiple arcs : with many other cu-
rious or useful improvements, which are found in the
great multitude of his papers, that are printed in the
different volumes of the Memoirs of the Academy of
Sciences, viz, in almost every volume, from the year
1699 to 1729.

LAGRANGE (Joseph Louis), a most illustrious ma-
thematician and philosopher, was born at Turin, Nov. 25,
1736, and died at Paris, April 1813. His father was
treasurer of war; but falling into poverty through unfor-
tunate speculations, induced our author to endeavour to
render himself distinguished in literature. His taste how-
ever for mathematics did not appear early. He was pas-
sionately devoted to Cicero and Virgil, before he could
relish Archimedes and Newton. He soon however be-
came an enthusiastic admirer of the geometry of the an-
cients, which he at first preferred to the modern analysis.
But a memoir which the celebrated Halley had composed

to demonstrate the superiority of the analytic method, had
the effect of teaching him his true path to glory. He de-
voted himself to this new study with the same success that
he had had to synthesis; and his application was so de-
cided, that at the age of 16, he became professor of ma-
thematics in the royal military school at Turin.

The youth of the professor however had no effect in
hindering the pupils, all older than himself, from profiting
by his lessons. Some of them he distinguished as his'
friends ; and this early association laid the- foundation of
an important institution, the Academy of Turin, which
published, in 1759, a first volume, under the title of Actes
de la Societe Privee. In those acts it appears that the
young Lagrange directed the philosophical researches of
Cigna the anatomist, and the labours of the Chevalier de
Saluces. He furnished to Foncenex the analytical part of
his memoirs, leaving to him the task of developing the rea-
soning upon which the formulas depended. But Lagrange,
while he abandoned insulated theorems, published at the
same time, under his own name, theories which he promised
to develope further. Thus, alter giving new formulas of
maxima and minima, in all cases, after having shown the
insufficiency of the known methods, he announced that he
would treat this subject, which he considered as important,
in a work which he was preparing, in which would be de-
duced, from the same principles, all the mechanical pro-
perties of bodies, whether solid or fluid, thereby laying at
the age of 23, the foundation of those great works which
constitute the admiration of philosophers.

Newton had undertaken to submit the motions of fluids
to calculation, and had made researches on the propaga-
tion of sound ; but his principles were insufficient, and his
suppositions sometimes erroneous. Lagrange demonstrated
this, and founded his researches on the known laws of
dynamics, by considering only in the air the particles
which are in a straight line, thereby reducing the problem
to that of vibrating chords. He demonstrated, that, what-
ever figure is given to the chord, the duration of the oscil-
lations is always the same: a truth derived from experi-
ment, but which Dalembert considered as very difficult,
if not impossible, to demonstrate. He then passed to the
propagation of sound, treated of simple and compound
echoes, of the mixture of sounds, of the possibility of their
spreading in the same space without interfering with each
other, and demonstrated rigorously the generation of har-
monious sounds

Euler saw the merit of the new method, and adopted it
as an object of his profoundest meditations; but Dalem-
bert did not yield the point in dispute. In his private
letters, as well as in his printed memoirs, he proposed nu-
merous objections, to which Lagrange afterwards answer-
ed. The first notice of Euler was to make Lagrange an
associate of the Berlin academy. When he announced to
him this nomination on the 20th of October, 1759, he
said, " Your solution of the problem of isoperi metre's
leaves nothing to desire; and I am happy that this subject,
with which I have been almost solely occupied since my
first attempts, has been carried by you to the highest degree
of perfection. The importance of the point has induced
me to draw up, with your assistance, wn analytical solu-
tion of it. But I shall not publish it till you have pub-
lished the sequel of your researches, that I may not de-
prive you of any part of the glory which is your due."

Dalembert considered it impossible to subject to calcu-
lation the motions of a fluid inclosed in a vessel, unless

LAG

[ 707 ]

L A «

this vessel had a aertain figure. Lagrange demonstrated
the contrary; except in the case when the fluid divides
itself into different masses. But even then we may deter-
mine the places where the fluid divides itself into different
portions, and determine the motion of each as if it were
alone. Dalembert thought, that in a fluid mass, such as
the earth may have been at its origin, it was not necessary
for the different beds to be on a level; butLagrange shows,
that the equations of Dalembert are themselves equations
of beds on a level. In combating Dalembert with all the
respect due to a mathematician of his rank, he often em-
ployed very beautiful theorems, for which he had been in-
debted to his adversary. Dalembert, on his side, added
to the delicacy of Lagrange. " Your problem appeared
to me so beautiful," says he in a letter to Lagrange, " that
I have sought for another solution of it. I have found a
simpler method of arriving at your elegant formula."

The Academy of Sciences of Paris proposed, at this
time (176'4), as the subject of a prize, the theory of the
libration of the moon, i. e. they demanded the cause why
the moon, in revolving round the earth, always turns the
same face to it, some variations excepted, observed by
astronomers, and of which Cassini had first explained the
phenomenon. The point was to calculate all the pheno-
mena, and to deduce them from the principle of universal
gravitation. Such a subject was an appeal to the genius
of Lagrange, an opportunity furnished to apply his analy-
tical principles and discoveries. The hope of Dalembert
was not disappointed, and the memoir of Lagrange is one
of his finest pieces. We see in it the germ of his Meca-
nique Analvtique. Dalembert wrote to him : " I have
read with as much pleasure as advantage, your excellent
paper on the moon's librations, so worthy of the prize
which it has obtained."

This success encouraged the Academy to propose, as
a prize, the theory of the satellites of Jupiter. Euler,
Clairaut, and Dalembert, employed themselves about the
problem of these bodies on the principle of the movements
of the moon. Bailly applied the theory of Clairaut to the
problem of ihe satellites, and it led him to some interesting
results. But his theory was insufficient; the earth has
only one moon, while Jupiter has four, which continue to
act upon each other, and vary their positions in their re-
volutions. The problem was one of six bodies : but La-
grange attacked the difficulty and overcame it, demon-
strating the cause of the inequalities observed by astrono-
mers, and pointing out some others too trivial to be as-
certained by observations. The shortness of the time
allowed, and the multiplicity of the calculations, analy-
tical and numerical, did not permit him to exhaust the
subject entirely in a first memoir. He was sensible of this
himself, and promised further results, which his other
labours always prevented him from completing; but
twenty-four years after, M. Laplace took up that difficult
theory, and completed it.

About the same time a problem of a different kind
drew the attention of M. Lagrange. Fermat, a great ma-
thematician, had left some remarkable theorems respect-
ing the properties of numbers, which it seems he discovered
by induction. He promised the demonstrations of them,
but at his death no trace of them could be found. Many
mathematicians employed themselves on the theorems of
Fermat; but none were successful. Euler alone had pe-
netrated into that difficult path ; but Lagrange, in demon-
strating or rectifying some opinions of Euler, resolved a

problem which gave him a key to all the others; and from
which he deduced the complete solution of equations of
the second degree, with two indeterminates which must
be whole numbers.

His residence at Turin was not agreeable to him. He
saw no person who cultivated the mathematics with suc-
cess. He was impatient to visit Paris. M. de Caraccioli,
with whom he lived in the greatest intimacy, was appointed
ambassador to London, and was to pass through Paris on
his way, where he intended to spend some time. He pro-
posed this journey to M.Lagrange, who accepted it with
joy, and was received at Paris, as he had a right to ex-
pect, by Dalembert, Clairaut, Condorcet, Fontaine, Nol-
let, Marie, and the other mathematicians. Falling dan-
gerously ill after a dinner, in the Italian style, given by
Nollet, he was not able to accompany his friend to Lon-
don, who was obliged to leave him in a furnished lodging
under the care of an agent.

This incident changed his projects, and he thought of
returning to Turin, when he understood that the academy
of Berlin was threatened with the loss of Euler, who in-
tended to return to Petersburg. Dalembert suggested
the idea of putting Lagrange in the place of Euler, and
Euler himself pointed out Lagrange as the only man ca-
pable of filling his place. Lagrange was therefore ap-
pointed, and he received a pension of 1500 Prussian
crowns, about 250/. with the title of director of the aca-
demy in regard to the physico-mathematical sciences.
Euler and Lagrange, in the place of Maupertuis, received
only half his salary, but Frederick had no respect for the
sciences, though he considered himself obliged to protect
them as a king.

Lagrange took possession of his new situation in 1765.
He was well received by the king; but he soon perceived
that the Germans do not like foreigners to occupy situa-
tions in their country. He applied to the study of their
language, and devoted himself to mathematics. " The
king," said he, " treated me well; I thought that he pre-
ferred me to Euler, who was something of a devotee, while
I took no part in the disputes about worship ; and did not
contradict the opinion of any one."

It will not be expected that I should follow Lagrange
in the important researches with which he filled the Ber-
lin Memoirs; and even some volumes of the Memoirs of
the Turin Academy, which was indebted to him for its
existence. All the space that could be devoted to this
biography, would be insufficient to give even an imperfect
idea of his labours. Some of these memoirs are of such
extent and importance, that they might pass for great se-
parate works, yet they constituted a part only of what
these 20 years enabled him to produce. He now com-
posed his Mecanique Analytique, and he was anxious to
have it printed at Paris ; but he experienced great diffi-
culty in finding a bookseller who would undertake to pub-
lish a work which, the more sublime the theory, thefewe'r
readers would be found capable of appreciating. At
length one was found, and the work was ably edited by
M. Legend re, who was repaid by the esteem and the gra-
titude of the author. The death of Frederick occasioned
great changes in Prussia, and still greater were to be ap-
prehended. Philosophers were no longer respected as
heretofore, and these causes, with the approaching publi-
cation of the Mecanique Analytique, were sufficient to
bring him back to Paris.

The successor of Frederick, though he did not much
4X2

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interest himself in the sciences, made some difficulty in
allowing a philosopher to depart, whom his predecessor
had invited, and whom he had honoured with his particu-
lar esteem. After some delay, M. Lagrange obtained
liberty to depart, but it was stipulated that he should
still give some memoirs to the Berlin academy, and the
volumes of 1792, 1793, and 1803, show that he kept his
promise.

In 1787 then, Lagrange returned to Paris, to take his
seat in the Academy of Sciences, of which he had been a
foreign member for 15 years. To give him the right of
voting in all their deliberations, this title was changed into
that of Veteran Pensionary. The queen treated him with
regard, and considered him as a German. He had been
recommended to her from Vienna; and he obtained, in
consequence, a lodging in the Louvre, wheje he lived
happy till the revolution.

The satisfaction he enjoyed in his new residence, did
not however appear outwardly. He was affable and kind
when interrogated, but he spoke little, and appeared ab-
sent and melancholy. In companies which must have
been suited to his taste, among the most distinguished men
of all countries who met at the house of the illustrious
Lavoisier, he was seen musing, with his head reclined
against a window, where nothing seemed to attract his
attention. He acknowledged, himself, that his enthusiasm
was gone, that he had lost even his taste for mathematics.
When informed that any mathematician was employed on
a particular object, " so much the better," he wouldvsay,
" I had begun it, it will now be unnecessary for me to
finish it." But he had merely changed the object of his
studies. Metaphysics, the history of human nature, of
different religions, the general theory of languages, medi-
cine, and botany, divided his leisure hours. Surrounded
by chemists who were reforming the theory and language
of the science, he made himself acquainted with their dis-
coveries, which gave to facts formerly isolated, that con-
nexion which distinguishes the different parts of mathe-
matics. In this philosophical repose he passed his time
till the revolution, without adding anything to his mathe-
matical discoveries.

The revolution gave philosophers the power of making
a great and difficult innovation in the establishment of a
system of weights and measures, founded on nature, and
perfectly analogous to the decimal scale of numbers. La-
grange was one of the commissioners whom the Academy
charged with the execution of that task, and he was one
of its most ardent promoters, for he wished to see the de-
cimal system adopted in all its simplicity.

When the academy was suppressed, the commission
charged with the establishment of the new system was in-
terrupted for a time. In order to purify it, the names of
Lavoisier, Borda, Laplace, Coulomb, Brisson, and De-
lambre, were struck out. Lagrange was retained, as pre-
sident ; though he could not help expressing his regret at
the dismission of so many of his associates ; but unless the
suppression had been total, it could scarcely have ex-
tended to him. Besides, he was known to be wholly de-
voted to the sciences ; he had no place either in the civil
department or the administration ; and the moderation of
his character had prevented him from expressing what he
could not but lliink in secret. The horrors, however,
attending the revolution, made a strong impression on his
sensible mind. In a conversation with M. Delambre, the

trary to every thing like justice, had thrown all lovers of the
sciences into mourning, by cutting oft" the most illustrious
philosopher in Europe, Lavoisier: " It has cost them but a
moment," said he, " to cut off that head; but a hundred
years perhaps will not be sufficient to produce another
like it." Some months before they had had a similar con-
versation in the cabinet of Lavoisier, on account of the
death of the unfortunate Bailly. He lamented the dread-
ful consequences of the dangerous experiment which the
French had attempted. The projection of their plans of
amelioration appeared to him no certain proofs of great-
ness in the human mind : " If you wish tosee it truly
great," added he, " enter into the cabinet of Newton,
employed in decomposing light, or in explaining the sys-
tem of the universe."

For some time he regretted that he had not listened to
the advice of his friends, who at the commencement of the
troubles had recommended him to seek an asylum, which
it would have been so easy to find. As long as the revo-
lution seemed only to threaten the pension which he en-
joyed in France, he had neglected that consideration,
from a feeling of curiosity to be upon the spot of one of
those great convulsions which it is always more prudent to
observe at a distance. " It was your own choice," said
he several times to himself. It was to no purpose that a
special decree of the Constituent Assembly had ensured
the payment of his pension, because the depreciation of
the paper currency rendered it illusory. He had been
named member of the board of consultation, appointed to
examine and reward useful inventions, and he had been
appointed one of the administrators of the Mint. This
commission offered him few objects to fix his attention,
and in no degree removed his apprehensions. It was again
proposed to draw him to Berlin, and to restore him to his
former situation, and he agreed to the proposal. Herault
de Sechelles, to whom he applied for a passport, offered
him, for greater security, a mission to Prussia; but Ma-
dame Lagrange, his wife, would not consent to quit her
country; a repugnance which at that time he considered
as a misfortune, though it proved a source of fortune and
new glory.

The Norman school, of which he was named professor,
but which had only an ephemeral existence, scarcely gave
him time to explain his ideas respecting the foundation of
arithmetic and algebra, and their application to geometry.
The Polytechnic school, a happier idea, had a more du-
rable success : and, among the best effects which it pro-
duced, we may place that of having restored Lagrange to
analysis. There he had an opportunity of developing those
ideas, the germ of which was to be found in two memoirs
published in 1772, and the object of which was to explain
the true metaphysics of the differential and integral calcu-
lus. He composed his Analytical Functions, and his Lec-
tures on that Calculus, of which he published several edi-
tions. He likewise published his Treatise on the Nume-
rical Solution of Equations, with notes on several points
of the theory of algebraic equations.

The desire of multiplying useful applications induced
him to undertake a new edition of the Mecanique Analy-
tique. His project was to'dcvelope the most useful parts
of it. He laboured on it with all the ardour and intellec-
tual power which he could have applied at any period of
his life. But this application occasioned a degree of
fatigue which threw him into a fainting fit, and he was

day after the atrocious and absurd sentence which, con- found in that statu by Madame Lagrange. Ilis head iu

LAG

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LAG

falling had struck against the corner of a table, and the
shock had not restored him to his senses. This was a
warning to take more care of himself, and he thought so
at first ; but he was too anxious to finish his work, the
printing of which was not completed at the period of his
decease. The first volume had appeared some time before
his death. It had been followed by a new edition of his
Fonctions Analytiques. So much labour exhausted him.
Towards the end of March (1813) a fever came on, he lost
his appetite, his sleep was uneasy, and his waking was ac-
companied by alarming swoonings. He perceived his
danger ; but, preserving his serenity, he studied what
passed within him, and, as if he were assisting at a great
and uncommon experiment, he bestowed all his atteution
on it. Friendship conducted to his house, on the 8th of
April, MM. Lacepede, Monge, and Chaptal, who wrote
down the principal points of a conversation which was his
last ; and the following passages, within inverted commas,
are copied from the manuscript of M. Chaptal.

" He received them with tenderness and cordiality. I
was very ill, my friends (said he), the day before yester-
day ; I perceived myself dying, my body became weaker,
my mental and physical powers were gradually declining;
I' observed with pleasure the gradual diminution of my
strength ; and I arrived at the point without pain, without
regret, and by a very gentle declivity. Death is not to
be feared, and when it comes without violence, it is a last
function which is neither painful nor disagreeable." He
then explained to them his ideas respecting life, the seat of
which he considered as spread over the whole body, in
every organ, and all parts of the machine, which in his
case became equally feebler in every part by the same de-
gree. " A little longer, and there would have been no
functions, death would have overspread the whole body,
for death is merely the absolute repose of the body ; I
wished to die," added he with greater force, " I found a
pleasure in it ; but my wife did not wish it. I should
have preferred at that time a wife less kind, less eager to
restore in\ strength, and who would have allowed me
gently to have finished my career. I have performed my
task, I have acquired some celebrity in the mathematics,
I have hated nobody, I have done no ill ; it is now proper
to finish."

As he exerted himself too much, especially in these last
words, his friends, notwithstanding the interest with which
they had listened to him, proposed to retire; but he re-
tained them, and began to relate to them the history of his
life, his labours, his success, his residence at Berlin, (where
he had often told us what he had seen near a kin»,) of his
arrival at Paris, the tranquillity he had enjoyed at first,
the anxiety which he suffered during the revolution, and
how he had been finally rewarded by a powerful monarch,
capable of appreciating his worth, who had loaded him
with honours and dignities, and who had even lately sent
him the urand ribbon of the Imperial Or ler of Re-Union.
And, it may be added, who, after having given him during
his life the most unequivocal proofs of the highest esteem,
has since, says M. Delambre, done more for his widow
and his brother, than Frederick had done for him all the
time he was director of his academy.

He had been ambitious neither ot riches nor honour; but
he had received both with respectful gratitude, and he re-
joiced at the acquisition for the glory of the sciences. He
had not lost all hope of cure; he believed only that his
convalescence would be long. He offered, when he reco-

vered his strength, to go and dine at M. Lacepede's coun-
try-house, with MM. Monge and Chaptal, and proposed
to give them details respecting his life, which could no
where else be found. During this conversation, which
lasted more than two hours, his memory often failed him;
he made vain efforts to recover names and dates, but his
discourse was always connected, full of strong thoughts
and bold expressions. This exercise of his faculties wasted
the whole remains of his strength. Scarcely had his
friends left him, when he fell into a fainting fit, and he
died two days after, on the 10th of April, 1813.

Lagrange was of a delicate but good constitution. His
tranquillity, his moderation, and an austere and frugal
regimen, from which he rarely deviated, prolonged his
life to the age of 76 years, 4 months, and 15 days. He
was twice married : first at Berlin, for the purpose of
being on a footing with the rest of the academicians, none
of whom were bachelors. He afterwards married, in
France, Mademoiselle Lemonnier, daughter of the cele-
brated astronomer of that name. The countess Lagrange,
daughter, grand-daughter, and niece,, of members of the
Academy of Sciences, was deserving of the name which
he gave her. This advantage, in her eyes, made up for
the difference of their ages, and she soon felt for him the
tenderest regard. He was so grateful that he could
scarcely bear to be separated from her, and it was on
her account alone that he felt anv regret at relinquishing
life, ' S

Though he had a venerable figure, indicating his excel-
lent characteristics, he would never allow his portrait to
be drawn. More than once, by a very excusable piece of
address, persons have been introduced during the meeting
of the Institute to take a sketch of him without his know-
ledge.— Gentle and even timid in conversation, he took a
pleasure in asking questions, either to draw out others, or
to add their reflections to his own vast knowledge. When
he spoke, it was always in a tone of doubt, and his first
words usually were, "1 do not know." He respected the
opinions of others, and was very far from laying down his
own as a rule. Yet it was not easy to make him change
them. Sometimes he even defended them with a degree
of heat which continued to increase till he was sensible of
some alteration in himself; he then immediately resumed
his usual tranquillity.

Among the master-pieces which the world owes to his
genius, his Mecanique is the most remarkable and the-
most important. His Fonctions Analytiques hold only
the second place, notwithstanding the capability of the
principal idea, and the beauty of the developements. A no-
tation less commodious, and calculations more embarrass-
ing, though more luminous, will prevent mathematicians
from employing, except in difficult and doubtful cases, his
symbols and names. It is sufficient that he has proved
the legitimacy of the more expeditious processes of the
differential and integral calculus, and he has himself fol-
lowed the ordinary notation in the second edition of his
Mecanique. This great work is entirely founded on the
calculus of variations, of which he was the inventor. The
whole flows from a single formula, and from a principle
known before his time; but the utility of it was far from
being suspected. This sublime composition includes all
his preceding labours which could be connected with it.
It is distinguished likewise by the philosophical spirit
whifch reigns from one end of it to the other. The geo-
metrical law of the celestial motions are deduced from

L A L

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simple mechanical and analytical considerations. From sance des Temps, from the year 7 to 12, or 1799 to 1804,
those problems, which serve to analyse the true system of contains his new catalogue of more than 12,000 stars

the world, the author passes to questions more difficult
and complicated, which show the extent of his resources.
We see there his new theory of the variations of arbitrary
constant quantities in the motion of the planets, which so
much distinguished the Memoirs of the Institute, where it
proved that the author, at the age of 75, had not sunk
from the rank which he had filled during so long a period
in the opinion of all mathematicians.

Finally, it is remarked, that Legrange was successively
the founder of the Academy of Turin, director during 20
years of the Berlin Academy, foreign associate of the Aca-
demy of Sciences of Paris, member of the Imperial Insti-
tute, and of the Board of Longitude ; and he was besides,
senator and count of the empire, grand officer of the Le-
gion of Honour, and Grand Cross of the Imperial Order of
Re-Union.

Abridged from Delambre's Eloge of Lagrange.
LAKE, a collection of water, inclosed in the cavity of
some inland place, of a considerable extent and depth ;
as the lake of Geneva, &c.

LALANDE (Joseph Jerome le Francais de), an
eminent professor of astronomy in the college of France,
was born at Bourg, in the department of l'Aix, July 11,
1732, and he died at Paris, April 4, 1807, in the 75th
year of his age. He was intended for the bar, and went to
Paris for the study of Jurisprudence, when the view of the
observatory inspired him with notions which deranged his
former plan, and became the ruling passion of his life.
He was kindly received by the celebrated astronomer le
Monnier, by whose lessons he made such a rapid advance,
that at the age of 19 he was nominated commissioner
from the Academy of Sciences, to go to Berlin, to deter-
mine the parallax of the moon, while M. Lacaille was to
perform the same operation at the Cape of Good-Hope.
Soon after his return, he was appointed a member of the
Academy of Sciences; from which time, every volume of
their Memoirs contained some of his compositions, which
were not limited to astronomical observations. He pub-
lished the French edition of Halley's Tables, and the his-
tory of the comet of 1759, besides furnishing Clairaut
with many calculations for the theory of the same. Being
charged in 1760 with the compilation of the Connoissance
des Temps, he quite changed the form of the work, giving
it that which it bears at present ; of which collection, he
edited the 32 volumes from 1775 to 1807-

In 1764 appeared the first edition of his great work,
the Traite Astronomique, a classical work, which he has
since perfected, in 4 vols 4to. He composed all the
astronomical articles for the Yverdun Encyclopedia, and
new cast the whole for the Encyclopedie Methodique.

To his written instructions he joined, during 46 years,
oral lessons. In 176l he succeeded Delille in the astro-
nomical chair at the college of France. In the midst of
his labours, he drew up his Voyage d'ltalie, his Trait*
des Canaux, and his Bibliographic Astronomique, a vast
catalogue of all the works which have appeared on this
science. In 1802, M. Lalancle published the latter part
of the new edition of Montucla's History of Mathematics,
in 4 vols 4to ; most of the last two volumes being prepa-
red from Montucla's papers, with the assistance of La-
place, Lacroix, and other mathematicians.

In the year 1793, he published Abrege de Navigation,
containing many valuable rules and tables. The Connois-

A few. years before his death, viz, in 1802, he published
a small collection for the pocket, of Tables of Logarithms,
Sines, and Tangents, on the plan of Lacaille, but much
inferior. And during the latter years of his life, he pub-
lished annually a concise history of astronomy for the
current year, which exhibited a summary of the most re-
markable facts, discoveries, and inventions connected with
the subject.

Lalande was certainly an excellent astronomer,' and an
active promoter of that science; but he had very little
taste, and a very confined knowledge of mathematics
in general. He considered himself, however, as the father
of the mathematical sciences generally, and at his death,
he founded the prize of a medal, which the Institute an-
nually awards to the author of the best astronomical me-
moir, or the maker of the most curious observations. To
the extraordinary ardour and activity of his character, he
united a love for truth that approached the borders of
fanaticism. To him every degree of concealment appear-
ed unworthy of an honest man; and he therefore, without
any reserve., uttered whatever might be his sentiments on
all occasions. The high consideration which he obtained,
would probably, with prudence and circumspection, have
secured him an enviable respect to the end of his days :
but the habit of speaking his mind, which he did not re-
frain from in the most stormy times, and on topics when
he might, and indeed ought to have been silent, together
with his bluntness in sometimes opposing the established
systems of others, animated against him a crowd of detrac-
tors and enemies. His long and durable services in mat-
ters of science, were thus in a measure forgotten o» ne-
glected, in the contemplation of his dangerous specula-
tions, and of the imprudent zeal with which he promul-
gated his opinions.

By his will, M. Lalande ordered his body to be dis-
sected, and the skeleton to be placed in the Museum of
Natural History. His relations however, regardless of
the injunction, caused the body to be interred a few days
after his death; on which occasion, the funeral was at-
tended by the members of the National Institute.

LAMBERT (John Henry), an eminent mathematician,
was a native of Mulhausen, in Alsace, where he was born,
April28, 1728. Hiseducation was rather confined, but he
gave early indications of genius. At a very early period, he
invented an arithmetical machine, a mercurial pendulum
that vibrated for 27 minutes, arithmetical scales, and a
machine for drawing in perspective. From 174S to 175fJ,
he was employed as tutor to the grandsons of M. le comte
de Salis, at Coire. In 1757, the tutor and his pupils had
passed the year in Holland, and he there published his
treatise on the passage of Light. They afterwards tra-
velled to Paris, where he was introduced to Dalembert.
In 1759, he published his Perspective at Zurich. From
1753 to 176'0, he had several pieces published in the Acta
Helvetica. In 1759, his Photometry was printed at Augs-
burg. In 1760, he collected the different pieces, still in
a fugitive state, of his Novum Organum; but which was
not published till 176'4. In 1761, his treatise on the
Properties of the Orbits of Comets was printed at Augs-
burg, lie was admitted a pensioner of the Berlin Aca-
demy in 176'5 ; from which time, to his death, in 1777,
he furnished many memoirs for the several volumes of
that academy. — The scientific knowledge of this gentleman

LAN

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LAN

was very considerable ; and he had great independence of ciples which his new analysis was founded on, he applied

it, in a variety of problems, to drawing tangents, and find-
ing the properties of curve lines; to describing their in-
volutes and evolutes, finding the radius of curvature, their
greatest and least ordinates, and points of contrary flexure;
to the determination of their cusps, and the drawing of
asymptotes: and he proposed, in a second book, to extend
the application of this new analysis to a great variety of
mechanical and physical subjects. The papers which were
to have formed this book lay long by him; but he never
tound leisure to put them in order for the press.

In the year 1766, Mr. Landen was elected a Fellow of
the Royal Society. And in the 58th volume of the Philo-
sophical Transactions, for the year 1768, he gave a spe-
cimen of a new method of comparing curvilinear areas;
by means of which many areas are compared, that did not
appear to be comparable by any other method : a circum-
stance of no small importance in that part of natural phi-
losophy which relates to the doctrine of motion.

In the 60th volume of the same work, for the year 1770,
he gave some new theorems for computing the whole areas

mind, with a very warm and vivid imagination.

LAMMAS-Day, the 1st of August; so called, accord-
ing to some, because lambs then grow out of season, as
being too large. Others derive it from a Saxon word, sig-
nifying loaf-mass, because on that day our forefathers
made an offering of bread prepared with new wheat. It is
celebrated by the Romish church in memory of St. Peter's
imprisonment.

LAMPiEDIAS, a kind of bearded comet, resembling a
burning lamp, being of several shapes; for sometimes its
flame or blaze runs tapering upwards like a sword, and
sometimes it is double or treble pointed.

LANDEN (John), an eminent mathematician, was
born at Peakirk, near Peterborough in Northamptonshire,
in January 1719- He became very early a proficient in
the mathematics, for we find him a very respectable con-
tributor to the Ladies' Diary in 1744; and he was soon
among the foremost of those who then contributed to the
support of that small but valuable, publication, in which
almost every English mathematician, who has arrived at

any degree of eminence for the best part of that century, of curve lines, where the ordinates are expressed by frac

has contended for fame atone time or other of his life. Mr
Landen continued his contributions to it at times, under
various signatures, till within a few years of his death.

It has been frequently observed, that the histories of li-
terary men consist chiefly of the history of their writings ;
and the observation was never more fully verified, than in
the present article concerning Mr. Landen.

In the 48tb volume of the Philosophical Transactions,
for the year 1754, Mr. Landen gave An investigation of
some theorems which suggest several very remarkable
properties of the Circle, and are at the same time of con-
siderable use in resolving fractions, the denominators of
which are certain multinomials, into more simple ones, and
by that means facilitate the computation of fluents.

In the year 1755, he published a volume of about I GO
pages, entitled Mathematical Lucubrations. The title to
this publication was made choice of, as a means of inform-
ing the world, that the study of the mathematics was at that
time rather the pursuit of his leisure hours, than his prin-
cipal employment : and indeed it continued to be so, dur-
ing the greatest part of his life ; for about the year 1762 he
was appointed agent to Earl Fitzwilliam, an employment
which he resigned only two years before his death. These
lucubrations contain a variety of tracts relative to the rec-
tification of curve lines, the summation of series, the find-
ing of fluents, and many other points in the higher parts of
the mathematics.

About the latter end of the year 1757, or the beginning
of 1758, he published proposals for printing by subscrip-
tion The Residual Analysis, a new branch of the algebraic
art: and in 1758 he published a small tract, entitled A
Discourse on the Residual Analysis ; in which he resolved
a variety of problems, to which the method of fluxions had
usually been applied, by a mode of reasoning entirely new:
he also compared these solutions with others derived from
the fluxionary method; and showed that the solutions by
his new method were commonly more natural and elegant
than the fluxionary ones.

In the 51st volume of the Philosophical Transactions, for
the year 1760, he gave a new method of computing the
sums of a great number of infinite series.

In 1764, he published the first book of The Residual
Analysis. In this treatise, besides explaining the prin-

tions of a certain form, in a more concise and elegant man-
ner than had been done by Cotes, De Moivre, and others
who had considered the subject before him.

In the 6lst volume, for 1771, he has investigated se-
veral new and useful theorems for computing certain
fluents, which are assignable by arcs of the conic sections;
a subject which had been considered before, both by Mac-
laurin and Dalembert; but some of the theorems that
were given by these celebrated mathematicians, being in
part expressed by the difference between an hyperbolic
arc and its tangent, and that difference being not directly
attainable when the arc and its tangent both become in-
finite, as they will do when .the whole fluent is wanted,
though such fluent be finite ; these theorems therefore fail
in such cases, and the computation becomes impracticable
without further help. This defect Mr. Landen removed,
by assigning the limit of the difference between the hyper-
bolic arc and its tangent, while the point of contact is sup-
posed to be removed to an infinite distance from the ver-
tex of the curve. And he concludes the paper by stating
a curious and remarkable property relating to pendulous
bodies, which is deducible from those theorems. In the
same year he published Animadversions on Dr. Stewart's
Computation of the Sun's Distance from the Earth.

In the 6'5th volume of the Philosophical Transactions,
for 1775, he gave the investigation of a general theorem,
which he had promised in 1771, for finding the length of
any curve of a conic hyperbola by means of two elliptic
arcs: and he observes, that by the theorems there investi-
gated, both the elastic curve and the curve of equable re-
cess from a given point, may be constructed in those cases
where Maclaurin's elegant method fails.

In the 67th volume, for 1777, he gave a new theory of
the motion of bodies revolving about an axis in free
space, when that motion is disturbed by some extraneous
force, either percussive or accelerative. At this time he
did not know that the subject had been treated by any per-
son before him, and he considered only the motion of a
sphere, spheroid, and cylinder. After the publication of
this paper however he was informed, that the doctrine of
rotatory motion had been considered by Dalembert ; and
on procuring that author's Opuscules Mathematiques, he
there learned that Dalembert was not the only one who

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had considered the matter before him; for Dalembert
there speaks of a mathematician, though he does not men-
tion his name, who, after reading what had been written on
the subject; doubted whether there could be any solid
whatever, besides the sphere, in which any line, passing
through the centre of gravity, will be a permanent axis of
rotation. In consequence of this, Mr. Landen took up
the subject again; and though he did not then give a so-
lution to the general problem, viz, " to determine the mo-
tions of a body of any form whatever, revolving without
restraint about any axis passing through its centre of
gravity," he fully removed every doubt of the kind which
had been started by the person alluded to by Dalembert,
and pointed out several bodies which, under certain di-
mensions, have that remarkable property. This paper is
given, among many others equally curious, in a volume of
Memoirs, which he published in the year 1780. That vo-
lume is also enriched with a very extensive appendix, con-
taining theorems for the calculation of fluents ; which are
more complete and extensive than those that are found in
any author before him.

In 1781, 1782, and 1783, he published three small
tracts on the summation of converging series ; in which
he explained and showed the extent of some theorems
which had been given for that purpose by De Moivre, Stir-
ling, and his »ld friend Thomas Simpson, in answer to
some things which he thought had been written injurious
to the fame of those excellent mathematicians. It was the
opinion of some, that Mr. Landen did not show less ma-
thematical skill in explaining and illustrating these theo-
rems, than he has done in his writings on original sub-
jects; and that the authors of them were as little aware
of the extent of their own theorems, as the rest of the
world were, before Mr. Landen's ingenuity made it obvious
to all.

About the beginning' of the year 1782, Mr. Landen had
made such improvements in his theory of rotatory motion,
as enabled him, he thought, to give a solution of the ge-
neral problem mentioned above; but finding the result of
it to differ very materially from the result of the solution
which had been given of it by Dalembert, ana1 not being
able to .see clearly where that gentleman in his opinion had
erred, he did not venture to make his own solution public.
In the course of that year, however, having procured the
Memoirs of the Berlin Academy for 1757, which contain
M. Euler's solution of the problem, he found 'that this
gentleman's solution gave the same result as had been de-
duced by Dalembert ; but the perspicuity of Euler's man-
ner of writing enabled him to discover where he had dif-
fered from'his own, which the obscurity of the other did
not do. The agreement, however, of two writers of such
established reputation as Euler and Dalembert made him
long dubious of the truth of his own solution, and induced
him to revise the process again and again with the utmost
circumspection; and being every time more convinced
that his own solution was right, and theirs wrong, he at
length gave it to the public, in the 75th volume of the Phi-
losophical Transactions, for 17S5.

The extreme difficulty of the subject, joined to the con-
cise manner in which Mr. Landen had been obliged to
give his solution, to confine it within proper limits for the
Transactions, rendered it too difficult, or at least too la-
borious a task for most mathematicians to read it ; and
this circumstance, joined to the established reputation of
Eulcr and Dalembert, induced many to think that their

solution was right, and Mr. Landen's wrong ; an'd there
did not want attempts to prove it; particularly a long and
ingenious paper by the learned Mr. Wildbore, a gentle-
man of very distinguished talents and experience in such
calculations; this paper is given in the SOth volume of the
Philosophical Transactions, for the year 1790, in which he
agrees with the solutions of Euler and Dalembert, and
against that of Mr. Landen. This determined the latter
to revise and extend his solution, and give it at greater
length, to render it more generally understood. About
this time also he met by chance with the late Frisi's Cos-
mographise Physicae et Mathematicas ; in the second part
of which there is a solution of this problem, agreeing in
the result with those of Euler and Dalembert. Here Mr.
Landen learned that Euler had revised the solution which
he had given formerly in the Berlin Memoirs, and given it
another form, and at greater length, in a volume published
at Rostoch and Gryphiswald in 1765, entitled, Theoria
Motiis Corporum Solidorum seu Rigidorum. Having
therefore procured this book, Mr. Landen found the same
principles employed in it, and of course the same conclu-
sion resulting from them, as in M. Eulerjs former solution
of the problem. But notwithstanding that there were
thus a coincidence of at least four most respectable ma-
thematicians against him, Mr. Landen was still persuaded
of the truth of his own solution, and prepared to defend
it. And as he was convinced of the necessity of explain-
ing his ideas on the subject more Fully, so he now found it
necessary to lose no time in setting about it. He had for
several years been severely afflicted with the stone in the
bladder, and towards the latter part of his life to such a
degree as to be confined to his bed for more than a month
at a time: yet even this dreadful disorder did not ex-
tinguish his ardour for mathematical studies; for the se-
cond volume of his Memoirs, lately published, was written
and revised during the intervals of his disorder. This vo-
lume, besides a solution of the general problem concerning
rotatory motion, contains the resolution of the problem
relating to the motion of a top ; with an investigation of
the motion of the equinoxes, in which Mr. Landen has
first of any one pointed out the cause of Sir Isaac New-
ton's mistake in his solution of this celebrated problem;
and some other papers of considerable importance. He just
lived to see this work finished, and received a copy of it
the day before his death, which happened on the 15th of
January 1790, at Milton, near Peterborough, in the 71st
year of his age.

LANTERN (Magic). See Magic Lantern.
LARBOARD, the left-hand side of a ship, when a per-
son stands with his face towards the head.

LARMIER, in Architecture, a flat square member of
the cornice below the cimasium, and jets out farthest ;
being so called from its use, which is to disperse the wa-
ter, and cause it to fall at a distance from the wall, drop
by drop, or, as it were, by tears; larme in French signi-
fying a tear.

LATENT Heat, is that which is insensible, inactive, dor-
mant, &c.

LATERAL Equation, in Algebra, is the same with
simple equation. It has but one root, and may be con-
structed by right lines only.

LATITUDE, in Geography, or Navigation, the distance
of a place from the equator; or an arch of the meridian,
intercepted between its zenith and the equator. Hence the
latitude is either north or south, according as the place is

L A T

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UT

on the north or south side of the equator : thus London is
said to be in 51° 31' of north latitude. Circles parallel to
the equator are called parallels of latitude, because they
show the latitudes of places by their intersections with the
meridian. The latitude of a place is equal to the elevation
of the pole above the horizon of the place: and hence
these two terms are used indifferently for each other.
This will be evident fromthe figure,
where the circle zhqp is the meri-
dian, z the zenith of the place, ho
the horizon, eq the equator, and P
the pole, then is ze the latitude,
and po the elevation of the pole
above the horjzon. And because
pe = zo, .being each a quadrantal
arc, if the common part pz be taken from both, there will
remain the latitude ze = po the elevation of the pole. —
Hence we have a method of measuring the circumference
of the earth, or of determining the quantity of a degree on
its surface ; for by measuring directly northward or south-
ward, till (lie pole be one degree higher or lower, we
have the number of miles in a degree of a great circle on
the surface of the earth ; and consequently multiplying
that by 360, will give the number of miles round the
whole circumference of the earth. The knowledge of the
latitude of the place, is of the utmost consequence, in geo-
graphy, navigation, and astronomy; it may be proper
therefore to lay down some of the best modes of determin-
ing it, both by sea and land.

1st. One method is, to find the height of the pole, to
which the latitude is equal, by means of the pole star, or
any other circumpolar star, thus : Either draw a true me-
ridian line, or find the times when the star is on the meri-
dian, both above and below the pole ; then at these times,
with a quadrant, or other fit instrument, take the altitudes
of the star ; or take the same when the star comes upon
your meridian line; which will be the greatest and least
altitude of the star: then shall half the sum of the two be
the elevation of the pole, or the latitude sought. — For, if
abc be the path of the star about the pole p, z the zenith,
and ho the horizon : then is ao the altitude of the star
upon the meridian when above the pole, and co the same
when below the pole ; hence, because ap = cp, therefore
co 4- co = 2op, hence the height of the pole op, or lati-
tude of z, is equal to half the sum of ao and co.

2d. A second method is by means of the declination1' of
the sun, or a star, and one meridian altitude of the same,
thus: Having with a quadrant, or other instrument, ob-
served the zenith distance zd of the luminary ; or else its
altitude nd, and taken its complement zd; then to this
zenith distance, add the declination d'E when the luminary
and place are on the same side of the equator, or subtract
it when on different sides, and the sum or difference will
be the latitude ez sought. But note, that all altitudes
observed, must be corrected for refraction and the dip of
the horizon, and for the semidiameter of the sun, when
that is the luminary observed.

Many other methods of observing and computing the
latitude may be seen in Robertson's Navigation ; see book
5 and book 9- See also the Nautical Almanac for 1771 ;
and Mr. Brinkley's tables and rules for the same purpose,
in the Nautical Almanac for the year 1797. Mr. Richard
Graham contrived an ingenio.us instrument for taking the
latitude of a place at any time of the day- See Philos.
Trans. No. 435, ox my Abr. vol. 7. pa. 673.

Vol. I.

Latitude, in Astronomy, as of a star or planet, is its
distance from the ecliptic, being an arch of a circle of
latitude, reckoned from theecliptic towards its poles, either
north or south. Hence, the astronomical latitude is quite
different from the geographical, the former measuring
from the ecliptic, and the latter from the equator, so t,hat
this latter answers to the declination in astronomy, which
measures from the equinoctial. The sun has no latitude,
being always in the ecliptic ; but all the stars have their
several latitudes, and the planets are continually changing
their latitudes, sometimes north and sometimes south, cros-
sing the ecliptic from the one side to the other ; the points
in which they cross the ecliptic being called thenodes of the
the planet, and in these points it is that they may appear to
pass over the face of the sun, or behind his body, viz, when
they come both to this point of the ecliptic at the same time.

Circle of Latitude, is a great circle passing through
the poles of the ecliptic, and consequently perpendicular
to it, as the meridians are perpendicular to the equator,
and pass through its poles.

Latitude, of the Moon, North ascending, is when she
proceeds from the ascending node towards her northern
limit, or greatest elongation.

LATiTUDE,IVort/i descending, is when the moon returns
from her northern limit towards the descending node*.

Latitude, South descending,is when she proceeds from
the descending node towards her southern limit.

Latitude, South ascending, is when she returns from
her southern limit towards her ascending node. And the
same is to be understood of the planets.

Heliocentric Latitude of a planet, is its latitude, of
distance from the ecliptic, such as it would appear from,
the sun. — This, when the planet comes to the same
point of its orbit, is always the same, or unchangeable.

Geocentric Latitude of a planet, is its latitude as seen
from the earth. — This, though the planet be in the same
point of its orbit, is not always the same, but alters ac-
cording to the position of the earth, in respect to the
planet. The latitude of a star is altered only by the aber-
ration of light, and the secular variation of latitude.

Difference of Latitude, is an arc of the meridian, or
the nearest distance between the parallels of latitude of
two places. When the two latitudes are of the same name,
either both north or both south, subtract the less latitude
from the greater, to give the difference of latitude; but
when they are of different names, add them together for
the difference of latitude.

Middle Latitude, is the middle point between two la-
titudes or places ; and is found by taking half the sum of
the two.

Parallax of Latitude. See Parallax.

Refraction of Latitude. See Refraction.

LATUS Rectum, in Conic Sections, the same with pa-
rameter ; which see.

Latus Transversum, of the hyperbola,
is the right line between the vertices of
the two opposite sections; or that part
of their common axis lying between the
two opposite cones ; as the line de. It
is the same as the transverse axis of the
hyperbola, or opposite hyperbolas.

Latus Primarium, a right line, DD,or
ee, drawn through the vertex of the sec-
tion of a cone, within the same, and pa-
rallel to the base.

4- Y

LEI [714

LEAGUE, an extent of three miles in length. A nau-
ticalleague, or three nautical miles, is the 20th part of a
degree of a great circle.

LEAP- Year, the same as Bissextile ; which see. It
is so called from its leaping a day more that year than in a
common year; consisting of 366 days, and a common
year only of 365. This happens every 4th year, except
only such complete centuries as are not exactly divisible
by 4; such as the 17th, 18th, 19th, 21st &c, centuries,
because 17, 18, 10, 21, &c, cannot be divided by 4 with-
out a remainder.

To find Leap-Year, Ifc. Divide the number of the year
by 4; then if 0 remain, it is leap-year; but if 1, 2, or 3
remain, it is so many after leap-year. Or the rule is
sometimes thus expressed, in these two memorial verses:
Divide by 4; what's left shall be,
For leap-year 0 ; for past 1, 2, or 3.
Thus if it be required to know what year 1814 is :
then 4) 1314(453

2 remains :
so that 2 remaining, shows that 1814 is the 2d year after
leap-year. And to find what year 1816 is :

then 4) 1816 (454
here 0 remaining, shows that 18l6is a leap-year.

LEE, a term in Navigation, signifying that side, or
quarter, which is from the wind.

Lee- Way, of a ship, is the angle made by the point of
the compass steered on, and the real line of the ship's way,
occasioned by contrary winds and a rough sea. All ships
are apt to make some lee-way ; so that something must be
allowed for it, in casting up the log- board. But the lee-
way made by different ships, under similar circumstances
of wind and sails, is different; and even the same ship, with
different lading, and having more or less sail set, will
have more or less lee-way. The usual allowances for it
are these, as they were given by Mr. John Buckler, to
the late ingenious Mr. William Jones, who first published
them in 1702 in his Compendium of Practical Navigation.
1st, When a ship is close-hauled, has all her sails set, the
sea smooth, and a moderate gale of wind, it is then sup-
posed she makes little or no lee-way. 2d, Allow 1 point,
when it blows so fresh that the small sails are taken in.
3d, Allow 2 points, when the top-sail must be close reefed.
4th, Allow 2\ points, when one topsail must be handed.
5th, Allow 3| points, when both topsails must be taken
in. 6th, Allow 4 points, when the fore-course is handed.
7th, Allow 5 points, when trying under the mainsail only.
8th, Allow 6 points, when both main and fore-courses
are taken in. 9th, Allow 7 points, when the ship tries a
hull, or with all sails handed.

When the wind has blown hard in either quarter, and
shifts across the meridian into the next quarter, the lee-
way will be lessened. But in all these cases, respect must
be had to the roughness of the sea, and the trim of the
ship. And hence the mariner will be able to correct his
course.

LEGS, of a Triangle. When one side of a triangle is
taken as the base, the other two are sometimes called the
legs. The term is more frequently used for the base and
perpendicular of a right-angled triangle, or the two sides
about the right angle. „

Hyperbolic Legs, arc the parts of a curve line that par-
take of the nature of the hyperbola, or having asymptotes.
LEIBNITZ (Godfrey-William), an eminent mathe-
matician and philosopher, was born at Lcipsic in Saxony

L E I

in l6*6. At the age of 15., he applied himself te the
study of the mathematics at Leipsic and Jena; and in 1663.
maintained a thesis De Principiis Individuation^. The
year following he was admitted master of arts. He read
with great attention the Greek philosophers ; and endea-
voured to reconcile Plato with Aristotle, as he afterwards
did Aristotle with Descartes. But the study of the law
was his principal view 5 in which faculty he was admitted
bachelor in 1.665. The year following he would have
taken the degree of doctor; but was refused it on pretence
that he was too young, though in reality because he had
raised himself several enemies by rejecting the principles of
Aristotle and the schoolmen. On this he repaired to
Altorf, where he maintained a thesis De Casibus Per-
plexis, with such applause, that he had the degree of doc-
tor conferred on him.

In 1672 he went to Paris, to manage some affairs at the
French court for the baron Boinebourg. Here he became
acquainted with all the literati, and made farther and
considerable progress in the study of mathematics and
philosophy, chiefly, as he says, by the works of Pascal,
Gregory St. A^incent, and Huygens. In this course, hav-
ing observed the imperfection of Pascal's arithmetical ma-
chine, he invented anew one, as he called it, which was
approved of by the minister Colbert, and the Academy of
Sciences, in which he was offered a seat as a member, but
refused the offers made to him, as it would have been ne-
cessary to embrace the Catholic religion.

In 1673 he came over to England; where he became
acquainted with Mr. Oldenburg, secretary of the Royal
Society, and Mr. John Collins, a distinguished member
of the Society ; from whom it seems. he received some
hints of the method of fluxions, which had been invented,
in 1664 or 1665, by the then Mr. Isaac Newton, which
probably laid the foundation of his Differential Calculus.
The same year he returned to France, where he resided till
I676, when he again passed through England and Hol-
land, in his journey to Hanover, where he proposed to
settle.

In 1700 he was admitted a member of the Royal Aca-
demy of Sciences at Paris : and in the same year the
elector of Brandenburg, afterwards king of Prussia, founded
an academy at Berlin by his advice ; of which he was,
appointed perpetual president, though his affairs would
not permit him to reside constantly at that place. Besides
the office of privy counsellor of justice, which the elector
of Hanover had given him, the emperor appointed him,
in 1711, aulic counsellor; and the czar made him privy
counsellor of justice, with a pension of 1000 ducats.
Leibnitz undertook at the same time to establish an aca-
demy of sciences at Vienna ; but the plague prevented the
execution of it. However, the emperor, as a mark of his
favour, settled a pension on him of 2000 florins, and pro-
mised him one of 4000 if he would come and reside at
Vienna; an offer he was inclined to comply with, but
was prevented by his death.

The elector of Hanover being raised to the throne of
Great Britain, Leibnitz visited that place the latter end of
1714, when he received particular marks of friendship
from the king, and was frequently at court. He now was
engaged in a dispute with Dr. Samuel Clarke, on the sub-
jccVof free will, the reality of space, and other philoso-
phical subjects ; which was conducted with great candour
and learning; and the papers that were published by
Clarke, will ever be esteemed by men of genius and learn-

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ing. The controversy ended only with the death of Leib-
nitz, Nov. 14, 17l6, which was occasioned by the gout and
stone, in the 70th year of his age.

In person he was of a middle stature, and a thin habit
of body. He had a studious air, and a sweet aspect,
though near-sighted. He was indefatigably industrious
to the end of his life: he ate and drank little : hunger
alone marked the time of his meals, and his diet was plain
and strong. He had a very good memory, and it was
said could repeat the iEneid from beginning to end. What
he wanted to remember, he wrote down, and uever read it
afterwards. He was never married, nor ever seems to
have entertained the idea but once, when he was about 50
years old ; and the lady desiring time to consider of it,
gave him an opportunity of doing the same : he used to
say, " that marriage was a good thing, but a wise man
ought to consider of it all his life."

Leibnitz was author of a great multitude of writings ;
several of which were published separately, and many
others in the memoirs of different academies. He invented
a binary arithmetic, and many other ingenious matters.
His claim to the invention of fluxions, has been spoken of
under that article. Hanschius collected, with great care,
every thing that Leibnitz had said, in different passages of
his works, on the principles of philosophy ; and formed
of them a complete system, under the title of G. G. Leib-
nitzii Principia Philosophic more geometrico demon-
strata &c, 1728, in 4to. A collection of our author's
letters appeared in 1734 and 1735, entitled, Epis-
tolae ad diversos theologici, juridici, medici, philo- ,
sophici, mathematici, historici, et philologici argumenti
e mss. auctoris : cum annotationibus suis primum di-
vulgavit Christian Cortholtus. But all his works were
collected, distributed into classes by M. Dutens, and
published at Geneva in 6 large volumes 4to, in 1768,
entitled, Gothofredi Guillelmi Leibnitii Opera Om-
nia &c.

Leibnitzian Philosophy, or the philosophy of
Leibnitz, is a system formed and published by its author
in the last century, partly in emendation of the Cartesian,
and partly in opposition to the Newtonian doctrine. In
this philosophy, the author retained the Cartesian subtile
matter, with the vortices and universal plenum ; and he
represented the universe as a machine that should pro-
ceed for ever, by the laws of mechanism, in the most per-
fect state, by an absolute inviolable necessity. After
Newton's philosophy was published, in 16187, Leibnitz
printed an essay on the celestial motions in the Act. Erud.
1689, where he admits the circulation of the ether with
Descartes, and of gravity with Newton ; though he has
not reconciled these principles, nor shown how gravity
arose from the impulse of this ether, nor how to account
for the planetary revolutions in their respective orbits.
His system is also defective, as it does not reconcile the
circulation of the ether with the free motions of the comets
in all directions, or with the obliquity of the planes of the
planetary orbits; nor resolve other objections to which the
hypothesis of the vortices and a plenum is liable.

Soon after the period just mentioned, the dispute com-
menced concerning the invention of the method of
fluxions, which led M. Leibnitz to take a very decided
part in opposition to the philosophy of Newton. From
the goodness and wisdom of the Deity, and his principle of
a sufficient reason, he concluded, that the universe was a
perfect work, or the best that could possibly have been

made ; and that other things, which are evil or incommo-
dious, were permitted as necessary consequences of what
was best : that the material system, considered as a per-
fect machine, can never fall into disorder, or require to
be set right ; and to suppose that God interposes in it, is
to lessen the skill of the author, and the perfection of his
work. He expressly charges an impious tendency on the
philosophy of Newton, because he asserts, that the fabric
of the universe and course of nature could not continue
for ever in its present state, but in process of time would
require to be re-established or renewed by the hand of its
first framer. The perfection of the universe, in conse-
quence of which it is capable of continuing for ever by
mechanical laws in its present state, led M. Leibnitz to
distinguish between the quantity of motion and the force
of bodies ; and, while he owns, in opposition to Descartes,
that the former varies, to maintain that the quantity of
force is for ever the same in the universe ; and to measure
the forces of bodies by the squares of their velocities.

M. Leibnitz proposes two principles as the foundation
of all our knowledge; the first, that it is impossible for a
thing to.be, and not to be at the same time, which he says
is the foundation of speculative truth ; and secondly, that
nothing is without a sufficient reason why it should be so,
rather than otherwise ; and by this principle he says we
make a transition from abstracted truths to natural philo-
sophy. Hence he concludes that the mind is naturally
determined, in its volitions and elections, by the greatest
apparent good, and that it is impossible to make a choice
between things perfectly like, which he calls Indiscerni-
bles ; whence he infers, that two things perfectly like
could not have been produced even by the Deity himself:
and one reason why he rejects a vacuum, is because the
parts of it must be supposed perfectly like to each other.
For the same reason too, he rejects atoms, and all similar
parts of matter, to each of which, though divisible ad in-
finitum, he ascribes a Monad (Act. Lipsias 1698, pa. 435)
or active kind of principle, endued with perception and
appetite. The essence of substance he places in action or
activity, or, as he expresses it, in something that is be-
tween acting and the faculty of acting. He affirms that
absolute rest is impossible, and holds that motion, or a
sort of nisus, is essential to all material substances.
Each monad he describes as representative of the whole
universe from its point of sight ; and yet he tells us,' in one
of his letters, that matter is not a substance, but a sub-
stantiatum, or phenomene bien fonde. See also Maclau-
rin's View of Newton's Philosophical Discoveries, book 1,
chap. 4.

LEMMA, is a term chiefly used by mathematicians,
and signifies a proposition, previously laid down to pre-
pare the way for the more easy apprehension of the de-
monstration of some theorem, or the construction of some
problem.

LEMNISCATE, the name of a curve
in the form of the figure of 8. If we call
ap, x ; pq, y, and the constant line ab
or ac, a; theequationoy = x^/ici1 — x2),
or (ry* = cfx1 — .r4, expressing a line of
the 4th degree, will denote a lemniscate,
having a double point in the point a.
There may be other lemuiscates, as the el-
lipse of Cassini, &c ; but that above defined
is the simplest of them.

It easily appears that this curve is quadrable.
4Y2

For

L E N

[ 716 J

L E N

since ay = x*/(az — x"), therefore the fluxion of the perpendicular ae. In the same manner, another ray fall-
ing on B, at an equal distance on the other side of the

curve or yk is = -£ ^/(a — x°) ; the fluent of which is
■s! .(a2 — x2) * for the general area of the curve ;

3 3a v ' °

which, when x is = a, becomes barely |a = aqb.

LENS, a piece of glass or other transparent substance,
having its two surfaces so formed that the rays of light, in

vertex c, and parallel to the former ray da, will be re-
fracted into the same point t. And it will also be found,
that all the intermediate parallel rays will converge to the
same point, very nearly.

On the other hand, if the rays fall parallel on the inside
of this denser medium, as in the figure below, they will

passing through it, have their direction changed, and made telKj from lne perpendicular ea/; and converge to a point
to converge and tend to a point beyond the lens,_or to be- T jn the air, or any rarer medium. Also the ray incident
come parallel after converging or diverging, or lastly to on B) at the same distance from the vertex c, will converge
diverge as if they had proceeded from a point before the to tne same place t, together with all the intermediate
lens. Some lenses are convex, or thicker in the middle; parallel rays,
others concave, or thinner in the middle ; while others are
plano-convex, or.plano- concave ; and some again are con-
vex on one side and concave on the other, which are call-
ed meniscuses, the properties of which see under that
word. When the particular figure is not considered, a
lens that is thickest in the middle is called, a convex lens;
and that which is thinnest in the middle is called a con-
cave lens, without further distinction. These several
c t, . , • .i jc„„,.„. Since therefore rays are made to converse when thev

forms of lenses are represented in the annexed figure : ., . J - s *•? >-"*y

r a . pass either from a rarer or a denser medium terminated

by a convex surface, and converge again when they pass
from the same medium convex towards the rarer, a lens
which is convex on both sides must, on both accounts,
make parallel rays converge to a point beyond it. Thus
the parallel rays between a and b, falling on

where a, b are convex lenses, and c, d, e are concave
ones; also a is a plano-convex, b is convexo-convex,
c is plano-concave, n is concavo-concave, and e is a me-
niscus.

In every lens, the right line perpendicular to the two
surfaces, is called the Axis of the lens, as f g; the points
where the axis cuts the surface, are called the Vertices of
the lens ; also the middle point between them is called the
Centre; and the distance between them, the Diameter.
Some confine lenses within the diameter of half an inch;
and such as exceed that thickness, they call Lenticular
Glasses. — Lenses are either blown or ground.

Blown Lenses, are small globules of glass, melted in
the flame of a lamp or taper. See Micuospope.

Ground Lenses, are such as are ground or rubbed into
the desired shape, and then polished. For a method of
grinding them, and description of a machine for that pur-
pose, see Philos. Trans, vol. xli. pa. 555, or my Abr.
viii, 451.

Maurolyc first delivered something relative to the nature
of lenses; but we are chiefly indebted to Kepler for ex-
explaining the doctrine of refraction through mediums of
different forms, the chief substance of which may be com-
prehended in the cases following: •

the convex surface of the glass AB, would in that dense
medium have converged to t ; but that medium being ter-
minated by another convex surface, they will be made more
converging, and be collected at some place F, nearer to the
lens.

Again, to explain the effects of a concave glass, let ab
be the concave side of a dense medium, the centre of con-
cavity being at e. In this case, da will be refracted to-
wards the perpendicular ea ; and so likewise will the ray
incident at b; in consequence of which they will diverge
from one another within the dense medium. The inter-
mediate rays will also diverge more or less, as they recede
from the axis tc ; which, being in the perpendicular, will
so straight on.

Let da be a ray of light falling on a convex dense me-
dium, having its centre at E. When the ray arrives at a,
it will not proceed in the same direction At; but it will If the rays be parallel within the dense medium, tlicy

U. there bent, and thrown into a direction at, nearer the will diverge when they pass from thence into a rarer me-

LEN

[ 717 ]

LEN

rays, in proportion to their distance from the axis or cen-
tral ray tc.

„y ■„,. ;. i„„d«„. ., ,, «*■„„ „i,h .„ ,he ,„,„„«„„„ .f„r,re,ht° s teS™ * JKtat'rs

glass, above that of the focus, is negative, which is the
same distance taken the contrary way. Montucla vol 2
pa. 177. ' '

And from the principle above-mentioned, it will not be
difficult to understand the application of lenses, in the
rationale of telescopes and microscopes. And on these
principles also is founded the structure of refracting burn-
ing glasses, by which the sun's light and heat are exceed-
ingly augmented in the focus of the lens, whether convex
or plano-convex; since the rays, falling parallel to the
axis of the lens, are reduced into a much narrower com-
pass; so that it is no wonder they burn some bodies, melt
others, and produce other extraordinary phenomena.

In the Philos. Trans, vol. xvii. 960, or my Abr. iii, 593,
Dr. Halley gives an ingenious investigation of the foci of
rays refracted through any lenses, nearly as follows:

.Therefore, if a dense medium, as the glass ab, be ter-
minated by two concave surfaces, parallel rays passing
through it will be made to diverge by both the sides of it.
Thus the first surface ab will make them diverge as if
they had come from the point t ; and with the effect of
the second surface added to this, they will diverge as from
a nearer point, f.

It was Kepler, who by these investigations first gave a clear
explanation of the effects of lenses, in making the rays of
a pencil of light converge or diverge. He showed that a
plano-convex lens makes rays, that were parallel to i\,s
axis, meet at the distance of the diameter of the sphere o f
convexity ; but that if both sides of the lens be equally
convex, the rays will have their focus at the distance of
the radius of the circle corresponding to that degree of
convexity. But he did not investigate any rule for the
foci of lenses unequally convex. He only says, in gene-
ral, that they will fall somewhere in the medium, between
the foci belonging to the two different degrees of convexity.
It is to Cavalerius that we owe this investigation; who
laid down this rule: As the sum of both the diameters is
to one of them, so is the other to the distance of the focus.
And it is to be noted that all these rules, concerning con-
vex lenses, are applicable to those that are concave, with
this difference, that the focus is on the contrary side of the
glass. See Montucla, vol. 2, pa. 17 6 ; or Priestley's Hist,
of Vision, pa. 65, 4to.

On this principle it was not difficult to find the foci of
pencils of rays issuing from any point in the axis of the
lens; since those that are parallel will meet in the focus;
and if they issue from the focus, they will be parallel on
the other side. If they issue from a point between the
focus and the glass, they will continue to diverge after
passing the lens, but less than before; while those that
come from beyond the focus, will converge after passing
the glass, and will meet in a place beyond the opposite
focus. This philosopher particularly observed, that rays
which issue from twice the distance of the focus, will
meet at the same distance on the other side. The most
important of these observations have been already illus-
trated by proper figures, and from them the rest may be
easily conceived. Later optical writers have assigned
the distances at which rays will meet, that issue from any
other place in the axis of a lens; but Kepler was too
much intent upon his astronomical and other pursuits, to
pay much attention to geometry. Montucla gives the fol-
lowing rule concerning this subject : As the excess of the
distance of the object from the glass, above the distance
of the focus, is to the distance of the focus ; so is this dis-
tance, to the place of convcrgency beyond the glass. And

Let bel, be a double convex lens, c the centre of the
segment eb, and k the centre of the segment el; bl the
thickness or diameter of the lens, and d a point in the
axis ; it is required to find the point f, or focus, where
the rays proceeding from d shall be collected, after being
refracted through the lens at a and a, points very near to
the axis bl. Put the distance da or db = d, the radius
ca or cb=7-, and the radius Ka or ki = e; also the
thickness of the lens bl = (, and m to n the ratio of the
sine of the angle of incidence dag to the sine of the re-
fracted angle hag or cam ; or m to n will be the ratio of
those angles themselves nearly, since very small angles are
to each other in the same ratio as their sines. Hfince
m is as the angle dag or dac,
n is as the angle hag or mac,
and because in this case the sides are as their opposite
angles, therefore DC : da : : iDAC : Z.C, or d -+-r:d: :m:

which is as the z.c; from this take n or the ^mac,

d + r '

. . . dm — dn — m . ,

and there remains — — ■ as the /.m; hence again

mdr

ca : ma or mb, that is

dm
d + r '

= ma or MB;

d + r

which shows in what

(to — 11) .d — nr

point the rays would be collected after one refraction,
viz, when nr is less than [m — n) . d. But when nr is
= (m — n~) . d, the point would be at an infinite distance,
or the rays will be parallel to the axis; and when nr is
greater than (m — n) .d, then mb is negative, or m falls
on the other side of the lens beyond d, and the rays still
continue to diverge after the first refraction.

The point m being now found, to or from which the rays
proceed after the first refraction, and bm — bl being
thus given, which call d, by a process like the former it
follows that fl, or the focal distance sought, is equal to

; , •-■ — — -=/• And here, instead of d substituting,

\m — 11) , D + mu ' ^

MB — LB Of

the same theorem will become

(mptlr — ndt + vprl)

L E N i 718

t, and putting p for ■ _ ,

]

L E N

— = /, the focal dis-

mdr + md R — mpr R — (m — n) .dt + ml

tance sought, in its most general form, including the thick-
ness of the lens; being the. universal rule for the foci of
double convex glasses exposed to diverging rays.

But if * the thickness of the lens be rejected, as not
sensible, the rule will be much shorter, viz,

pdr R /.

dr + i/R- prR ...

If therefore the lens consist of glass, whose refraction is

as 3 to 2, it will be ^ + *^*_ _ — jfl And if it be of wa-

SdrR
ter, whose refraction is as 4 to 3, it will be jr + dR_3rR

= /. But, if the lens could be made of diamond, whose

refraction is as 5 to 2, it would be ^ + 3rfR _ ;^ = /. ,

If the incident rays, instead of diverging, be converging,
the distance db or d will be negative, and then the theo-
rem for a double convex glass lens will be _ dr _ dR _ —

nr _._?*? =/. in which case therefore the focus is

dr + dR + 2rR
always on the other side of the glass.

And if the rays be parallel, as coming from an infinite
distance, or nearly so, then will d be negative, as well as
the terms in the theorem in which it is found; and there-
fore the other term prR will be nothing, in respect to those
infinite terms; and by omitting it, the theorem will be

>2_ ='-PH. =/' «Je -^ =/for glass.
dr+ dR r + R r-t-R

And here if r == it, or the two sides-of the glass be of

equal convexity, this last will become barely — or only
r =/.the focus, which therefore 'is in the centre of the
convexity of the lens.

If the lens be a meniscus of glass ; then, making r ne-
gative, the theorem is

= /for diverging rays,

The theorems for parallel rays, as coming from an in
finite distance, take place in the common refracting teles-
copes. And those for converging rays are chiefly of use
to determine the focus resulting from any sort of lens
placed in a telescope, between the focus of the object-glass
and the glass itself; the distance between the said focus
of the object-glass and the interposed lens being made
= — d; while those for diverging rays are chiefly of use
in microscopes, reading-glasses, and other cases in which
near objects are viewed.

It is evident that the foregoing general theorem will
serve to find any of the other circumstances, as well as the
focus, by considering this as given. Thus, for instance,
suppose it be required to find the distance at which an ob-
ject being placed, it shall by a given lens be represented as
large as the object itself; which is of singular use in view-
ing and drawing them, by transmitting the image through
a glass in a dark room, as in the camera obscura, which
gives not only the true figure and shades, but the colours
themselves as vivid as the life. Now in this case d is =/,
which makes the theorem become/>d>-R = dV-t- dzR—pdrR,

and this gives d = . But if the. two convexities be-

T + R ,

long to equal spheres, so as that r = r, then it is d = pi;
or == 2r when the lens is glass. So that if the object be
placed at the diameter of the sphere distant from the lens,
then the focus will be as far distant on the other side, and
the image as large as the object. But if the glass were a
plano-convex, the same distance would be just twice as
much.

Again, recurring to the first general theorem, including
t, the thickness of the lens; let the lens be a whole sphere;
then t = 2r, and r = r ; and hence the theorem reduces

mpdr — IrMr — 2rcprJ

2«d -

— dr -t- dR -+- 2rR
— 2rfrR

dr - dR -
2*R

dr + dR — arR dr — dR + 2rR

2rR 2rR

—/for converging rays,

/for parallel rays.

and

If the lens be a double concave glass, r and R will be
both negative, and then the theorem becomes

— 2drR ~ e T i

— = / for diverging rays, always negative;

dr + dR + 2rR a J J °

— -IdrR

dr + dR — 2rK
— 2rR

:/ for converging rays;

and — — = / for parallel rays.

And here, if the radii of curvature r and u be equal,
this last will be barely — r =/for parallel rays falling
on a double concave glass of equal curvature.

Lastly, when the lens is a plano-convex glass ; then, r
buing infinite, the theorem becomes

-1—L =/for diverging rays,

2dR

=/ior converging rays,

and 2r = / for parallel rays.

■ mpr

= /•

And here if d be infinite, the theorem contracts to

mp — 2;j SB — m , , .

— r or r =/; or for glass \r =f: showing

2ra 2m — 2k o-'.' o

that a sphere of glass collects the sun's rays at half the
radius of the sphere without it. And for a sphere of
water, the focus is at the distance of a whole radius.

For another example; when a hemisphere is exposed to
parallel rays ; then d and r being infinite, and t = r, the

theorem becomes r, = — r ==/. That is, in

m mJ — mn

glass it is |r, and in water fr.

Several other corollaries njay be deduced from the fore-
going principles. As follows t

1st. That the thickness of the lens, being very small, the
focus will remain the same, when cither side is exposed
to the rays.

2d. If a luminous body be placed in a focus behind a
lens, whether plano-convex, or convex on both sides; or
whether equally or unequally so ; the rays become parallel
after refraction, as the refracted rays become, what were
before the incident rays. And hence, by means of a con-
vex lens, or a little glass bubble full of water, a very in-
tense light may be projected to a great distance. Which
furnishes us with the structure of a lamp or latltern, to
throw an intense light to an immense distance: for a lens,
convex on both sides, being placed opposite to a concave
minor, and a lighted candle or wick being placed in the
common focus of both, the rays reflected back from the
mirror to the lens will be parallel to each other; and
after refraction will converge, till they concur at the dis-

L E N

[ 719 ]

LEV

tance of the radius, after which they will again diverge.
But the candle being likewise in the focus of the lens, the
rays it throws on the lens will be parallel; and therefore a
very intense light meeting with another equally intense,
at the distance of the diameter from the lens, the light
will be surprising: and though it afterwards decrease, yet
the parallel and diverging rays going a long way together,
it will be very great at a great distance. Lanterns of this
kind are of considerable service in the night time, to dis-
cover remote objects.

If it be required to have the light, at the same time,
transmitted to several places, as through several streets, &e,
the number of lenses and mirrors must be increased.

3. The images of objects are shown inverted in the focus
of a convex lens: nor is the focus of the sun's rays any
thing else, in effect, but the image of the sun inverted.
Hence, in solar eclipses, the sun's image, eclipsed, as it is,
may be burnt by a large lens on a board, &c, and exhibit
a very entertaining phenomenon.

4th. If a concave mirror be so placed, as that an in-
verted image, formed by refraction through a lens, be
found between the centre and the focus, or even beyond
the centre, it will again be inverted by reflection, and so
appear erect; in the first case beyond the centre, and in
the latter between the centre and the focus. And on these
principles the camera obscura is constructed.

5th. The image of an object, delineated beyond a con-
vex lens, is of the same magnitude, as it would be of, were
the object to shine into a dark room through a small hole,
upon a wall, at the same distance from the hole, as the
focus is from the lens. — When an object is less distant from
a lens than the focus of parallel rays, the distance of the
image is greater than that of the object; otherwise, the
distance of the image is less than that of the object : in the
former case therefore, the image is larger than the object;
in the latter, it is less.

When the images are less than the objects, they appear
more distinct and vivid ; because then more rays arc ac-
cumulated into a given space. But if the images be made
greater than the objects, they will not appeal distinctly ;
because in that case there are fewer lays which meet after
refraction in the same point; whence it happens, that rays
proceeding from different points of an object, terminate in
the same point of an image, which is (he cause of confusion.
Hence it appears that the same aperture of a lens may be
admitted in every case, if we would keep off the rays which
produce contusion. However, though the image be then
more distinct, when no rays are admitted but those near
the axis, yet for want of rays the image is apt to be dim.

6. If the eye be placed in the focus of a convex lens,
an object viewed through it, appears erect, and enlarged in
the ratio of the distance of the object from the eye, to
that of the eye from the lens, if it be near; but infinitely
if remote.

7th. An object viewed through a concave lens, appears
erect, and diminished in a ratio compounded of the ratios
of the space in the axis between the point of incidence,
and the point to which an oblique ray would pass without
refraction, to the space in the axis between the eye and
the middle of the object; and the space in the same axis
between the eye and the point of incidence, to, the space
between the middle of the object and the point to which
the oblique ray would pass without refraction.

Finally, it may be observed, that the very small magni-
fying glasses used in microscopes, most properly come un-

der the denomination of lens, as they most approach (o
the figure of the lentil, a seed of the vetch or pea kind,
from whence the name is derived ; but the reading glasses,'
and burning glasses, and all that magnify, come under the
same denomination ; lor their surfaces are convex, though
less so. A drop of water is a lens, and it will serve as
one ; and many have used it by way of lens in their micro-
scopes. A drop of any transparent fluid, enclosed be-
tween two concave glasses, acquires the shape of a lens, and
has all its properties. The crystalline humour of the eye
is a lens exactly of this kind ; it is a small quantity of a
translucent fluid, contained between two concave and
transparent membranes, called the coats of the eye; and
it acts as the lens made of water would do, in an equal de-
gree of convexity.

LEO, the Lion, a considerable constellation of the
northern hemisphere, being one of the 48 old constella-
tions, and the 5th sign of the zodiac. It is marked thus
Si, as a rude sketch of the animal. The Greeks fabled
that this was the Nemsan lion, which had dropped from
the moon, but being slain by Hercules, was raised to the
heavens by Jupiter, in commemoration of the dreadful
conflict, and in honour of that hero. But the hiero<dy-
phical meaning of this sign, so depicted by the Egyptians
long before the invention of the fables of Hercules, was
probably no more than to signify, by the fury of the lion,
the violent heats occasioned by the sun when he entered
that part of the ecliptic. — The stars in the constellation
Leo, in Ptolemy's catalogue are 27, besides 8 unformed
ones, now counted in the constellation Coma Berenices;
in Tycho's 30, in that of Hevelius 49, and in Flamsteed's
95; one of them, of the first magnitude, in the breast of
the Lion, is called Regnlus, and Cor Leonis, or Lion's
Heart.

Leo Minor, the Little Lion, a constellation of the north-
ern hemisphere, and one of the new ones that were formed
out of what were left by the ancients, under the name of
stelte informes, or uniformed stars, and added to the 48 old
ones. It contains 53 stars in Flamsteed's catalogue.

Cor Leonis, Lion's heart, a fixed star of the first magni-
tude, in the sign leo; called also regulus, basilicus, &c.

LEPUS, the Hare, a constellation of the southern he-
misphere, and one of the 48 old constellations. The
Greeks fabled, that this animal was placed in the heavens,
near Orion, as being one of the animals which he hunted.
But it is probable their masters, the Egyptians, had some
other meaning in this hieroglyphic. The stars in the con-
stellation lepus, in Ptolemy's catalogue are 12, in Tycho's
13, and in Flamsteed's 19.

LEUC1PPUS, a celebrated Greek philosopher and ma-
thematician, who flourished about the 42Sth year before
Christ. He was the first author of the famous system of
atoms and vacuums, and of the hypothesis of storms ; since
attributed to the moderns.

LEVEL, an instrument used to make a line parallel to-
the horizon, and to continue it out at pleasure; and by this
means to find the true level, or the differc/ice of ascent or
descent between two or more places, for conveying water,
draining fens, &c. There are several instruments, of dif-
ferent constructions and materials invented for the per-
fection of levelling, as may be seen in Lahire'sand Picard's
treatises of levelling, in Biron's treatise on mathematical
instruments, as also in the Philos. Trans, and the Memoircs
de l'Acad. &c. But they may be reduced to the follow-
ing kinds.

LEV

[ 720 ]

LEV

Wa/er-LEVEL, that which shows the horizontal line by
means of a surface of water or other fluid ; founded on this
principle, that water always places itself level or horizon-
tal. The most simple kind is made of a long wooden
trough or canal ; which being equally filled with water,
its surface shows the. line of level. And this is the cho-
robates of the ancients, described by Vitruvius, lib. vm,
cap. 6. The water-level is also made with two cups fitted
to the two ends of a straight tube, about an inch diameter,
and 3 or 4 feet long, by means of which the water com-
municates from the one cup to the other; and this pipe
being moveable on its stand by means of a ball and socket,
when the two cups show equally full of water, their two
surfaces mark the line, of level.

This instrument, instead of cups, may also be made with
two short cylinders of glass 3 or 4 inches long, fastened to
each extremity of the pipe with wax ,or mastic. The pipe
is filled with common or coloured water, which shows
itself through the cylinders, by means of which the line of
level is determined ; the height of the water, with respect
to the centre of the earth, being always the same in both
cylinders. This level, though very simple, is however
very commodious for levelling small distances. See the
method of preparing .and using a water-level, and a mer-
curial level, annexed to Davis's quadrant, ior the same
purpose, by Mr. Leigh, in Philos. Trans, vol. xl. 417, or
my Abr. viii, 260.

^jV-Level, is that which shows the line of level by
means of a bubble of air, enclosed with some fluid in a
glass tube, of an indeterminate-length and thickness, and
having its two ends hermetically scaled : an invention, it
is said, of M. Thevenot. When the bubble fixes itself at
a certain mark, made exactly in the middle of the tube, the
case or ruler in which it is fixed, is then level; and when
it is not level, the bubble will rise to one end. This glass-
tube may be set in another of brass, having an aperture in
the middle, where the bubble of air may be observed. The
liquor with which the tube is filled, is usually either oil of
tartar, or aqua secunda; those not being liable to freeze
as common water, nor t6 rarefaction and condensation as
spirit of wine is.

There is one of these instruments with sights, being an
improvement on that last described, which, by the addition

Mr. Huygens. It is like the last; with only this difference,
that instead of plain sights, it carries a telescope, to deter-
mine exactly a point of level at a considerable distance.
The screw 3, is for raising or lowering a little fork, which
carries the hair, and making it agree with the bubble of
air when the instrument is level; and the screw 4, is for
making the bubble of air, d or n, agree with the telescope.
The whole is fitted to a ball and socket, or otherwise moved
by joints and screws. — It may be observed that a tele-
scope may be added to any kind of level, by applying it
upon, or parallel to, the base or ruler, when there is occa-
sion to take the level of remote objects: and it possesses
this advantage, that it may be inverted by turning the ruler
and telescope half round; and if then the hair cut the
same point that it did before, the operation is just. Many
varieties and improvements of this instrument have been
made by the more modern opticians.

Dr. Desaguliers proposed a machine for taking the dif-
ference of level, which contained the principles both of a
barometer and thermometer ; but it is not accurate in prac-
tice: Philos. Trans, vol. xxxiii, pa. 165, or my Abr.vol.vii.,
49. Fig. 3, 4, 5, 6.

Mr. Hadley too has contrived a SpiritLevel to be fixed
to a quadrant, for taking a meridian altitude at sea, when
the horizon is not visible. See the description and figure
of it in the Philos. Trans, vol. xxxviii, 167, or Abr. vii,
620. Various other spirit levels, and mercurial levels, are
also invented and used on different occasions.

Reflecting Level, that made by means of a pretty long
surface of water, representing the same object inverted,
which we see erect by the eye; so that the point where these
two objects appear to meet, is on a level with the place
where the surface of the water is found. This is the in-
vention of M. Mariotte.

There is another reflecting level, consisting of a po-
lishod metal mirror, placed a little before the object glass
of a telescope, suspended perpendicularly. This mirror
must be set at an angle of 45 degrees ; in which case the
perpendicular line of the telescope becomes a horizontal
line, or a line of level. Which is the invention of M.
Cassini.

Artillery Foo£-Level, is in form of a square (fig. 7),
having its two legs or branches of an equal length ; at the

of other apparatus, becomes more exact and commodious, junction of which is a small hole, by which hangs a plum

It consists of an air-level, No. 1, (Jig. \, pi. 18) about S
inches long, and about two thirds of an inch in diameter,
set in a brass tube, 2, having an aperture in the middle c.
The tubes are carried in a strong straight ruler, of a foot
long; at the ends of which are fixed two sights, 3, 3, ex-
actfy perpendicular to the tubes, and of an equal height,
having a square hole, formed by two fillets of brass crossing
each other at right anglqs; in the middle of which is
drilled a very small hole, through which a point on a level
with the instrument is seen. The brass tube is fastened
to the ruler by means of two screws ; the one of which,
marked 4, serves to raise or depress the tube at pleasure,
for bringing it towards a level ; the top of the ball and
socket is rivetted to a small ruler that springs, one end of
which is fastened with springs to the great ruler, and at
the other end is a screw, 5, serving to raise and depress the
instrument when nearly level. But this instrument is less
commodious than the following one : for though the holes
be ever so small, yet they will still take in too great a space
to determine the point of level precisely

met playing on a perpendicular line in the middle of a
quadrant, which is divided both ways from that point into
45 degrees. This instrument may be used on other occa-
sions, by placing the ends of its two branches on a plane;'
for when the plummet plays perpendicularly over the mid-
dle division of the quadrant, the plane is then level. To
use it in gunnery, place the two ends on the piece of ar-
tillery, which may be raised to any proposed height, and
the plummet will cut the degree above the level. But this
supposes the outside of the cannon is parallel to its axis,
which is not always the case; and therefore another in-
strument is employed either to set the piece level, or ele-
vate it at any angle : namely a small quadrant, with one
of its radii continued out pretty long, which being put
into the inside of the cylindrical bore, the plummet shows
the angle of elevation, or the line of level. See Gunner's
Quadrant.

Carpenter's, Bricklayer's, or Pavior's Level, consists of
a long ruler, in the middle of which is fitted at right angles
another broader piece, at the top of which is fastened a

Fig. 2, is a Levetwilh Telescopic Sights, first invented by plummet, which when it hangs over the middle line of the

L E V

[ 72i ]

LEV

2d or upright piece, shows that the base or long ruler is ho-
rizontal or level. Fig. S.

Mason's Level, is composed of 3 rules, so jointed as to
form an isosceles triangle, somewhat like a Roman a ; from
the vertex of which is suspended a plummet, which hangs
directly over a mark in the middle of the base, when this
is horizontal or level. Fig. 8.

Plumb or Pendulum Level, said to be invented by M.
Picard; fig. 10. This shows the horizontal line by means
of another line perpendicular to that described by a plum-
met or pendulum. It consists of two legs or branches,
joined at right angles, the one of which, of about IS inches
long, carries a thread and plummet; the thread being hung
near the top of the branch, at the points. The middle of
the branch where the thread passes is hollow, so that it may
hang free every-where: but towards the bottom, where
there is a small blade of silver, on which a line is drawn
perpendicular to the telescope, the said cavity is covered
by two pieces of brass, with a piece of glass a, to see the
plummet through, forming a kind of case, to prevent the
wind from agitating the thread. The telescope, of a
proper length, is fixed to the other leg of the instrument,
at right angles to the perpendicular, and having a hair
stretched horizontally across the focus of the object-
glass, which determines the point of level, when the
string of the plummet hangs against the line on the silver
blade : the whole being fixed by a ball and socket to its
stand.

Fig. 12, is a Balance Level; which being suspended by
the ring, the two sights, when in equilibrio, will be hori-
zontal, or in a level..

Some other leve'is are also represented in pi. IS.
LEVELLING, the art or act of finding a line parallel
to the horizon ».t one or more stations, to determine the
height or depth of one place with respect to another; for
laying out grounds even, regulating descents, draining
morasses, conducting water, &c.

Two or more places are on a true level when they are
equally distant from the centre of the earth. Also one
place is higher than another, or out of level with it, when
it is farther from the centre of the earth: and a line equally
distant from that centre in all its points, is called the line
of True Level. Hence, because the earth is round, that
line mus'c be a curve, and make a
part of t'ne earth's circumference, or
at least parallel to it, or concentrical f-y ~}iv

with it ; as the line bceg, which has
all its points equally distant from a
the ce ntre of the earth ; considering
it as 'a perfect globe.

B'ut the line ofsight bdf &c, given
by t .he operations of levels, is a tan-
gen t, or a right line perpendicular to

thf • semidiameter of the earth at the point of contact B>
rif ,ing always higher above the true line of level, the farther
il le distance is, this being called the Apparent Line of
J^evel. Thus, CD is the height of the apparent level
above the true level, at the distance bc or bd; also ef
is the excess of height at E ; and gh at g; &c. The dif-
ference, it is evident, is always equal to the excess of
the secant of the arch of distance above the radius of the
earth.

The common methods of levelling arc sufficient for lay-
ing pavements of walks, or for conveying water to small
distances, &c : but in more extensive operations, as an
Vol. I.

levelling the bottoms of canals, which are to convey water
to the distance of many miles, and such like, the difference
between the true and the apparent level must be taken into
the account.

Now the difference CD between the true and apparent
leve), at any distance bc or bd, may be found thus : By a
well-known property of the circle 2ac h- cd : bd : : bd :
CD ; or because the diameter of the earth is so great with
respect to the line CD at all distances to which an opera-
tion of levelling commonly extends, that 2a c may be safely
taken for 2ac -+- cd in that proportion without any sen-
sible error, it will be 2ac : bd : : bd : CD, which there-

c .. ED2 BC3 ,,.,,.„

tore is = -— or — - nearly ; that is, the difference be-

2AC 2AC J

tween the true and apparent level, is equal to the square
bf the distance between the places, divided by the diame-
ter of the earth ; and consequently it is always propor-
tional to the square of the distance.

Now the diameter of the earth beingnearly 7914 miles ;
if we first take bc = 1 mile, then the excess — becomes

' 2AC

—^ of a mile, which is S inches, the height of the appa-
rent above the true level at the distance of one mile.
Hence, proportioning the excesses in altitude according to
the squares of the distances, the following table is obtain-
ed, which shows the height of the apparent above the true
level for every 100 yards of distance on the one hand, and
for every mile on the other.

Dif. of level,

or CD.

Yards.

Inches.

100

0-026

200

0-103

300

0-231

400

0-411

500

0-643

600

0-925

700

1-260

800

1-654

900

2-081

1000

2-570

1100

3-110

1200

3-701

1300

4-344

1400

5-038

1500

' 5-784

1600

6-580

1700

7-425

Dist. or BC

Dif. of level,
of CD.

Miles.

Feet. Inches.

r.

0 0£

\

0 2

3_

0 4-£

1

0 8

2

2 8

3

6 0

•1

10 7

5

16 7

6

23 11

7

32 6

8

42 6

9

53 9

10

66 4

11

80 3

12

95 7

13

112 2

14

130 1

By means of these tables of reductions, we can now
level to almost any distance at one operation, which the
ancients could not do but by a great multitude ; for,
being unacquainted with the correction answering to par-
ticular distances, they only levelled from one 20 yards to
another, when they had occasion to continue the work to
any considerable extent.

This table will answer several useful purposes. Thus,
first, to find the height of the apparent above the true
level, at any distance. If the given distance be contained
in the table, the correction of level is found on the same
line with it : thus at the distance of 1000 yards, the cor-
rection is 2-57, or two inches and a half nearly ; and at
the distance of 10 miles, it is 66 feet 4 inches. But if
4Z

L E V

the exact distance be not found in thetahle, then multiply
the square of the distance in yards by 2"57, and divide by
1,000,000, or cut off 6' places on the right for decimals,
and the rest will be inches : or multiply the square of the
distance in miles by 66' feet 4 inches, and divide by 100.
2ndly, To find the extent of the visible horizon, or how
far can be seen from any given height, on a horizontal
plane, as at sea, &c : Suppose the eye of an observer, on
the top of a ship's mast at sea, be at the height of 1 30 feet
above the water, he will then see about 14 miles all
around. Or from the top of a cliff by the sea-side, the
height of which is 66 feet, a person may see to the di-
stance of near 10 miles on the surface of the sea. Also,
when the top of a hill, or the light in a lighthouse, or
such like, whose height is 130 feet, first comes into the
view of an eye on board a ship ; the table shows that the
distance of the ship from it is 14 miles, if the eye be at-
the surface of the water; but if the height of the eye. in
the ship be 80 feet, then the distance will be increased by
near 11 miles, making in all about 25 miles, distance.

3dly, Suppose a spring to be on one side of a hill, and
a house on an opposite hill, with a valley between them ;
and that the spring seen from the house appears, by a le-
velling instrument, to be on a level with the foundation
of the house, which suppose is at a mile distance from it;
then is the spring "8 inches above the true level of the
house ; and this difference would be about sufficient for
the water to be brought in pipes from the spring to the
house.

4th, If the height or distance exceed the limits of the
table : Then, first, if the distance be given, divide it by 2,
or by 3, or by 4, &c, till the quotient come within the
distances in the table ; then take out the height answering
to the quotient, and multiply it by the square of the di-
visor, that is by 4, or 9, or 16, &c, for the height re-
quired : So, if the top of a hill be just seen at the distance
of 40 miles; then 40 divided by i gives 10, to which in
the table answers 66% feet, wlych being multiplied by 16,
the square of 4, gives IO614 feet for the height of the hill.
But when the height is given, divide it by one of these
square numbers 4, 9> 16, 25, &c, till the quotient come
■within the limits of the table, and multiply the quotient by
the square root of the divisor, that is by 2, or 3, or 4, or
5, &c, for the distance sought : So when the top of the
peak of Teneriff, said to be almost 3 miles or 15,840 feet
hit»h, just comes into view at sea ; divide 15,840 by 225,
or the square of 1 5, and the quotient is 70 nearly ; which
answers in the table, by proportion, to about lOf. miles;
then multiplying 10f by 15, gives 154-f- miles, for the di-
stance of the hill.

On the Practice of Levelling.

The operation of levelling is as follows. Suppose the
height of the point a on the top of a mountain, above that
of b, at the foot of it, be required. Place the level about
the middle distance at D, and set up pickets, poles, or
staffs, at a and b, where persons must attend with signals
for raising and lower-

[ 722 ] LEV

there toraise or lower the mark, till it appear through
the telescope, or sights, &c, at e : then measure exactly
the perpendicular height of the point e above the point a,
which suppose 5 feet S inches, set it down in your book.
Then direct your sight the other way, towards the pole b,
and cause the person there to raise or lower his mark,
till.it appear in the visual line as before at c; and mea-
suring the height of c above b, which suppose 15 feet 6
inches; set this down in your , book also, immediately
above the number of the first observation. Then subtract
the one from the other, and the remainder 9 feet 10
inches, will be the difference of level between a and B, or
the height of the point a above the point b.

It the point d, where the instrument is fixed, be ex-
actly in the middle between the points a and b, there
will be no necessity for reducing the apparent level to the
true one, the visual ray on both sides being raised equally
above the true level. But if not, each height must be cor-
rected or reduced according to its distance, before the one
corrected height is subtracted from the other; as in the
case following.

ing, on the said poles,
little marks of paste-
board or other matter.
The level having been
placed horizontally by
the bubble, &c, look
towards the staff ae,
and cause the person

y

When the distance is very considerable, or irregular, so
that the operation cannot be effected at once placing of
the level; or when it is required to know if there be a
sufficient descent for conveying water from the spring a
to the point B ; it will be necessary to perform this at
several operations. Having chosen a proper place for the
first station, as at I, fix a pole at the point a near the
spring, with a proper mark to slide up and down it, as l;
and measure the distance from a to 1. Now the level being
adjusted in the point 1, let the mark l be raised or lowered
till it is seen through the telescope or sights of the level,
and measure the height al. Then having fixed another
pole at H, direct the level to it, and cause the mark g to
be moved up or down till it appear through the instru-
ment : then measure the height gh, and the distance from
1 to h ; noting them down in the book. This done, re-
move the level forwards to some other eminence as e,
from which the pole h may be viewed, as also another
pole at d; and having adjusted the level in the point e,
look back to the pole 11 ; managing the mark as before,
till the visual ray will give the point F ; then measuring
the distance he and the height HP, note them down in the
book. And here turn the level to look at the next pole
d, and the visual ray will give the point d ; there measure
the height of d, and the distance eb, entering them in the
book as before. And thus proceed from one station to
another, till the whole is completed.

But all these heights must be corrected or reduced by
the foregoing table, according to their respective di-
stances ; and the whole, both distances and heights, with
their corrections, entered in the book in the following
manner.

L E V

[ 723 ]

LEV

Buck-rights.

Fore-sialtts.

Dists.

yds

EB

000

2165

2590

Dis

.-1755

His.

ft.

27

6

6-1

20

n-i

21

o-s

Having summed up all the columns, add those of the
distances together, and the whole distance from a to b is
4755 yards, or 2 miles ,and 3 quarters nearly. Then,
the sums of the corrections taken from the sums of the
apparent heights, leave the two corrected heights; the
one of which being taken from the other, leaves 5 feet
If) inc. for the true difference of level sought between
the two places a and b, which is at the rate of an inch and
half nearly to every 100 yards, a quantity more than suffi-
cient to cause the water to run from the spring to the house.

Or, the operation may be otherwise performed, thus :
Instead of placing the level between every two poles, and
taking both back-sights and fore-sights; plant it first at
the spring a, and thence observe the level to the first pole;
then remove it to this pole, and observe the 2d pole ;
next move it to the 2d pole, and observe the 3d pole ; and
so on, from one pole to another, always taking forward
sights or observations only. Then at the last, add all the
corrected heights together, and the sum will be the whole
difference of level sought.

Dr. Halley suggested a new method of levelling, per-
formed wholly by means of the barometer, the mercury
being suspended at so much the less height, as the place
is farther remote from the centre of the earth ; and hence
the different heights of the mercury in two places, give
the difference of level. This method is, in fact, no other
than the method of measuring altitudes by the barometer,
which has lately been so successfully practised and per-
fected by M. Deluc and others; but though it serves very
well for the heights of hills, and other considerable alti-
tudes, it is not accurate enough for determining small al-
titudes, to inches and parts. See the Barometrical Mea-
surement of Altitudes.

Levelling Poles, or Staves, are instruments used in
levelling, for carrying the marks to be observed, and at
the same time to measure the heights of those marks from
the "round. They usually consist each of two long wooden
rulers, made to slide over each other, and divided into
feet and inches, &c.

LEVER, a straight bar of iron or wood, &c, supposed
to be inflexible, supported on a fulcrum or prop by a
single point, about which all the parts are moveable. The
lever is the first of those simple machines called mecha-
nical powers, being the simplest of them all ; and is
chiefly used for raising great weights to small heights.

The lever is of three kinds. First the common sort,
where the weight intended to be raised is at one end of it,-
our strength or another weight called the power is at the
other end, and the prop or fulcrum is between them
both. In stirring up the fire with a poker, we make use
of this lever ; the poker is the lever, it rests on one of the
bars of the grate as a prop, the incumbent fuel is the
weight to be overcome, and the pressure of the hand on
the other end is the force or power. In this,.as in all the
other machines, we have only to increase the distance
between the force and the prop, or to decrease the di-
stance between the weight and the prop, to give the ope-
rator the greater power or effect. To this kind of lever
may also be referred all scissars, pincers, snuffers, &c.
The steel-yard and the common balance are also levers of
this kind.

In the lever of the 2d kind the prop is at one end, the
force or power at the other, and the weight to be raised
is between them. Thus, in raising a -water-plug in the
streets, the workman puts his iron bar or lever through
the ring or hole of the plug, till the end of it reaches the
ground on the other side; then making that the prop, he
lifts the plug by his force or strength applied at the other
end of the lever. In this levertoo, the nearer the weight
is to the prop, or the farther the power from the prop, the
greater is the effect. To this 2d kind of lever may also
be referred the oars and rudder of a boat, the masts of a
ship, cutting knives fixed at one end, and doors, whose
hinges serve as a fulcrum.

In the lever of the third kind, the power acts between
the weight and the prop; such as a ladder raised by a
man somewhere between the two ends, to rear it against
a wall, or a pair of tongs, &c.

*\

JL

It is by this kind of lever also that the muscular mo-
tions of animals are performed, the muscles being inserted
much nearer to the centre of motion, than the point where
is placed the centre of gravity of the weight to be raised ;
so that the power of the muscle is many times greater
than the weight it is able to sustain. And in this third
kind of lever, to produce a balance between the power and
weight, the power or force must exceed the weight, in the
same proportion as it is nearer the prop than the weight
is ; whereas, in the other two kinds, the power is less
than the weight, in the same proportion as its distance is
greater; that is, universally, the power and weight air

4 Z2

LEV

I 72* ]

LEV

each of them reciprocally as their distance from the prop; found that this demonstration serves also for the other

as is demonstrated below. kinds of levers, by drawing the lines as directed. Hence,

Some authors make a 4th kind, of what is called a if any given force p be applied to a lever at a ; its effect

bended lever; such as a hammer in drawing a nail, &c. upon the lever, to turn it about the centre of motion c, is

In all levers, the universal property is, that the effect as the length of the arm ca, and the sine of the angle of

of either the weight or the power, to turn the lever about direction cae. For the perp. ce is as ca x sin. Z. a.

the fulcrum, is directly as its force and its distance from It having been objected to the principle of this demon-

the orop, that is as df, where d denotes the distance, and stration, that it inferred the stability of the lever, or the

/the force, strength, or weight, &c, of the agent. For line pw, from the theorem that three forces meeting in a

since it is proved that at a double distance it will have a point, and keeping that point, or a body there, stable or

double effect, at a triple distance a triple effect, and so fixed, are proportional to the three sides of a, triangle

on ; also since a double force produces a double effect, a which are parallel to their directions ; which is a miscon-

triple force a triple effect, and so on ; therefore universally ception. For, it is only meant that the point o is stable

the effect is as df the product of the two. In like manner, by that theorem. As to the stability of the lever pw,that

ifD be the distance of another power or agent, whose in- is inferred from its being perp. to the given line oc, and

tensity is F, then is df the effect of this also to move the limited by the two lines op, ow, given in position. — Since

lever. And if these two agents act against each other on the time when the above demonstration was printed in the

the lever, and their effects be supposed equal, or the first edition of this Dictionary, I have seen another similar

lever kept in equilibrio by the equal and contrary effects one that had been given in Dr. Hamilton's Essays.

of these two agents ; then is df = df, which equation re- In any analogy, because the product of the extremes is

solves into this analogy, viz, D : d : if : F ; that is, the equal to that of the means ; therefore the product of the

distances of the agents from the prop, are reciprocally or power by the distance of its direction is equal to the pro-

inversely as their forces, or the power is to the weight, as duct of the weight by the distance of its direction. That

the distance of the latter from the prop is to the distance is> p x ce = w x cd.

of the former. If the lever, with the two weights fixed to it, be made

Writers on mechanics commonly demonstrate this pro- to move about the centre c ; the momentum of the power

portion in a very absurd manner, viz, by supposing the wjll be equal to that of the weight; and the weights will

lever put into motion about the prop, and then inferring be reciprocally proportional to their velocities,

that, because the momenta of two bodies are equal, when When the two forces act perpendicularly on the lever,

as two weights &c ; then, in case of an equilibrium, E

coincides with p, and d with w ; and the distances ci>,
cw, taken on the lever, or the distances of the power and
weight, from the fulcrum, are reciprocally proportional to
the power and weight.

In a straight lever, when kept in equilibrio by a weight
and power acting perpendicularly upon it ; then, of these
three, the power, weight, and pressure on the prop, any
one is as the distance of the other two.

And hence too p +'w : P :': ed : CD,
and r ■+- w : w : : ed : cp;
that is, the sum of the weights is to either of them, as the
sum of their distances is to the distance of the other.

placed upon the lever at such distances, that these dis-
tances are reciprocally proportional to the weights of the

bodies, and that therefore this is also the proportion in

case of an equilibrium; which is an attempt absurdly to

demonstrate a thing, supposing the contrary, that a body

is at rest, by supposing it to be in motion. I shalLthere-

fore give here a universal demonstration of the property,

on the pure principles of rest and

pressure, or force only. Thus, let

pw be a lever, c the prop, and p and

w any two forces acting on the lever

at the points p and w, in the direc-
tions po, wo ; then if ce and cd be

the perpendicular distances of the
directions of these forces from the

prop c, it is to be demonstrated that
p : w : : cd : ce. In order to which

join co, and draw cb parallel to wo, °

and cf parallel to po. Then will co, the perpendicular to
pw, be the direction of the pressure on the prop, other-
wise there could be no equilibrium, for the directions of
three forces that keep each other in equilibrium, must ne-
cessarily meet in the same point. And because any three
forces that keep each other in equilibrium, are propor-
tional to the three sides of a triangle formed bv drawing lever, and keep it in equilibrio ; then the sum oi the pro-
lines parallel to the directions of these forces /therefore ducts on one side of the prop, will be equal to the sum on
the forces on p, c, and w, are as the three lines no, co, the other side, made by multiplying each weight by its
cb, which are in the same direction, or parallel to them ; distance from the prop; viz, p . ac + Q . bc _ n . dc + s .
that is, the force p is to the force w, as bo or its equal cf ec + &c
is to cb. But the two triangles cdf, ceb are equiangular,
and have their like sides proportional,

viz, cf : cb : : CD : ce;
and because it was cf : cb : : p : w ;
therefore by equality p : w : : cd : ce ;
that is, each force is reciprocally proportional to the dis-
tance of its direction from the fulcrum. And it will be

6,

©a.

Also, if several weights P, q, r, s, &c, act on a straight

Hitherto the lever has been considered as a mathema-
tical line void of weight or gravity. But when its weight
is considered, it is to be done thus: Find the weight and
the centre of gravity of the lever alone, and then consider
it as a mathematical line, but having an equal weight sus-
pended by that centre of gravity ; and so combine its
effect with those of the other weights, as above.

L E Y

[ 7.25 ]

L I B

On the foregoing principles depends the nature of scales
and beams for weighing all bodies. For, if the distances
be equal, then will the weights be equal also ; which gives
the construction of the common scales. And the Roman
statera, or steel-yard, is also a lever, but of unequal arms
or distances, so contrived that one weight only may serve
to weigh a great many, by sliding it backwards and for-
wards to different distances upon the longer arm of the
lever. See Balance, &c.

Also, on the principle of the lever depend almost all
other mechanical powers and effects. See Wheel-asd-1
axle, Pulley, Wedge,. Screw, &c.

LEVITY, the privation or want of weight in any body,
when compared with another that is heavier ; in which
sense it stands opposed to gravity. Thus cork, and most
sorts of wood that float in water, have levity with respect
to water, that is, are less heavy. The schools maintained
that there was such a thing as positive and absolute le-
vity; and to this they imputed the rise and buoyancy of
bodies lighter in specie than the bodies in which they rise
and float. But it is now well known that this happens
only in consequence of the heavier and denserfluid, which,
by its superior gravity, gains the lowest place, and raises
up the lighter body by a force which is equal to the dif-
ference of their gravities. It was demonstrated by Archi-
medes, that a solid body will float any-where in a fluid of
the same specific gravity as itself; and that a lighter body
will always be raised up in it.

LEUWENHOEK (Antony), a celebrated Dutch phi-
losopher, was born at Delft in 1632; and acquired a
great reputation throughout all Europe, by his experi-.
ments and discoveries in natural history, by means of the
miscroscope. He particularly excelled in making glasses
for microscopes and spectacles; and he was a member
of most of the literary societies of Europe; to whom he
sent many memoirs. Those in the Philosophical Trans-
actions, and in the Paris Memoirs, extend through
many volumes ; the former were extracted, and publish-
ed at Leyden, in 1722. He died in 1723, at 91 years
of age.

LEYBOURN (William), a respectable and very useful
mathematician, of the 17th century, died about the year
1690. He was, it seems, originally a printer in London;
but by industry became a considerable proficient in ma-
thematics, and published a number of very useful and po-
pular books, on the practical parts of those sciences.
Those of them that are in my possession, are the follow-
ing: 1. Compleat Surveyor, in folio, J6>53, afterwards
re-published and improved by Cunn. — 2. Nine Geome-
trical Exercises, in 4to, 1669- — 3. Dialling, in 4to, 16S7;
the same in folio, 1700. — 4. Cursus Mathematicus, 2 vols
folio, 1690, then held in much esteem. — 5. Recreations,
in folio, 1694. — 6. Arithmetic, 8vo, 1700. — 7. On the
Gunter's Quadrant, 12mo, 1731. There may be other edi-
tions of these works, and probably earlier ones of several
of them, as well as of other works which I have not seen,
such as one entitled the Trader's Guide.

LEYDEN Piiial, in Electricity, is a glass phial or
jar, coated both within and without with tin-foil, or some
other conducting substance, that it may be charged, and
thus employed in a variety of useful and entertaining ex-
periments. Or even flat glass, or any other shape, so
coated and used, has also received the same denomination.
Also a vacuum produced in such a jar, &c, has been
Mamed the Leyden Vacuum.

The Leyden phial has been so called, because it is said
that M. Cunseus, a native of Leyden, first contrived, about
the close of the year 1745, to accumulate the electrical
power in glass, and to use it in the same manner as at
present. But Dr. Priestley asserts that this discovery was
first made by Von Kleist, dean of the cathedral in Camin;
who, on the 4th of November 1745, sent an account of
it to Dr. Lieberkuhn at Berlin : hqwever, those tp whom
Kleist's account was communicated, could not succeed in
performing his experiments. The chief circumstances of
this discovery are slated by Dr. Priestley in his Hist, of
Electricity, vol. 1, pa. 191, &c.
LIBRA. See Balance.

Libra is aiso one of the 4S old constellations, and the
7th sign of the zodiac, being opposite to Aries, and marked
like a part of a pair of scales, thus =£>:. The figure of the
balance was probably given to this part of the ecliptic,
because when the sun arrives at this part, which is at the
time of the autumnal equinox, the days and nights are
equal, as if weighed in a balance. The stars in this con-
stellation are, according to Ptolemy 17, Tycho 10, Heve-
lius 20, and Flamsteed 51.

Libra also denotes the ancient Roman pound, which
was divided into 12 unciae, or ounces, and the ounce into
24 scruples. It seems the mean weight of the scruple
was nearly equal to 17! grains Troy, and consequently the
libra, or pound, equal to 5040 grains. It was also the
name of a gold coin, equal in value to 20 denarii. See
Philos. Trans, vol. 6l, pa. 462.

The French livre is derived from the Roman libra, as
this was from the Sicilians, who called it litra. This was
used in France for the proportions of their coin till about
the year 1100, their sols being so proportioned as that 20
of them were equal to the libra. By degrees it became a
term of account, and every thing of the value of 20 sols
was called a livre.

LIBRATION, of the Moon, is an apparent irregularity
in her motion, by which she seems to librate, or waver,
about her own axis, at one time towards the east, and at
another towards the west. See Moon, and Evection.
Hence it is that some parts, near the moon's western edge,
at one time recede from the centre of the disc, while those
on the other or eastern side approach nearer to it; and,
on the contrary, at another time the western parts are
seen to be nearer the centre, and the eastern parts farther
from it: by which means it happens that some of those
parts, which were before visible, set and hide themselves
in the hinder orinvisible side of the moon, and afterwards
return and appear again, on the nearer or visible side.

This libration of the moon was first discovered by He-
velius, in the year l6"54; and it is owing to her equable
rotation round her own axis, once in a month, in conjunc-
tion with her unequal motion in the perimeter of her orbit
round the earth. For if the moon moved in a circle,
having its centre coinciding with the centre of the earth,
while it turned on its axis in the precise time of its period
round the earth, then the plane of the same lunar meri-
dian would always pass through the earth, and the same
face of the moon would be constantly and exactly turned
towards us. But since the real motion of the moon is
about a point at a considerable distance from the centre
of the earth, that motion is very unequal, as seen from
hence, the plane of no one meridian constantly passing
through the earth.
The libration of the moon is of three kinds.

L I V

[ 726 ]

L I F

1st, Her libration in longitude, or a seeming vibrating
motion according to the order of the signs of the zodiac.
This libration is nothing twice in each periodical month,
viz, when the moon is in her apogcum, and when in her
perigeum ;-for in both these cases the plane of her meri-
dian, which is turned towards us, is directed alike towards
the earth.

2d, Her libration in latitude; which arises from hence,
that her axis not being perpendicular to the plane of her
orbit, but inclined to it, sometimes one- of her poles, and
sometim.es the other will decline, as it were, or dip a little
towards the earth, and consequently she will appear to li-
brate a little, and to show sometimes more of her spots,
and sometimes less of them, towards each pole; Which
libration, depending oil the position of tin- moon, in re-
spect to the 'nodes of ner orbit, and her axis being nearly
perpendicular to the plane ot the ecliptic, is very properly
said to be in latitude : and this also is completed in the
space of the moon's periodical month, or rather while the
moon is returning again to the same position, in respect
of her nodes.

3d, There is also a third kind of libration; by which
it happens that though another part of the moon be not
reall) turned to the earth, as in the former libration, yet
another is illuminated by the sun. For since the moon's
axis is nearly perpendicular to the plane of the ecliptic,
when she is most southerly, in respect of the north pole of
the ecliptic, some parts near to it will be illuminated by
the sun; while, on the contrary, the south pole will be in
darkness. In this case therefore, if the sun be in the same
line with the moon's southern limit, then, as she proceeds
from conjunction with the sun towards her ascending
node, she will appear to dip her northern polar, parts a
little into the dark hemisphere, and to raise her southern
polar parts as much into' the light one : and the contrary
to this will happen two weeks after, while the new moon
is descending from her northern limit; for then her nor-
thern polar parts will appear to emerge out of darkness,
and the southern polar parts to dip into it: which seeming
libration, or rather these effects of the former libration in
latitude, depending on the light of the sun, will be com-
pleted in the moon's synodical month. Greg. Astrsn.
lib. 4-, sect. 10.

Libration of the Earth, is a term applied by some
astronomers to that motion, by which the earth is so re-
tained in its orbit, that its axis -continues constantly pa-
rallel to the axis of the world. This Copernicus calls the
motion of libration, which may be thus illustrated : Sup-
pose a globe, with its axis parallel to that of the earth,
painted on a flag at the head of a mast, moveable on its
axis, and constantly driven by an east wind, while it sails
round an island, it is evident that the painted globe will
be so librated, as that its axis will be parallel to that of
the world, in every situation of the ship.

LIFE-ANNUITIES, are such periodical payments as
depend on the continuance of some particular life or
lives; and they may be distinguished into annuities that
commence immediately, and such as commence at some
future period, called reversionary life-annuities.

The value, or present worth, of an annuity for any pro-
posed life or lives, it is evident, depends on two circum-
stances, the interest of money, and the chance or expec-
tation of the continuance of life. On the former only, it
has been shown, under the article Annuities, depends
the value or present worth of an annuity certain, or that
is not subject to the continuance of a life, or othei con-
tingency ; but the expectation of life being a thing not
certain, but only possessing a certain chance, ii ib evident
that the value of the certain annuity, as stated above,
must be diminished in proportion as the expectancy is be-
low'certainly : thus, if the present value of an annuity
certain be any sum, as suppose 100/. and the value or ex-
pectancy of the life be f, then the value of the life-an-
nuity will be only half of the former, or 50/; and if the
value of the life be only i, the value of. the life-annuity
will be but i of 100/, that is 33/. 6*. 8d ; and so on.

The measure of the value or expectancy of life, depends
on the proportion of the number of persons that die, out
of a given number, in the time proposed ; thus, if 50 per-
sons die, ou,t of 100, in any proposed time, then, half the
number only remaining alive, any one person has an equal
chance to live or die in that time, or the value of his life
for that time is J; but if \ of the number die in the time
proposed, or only -J- remain alive, then the value of any
one's life is ■§■; and if |. or the number die, or onlv i re-
man, alive, then the value of any life is buti; and so on.
In 'se proportions then must the value of the annuity
cerl ii iiiminibhed, to give the value of the like lile

annuity.

It is plain therefore that, in this business, it is necessary
to know the value of life at all the different ages, from
some table of observations on the mortality of mankind,
which may show the proportion of the persons living, out
of a given number, at the end of any limited time; or
from some certain hypothesis, or assumed principle. Now
various tables and hypotheses of this kind have been given
by the writers on this subject, as Dr. Halley, Mr. De-
moivre, Mr. Thomas Simpson, Mr. Dodson, Mr. Kersse-
boom, Mr. Parcieux, Dr. Price, Mr. Morgan, Mr. Baron
Maseres, Mr. Baily, and many others. But the same
table of probabilities of life will not suit all places; for
long experience has shown that all places are not equally
healthy, or that the proportion of- the number of persons
that die annually, is different for different places. Dr. Hal-
ley computed a table of the annual deaths as drawn from
the bills of mortality of the city of Breslaw in Germany :
Mr. Smart and Mr. Simpson from those of London; Dr.
Price fiom those of Northampton; Mr. Kersseboom from
those of the provinces of Holland and West-Friesland ;
and M. Parcieux from the lists of the French tontines, or
long annuities; and all these are found to differ from one
another. It may not therefore be improper to insert here
a comparative view of the principal tables that have been
given of this kind, as below ; where the first column shows
the age, and the other columns the number of persons
living at that age, out of 1000 born, or of the age 0, in the
first line of each column.

L I F

[ 727 ]

L I F

Table I. — Showing the Number of Persons living at all Ages, out of 1O0O that had been born at
several Places, viz.

Ages.

Vienna.

Berlin.

London.

Vorwich.

North-
ampton.

Breslaw.

Branden-
burg.

Holy-
Cross.

Holland.

France.

Vaud,
Switzerland.

0

1000

1000

1000

1000

1000

1 000

1000

1000

1000

1000

1000

1

542

633

680

798

738

769

775

882

804

805

811

2

47 1

528

548

651

628

658

718

702

768

777

765

3

430

4s5

492

595

585

614

687

7 17

736

750

735

4

400

434

452

566

562

5S5

604

682

709

727

715

5

377

403

426

544

544

563

642

659

689

711

701

6

357

387

410

526

530

540

622

636

670

697

688

7

344

376

397

511

518

532

607

618

664

686

677

8

337

367

388

500

510

523

595

604

652

676

667

9

331

361

380

490

504

515

585

595

646

" 667

659

10

326

356

373

4S1

498

508

577

589

639

660

653

11

322

3.33

367

474

493

502

570

585

633

654

648

12

318

350

361

469

488

497

564

581

627

649

643

13

314

347

356

464

484

492

559

577

621

644

639

U

310

344

351

460

480

488

554

573

616

639

635

1.5

306

341

347

455

475

483

549

5,69

611

635

631

16

302

338

343

451

470

479

544

565

606

631

626

17

299

335

33S

446

465

474

539

560

601

626

622

IS

295

332

. 334

442

459

470

535

555

596

621

618

19

291

328

329

437

453

465

531

550

590

616

614

20

287

324

325

432

447

461

527

545

5S4

610

610

21

28 +

320

321

426

440

450

522

539

577

604

606'

22

280

315

316

421

433

451

517

532

571

598

602

23

276

310

310

415

426

446

512

525

566

592

597

24

273

305

305

409

419

441

507

518

559

580

592

25

269

297

299

404

412

436

502

512

' 551

580

587

20

265

293

294

398

405

431

498

506

543

574

582

27

261

.2S7

288

392

398

426

495

501

535

568

577

28

256

281

283

385

391

421

492

490

526

562

572

29

251

275

27 8

378

384

415

489

491

517

556

567

30

247

269

272

372

378

409

486

486

508

550

563

31

243

264

206

366

372

403

482

481

499

544

558

32

239

259-

260

361

366

397

477

476

490

538

553

33

235

254

254

355

360

391

472

47 1

4S2

532

548

34

23 1

249

248

350

354

384

467

466

474

526

544

35

226

243

242

344

348

377

462

460

467

520

539

36

221

237

230

338

342

370

456

454

460

514

533

37

216

230

230

333

336

363

450

447

453

508

527

38

211

223

224

327

330

356

444

440

446

503

520

39

205

216

218

322

324

349

438

433

439

497

513

40

199

209

214

317

317

342

432

426

432

492

506

41

194

203

207

311

310

335

427

41S

425

487

500

42

189

197

201

306

303

328

422

410

419

482

494

43

185

192

194

300

296

321

417

401

413

476

4S8

4t

181

187

187

294

289

314

412

393

407

471

482

45

176

182

180

287

282

307

407

.386

400

466

476

46

171

177

174

281

275

299

400

379

393

460

469

47

105

172

167

274

268

291

394

372

386

455

461 -

48

159

167

159

268

261

2S3

388

365

378

449

451

49

153

162

153

261

254

275

381

359

370

443

441

50

147

157

147

255

247

267

• 374

353

362

436

431

51

1 12

152

141

248

239

259

367

347

354

429

422

62

137

147

135

242

232

250

359

340

345

422

414

53

133

142

130

235

225

241

351

333

330

414

406

54

123

137

125

228

218

232

343

326

327

400

397

55

1 23

132

120

221

211

224

334

318

318

397

388

56

1 17

127

116

2 1 3

204

216

324

310'

309

388

377

57

1 1 1

121

1 1 1

200

197

209

314

301

300

379

. 364

58

106

115

106

199

190

201

304

292

291

309

348

.59

101

109

101

191

183

193

293

283

282

359

331

LIF

L W 3

L I F

Ages.

Vienna.

Berlin.

London.

Norwich.

Norih-

cunpion.

Breslaw.

Branden-
burg.

Holy-
Cross.

Holland.

France.

Vaud,

Switzerland.

60

96

103

96

184

176

186

2S2

273

273

349

314

61

91

97

92 .

177

169

178

271

263

264

339

299

62

87

92

87

169

162

170

260

253

255

329

286

63

82

8S

83

161

155

163

248

243

245

318

274

64

77

S4

78

153

148

155

236

233

235

307

262

65

72

80

74

144

141

147

224

223

225

296

250

66

67

75

70

136

134

140

213

213

215

285

236

67

62

70

65

128

127

132

202

203

205

273

220

68

57

65

61

119

120

124

190

193

195

260

202

69

52

60

56

111

113

1 17

178

IS2

185

246

184

70

■ 48

55

52

103

106

109

166

171

175

232

168

71

44

51

47

94

99

101

153

161

165

21S

153

72

40

47

43

86

92

93

138

151

155

195

140

73

36

43

39

79

85

85

122

142

145

1SS

129

74

33

39

35

71

78

77

107

134

135

173

119

75

30

35

32

64

71

69

93

126

125

15S

109

76

27

32

28

57

64

61

80

119

114

144

98

77

24

29

25

50

58

53

6S

112

103

129

S5

78

21

26

22

43

52

45

59

105

92

115

71

79

18

23

19

37

46

38

51

98

82

102

58

SO

16

20

17

32

40

32

44

90

72

88

46

81

14

18

14

27

34

26

38

81

62

75

36

82

12

16

12

23

28

22

32

71

53

63

29

83

10

14

10

19

23

IS

25

61

45

53

24 .

84

8

12

8

16

19

15

21

51

38

44

20

85

7

10

7

13

16

12

15

41

31

36

17

86

6

8

6

10

13

9

11

32

25

28

14

87

5

7

5

8

11

6

S

24

19

21

11

88

4

6

4

6

8

4

6

17

14

16

9

89

3

5

3

5

6

2

4

11

10

12

7

90

2

4

2

4

4

1

3

7

7

8

5

These tables show that the mortality and chance of life
are various in different places ; and that therefore, to ob-
tain a sufficient accuracy in this business, it is necessary
to adapt a table of probabilities or chances of life, to every
place for which annuities are to be calculated ; or at
least one set of tables for large towns, and another for
country places, as well as for the supposition of different
rates of interest.

Several of the foregoing tables, as they commenced with
different numbers, are here reduced to the same number
at the beginning, viz, 1000 persons, by which means we
are enabled by inspection, at any age, to compare, the
numbers together, and immediately perceive the relative
decrees of vitality at the several places. The tables are
also arranged according to the degree of vitality among
them ; the least, or that at Vienna, first; and the rest in
their order, to the highest, which is the province of Vaud
in Switzerland. The authorities on which these tables
depend, are as they here follow. The first, taken from
Dr. Price's Observations on Reversionary payments, is
formed from the bills at Vienna, for 8 years, as given by
Mr. Susmilch, in his Gottliche Ordnung ; the 2d, for
Berlin, from the same, as formed from the bills there for 4
years, viz, from 1752 to 1755; the 3d, from Dr. Price,
showing the true probabilities of life in London, formed
from the bills for ten years, viz, from 1759 to 1?6S; the
4th, for Norwich, formed by Dr. Price from the bills for

30 years, viz, from 1740 to 1?6"9 > the 5th, D3' the same,
from the bills for Northampton ; the 6th, as deduced by
Dr. Halley, from the bills of mortality at Breslaw; the
7th shows the probabilities of life in a county parish in
Brandenburg, formed from the bills for 50 years, viz, from
1710 to 1759, as given by Mr. Susmilch; the Sth shows
the probabilities of life in the parish of Holy-Cross, near
Shrewsbury, formed from a register kept by the Rev. Mr.
Garsuch, for 20 years, from 1750 to 1770; the £)th, for
Holland, was formed by M. Kersseboom, from the regis-
ters of certain annuities for lives granted by the govern-
ment of Holland, which had been kept there for 125 years,
in which the ages of the several annuitants dying during
that period had been truly entered ; the 10th, for France,
were formed by M. Parcieux, from the lists of the French
tontines, or long annuities, and verified by a comparison
with the mortuary registers of several religious houses for
both sexes ; and the 1 1th, or last, for the district of Vaud
in Switzerland, was formed by Dr. Price from the regis-
ters of 43 parishes given by M. Muret, in the Bern Me-
moirs for the year 1766*.

Now from such lists as the foregoing, various tables have,
been formed for the valuation of annuities on single and
joint lives, at several rates of interest, in which the value
is shown by inspection. — The following are those that are
given by Mr. Simpson, in his Select Exercises, as deduced
from the London bills of mortality.

L 1 F

|[ 759 ]

L I F

Table II. — Showing the Value of an Annuity on One Life, or Number of Years Annuity in the Value, supposing
Money to bear Interest at the several Rates of 3, 4, and 5 per cent;

A"e,

Years value at

Years value at

Years value at

Age.

Years value at

Years value at

Years ralue at

3 per cent.

4 per cent.

5 per cent.
14-1

3 per cent.

4 per cent.

5 per cent.

6

18-S

16-2

41

13-0

11-4

10-2

7

18-9

16-3

14-2

42

12-8

11-2

10-1

8

190

16-4

14 3

43

12-6

11-1

10-0

9

19-0

16-4

14-3

44

12-5

11-0

9-9

10

19'0

16-4

14-3

45

12-3

10-8

9-8

11

19-0

164

14-3

46

12-1

10-7

9-7

12

■18-9

16-3

14-2

47

11-9

10-5

9-5

13

18-7

162

14-1

48

11-8

10-4

9-4

14

18-5

16-0

14-0

49

11 '6

10-2

9-3

15

18-3

158

13-9

50

11-4

10-1

9-2

16

18-1

15-6

13-7

51

11-2

9-9

9-0

17

17-9

15-4

13-5

52

1 10

9-8

,8-9

18

17-6

15-2

13-4

53

10-7

9-6

8-8

19

17-4

15-0

132

54

10-5

9-4

8-6

20

17-2

14-3

13-0

55

10-3

9-3

8-5

21

17-0

14-7

12-9

56

10-1

9-1

8-4

22

16-8

14-5

12'7

57

9-9

8-9

8-2

23

16-5

14-3

12-6

58

9-6

S-7

8-1

24

16-3

14-1

1-2-4

59

9-4

8-6

8-0

25

16-1

14-0

12-3

60

9-2

8-4

7-9

26

15-9

13-8

12-1

61

8-9

8-2

7-7

27

15-6

13-6

120

62

8-7

8-1

7-6

28 •

15-4

13-4*

11-8

63

8-5

7-9

7-4

s 29

15-2

13-2

11-7

64

8-3

7-7

7-3

30

15-0

13-1

11-6

65

8-0

7-5

7-1

31

14-8

12-9

11-4

66

7-8

7-3

6-9

32

14-6

12-7

11-3

67

7-6

7-1

6-7

33

14-4

12-6

11-2

68

7-4

6-9

6-6

34

14-2

12-4

u-o

69

7-1

6-7

6-4

35

14-1

123

109

70

6-9

6-5

6-2

36

13 9

121

10-8

71

6-7

6-3

6-0

37

13-7

11-9

10'6

72

6-5

6-1

5-S

38

13-5

11-8

10-5

73

6-2

5-9

5-6

39

13-3

11-6

1.0-4

74

5-9

5-6

5-4

40

13-2

11-5

10-3

75

5-6

5-4

5-2

Table III. — Showing the Value of an Annuity for Two Joint Lives, that is, for as long as they exist together.

Age of

Age of

Value at

Value at

Value at

Age of

Age of

Value at

Value at

%% Value at

Younger.

Elder.

3 per cent.

4 per cent.

5 per cent.

Younger.

Elder.

3 percent.

4 per cent.

5 per cent.

10

14-7

13-0

11-6

45

9-6

8-8

8-0

15

14-3

12-7

11-3

50

8-9

8-2

7-5

20

13-8

12-2

10-8

55

8-2

7-6

7-0

25

13-1

11-6

10-2

15

60

7-5

7-0

6-5

30

12-3

10-9

9-7

65

6-S

6-4

6-0

. 35

11-5

10-2

9-1

70

6-0

5-7

5-4

10

40

45

10-7
10-0

9-6
9-0

8-6
8-1

75

5-2

5-0

4-8

20

12-8

11-3

10-1

50

9-3

8-4

7-6

25

12-2

10-8

9-7

55

6-6

7-8

7-1

30

11-6

10-3

9-2

60

7-8

7-2

6-6

35

10-9

9-8

8-8

65

6-9

C-5

6-1

40

10-2

9-2

8-4

70

6-1

5-8

5-5

45

9-5

8-6

7-9

15

5-3

5-1

4-9

20

50
55

8-8
8-1

8-0
7-5

7-4

15

13-9

12-3

11-0

6-9

20

13-3

11-8

10-5

60

7-4

6-9

6-4

15

25

12-6

11-2

10-1

65"

6-7

6-3

5-9

30

] 1-9

10-6

9-5

70

6-0

5-7

5-4

35

11-2

10-0

9-0

75

5-2

5-0

4-8

40

10-4

9-4

8-5

Vol. I.

5A

L IF

[ 730 ]

L I F

Age of

Age of

Value at

Value at

Value at

Age of

Age of I

Value at 1

Value at

Value at

Younger.

Elder.

3 percent.

4 per cent.

5 per cent.

Younger.

Elder.

3 per cent. 1

4 per cent.

5 per cent.

25

11-S

10-5

9-4

60

7-0

6-4

6-0

30

11-3

10-1

9-0

40

65

6-4

5-9

5-5

35

10-7

9-6

8-6

70

5-7

5-4

5-1

40
45

10-0
9-4

9-1
8-5

8-2

1 7-8

75

5-0

4-8

4-6

45

S-3

7-4

d-7

25

50

8-7

7-9

7-3

50

7-9

7-1

6-5

55

8-0

7-4

6-S

55

7-4

6-7

6-2

60

7-3

6-8

6-3

45

60

6-3

6-3

5-8

65

6-6

6-2

5-8

65

6-3

5-8

5-4

70

5-9

5-6

5-3

70

5-6

5-3

5-0

75

5-1

4-9

4-7

75

- 4-9

4-7

4-5

30

10-8

9-6

8-6

50

7-6

6-8

6-2

35

10-3

9-2

8-3

55

7-2

6-5

6-0

40

9-7

8-8

8-0

60

. 6-7

6-1

5-7

45

9-1

8-3

7-6

50

65

6-2

5-7

5-3

30

50

8-5

7-8

7-2

70

5-5

5-2

4-9

55

7-9

7-3
6-7

6-7
6-2

75

4-8

4-6

4-4

60

; "2

55

6-9

6-2

5-7

65

6-5

6-1

5-7

'60

6-5

5-9

5-5

70

5-8

5-5

~ 5-2

55

65

6*0

5-0

5-2

75

5-1

4-9

4-7

70

75

5-4
4-7

5-1
4-5

4-8

35

9-9

8-8

8-0

7-7

4-3

40

9*4

60

6-1

5-6

5'2

45

8-9

8-1

7-4

65

5-7

5-3

4-9

50

8-3

7-6

7-0

60

70

5-2 .

4-9

4-6

35

55
60
65

7-7
7-1
6-4

7-1
6-5

6-6
6-1

75

4>6

4-4

4-2

6-0

5-6

65

5-4

5-0

4-7

70

5-7

5-4

5-1

65

70

4-9

4-6

4-4

75

5-0

4-8

4-6

75

4-4

4-2 »

4-0

40

9-1

8-1

7-3

70

70

4-6

4'4

4-2

45
50

8-7
8-2

7-3
7-4

7-1
6-8

75

4-2

4-0

3-9

40

75

75

3-8

3-7

3-6

55

7-6

6-9

6-4

Table IV. — Far the Value of an Annuity on the Longer of Two Given Lives.

Age of

Age of

Value at

Value at

Value at

Age of

Age of

Value at

Value at

Value at

Younger.

Elder.

3 per cent.

4 per cent.

.5 per cent.

Younger.

Elder.

3 per cent.

4 per cent.

5 per cent.

10

23-4

19-9

17-1

50

20-7

17-6

15-4

15

22-9

19-5

. 16-8

55

20-4

17-4

15-3

20

22-5 19-1

16-6

15

60

20-1

17-2

15-2

25

22-2 1S--8 v

16-4

65

19-8

16-9

15-0

30

21-9 18-6

16-2

70

19-4

16-6

14-7

35
40

21-6 18-4
21-4 18-3

16-1
16-0

75

18-9

16-3

14-4

10

. 20

21-6

18-3

15-8

45

21-2 18-2

15-9

25

21-1

17-9

15-5

50

20-9

18-0

15-8

30

20-7

17-6

15-3

55

20-7

17-8

15-7

3.5

20-4

17-4 .

15-1

60

20-4

17-6

15-5

40

20-1

17-2

15-0

65

20-1

17-4

15-3

45

19-9

17-0

14-9

70

19-8

17-2

15-1

20

50

19-6

16-8

14-7

75

19-5

16-9

14-8

55
60

19-4
19-1

16-6
16-3

14-5

15

22-8

19-3

16-7

14-3

20

22-3

18-9

16-4

65

18-7

16-0

14-1

25

21-9

18-6

16-2

70

18-2

15-7

13-8

15

30
35

21-6
21-3

18-3
18-1

16-0
15-9

75

17-7

15-3

13-5

25

20-3

17-4

15-1

40

21-1

17-9

15-7

25

30

1 9-8

17-0

14-9

45

20-9

17-8

15-6

35

19*4

16-7

14-7

L I F

C 731 ]

L I F

Age of

Age of

Value at

Value at

Value at

Age of

Age of

Value at

Value at

Value at

Younger.

Elder.

3-percent.

4 per cent.

5 per cent!

Younger.

Elder.

3 per cent.

4 per cent.

5 per cent.

40

19-2

16-5

14-5

65

14-9

13-1

1 1-8

45

J8-9

1 6-3

14-3

40

70

14-5

12-7

1 1-4

50
55

1S-7
18-4

16-1

14-2 •

75

14-0

12-3

11-0

25

15 "9

140

45

16-2

14-2

12-8

60

18-0

,15-6 -

13-S :

50

15-7

13-8

12-5,

65

17-6

15-3

13-6

55

15-2

1 3-4

12-1

70

17-2

15-0

13-3

45

60

14-7

12-9

11-7

75

16-7

14-6

12-9

65

70

14-1
13-6

12-3
12-0

11-4

30

19-3 •

16-6

14-5

11-0

35
40

18-8

18-4

16-2
1.5-9

14-2
14-0

75

13-1

11-6

10-6

50

15-0

13-3

12-1

45

] 8- 1 '

15-6

13-8

55

14-5

12-9

11-7

30 i

50

17-8

15-4

13-6

60

13-9

12-4

11-3

55

17-4

15-1

13-4

50

65

13-3

12-0

10-9

60

17-0

14-8

13-2

70

12-8

11-5

10-5

65 :

16-6

14-5

12-9

75

12-3

11-0

10-1

70
75

12-6
12-2

15-6

13-7

55

13-6

12-4

ii-3

60
65

13-0
, 12-4

11-9
11-3

10-9
10-5

35

J 8-3

15-8

13-8

55

40

17-8

15-4

13-5

70

11-8

10-8

10-0

45

17-4

15-1

13-3

75

11-3

10-3

9-5

50
55

17-1
16-7

14-8
14-5

13-1
12-9

35

,

60

12-2

11-2

10-5

60

16-3

14-2

12-7

60

65

11-5

10-6

10-0

65

15-8

13-8

12-4

70

10-9

10-1

9-5

70
75

15-3 •

14-8

13-4
13-0

12-0
11-6

75

10-3

9-5

9-0

65

65
70

10-7
10-0

10-0

9-4

9.-4

40

17-3

15-0

13-3

8-9

45
.50

16-8
16-3

14-6
14-2

13-0
12-7

75

9-3

8-7

8-3

40

70

70

y-2

S-6

8-2 -

55
60

15-9
15-4

13-9
13-5

12-4
12-1

75

S-4

7-9

7-6

75

75

7-6

7-2

6-9

The uses of these tables may be exemplified in the fol-
lowing problems.

Prob. 1. To find the Probability or Proportion of Chance,
that a Person of a Given Age continues in being a proposed
number of years. — Thus, suppose the age be 40, and the
number of years proposed 15; then, to calculate by the
table of the probabilities for London, in tab. 1. against 40
years stands 214, and against 55 years, the age to which
the person must arrive, stands 120, which shows that, of
214 persons who attain to the age of 40, only 120 of them
reach the age of 55, and consequently 94 die between the
ages of 40 and 55 : It is evident therefore that the odds for
attaining the proposed age of 55, are as 120- to 94, or as
9 to 7 nearly.

Prob. 2. To find the Value of an Annuity for a proposed
Life. — This problem is resolved from tab. 2, by looking
against the given age, and under the proposed rate of in-
terest ; then the corresponding quantity shows the num-
ber of years purchase required. For example, if the
given age be 36, the rate of interest 4 per cent, and the
proposed annuity 250 : then in the table it appears that
the value is 12'1 years purchase, or I2'l times 250/, that
is 3025/.

After the same manner the answer will be found in any
other case falling within the limits of the table. But as
there may sometimes be occasion to know the values of
lives computed at higher rates of interest than those in the
table, the two following practical rules are subjoined ;
by which the problem is resolved independently of tables.

• Rule 1. When the given age is not less than 45 years
nor greater than 85, subtract it from 92; then multiply
the remainder by the perpetuity, and divide the product by
thesaid remainder added to2{ times the perpetuity; so shall
the quotient be the number of years purchase required.
Where it is to. be noted, that by the perpetuity is meant
the number of years purchase of the fee-simple ; found
by dividing 100 by the rate per cent at which interest is
reckoned.

Ex. Let the given age be 50 years, and the rate of in-
terest 10 per cent. Then subtracting 50 from 92, there
remains 42 ; which multiplied by 10 the perpetuity, gives
420; and this divided by 67, the remainder increased by
2| times 10 the perpetuity, gives 6'3 nearly, for the num-
ber of years purchase. Therefore, supposing the annuity
to be 100/, its value in present money will be 630/.

Rule 2. When the age is between 10 and 45 years;
take 8 tenths of what it wants of 45, which divide by the
rate per cent increased by 1'2; then if the quotient be
added to the value of a life of 45 years, found by the pre-
ceding rule, there will be obtained the number of years
purchase in this case. For example, let the proposed age
be 20 years, and the rate of interest 5 per cent. Here
then, taking 20 from 45, there remains 25 ; -^ of which
is 20 ; which divided by 6% gives 3"2 ; and this added
to 9"8, the value of a life of 45, found by the former rule,
gives 13 for the number of years purchase that a life of
20 ought to be valued at.

And the conclusions derived by these rules, Mr. Simp-
5 A 2

L I F

[ 732 ]

L I G

son adds, are so near the true values, computed from real
observations,- as seldom to differ from them by more than
T'5 or -5=5 of one year's purchase.

The observations here alluded to, are those which are
founded on the London bills of mortality: and a similar
method of solution, accommodated to the Breslaw ob-
servations, will be as follows, viz. " Multiply the difference
between the given age and 85 years by the perpetuity,
and divide the product by S tenths of the said difference
increased by double the perpetuity, for the answer."
Which, from S to 80 years of age, will commonly come
within less than | of a year's purchase of the truth.

PitOB.3. To find the Value of an Annuity for Tivo Joint
Lives, that is, for as long as they both continue in being toge-
ther.— In table 3, find the younger age, or that nearest to
it, in column 1, and the higher age in column 2; then
against this last is the number of years purchase in the
proper column for the interest. Ex. Suppose the two
ages be 20 and 35 years ; then the value

is 1 0'.9 years purchase at 3 per cent,
or 9'S - - at 4 per cent,
or S'8 - - at 5 per cent-

Prob.4. To find the Value of the Annuity for the Long-
est of Two Lives, that is, for as long as either of them con-
tinues in being. — In table 4, col. 1, find the age of the
youngest life, or the nearest to it, in col. 1, and the age
of the elder in col. 2 : then against this last is the answer
in the proper column of interest. — Ex. So, if the two ages
be 15 and -10 ; then the value of the annuity upon the
longest of two such lives

is 21 -1 years purchase at 3 per cent.

or \7'9 - - 4 per cent.

or J5-7 - - 5 per cent.

. In the last two problems, if the younger age, or the rate

of interest, be not exactly found in the tables, the nearest

to them may be taken, and then by proportion the value

for the true numbers will be nearly found.

Rules and tables for the values of three lives, &c, may
also ~be seen in Simpson, in Baron Mascres's, and in Bai-
ley's Annuities, &c. All these calculations have been
made from tables of the real mortuary registers, differing
unequally at the several ages : but rules have also been
given on other principles, as by Demoivre, on the suppo-
sition that the decrements of life are equal at all ages ; an
assumption not much differing from the truth, from 7 to
70 years of age.

Life-Annuities, payable half-yearly, !fc- — These are
worth more than those that are payable yearly, as com-
puted by the foregoing rules and tables, on the two fol-
lowing accounts: First, that parts of the payments are
received sooner; and 2dly, there is a chance of receiv-
ing some part or parts of a whole year's payment more
than when the payments are oVdy made annually. Mr.
Simpson, in his Select Exercises, pa. 283, observes, that
the value of these two advantages put together, will al-
ways amount to £ of a year's purchase for half-yearly
payments, and to -|- of a year's purchase for quarterly
payments; and Mr. Maseres, at page 233 &c of his An-
nuities, by a very elaborate calculation, finds the former
difference to be nearly J also. But Dr. Price, in an essay
in the Philos. Trans. vol.0"6', pa. 10<), states the same dif-
ferences only

at T\. for half-yearly payments,
and fV for quarterly payments:
And the doctor then adds some algebraical theorems for
such calculations.

Life-Annuities, secured by Land. — These differ from
other life-annuities only in this, that the annuity is to be
paid up to the very day of the death of the age in ques-
tion, or of the person on whose life the annuity is granted.
To obtain the more exact value therefore cf such an an-
nuity, a small quantity must be added to the same as
computed by the foregoing rules and observations, which
is different according as the paymenls are yearly, half-
yearly, or quarterly, &c ; and are thus stated by Dr. Price
in his essay quoted above ; viz, the addition

is — for annual payments,
or — for half-yearly payments,

or — for quarterly payments :

where n is the complement of the given age, or what it
wants of 86 years ; and y, h, q are the respective values
of an annuity certain for n years, payable yearly, half-
yearly, or quarterly. And, by numeral examples, it is.
found that the first of these additional quantities is about
TVi the second -^, and the 3d half a tenth of one year's
purchase.

Complement of Life. See Complement.

Expectation o/Lipe. See Expectation.

Insurance or Assurance on Lives. See Assurance.

LIGHT, that principle by which objects are made per-
ceptible to our sense of seeing; or the sensation occasioned
in the mind by the view of luminous objects. The na-
ture of light has been a subject of speculation from the
first dawnings of philosophy. Some of the earliest philo-
sophers doubted whether objects became visible by means
of any thing proceeding from them, or from the eye of the
spectator. But this opinion was qualified by Empedocles
and Plato, who maintained, that vision was occasioned by
particles continually flying off from the surfaces of bodies,
which meet with others proceeding from the eye; while
the effect was ascribed by Pythagoras solely to the parti-
cles proceeding from the external objects, and entering
the pupil of the eye. But Aristotle defines light to be the
act of a transparent body, considered as such : and he ob-
serves that light is not fire, nor any matter radiating
from the luminous body, and transmitted through the
transparent one.

The Cartesians have refined considerably on this notion:
they maintain thai light, as it exists in the luminous body,
is only a power or faculty of exciting in us a very clear
and vivid sensation ; or that it is an invisible fluid present
at all times and in all places, but requiring to be set in
motion, by a body ignited or otherwise properly qualified
to make objects visible to us.

Father Malbranche explains the nature of light from a
supposed analogy between it and sound. — Thus, he sup-
poses, all the parts of a luminous body are in a rapid mo-
tion, which, by very quick pulses, arc constantly com-
pressing the subtle matter between the luminous body and
the eye, and excites vibrations of pression : and as these
vibrations are greater, the body appears more luminous ;
but as they are quicker or slower, the body is of this or
that colour.

The Newtonians maintain, that light is not a fluid per
se, but consists of a great number of very small particles,
thrown off from the luminous body by a. repulsive power
with an immense velocity, and in all directions : and these
particles, they also assert, are emitted in right lines ;

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which rectilinear motion they preserve till they are turned
out of their path by some of the following causes, viz, by
the attraction of some other body near which they pass,
which is called Inflection; or by passing obliquely through
a medium of different density, which is called Refraction;
or by being turned aside by the opposition of some inter-
vening body, which is called Reflection ; or, lastly, by
being totally stopped by some substance into which they
penetrate, and which is called their Extinction. A suc-
cession of these particles following one another, in an ex-
act straight line, is called a Ray of Light; and this ray,
in whatever manner its direction may be changed, whether
by refraction, reflection, or inflection, always preserves a
rectilinear course till it be again changed ; neither, say
they, is it possible to make it move in the arch of a circle,
ellipsis, or other curve. For the above properties of the
rays of light, see the several words, Refraction, Re-
flection, &c.

The velocity of the particles and rays of light is truly
astonishing, amounting to near 2 hundred thousand miles
in a second of time, which is near a million times greater
than the velocity of a cannon-ball : and this amazing mo-
tion of light has been manifested in various ways,, but
chiefly from the eclipses of Jupiter's satellites. It was
first observed by Roemer, that the eclipses of those satel-
lites happen sometimes sooner, and sometimes later, than
the times given by the tables of them; and that the ob-
served time was before or after the computed time, ac-
cording as the earth was nearer to, or farther from Jupi-
ter. Hence Roemer andCassini both concluded that this
circumstance depended on the distance of Jupiter from
the earth ; and that, to account for it, they must suppose
that the light was about 14 minutes in crossing the earth's
orbit. This conclusion however was afterward abandoned
and attacked by Cassini himself. But Roemer's opinion
found an able advocate in Dr. Halley ; who removed Cas-
sini's difficulty, and left Roemer's conclusion in its full
force. Yet, in a memoir presented to the academy in
1707, M. Maraldi endeavoured to strengthen Cassini's
arguments; when Roemer's doctrine found a new defender
in Mr. Pound. See Philos. Trans, number 136, also my
Abridg. vol. 6', pa. 34-9 and 386, and Groves, Phys. Elem.
number 2636": and it has since been found, by repeated
experiments, that when the earth is exactly between Ju-
piter and the sun, his satellites are seen eclipsed about Si
minutes sooner than they could be according to the ta-
bles; but when the earth is nearly in the opposite point
of its orbit, these eclipses happen about 8£ minutes later
than the tables predict them. Hence then it is certain
that the motion of light is not instantaneous, but that it
takes up about l6f minutes of time to pass over a space
equal to the diameter of the earth's orbit, which is at least
igo millions of miles in length, or at the rate of near
200,000 miles per second, as above stated. Hence it is
easy to know the time in which light travels to the earth,
from the moon, or any of the other planets, or even from
the fixed stars if ever their distances be known ; but these
distances are so immensely great, that from, the nearest
of them, supposed to be Sirius, the dog-star, it requires
many years for light to travel to the earth: and it has
even been conjectured that there are many stars whose
light have not yet arrived at us since their creation.

It was Galileo who first conceived the notion of measur-
ing the velocity of light ; and a description of his contri-
vance for this purpose, is given in his Treatise on Me-

chanics, pa. 39. He had two men with lights covered ;
the one was to observe when the other uncovered his light,
and to exhibit his own the moment he perceived it. This
rude experiment was tried at the distance of a mile, but
without success, as may naturally be imagined: and the
members of the academy Del Cimento repeated the ex-
periment, and placed their observers, to as little purpose,
at the distance of 2 miles.

But our excellent astronomer, Dr. Bradley, afterwards
found nearly the same velocity of light as Roemer, from
his accurate observations, and most ingenious theory, to
account for some apparent motions in the fixed stars ; for
an account of which, see Aberration of light. By a
long series of these observations, he found the difference
between the true and apparent place of several fixed stars,
for different times of the year ; which difference could no
otherwise be accounted for, than from the progressive mo-
tion of the rays of light. From the mean quantity of this
difference he found, that the ratio of the velocity of light
to the velocity of the earth in its orbit, was as 10313 to 1,
or that light moves 10313 times faster than the earth
moves in its orbit abput the sun ; and as this latter mo-
tion is at the rate of 18-J4 miles per second nearly, it fol-
lows that the former, or the velocity of light, is at the
rate of about 195000 miles in a second; a motion ac-
cording to which it will require just 8' 7" to move from the
sun to the earth', or about 95 millions of miles.

It was also inferred, from the foregoing principles, that
light proceeds with the same velocity from all the stars.
And hence it follows, that the motion of light, in its path
through the immense space above our atmosphere, is
equable or uniform. And since the different methods of
determining the velocity of light thus agree in the result,
it is reasonable to conclude that, in the same medium,
light is propagated, after it has been reflected, with the
same velocity as before. For an account of Mr. Melville's
hypothesis of the different velocities of differently coloured
rays, see Colour.

To the doctrine concerning the materiality of light, and
its amazing velocity, several objections have been made ;
of which the most considerable 'is, That as rays of light
are continually passing in different directions from every
visible point, they must necessarily interfere with each
other in such a manner, as entirely to confound all di-
stinct perception of objects, if not quite to destroy the
whole sense of seeing : not to mention the continual waste
of substance which a constant emission of particles must
occasion in the luminous body, and thereby since the
creation must have greatly diminished the matter in the
sun and stars, as well as increased the bulk of the 'earth
and planets by the vast quantity of particles of light ab-
sorbed by them in so long a period of time.

But it" has been replied, that if light were not a body,
but consisted in mere pression or pulsion, it could never
be propagated in right lines, but would be continually in-
flected ad umbram. Thus Sir I. Newton: " A pressure
on a fluid medium, i. e. a motion propagated by such a
medium, beyond an obstacle,' which impedes any part of
its motion, cannot be propagated in right lines, but will
be always inflecting and diffusing itself every way, to the
quiescent medium beyond that obstacle. The power of
gravity tends downwards ; but the pressure of water arising
from it tends every way with an equable force, and is pro-
pagated with equal ease and equal strength, in curves, as
in straight lines. Waves, on the surface of the water, gliding

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by the extremes of any very large obstacle, inflect and di-
late themselves, still diffusing gradually into the quiescent
water beyond that obstacle. The waves, pulses, or vibra-
tions of the air, wherein sound consists, are manifestly
inflected, though not so considerably as the waves of
water ; and sounds are propagated with equal ease, through
crooked tubes, and through straight lines ; but light was
never known to move in any curve, nor to inflect itself ad
umbram."

it must be acknowledged however, that many philoso-
pher?, both English and foreigners, have recurred to the
opinion, that light consists of vibrations propagated from
the luminous body, through a subtle etheriai medium.
The ingenious Dr. Franklin, in a letter dated April 23,
1752, expresses his dissatisfaction with the doctrine of
light consisting of particles of matter continually driven
off from the. sun's surface, with such an astonishing velo-
city. " Must not," says he, " the smallest portion con-
ceivable, have, with such a motion, a force exceeding that
of a 24 pounder discharged from a cannon ? Must not the
sun diminish exceedingly by such a waste of matter ; and
the planets, instead of drawing nearer to him, as some
have feared, recede to greater distances through the dimi-
nished attraction ? Yet these particles, with this amazing
motion, will not drive before them, or remove, the least
and slightest dust they meet with ; and the sun appears to
continue of his ancient dimensions, and his attendants
move in their ancient orbits." He therefore conjectures
that all the phenomena of light may be more properly
solved, by supposing all space filled with a subtle elastic
fluid, which is not visible when at rest, but which, by its
vibrations, affects that fine sense in the eye, as those of
the air affect the grosser organs of the ear ; and even that
different degrees of the vibration of this medium may cause
the appearances of different colours. Franklin's Exper.
and Observ. 1769, pa. 264.

The celebrated Euler has also maintained the same hy-
pothesis, in his Theoria Lucis et Colorum. In the sum-
mary of his arguments against the common opinion, re-
cited in Acad. Berl. 1752, pa. 271, besides the objections
above mentioned, he doubts the possibility, of particles
of matter, moving with the amazing velocity of light, and
penetrating transparent substances with so much ease. In
whatever manner they are transmitted, those bodies must
have pores, disposed in right lines, and in all possible di-
rections, to serve as canals for the passage of the rays :
but such a structure must take away all solid matter
from those bodies, and all coherence among their parts,
if they do contain any solid matter.

Doctor Horsley has taken considerable pains to obviate
the difficulties started by Dr. Franklin. Supposing that
the diameter of each particle of light does not exceed one
millionth of one millionth of an inch, and that the den-
sity of each particle is even 3 times that of iron, and also
that the light of the sun reaches the earth in 7', at the
distance of 22919 of the earth's semidiameters, he calcu-
lates thnt the momentum or force of motion in each par-
ticle of light coming from the sun, is less than that in an
iron ball of a quarter of an inch diameter, moving at the
rate of less than an inch in 12 thousand millions of mil-
lions of years. And hence he concludes, that a particle
of matter, which probably is larger than any particle of
light, moving with the velocity of light, has a force of
motion, which, instead of exceeding the force of a 24-
pounder discharged from- a cannon, is almost infinitely

less than that of the smallest shot discharged from a
pocket-pistol, or less than any that art can create. He
also thinks it possible, that light may be produced by a
continual emission of matter from the sun, without any
such waste of his substance as should sensibly contract his
dimensions, or alter the motions of the planets, within
any moderate length of time. In proof of this, he ob-
serves that, for the production of any of the phenomena
of light, it is not necessary that the emanation from the
sun should be continual, in a strict mathematical sense,
or without any interval ; and likewise that part of the
light which issues from the sun, is continually returned
to him by reflection from the planets, as well as other
light from the suns of other systems. He proceeds, by
calculation, to show, that in 385,130,000 years, the sun
would lose but the 13232d part of his matter, and conse-
quently of the gravitation towards him, at any given dis-
tance ; which is an alteration much too small to discover
itself in the motion of the earth, or of any of the planets.
He further computes that the greatest stroke which the
retina of a common eye sustains, when turned directly to
the sun in a bright day, does not exceed that which would
be given by an iron shot, a quarter of an inch diameter,
and moving only at the rate of l6-§ inches in a year;
whereas the ordinary stroke is less than the 20S4th part
of this. See Philos. Trans, vol. 60 and 6l.

In answer to the difficulty respecting the interference of
the particles of light with each other, Mr. Melville ob-
serves (Edinb. Ess. vol. 2), there is probably no physical
point in the visible horizon, that does not send rays to
every other point, unless where opaque bodies interpose.
Light, in its passage from one system to another, often
passes through torrents of light issuing from other suns
and systems, without ever interfering, or being diverted
from its course, either by it, or by the particles of that
elastic medium, which has been supposed by some diffused
through all the mundane space. To account for this fact,
he supposes that the particles of light are incomparably
rare, even when they are the most dense, or that their
diameters are incompan.b^- less than their distance from
one another : which obviates the objection urged bv Euler
and others against the materiality , of light, >from its influ-
ence in disturbing the freedom and perpetuity of the ce-
lestial motions. Boscovich and some others solve the dif-
ficulty concerning the non-interference of tiie particles of
light,' by supposing that each particle is endued with an
insuperable impulsive force; but in this case, their spheres
of impulsion would be more likely to interfere, and on
that account they would be more liable to disturb one
another.

Mr. Canton shows (Philos. Trans, vol. 58, pa. 344), that
the difficulty of the interference will vanish, if a very small
portion of time be allowed between the emission of every
particle and the next that follows in the same direction.
Suppose, for instance, that a lucid point in the sun's sur-
face emits 150 particles in a second of time, which, he
observes, will be more than sufficient lo give continual
light to the eye, without the least appearance of inter-
mission ; yet still the particles of such a ray, on account
of their great velocity, will be more than 1000 miles be-
hind each oilier, a space sufficient to allow others to pass
in all directions without any perceptible interruption.
And if we adopt the conclusions drawn from the experi-
ments on the duration of the sensations excited by light,
by the chevalier D'Arcy, in the Acad. Scienc. 1765, who

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_states it at the 7th part of a second, an interval of more
than 20,000 miles may be admitted between every two
successive particles.

The doctrine of the materiality of light is further con-
firmed by those experiments, which show, that the colour
and inward texture of some bodies are changeel by being
exposed to the light.

Of the Momentum, or Force, of the Particles of Light.
Some writers. have attempted to prove the materiality of
light, by determining the momentum of their component
particles, or by showing that they have a force, so as, by
their impulse, to give motion to light bodies. M. Hom-
berg, Ac. Par. 1708, Hist. pa. 25, imagined, that he could
not onl3> disperse pieces of amianthus, and other light sub-
stances, by the impulse of the solar rays, but also that by
throwing them upon the end of a kind of lever, connected
with the spring of a watch, he could make it move sen-
sibly quicker; from which, and from other experiments,
lie inferred the weight of the particles of light. And
Hartsoeckc r made pretensions of the same nature. But M.
Du Fay and M. Mairan made other experiments of a more
accurate kind, without the effects which the former had
imagined, and which even proved that the effects men-
tioned by them were owing to currents of heated air pro-
duced by the burning glasses used in their experiments, or
some other causes which they had overlooked.

Dr. Priestley, however, informs us, that Mr. Michell
endeavoured to ascertain the momentum of light with still
greater accuracy, and that his endeavours were not alto-
gether without success. Having found that the instrument
he used, acquired, from the impulse of the rays of light,
a velocity of an inch in a second of time, he inferred that
the quantity of matter contained in the rays falling upon
the instrument in that time, amounted to no more than
the 12 hundred millionth part of a grain. In the experi-
ment, the light was collected from a surface of about 3
square feet; and as this surface reflected only about the
half of what fell upon it, the quantity of matter contained
in the solar rays, incident upon a square foot and a half
ot surface, in a second of time, ought to be no more than
the 12 hundred millionth part of a grain, or upon one
square foot only, the 1 S hundred millionth part of a grain.
But as the density of the rays of light at the surface of the
sun, is 45000 times greater than at the earth, there ought
to issue from a square foot of the sun's surface, in one
second.of time, the 40 thousandth part of a grain of mat-
ter ; that is, a little more than 2 grains a day, or about
4,752,000 grains, which is about 670 pounds avoirdupois,
in 6"000 years, the time since the creation ; a quantity
which would have shortened the sun's semidiameter by no
more than about 10 feet, if it be supposed of no greater
density than water only.

The Expansion or Extension of any portion of light, is
inconceivable. Dr. Hooke shows that it is as unlimited
as the universe ; this he proves from the immense distance
ot many of the fixed stars, which only become visible to
the eye by the best telescopes. Nor, adds he, are they
only the great bodies of the sun or stars that are thus
liable to disperse their light through the vast expanse of
the universe, but the smallest spark of a lucid body must
do the same, even the smallest globule struck from a steel
by a flint.

Thclnlensity of different lights, or of thesame lightin diffe-
rent circumstances, affords a curious subject of speculation.
M.Bougucr, in hisTraitede Optique, found that when one

light is from 60 to 80 times less than another, its presence
or absence will not be perceived by an ordinary eye; that
the moon's light, when she is 19° 16' high above the ho-
rizon, is but about y of her light at 66° 11' high; and
when one limb just touched the horizon, her light is but
the 2000th part of her light at 66° 11' high ; and that
hence light is diminished in the proportion of 3 to 1 by
traversing 7-169 toises of dense air. lie found also, that
the centre of the sun's disc is considerably more luminous
than the edges of it ; whereas both the primary and se-
condary planets are more luminous at their edges than
near their centres: That, further, the light of the sun is
about 300,000 times greater than that of the moon ; and
therefore it is no wonder that philosophers have had so
little success in their attempts to collect the light of the
moon with burning-glasses; for, should one of the largest
of them even increase the light 1000 times, it will still
leave the light of the moon in the focus of the glass, 300
times less than the intensity of the common light of the sun.

Dr. Smith, in his Optics, vol. l, pa. 2Q, thought he
had proved that the light of the full moon would be only
the 90,900th part of the full day light, if no rays were lost
at the moon. But Mr. Robins, in his Tracts, vol. 2, pa.
225, shows that this is too great by one half. And Mr.
Michell, by a more easy and accurate mode of computa-
tion, found that the density of the sun's light on the sur-
face of the moon is but the 45,000th part of the density at
the sun ; and that therefore, as the moon is nearly of the
same apparent magnitude as the sun, if she reflected to us
all the light received on her surface, it would be only the
45,000th part of our day light, or that which we receive
from the sun. Admitting therefore, with M. Bouguer,
that the moon's light is only the 300,000th part of the day
or sun's light, Mr. Michell concludes that the moon re-
flects no more than between the 6-th and 7th part of what
she receives.

Dr. Gravesande says, a lucid body is that which emits
or gives fire a motion in right lines, and makes the dif-
ference between light and heat to consist in this, that to
produce the former, the fiery particles must enter the eye
in a rectilinear motion, which is not required in the latter:
on the contrary, an irregular motion seems more proper
for it, as appears from the rays coming directly from the
sun to the tops of mountains, which have not near that
effect with those in the valley, agitated with an irre-
gular motion, bv several reflections.

Sir I. Newton observes, that bodies and light act mu-
tually on each other; bodies on light, in emitting, re-
fleeting, refracting, and inflecting it; and light on bodies,
by heating them, and putting their parts into a vibrating
motion, in which heat principally consists. For all fixe^
bodies, he observes, when heated beyond a certain degree,
do emit light, and shine; which shining &c appears to be
owing to the vibrating motion of their parts; and all bo-
dies, abounding in earthy and sulphureous particles, if
sufficiently agitated, emit light, which way soever that
agitation be effected. Thus, sea water shines in a storm ;
quicksilver, when shaken in vacuo; cats or horses, when
rubbed in the dark ; and wood, fish, and flesh, when pu-
trefied.

Light proceeding from putrescent animal and vegetable
substances, as well as from glow-worms, is mentioned by
Aristotle. And Bartholin mentions four kinds of luminous
insects, two of which have wings : but in hot climates it
is said they are found in much greater numbers, and of

L I G [ 736 ]

different species. Columna observes, that their light is
not extinguished immediately on the death of the animal.
The first distinct account that occurs of light proceeding
from putrescent animal flesh, is that which is given by
Fabricius ab Aquapendente in 1592, de Visione &c, pa.
45. And Bartholin relates also a similar appearance in
his' treatise De Luce Animalium, which happened at Mont-
pel ier in l6ll.

Mr. Boyle speaks of a piece of shining rotten wood
which was extinguished in vacuo ; but on re-admitting
the air, it revived again, and shone as before ; though he
could not perceive that it was increased in condensed air.
But in Birch's History of the Royal Society, vol. 2, pa.
254, there is an account of the light of a shining fish,
which was rendered more vivid by putting the fish into a
condensing engine. Whitings were the fish commonly
used by Mr. Boyle in his experiments : though in a dis-
course read before the Royal Society in 16S1, it was as-
serted that, of all fishy substances, the eggs of lobsters,
after they had been boiled, shone the brightest. Birch's
Hist. vol. 2, pa. 70. In 1672 Mr. Boyle accidentally ob-
served light issuing from flesh meat; and, among other
remarks on this subject, he observes that extreme cold
extinguishes the light of shining wood ; probably because
extreme cold checks the putrefaction, which is the cause
of the light. The shell fish called pholas, is remarkable
for its luminous quality. The luminousness of the sea
has been also a subject of frequent observation. See Ignis
fatuus, Philosphorus, and Putrefaction, &c.

Mr. Hawksbee, and many writers on the subject of
electricity since his time, have produced a great variety of
instances of the artificial production of light, by the attri-
tion of bodies naturally not luminous ; as of amber rubbed
on woollen cloth in vacuo ; of glass on woollen, of glass on
glass, of oyster shells on woollen, and of woollen on
woollen, all in vacuo. On the several experiments of this
kind, he makes the following reflections : that different
sorts of bodies afford light of various kinds, with regard
both to colour and force ; that the effects of an attrition
are various, according to the different preparations and

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This action of bodies on light is found to exert itself at
a sensible distance, though it always increases as the dis-
tance is diminished ; as appears very sensibly in the
passage of a ray between the edges of two thin planes at
different apertures; which is attended with this peculiar
circumstance, that the attraction of one edge is increased
as the other is brought nearer to it. The rays of light,
in their passage out of glass into a vacuum, are not only
inflected towards the glass, but if they fall too obliquely,
they will revert back again to the glass, and be totally re-
flected. Now the cause of this reflection cannot be attri-
buted to any resistance of the vacuum, -but must be en-
tirely owing to some force or power in the glass, which
attracts or draws back the rays as 'they were passing into
the vacuum. And this appears to be further confirmed
from hence, that if you wet the back surface of the glass
with water, oil, honey, or a solution of quicksilver'then
the rays which would otherwise have been reflected, will
pervade and pass through that liquor; which shows that
the rays are not reflected till they come to that back sur-
face of the glass, nor even till they begin to go out of it;
for if, at their going out, they fall into any of the afore-
said mediums, they will not then be reflected, but will per-
sist in their former course, the attraction of the glass being
in this case counterbalanced by that of the liquor.

M. Maraldi prosecuted experiments similar to those of
Sir I. Newton on inflected light. And his observations
chiefly respect the inflection of light towards other bodies,
by which their shadows are partially illuminated. Acad.
Paris 1723, Mem. pa. 159. See also Priestley's Hist,
pa. 521, &c.

M. Mairan, without attempting the discovery of new
facts, endeavoured to explain the old ones, by the hypo-
thesis of an atmosphere surrounding all bodies ; and con-
sequently two reflections and refractions of light that im-
pinges on them, one at the surface of the atmosphere, and
the other at the surface of the body itself. This atmo-
sphere he supposed to be of a variable density and refrac-
tive power, like the air.

M. Dutour succeeded Mairan, and imagined that he

treatment of the bodies which are to endure it; and that -could account for all the phenomena by the help of an

bodies which have yielded a particular light, may be
brought by friction to yield no more of that light.

M.Bernoulli found by experiment, that mercury amal-
gamated with tin, and rubbed on glass, produced a con-
siderable light in the air; that gold rubbed on glass, ex-
hibited the same in a greater degree; but that the most
exquisite light of all was produced by the attrition of a
diamond, this being equally vivid with that of a burning
coal briskly agitated with the bellows. See Electri-
city, &c. '

Of the Attraction of Light. That the particles of light
are attracted by those of other bodies, is evident from nu-
merous experiments. This phenomenon was observed by
Sir I. Newton, who found, by repeated trials, that the rays
of light, in their passage near the edges of bodies, arc di-
verted out of the right lines, and always inflected or bent
towards those bodies, whether they be opaque or trans-
parent, as pieces of metals, the edges of knives, broken
glasses, &c. See Inflection and Raysv The cu-
rious observations that had been made on this subject by
Dr. Hooke and Grimaldi, led Sir I. Newton to repeat and
diversify their experiments, and to pursue them much
farther than they had done. For a particular account of
his experiment and observations, see his treatise on Optics,
pa. 293, &C.

atmosphere of a uniform density, but of a less refractive
power than the air surrounding all bodies. Dutour also
varied the Newtonian experiments, and discovered more
than three fringes in the colours produced by the inflection
of light. He further concludes thaUthe refracting atmo-
spheres, surrounding all kinds of bodies, are of the same
size; for when he used a great variety of substances, and of
different dimensions, he always found coloured streaks of
the same dimensions. - He also observes, that this hypothesis
contradicts an observation of Sir I. Newton, viz, that those
rays are the most inflected which pass the nearest to any
body. Mem. de Math, et de Phys. vol. 5, pa. 6"50, or
Priestley's Hist. pa. 531.

M. Lecat found that objects sometimes appear magni-
fied by means of the inflection of light; Looking at a
distant steeple, when a wire, of a less diameter than the
pupil of his eye, was held pretty near to it, and drawing it
several times between that object and his eye, he was sur-
prised to find that every time the wire passed before his
eye, the steeple seemed to change its place, and some hills
beyond the steeple seemed to have the same motion, just
as if a lens had been drawn between them and his eye.
This discovery led him to several others depending on the
inflection of the rays of light. Thus, he magnified small

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objects, as the head of a pin, by viewing them through a
small hole in a card ; so that the rays which formed the
image must necessarily pass so near the circumference of
the hole, as to be attracted by it. He exhibited also other
appearances of a similar nature. Traite des Sens, pa. 29'J.
Priestley, ubi supra, pa. 537-

Reflection and Refraction of Light. From the mutual
attraction between the particles of light and other bodies,
arise two other grand phenomena, besides the inflection of
light, which are called the reflection and refraction of light.
It is well known that the determination of bodies in mo-
tion, especially elastic ones, is changed by the interposition
of other bodies in their way : thus also light, impinging on
the surfaces of .bodies, should also be turned out of its
course, and beaten back or reflected, so as, like other
striking bodies,.to make the angle of. its reflection equal to
t-he angle of incidence. This, it is found by experience,
light does ; and yet the cause of this effect is different from
that just now assigned : for the rays of light are not re-
flected by striking on the very parts of the reflecting bodies,
but by some power, equally diffused over the whole surface
of the body, by which it acts on the light, either attracting
or repelling it, without contact: by which same power, in
other circumstances, the rays are refracted; and by which
also theravs are first emitted from the luminous body; as
Newton fully proves by a great variety of arguments. See
Reflection and Refraction.

That great author proves, that all those rays which are
reflected, do not really touch the body, though they ap-
proach infinitely near to it ; and that those which strike on
the parts of solid bodies, adhere to them, and are as it
were extinguished and lost. Since the reflection of the
rays is ascribed to the action of the whole surface of the
body without contact, if it be asked, how it happens that
all the rays are not reflected from every surface; but that,
while some are reflected, others pass through, and are re-
fracted ? the answer given by Newton is as follows: —
Every ray of light, in its passage through any refracting
surface, is put into a certain transient constitution or state,
which in the progress of the ray returns at equal intervals,
and disposes the ray at every return to be easily transmitted
through the next refracting surface, and between the re-
turns to be easily reflected by it: which alteration of re-
flection and transmission it appears is propagated from
every surface, and to all distances. What kind of action
or disposition this is, and whether it consists in a circula-
ting or vibrating motion of the ray, or the medium, or
something else, he does not enquire ; but allows those who
are fond of hypotheses to suppose, that the rays of light,
by impingeing on any reflecting or refracting surface,
excite vibrations in the reflecting or refracting medium,
and by that means agitate the solid parts of the body.
These vibrations, thus produced in the medium, move
faster than the rays, so as to overtake them ; and when
any ray is in that part of the vibration which conspires with
its motion, its velocity is increased, and so it easily breaks
through a refracting surface; but when it is in a contrary
part of the vibration, which impedes its motion, it is easily
reflected ; and thus every ray is successively disposed to be
easily reflected or transmuted by every vibration which
meets it. These returns in the disposition ofanyrayto be
reflected, he calls Fits of Easy Reflection ; and the returns
in the disposition to be transmitted, he calls fits of Easy
Transmission; also the space between the returns, the In-
terval of the Fits. Hence then the reason why the surfaces

Vol. I.

of all thick transparent bodies reflect part of the light in-
cident on them, and refract the rest, is, that some rays at
their incidence are in fits of easy reflection, and others of
easy transmission. For the properties of reflected Light, see
Reflection, Mirror, &c.

Again, a ray of light, passing out of one medium into
another of different density, and in its passage making an
oblique angle with the surface that separates the mediums,
will be refracted, or turned out of its direction; because
the rays are more strongly attracted by a denser than by
a rarer medium. That these rays are not refracted by
striking on the solid parts of bodies, but that this is ef-
fected without a real contact, and by the same force by
which they are emitted and reflected, only exerting itself
differently in different circumstances, is proved in a o,-eat
measure by the same arguments by which it is demonstrated
that reflection is performed -without contact. See Refrac-
tion, Lens, Colour, Vision, &c.

Chemical Properties of Light.— The chemical properties of
light are yet but very imperfectly Known, though there is
reason to believe that it is an agent of great importance in
many of the principal changes that are goingon in the visible
world, and even in the experiments of the laboratory its
effects are often clearly demonstrable.

The process of rapid combustion is always attended
with the evolution of light along with heat, the source of
both of which is ascribed by Lavoisier to the oxygen,
which is known to be essential to combustion; so that
according to this eminent philosopher, while any sub-
stance is burning, the oxygen alone unites with the com-
bustible substance, forming an oxyd, acid, &c, according
to circumstances; and both the light and heat which pre^
viously existed in the oxygen are now set at liberty, and
appear in the form of visible flame.

But there are many weighty objections to thelatter part
of this hypothesis (see Oxygen and Combustion), so that
many chemists have returned to the old opinion, that
the light is derived from the burning body, and not from
the oxygen that supports combustions; an opinion which
on the whole better agrees with the observed phenomena.

The sun's rays, the great source of light and heat to
our world, have been found by Dr. Herschel to contain,
besides all the common coloured rays, separable by the
prism, a number of rays of mere heat, giving no light, still
less refrangible than the red rays of the spectrum, and
therefore extending beyond them, and producing a greater
haat than any of the visible rays. Since that rime" it has
been discovered, as appears both by Dr. Wollaston,
Philosophical Trans, for 1S02, and Mr. Ritter, Philoso-
phical Journal, Svo, vols." 4 and 8, that beyonduhe oppo-
site, or violet side of the prism, are certain other rays, also
invisible, which do not effect the thermometer, but which
produce some of the chemical effects of the visible rays of
light, particularly of the violet rays to which they are con-
tiguous. These therefore appear to be, if the expression
may be allowed, invisible rays of light, or rays which,
though they do not illuminate, agr.ee in chemical proper-
ties with those that do, and appear to be more than any
other, separate from solar heat.

The most rapid and most unexceptionable chemical
effects of pure light hitherto discovered, has been the
change of colours of white lima cornea^ to a purplish
brown, or slate colour. This change takes place without
requiring the least increase of temperature, or in anyway
being affected by heat or cold, and it occurs equally well
5 B

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in close as in open vessels, and only on the surface imme-
diately exposed to the light. Akin's Cltem. Diet.

LIGHTNING, a large bright flame, shooting swiftly
through the atmosphere, of momentary or very short dura-
tion, and rommonly attended with thunder.

Some philosophers accounted for this awful natural
phenomenon in this manner, viz, that an inflammable
substance is formed of the particles of sulphur, nitre, and
other combustible matter, which are exhaled from the
earth, and carried into the higher regions of the atmo-
sphere, and that by the collision of two clouds, or other-
wise, this substance takes fire, and darts out into a train of
light, larger or smaller according to the strength and quan-
tity of the materials. And others have explained the phe-
nomenon of lightning by the fermentation of sulphureous
substances with nitrous acids. See Thunder.

But it is now universally allowed, that lightning is really
an electrical explosion or phenomenon. Philosophers had
not proceeded far in their experiments and enquiries on
this subject, before they perceived the obvious analogy
between lightning and electricity, and they produced many
arguments to evince their similarity. But the method of
proving this hypothesis beyond a doubt, was first proposed
by Dr. Franklin, who, about the close of the year 1749,
conceived the practicability of drawing lightning down
from the clouds. Various circumstances of resemblance
between lightning and electricity were remarked by this
ingenious philosopher, ant£ have been abundantly con-
firmed by later discoveries, such as the following: Flashes
of lightning are usually seen crooked and waving in the
air; so the electric spark drawn from an irregular body
at some distance, and when it is drawn by an irregular
body, or through a space in which the best conductors
are disposed in an irregular manner, always exhibits the
same appearance: lightning strikes the highest and most
pointed objects in its course, in preference to others, as
hills, trees, spires, masts of ships, &c; so all pointed con-
ductors receive and throw oft' the electric fluid more rea-
dily than those that are terminated by flat surfaces : light-
ning is observed to take and follow the readiest and best
conductor; and the same is the case with electricity in
the discharge of the Ley den phial; whence the doctor
infers, that in a thunder-storm, it would be safer to have
one's clothes wet than dry; lightning burns, dissolves
metals, rends some bodies, sometimes strikes persons blind,
destroys animal life, deprives magnets of their virtue, or
reverses their poles; and all these are well-known pro-
perties of electricity.

But lightning also gives polarity to the magnetic needle,
as well as to all bodies that have any thing of iron in
them, as bricks &c ; and by observing afterwards which
way the magnetic poles of these bodies lie, it may thence
be known in what direction the stroke passed. Persons
are sometimes killed by lightning, without exhibiting any
visible marks of injury; and in this case Sig. Beccaria
supposes that the lightning does not really touch them,
but only produces a sudden vacuum near them, and
the air rushing violently out of their lungs to supply
it, they cannot recover their breath again : and in proof
of this opinion he alleges, that the lungs of such persons
are found flaccid ; whereas these are found inflated when
the persons arc. really killed by the electric shock. This
hypothesis however is controverted by Dr. Priestley.

To demonstrate, by actual experiment, the identity of
the electric fluid with the matter of lightning, Dr. Frank-

lin contrived to bring lightning from the heavens, by
means of a paper kite, properly titled up for the purpose,
with a long fine wire string, and called aii electrical kite,
which he raised when a thunder-storm was perceived to
be coming on : and with the electricity thus obtained, he
charged phials, kindled spirits, and performed all other
such electrical experiments as are usually exhibited by
an excited glass globe or cylinder. This happened in
June 1752, a month after the electricians in France, in
pursuance of the method which he had -before proposed,
had verified the same theory. The. most active of these
were Messrs. Dalibard and Delor, followed by M. Mazeas
and M. Monnier.

In April and June 1753, Dr. Franklin discovered that
the air is sometimes electrified negatively, as well as some-
times positively ; and he even found that the clouds would
change from positive to negative electricity several times
in the course of one ihunder-gust. This curious and im-
portant discovery he soon perceived was capable of being
applied to practical use in life, and in consequence proposed
a method, which he soon accomplished, of securing build-
ings from being damaged by lightning, by means of Con-
ductors: which see.

Nor had the English philosophers been inattentive to
this subject : but, for want of proper opportunities of
trying the necessary experiments, and from some other un-
favourable circumstances, they had failed of success. Mr.
Canton, however, succeeded in July 1752; and in the
following month Dr. Bevis and Mr. Wilson observed nearly
the same appearances as Mr. Canton had done before.
By a number of experiments Mr. Canton also soon after
observed that some clouds were in a positive, while some
were in a negative state of electricity ; and that the elec-
tricity of his conductor would sometimes change, from one
state to the other, five or six times in less than halfan hour.

But Sig. Beccaria discovered this variable state of thun-
der clouds, before he knew that it had been observed by
Dr. Franklin or any other person; and he has given a
very exact and particular account of the external appear-
ances of these clouds. From the observations of his ap-
paratus within doors, and of the lightning abroad, he in-
ferred, that the quantity of electric matter in a common
thunder-storm, is inconceivably great, considering how
many pointed bodies, as spires, trees, &c, are continually
drawing it off, and what a prodigious quantity is repeat-
edly discharged to or from the earth. This matter is in
such abundance, that, he thinks it impossible for any cloud
or number of clouds to contain it all, so as cither to re-
ceive or discharge it. lie observes also, that during the
progress and increase of the storm, though the lightning
frequently struck to the earth, the same clouds were the
next moment ready to make a still greater discharge, and
his apparatus continued to be as much affected as ever ;
so that the clouds must have received at one part, in the
same moment when a discharge was made from them in
another. And from the whole he concludes, that the
clouds serve as conductors to convey the electric fluid
from those parts of the earth that are overloaded with it,
to those that are exhausted of it. The same cause by
which a cloud is first raised, from vapours dispersed in the
atmospheret draws to it those that are already formed, and
still continues to form new ones, till the whole collected
mass extends so far as to reach a part of the earth where
there is a deficiency of the electric fluid, and where the
electric niattci' will discharge itself on the earth. A chart-

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L I K

nel of communication being thus formed, a fresh supply
of electric matter is raised from the overloaded part, which
continues to be conveyed by the medium of the clouds,
till the equilibrium of the fluid is restored between the
two places of the earth. Sig. Beccaria observes, that a
wind always blows from the place from which the thun-
der-cloud proceeds ; and it is plain that the sudden accu-
mulation of such a prodigious quantity of vapours must
displace the air, and repel it on all sides. Indeed many
observations of the descent of lightning confirm his theory
of the manner. of its ascent; for it often throws before it
the parts of conducting bodies, and distributes them along
the resisting medium, through which it must force its pas-
sage ; and on this principle, the longest flashes of light-
ning seem to be made, by forcing into its way part of the
vapours in the air. One of the chief reasons why these
flashes make so long a rumbling, is that they are occasion-
ed by the vast length of a vacuum made by the passage of
the electric matter: for though the air collapses the mo-
ment after it has passed, and that the vibration, on which
the sound depends, commences at the same moment ; yet
when the flash is directed towards the person who hears
the report, the vibrations excited at the nearer end of the
track, will reach his ear much sooner than those from
the more remote end; and the sound will, without any
echo or repercussion, continue till all the vibrations have
successively reached him.

How it happens that particular parts of the earth, or
the clouds, come into the opposite states of positive and
negative electricity, is a question not absolutely determin-
ed : though it is easy to conceive that when particular
clouds, or different parts of the earth, possess opposite
electricities, a discharge will take place within a certain
distance; or the one will strike into the other, and in the
discharge a flash of lightning will be seen. Mr. Canton
queries whether the clouds do not become possessed of
electricity by the gradual heating and cooling of the air';
and whether air suddenly rarefied, may not give electric
fire to clouds and vapours passing through it, and air sud-
denly condensed receive electric fire from them. Mr.

Wilckc supposes, that the air contracts its electricity in
the same manner that sulphur and other substances do,
when they are heated and cooled in contact with various
bodies. Thus, the air being heated of cooled near the
earth, gives electricity to the earth, or receives it from it;
and the electrified air, being conveyed upwards by various
means, communicates its electricity to the clouds. — Others
have queried, whether, since thunder commonly happens
in a sultry state of the air, when it seems charged with
sulphureous vapours, the eletric matte'r then in the clouds
may not be generated by the fermentation of sulphureous
vapours with mineral or acid vapours in the air.

With regard to places of safety in times of thunder and
lightning, Dr. Franklin's advice is, to sit in a chair in the
middle of a room, provided it be not under a metal lustre
suspended by a chain, and laying the feet on another
chair. " It is still better," he says, " to bring two or three
mattresses or beds into the middle of the room, and folding
them double, to place the chairs upon them; for as they
are not so good conductors as the walls, the lightning will
not' be so likely to pass through them : but the safest place
of any, is in a hammock hung by silken cords, at an equal
distance from all the sides of a room. Dr. Priestley how-
ever observes, that the place of most perfect safety must
be a cellar, and especially the middle of it ; for when a

person is lower than the surface of the earth, the liohtnino
must strike it before it can possibly reach him. In the
fields, the most secure place is within a few yards of a tree,
but not quite near it. Beccaria cautions persons not al-
ways to trust too much to the neighbourhood of a hiohcr
or better conductor than their own body; since he has
repeatedly found that the lightning by no means descends
in one undivided track, but that bodies of various kinds
conduct their share of it at the same time, in proportion
to their quantity and conducting power. See Franklin's
Letters, Beccaria's Lettre dell' Ellettriccssimo, Priestley's
Hist, of Electric, and Lord Mahon's Principles of Elec-
tricity.

Lord Mahon observes that damage may be done by
lightning, not only by the main stroke and lateral explo-
sion, but also by what he calls the returning stroke-; by
which is meant the sudden violent return of that part of
the natural share of electricity which had been gradually
expelled from some body or bodies, by the superinduced
elastic electrical pressure of the electrical atmosphere of a
thunder-cloud.

Artificial Lightning, an imitation of real or natural
lightning by gunpowder, aurum fulminans, phosphorus,
&c, but especially the last, between which and lightning
there is much more resemblance than the others. Phos-
phorus, when newly made, gives a sort of artificial light-
ning visible in the dark, which would surprise those not
used to such a phenomenon. It is usual to keep this pre-
paration under water ; and if it is desired to see the cor-
ruscations to the greatest advantage, it should be kept in
a deep cylindrical glass, not more than three quarters
filled with water. At times the phosphorus will send up
corruscations, which will pierce through the incumbent
water, and' expand themselves with great brightness in the
upper or empty part of the glass, and much resembling
lightning. Thephosphorus, while burning, acts the part
of a corrosive, and when it goes out resolves into a men-
struum, which dissolves gold, iron, and other metals ; and
lightning, in like manner, melts the same substances.

LIKE Quantities, or Similar Quantities, in Algebra,
are such as are expressed by the same powers of the same
letters, or equally repeated in each quantity;' though the
numeral coefficients may be different.
Thus 4a and 5a are like quantities,
as are.also 3a1 and 12a%
and also 6bxy~ and \Gbiy1.
But 4a and 5b, or 3a"b and \Oa?b'1, &c, are unlike quan-
tities; because they have not the same dimensions through-
out, nor are the letters equally repeated. — Like quantities
can be united into one quantity, by addition or subtrac-
tion; but unlike quantities can only be added or sub-
tracted by placing the signs of these operations between
them.

Like Signs, in Algebra, are the same signs, either both,
positive or both negative. But when one is positive and
the other negative, they are unlike signs.
So, ■+- 3ab and -t- 5cd have like signs,
as have also — 2azc and — 2axz ;
but -+- 3ab and — 5crfhave unlike signs,-
asalso— 2ax and 3ax.

Like Figures, or Arcs, &c, are the same as similar
figures, arcs, &c. See Similar. All like figures have
their homologous lines in the same ratio. Also like plane
figures are in the duplicate ratio, or as the squares of their
homologous lines or sides ; and like solid figures are in the
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triplicate ratio, or as the cubes of their homologous lines
or sides.

LILLY (William), a noted English astrologer, born
in Leicestershire in 1602. His father was not able to give
him any farther education than common reading and
writing ; but young Lilly being of a forward temper, and
endued with shrewd wit, he resolved to push his fortune
in London, where he arrived in l6'20, and, for a present
support, articled himself as a servant to a mantua-maker
in the parish of St. Clement Danes. But in 1 624 he moved
a step higher, by entering into the service of Mr. Wright
in the Strand, master of the Salters company, who not
being able to write, Lilly among other offices kept his
books. On the death of his master, in 1 627, Lilly paid
his addresses to the widow, whom he married with a for-
tune of 1000/. Being now his own master, he pursued
the bent of his inclinations, which led him to follow the
puritanical preachers: but afterwards, turning his mind
to judicial astrology, in l632 he became pupil, in that
art, to one Evans, a profligate Welsh passon ; and the
next year gave the public a specimen of his skill, by an
intimation that the king had chosen an unlucky horoscope
for the coronation in Scotland. In 1 634, getting a ma-
nuscript copy of the Ars Noticia of Cornelius Agrippa,
with alterations, he followed the doctrine of the magic
circle, and the invocation of spirits, with great eagerness,
and practised it for some time ; after which he treated the
mystery of recovering stolen goods, &c with great con-
tempt, claiming a supernatural sight, and the gift of pro-
phetical predictions ; all which he well knew how to turn
to good advantage.

Mean while, he had buried his first wife, purchased a
moiety of 13 houses in the Strand, and married a second
wife, who, joining to an extravagant temper a termagant
spirit, which, notwithstanding his skill in magic, he could
not lay, made him unhappy, and greatly reduced his cir-
cumstances. With this uncomfortable yokemate he re-
moved, in 1636, to Hersham in Surrey, where he staid
till l6'4l ; when, seeing a prospect of fishing in troubled
waters, he returned to London. Here having purchased
several curious books in this art, which were found on
pulling down the house of another astrologer, he studied
them incessantly, finding"" out secrets contained in them,
which were written in an imperfect Greek character ; and
in l644, published his Meilinu's Anglicus, an almanac,
which he continued annually till his death, and several
other astrological works ; devoting his pen, and other la-
bours, sometimes to the king's paTty, and sometimes to
that of the parliament, hut mostly to the latter, raising
his fortune by favourable predictions to both parties, some-
times by presents, and sometimes by pensions: thus, in
1048, the council of state gave him in money 50/. and a
pension of 100/. per annum, ■which he received for two
years, and then resigned it on some disgust. By|his ad-
vice and contrivance, the king attempted several times to
make his escape from his confinement: lie procured and
sent the aquafortis and files to cut the iron bars of his prison
windows at Carisbrook 'castle ; but still advising and writ-
ing for the other party at'the same time. Mean while he
read public lectures on astrology, in 1048 and 16*49, for
the improvement of young students in that art ; and in
short, plied his business so well, that in 16*51 and 1(>V>2
he laid out near 2000/. for lands and a house at Hersham,
During the siege of Colchester, he and Booker were, sent
for thither, to encourage the soldiers ; which they did by

assuring them that the town would soon be taken J which
proved true in the event. — Having, in l650, written pub-
licly that the parliament should not continue, but a new
government arise ; agreeably to which, in his almanac for
l6o3, he asserted that the parliament stood upon a tick-
lish foundation, and that the commonalty and soldiery
would join together against them. Upon which he was
summoned before the committee of plundered ministers;
but, receiving notice of it before the arrival of the mes-
senger, he applied to his friend Leuthal the speaker, who
pointed out the offensive passages. He immediately al-
tered them ; attended the committee next morning, with
6 copies printed, which six alone he acknowledged to be
his; and by that means came off with only 13 days cus-
tody by the Serjeant at arms. This year he was engaged
in a dispute with Mr. Thomas Gataker. — in 1660 he was
indicted at Hicks's-hall, for giving judgment on stolen
goods ; but was acquitted. And in 16*59, he received
from the king of Sweden, a present of a gold chain and
medal, worth about 50/. on account of his having men-
tioned that monarch with great respect in his almanacs
of l6"57 and 165S. — After the Restoration, in 1660, being
taken into custody, and examined by a committee of the
house of commons, touching the execution of Charles the
1st, he declared, that Robert Spavin, then secretary to
Cromwell, dining with him soon after the fact, assured
him it was done by cornet Joyce. The same year he sued
out his pardon under the broad seal of England ; and af-
terwards continued in London till l66a ; when, on the
raging of the plague there, he 'retired to his estate at Her-
sham. Here he applied himself to the study of physic,
having, by means of his friend Elias Ashmole, procured
from archbishop Sheldon a licence to practise it, which
he did, as well as astrology, from thence till the time of
his death. — In October 1666 he was examined before a
committee of the house of commons concerning the fire of
London, which happened in September that year. A
little before bis death, he adopted for his son, by the
name of Merlin Junior, one Henry Coley, a taylor by
trade; and at the same time gave him the impression of
his almanac, which had been printed for 36" years suc-
cessively. This Coley became afterwards a celebrated
astrologer, publishing in his own name, almanacs, and
books of astrology, particularly one entitled A Key to
Astrology. Lilly died of a palsy in 16SI, at 79 years of
age; and his friend Mr. Ashmole placed a monument over
his grave in the church of Waiton-upon-Thames.

Lilly was author of many works. His Observations on
the Life and Death of Charles late King of England, if
we overlook the astrological nonsense, may be read with
as much satisfaction as more celebrated ly stories; Lilly-
being not only very well informed, but strictly impartial.
This work, with the Lives of Lilly and Ashmole, written
by themselves, were published in one volume, 8vo, in
1774, by Mr. Burman. His other works were principally
as follows: I. Merlinus Anglicus junior. — 2. Supernatural
Sight. — 3. The White King's Prophecy. — 4. England's Pro-
phetical Merlin : all printed in 14)44. — 5. The Starry
Messenger, 16*45. —6. Collection of Prophecies, 16-16. —
7- A Comment on the White King's Prophecy, 1646. — 8.
The Nativities of Archbishop Laud and Thomas Earl of
Strafford, 16*46'. — 0. Christian Astrology, 16*47: on this
piece he read his lectures in I 648, mentioned above.— 10.
The Third book of Nativities, I047— H. The World's
Catastrophe, 16*17. — 12. The Prophecies of Ambrose Mer-

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L I N

lin, with a "Key, 1647. — 13. Trithemius, or the Govern-
ment of the World by Presiding Angels, 16'47. — 14. A trea-
tise of the Three Suns seen in the winter of l6~47, printed
in 16'4S. — 15. Monarchy or no Monarchy, 1651.— l6\
Observations on (he Life and Death of Charles, late king
of England, 1 65 1 ; and aoain in 1051, with the title of
Mr. William Lilly's True History of King James and King
Charles the 1st, &c. — 17. Annus Tenebrosus; or, the Black
Year. This drew him into the dispute with Gataker, which
Lilly carried on in his Almanac in l6"54.

LI .MB, the outermost border, or graduated edge, of a
quadrant, astrolabe, or such like mathematical instru-
ment. The word is'also used for the arch of the primitive
circle, in any projection of the sphere in piano.

Limb also signifies the outermost border or edge of the
sun or the moon; as the upper limb, or edge ; the lower
limb; the preceding limb, or side ; the following limb. —
Astronomers observe the upper or lower limb of the sun
or moon, to.find their true height, or that of the centre,
which differs from the others by the semidiameter of the
disc.

LIMBERS, in Artillery, a sort of advanced train, joined
to the carriage of a cannon on a march. It is composed
of two shafts, wide enough to receive a horse between
them, called the fillet horse: these shafts are joined by
two bars of wood, and a bolt of iron at one end, and
mounted on a pair of rather small wheels. On the axle-
tree rises a strong iron spike, which is put into a hole in
the hinder part of the train of the gun-carriage, to draw
it by. But when a gun is in action, the limbers are taken
off, and run out behind it. — See the dimensions and figure
of it in Miiller's Treatise of Artillery, pa. 187-

LIMIT, is a term used by mathematicians, for some de-
terminate quantity, to which a variable one continually
approaches, and may come nearer to it than by any given
difference, but can never go beyond it; in which sense a
circle may be said to be the limit of all its inscribed and
circumscribed polygons : because these, by increasing the
number of their sides, may be made to approach so near
to the true area of the circle, that the difference shall be
less than any assignable quantity.

Limits of the roots of Equations, in Algebra, form a
part of that science, by means of which the solution of
equations is sometimes much facilitated, particularly in
those cases where we can only proceed by approximations;
for, knowing that a root must lie between certain limits,
that is, that it is greater than one known quantity, and
less than another, we are led to a near approximate value
in the first instance, which, without this consideration, we
should have perhaps discovered only after a tedious ope-
ration.

To find a limit greater than the greatest root of an equation.

Let a, b, c, ike, be roots of an equation; transform it
into another whose roots are a — e,b — e, c — e, &c; and
if by trial, such a value of e be found, that all the terms
of the transformed equation be positive, all its roots are
negative; and consequently e is greater than the greatest
root of the proposed equation.

Suppose, for example, it were required to find a number
greater than the greatest root of the equation x3 — 5 x7, h-
7x— 1 =o.

Assume x = y -+- e, and we have

I3 = y3 -+- 3ey* -t- 3 e~y -+- e'^J
— 5x* = — 5ya — 10ty — 5e- )>_„
* 7x = + 7y + 7«j

3e > = 0.
72)

in which equation, if 3 be substituted for e, each of the
quantities ei — be\ -+- 7c — 1, 3e2 — lOe -t- 7, 3e — 5, is
positive, or all the values of 7 are negative, therefore 3 is
greater ,than the greatest value of x.

Again, to find a limit less than the least root of the
equation x3 — 3x -h 72 =0.

When the signs of these roots are changed, this equation
becomes x3 — 3x — 72 = 0.

Assume x =y ->- e, then

X1 =ry3 -+- 3eyz -i- 3e*y •

— 3x = — 3 y ■

- 72 =

and here, if 5 be substituted fore, every term becomes
positive, consequently 5 is greater than tne greatest root
of the equation x3 — 3x — 72 = 0, and — 5 less than the
least root of the equation x3 — 3x -+- 72 = 0.

And in this manner may be found the limits of the roots
of equations of any dimensions. For more on this sub-
ject, see Maclaurin's Algebra, and Wood's Algebra.

Limit of Distinct Vision, in Optics. See Distinct Vision.

Limit of a Planet, has been sometimes used for its
greatest heliocentric latitude.

Limited Problem, denotes a problem that has but one
solution, or some determinate number of solutions: as, to
describe a circle through three given points that do not lie
in aright line, which is limited to one solution only; to
divide a parallelogram into two equal parts by a line pa-
rallel to one side, which admits of two solutions, accord-
ing as the line is parallel to the length or breadth of the
parallelogram ; or to divide a triangle in any ratio by a
line parallel to one side, which is limited to three solu-
tions, as the line may be parallel to any of the three sides.

LINE, in Geometry, a quantity extended in length only,
without either breadth or thickness. A line is sometimes
considered as generated by the flux or motion of a point;
and sometimes as the limit or termination of a superficies,
but not .as any part of that surface, however small.

Lines are either Right or Curved. A Right, or straight
line, is the nearest distance between two points, which are
its extremes or ends ; or it is a line which has in every
part of it the same direction or position. But a Curve
Line has in every part of it a different direction, and is not
the shortest distance between its extremes or ends.

Right Lines are all of the same species ; but curves are
of an infinite number of different sorts. As many may be
conceived as there are different compound motions, or as
many as there may be different relations between their or-
dinates, and abscisses. See Curves.

Again, Curve Lines are usually divided into geometrical
and mechanical.

Geometrical Lines are those which may be found ex-
actly in all their parts. See Geometrical Line.

Mechanical Lines are such as are not determined exactly
in all their parts, but only nearly, or tentatively.

Descartes, and his followers, define geometrical lines
to be those which may be expressed by an algebraical
equation of a determinate or finite degree, called its locus.
And mechanical lines, such as cannot be expressed by
such an equation. But others distinguish the same lines
by the name Algebraical and Transcendental.

Lines are also divided into orders, by Newton, accord-
ing to the number of intersections which may be made by
them and a right line, viz, the 1st, 2d, 3d, 4th, &c, order,
according as they may be cut by a right line, in 1, or 0,
or 3, or4, &c, points. In this way of considering them,
the right line only is of the 1st order, being but one in

L I N

[ 742 ]

LIS

number; the 2d order contains 4 curves only, being such saliant angles, called redents, redans, or flankers, towards
as maybe cut from a cone by a plane, viz, the circle, the the enemy. The distance between these angles is coin-
ellipse, the hyperbola, and the parabola ; the lines of the monly between the limits of 200 and 260 yards ; the or-
3d order have been enumerated by Newton, in a particular dinary flight of a musket ball, point blank, being corn-

treatise, who makes their number amount to 72; but
Mr. Stirling found 4 others, and Mr. Stone 2 more ;
though it is disputed by some whether the last two ought
to be accounted different from some of Newton's, or not.
See Newton's Enumer. Lin. Tertii Ordin. also Stirling's
Linea; Tert. Ordin. Newtqnianse Oxon. 1717, 8vo. and
Philos. Trans. No. 456", &c. Again,

Algebraical Lines are divided into different orders ac-
cording to the power or degree of their equations. So,
the simple equation a ■+■ by -+- ex = 0, or equation of the
1st degree, denotes the 1st order or right line; the equa-
tion a ■+- by ■+■ ex ■+- dy2, + exy -+-/x2 = 0, of the 2d de-
gree, denotes the lines of the 2d order ; and the equation
a -+■ by + ex -+- rfj/'2 ■+■ exy + fix'* -+■ gy3 ■+■ hxy* ■+■ ix~y ■+■
hx3 = 0, of the 3d degree, expresses the lines of the 3d
order ; and so on
Lignes Courbes.

Lines, considered with regard to their positions, are
either Parallel, Perpendicular, or Oblique. The con-
struction of which, see under the respective terms.

Line also denotes a French measure of length, being
the 12th part of an inch, or the 144th part of a foot.

In Astronomy,

monly within those limits ; though muskets a little elevated
will do effectual service at the distance of 36*0 yards.

Fundamental Line, is the first line drawn for the plan
of a place, and which shows its area.

Central Line, is the line drawn from the angle of the
centre to the angle of the bastion.

Line of Defence, &c. See Defence, &c.

Line of Approach, or Attack, signifies the work which
the besiegers carry on under cover, to gain the moat, and
the body of the place.

Line of Cir'cumvallation, is a line or trench cut by the
besiegers, within cannon-shot of the place, ranging round
the camp, aud securing its quarters against any relief to
be brought to the besieged.

Line of Contravallation, is a ditch bordered with a pa-
See Cramer's Introd. aT Analyse des rapet, serving to cover the besiegers on the side next the

place, and to stop the sallies of the garrison.

Lines of Communication are those which run from one
work to another.

Line of the Base, is that which joins the points of the
two nearest bastions.

To Line a work, signifies to face it, as with brick or
stone ; for example, to strengthen a rampart with a firm

Line of the Apses, or Apsides, the line joining the two wall, or to encompass a parapet or moat with good turf,

apses, or the longer axis of the orbit of a planet. &c.

Fiducial Line, the index line or edge of the ruler, Line, in Geography and Navigation, is emphatically

which passes through the middle of an astrolabe, or other used for the equator or equinoctial line,

instrument, on which the sights are fitted, and marking In Perspective,

the divisions. The Geometrical Line, is a right line drawn in any man-
Horizontal Line, a line parallel to the horizon. neron the geometrical plane.

Line of the Nodes, that which joins the nodes of the Terrestrial or Fundamental Line, is the common inter-
orbit of a planet, being the common section of the plane section of the geometrical plane and plane of the picture,
of the orbit with the plane of the ecliptic. Line of the Front, is any line parallel to the terrestrial
In Dialling, line.
Horizontal Line, is trie common section of the horizon Vertical Line, is the section of the vertical and draft

and the dial-plate.

Horary, or Hour Lines, arc the common intersections
of the hour-circles of the sphere with the plane of the dial.

Equinoctial Line, is the common intersection of the equi-
noctial and the plane of the dial.

In Fortification, Line is sometimes used for a ditch,
bordered with its parapet : and sometimes for a row of

planes.

Visual Line, is the line or ray conceived to pass from
the object to the eye.

Objective Line, is any line drawn on the geometrical
plane, whose representation is sought for in the draught
or picture.

Line of Measures, is used by Oughtred, and others, to

gabions, or sacks of earth, extended lengthwise on the denote the diameter of the primitive circle, in the pro
ground, to serve as a shelter against the enemy's fire, jection of the sphere in piano, or that line in which falls
When the trenches have been carried on within 30 paces the diameter of any circle to be projected.

of the glacis, they draw two lines, one on the right, and
the other on the left, for a place of arms.

Lines are commonly made to shut up an avenue or en-
trance to some place ; the sides of the entrance being co-
vered by rivers, woods, mountains, morasses, or other

Linear Numbers, aresuch as have relation to length
only ; such, for example, as express one side of a plane
figure ; and when the plane figure is a square, the linear
number is called a root.

Linear Problem, is one that can be solved geome-

obstructions, not easy to be passed over by an army, trically by the intersection of two right lines. This is
When they are constructed in an open country, they are called a simple problem, and is capable of only one so-
carried round the place to be defended, and resemble the lution.

lines surrounding a camp, called lines of circumvallation. LIQUID, a fluid which wets or smears such bodies as

Lines are also thrown up to stop the progress of an army; are immersed in it, arising from some configuration of its

but the term is mostly used for the line covering a pass particles, which disposes them to adhere to the surfaces

which can only be attacked in front. of bodies contiguous to them. Thus, water, oil, milk, &c,

When lines are made to cover a camp, or a large tract are liquids, as well as fluids ; bat quicksilver is not a li-

of land, where a considerable body of troops is posted,
the work is not made in one straight, or uniformly bend-
ing line ; but, at certain distances, the lines project in

quid, but simply a fluid

LISLE (William de), a learned French geographer,
was born at Paris in 1075. His father being much occu-

LOG

[ 743 ]

IOC

pied in the same pursuit, young Lisle began at 9 years of
age to draw maps, and soon made a considerable progress
in this art. In 1 699 he first distinguished himself to the
public, by giving a map of the world, and other pieces,
which procured him a place in the Academy of Sciences,
1702. He was afterwards appointed geographer to the
king, with a pension, and had the honour of instructing
the king himself in geography, for whose particular use he
drew up several works. Delisle's reputation was so
great, that scarcely any history or travels came out with-
out the embellishment of his maps. Nor was his name
less celebrated abroad than in his own country. Many
sovereigns in vain attempted to draw him out of France.
The Czar Peter, when at Paris on his travels, paid him a
visit, to communicate to him some remarks upon Musco-
vy; but more especially, says Fontenelle, to learn from
him, better than he could any-where else, the extent and
situation of his own dominions, Delisle died of an apo-
plexy in 1726, at 51 years of age. Besides the excellent
maps he published, he wrote many pieces in the Memoirs
of the Academy of Sciences.

LISLE (Joseph-Nicholas de), an eminent astrono-
mer and geographer, was born at Paris in 1688. He was
brother of the above-mentioned William, and had the
same education with him under their father, in geography
and astronomy. At the age of IS he made an accurate
observation of the great total eclipse of the sun in 1706;
which may be considered as the beginning of his public
career. He was admitted in the astronomical department
of the Academy in 1714. In 1724 he visited England,
and was much esteemed by Newton and Halley ; and he
was also chosen a Fellow of the Royal Society. In 1726
he was invited to Petersburg, where he had the charge of
the observatory, with a considerable pension. He re-
turned to 'Paris in 1747, where he filled the place of pro-
fessor to the Royal College, and where he formed the
astronomers Laland and Messier, illustrious pupils of so
great a master. M. Delisle died in 1768. His principal
works are, his History of Astronomy, 173S,in 2 vols 4to ;
and, 4 4 memoirs in the volumes of the Academy, from
the year 1714 to 1766.

LIST, or Listel, a small square moulding, serving to
crown or accompany larger mouldings ; or on occasion
to separate the flutings of columns.

LITERAL Algebra, See Algebra.

LIZARD, in Astronomy. See Lacerta.

LOADSTONE, or Magnet; which see.

LOCAL Problem, is one that is capable of an infinite
number of different solutions; because the point, which is
to solve the problem, may be indifferently taken within a
a certain extent; as suppose any>where in such a line,
within such a plane figure, &c, which is called a Geome-
trical Locus.

A local problem is Simple, when the point sought is in a
right line ; Plane, when the point sought is in the circum-
ference of a circle ; Solid, when it is in the circumference
of a conic section; or Sursolid, when the point is in the
perimeter of a line of a higher kind.

Local Motion, or Loco-Motion, the change of place:
See Motion.

LOCI, the plural of Locus, which see.
LOCI Plani, or Loca Plana, one of the last works of
Apollonius, that have, been restored by Schooten, in l656;
by Format, in 1679 ; and by Simson, in 1749.

LOCK, for Canals. See Canal.

LOCUS, is some line by which a local or indeterminate
problem is solved; or a line of which any point may
equally solve an indeterminate problem. Loci are ex-
pressed by algebraic equations of different orders, accord-
ing to the nature of the locus. If the equation, is con-
structed by a right line, it is called Locus ad Rectum ; if
by a circle, Locus ad Circulum ; if by a parabola, Locus
ad Parabolam; if by an ellipsis, Locus ad Ellipsim ; and
so on.
' The loci of such equations as are right lines or circles,
the ancients called Plane Loci ; and of those that are
conic sections, Solid Loci; but such as are curves of a
higher order, Sursolid Loci. The moderns distinguish the
loci into orders according to the dimensions of the equa-
tions by which they are expressed, or the number of the
powers of indeterminate or unknown quantities in any one
term: thus, the equation

ay = bx + c denotes a locus of the 1st order,
but y- = ax, or = ax — x2, &c, a locus of the 2d order,
and y3 = a2x, or = ax- — x3, &c, a locus of the 3d order,
and so on; where x and y are unknown or indeterminate
quantities, and the others known or determinate ones ;
also x denotes the absciss, and y the ordinate of the curve
or line which is the locus of the equation.

For instance, suppose two variable or indeterminate
right lines ap,
aq, making any
given angle paq
between them,
where they are
supposed to
commence, and
to extend inde-
finitely both

ways from the point a :

then calling any ap, x, and its
corresponding ordinate pq, y, continually changing its po-
sition by moving parallel to itself along the indefinite line
ap ; also in the line ap assume ab = a, and from b draw
bc parallel to pq and — b : then the indefinite line aq is
called in general a Geometrical Locus, and in particular

the locus of the equation y = — ; for whatever point q.is,

the triangles abc, apq. are always similar, and therefore
ab : bc : : ap : pq, that is a : b : : x : y, therefore ay = bx
is the equation to the right line aq, or aq is the locus 0^
the equation' ay = bx.

Again, if aq be a parabola, the
nature of which is such, that
ab : ap : : BCa : pq2, or a : x : : b2 :y%
and therefore ayx = b'x is the equa-
tion which has the parabola for its
locus, or the parabola is the locus to
every equation of this form ay1 = b2x.

Or-if aq be a circle, having its
radius ab = a, the nature of which
is this, that pq5 = ap . pd, or
y* = x . (2a — x) or = lax — x2 ;
therefore the locus of the equation
of this form y2 = 2«r — z% is a circle.

LOG [744

In like manner it will appear, that the ellipse is the
locus to the equation fyl = c~ x (tx — x2), and the hy-
perbola the locus to the equation Pi/2, == c'2 x (tx -+- x');
where t is the transverse, and c the conjugate axis of the
ellipse or hyperbola.

All equations whose loci are of the first order, may-
be reduced to one of the 4 following forms: 1st, y =

— ; 2d,^ = h f; 3d, y =- c; 4th, y = c — ';

where the letter c denotes the distance that the ordinates
commence from the line ap, either on the one side or
the other of it, according as the sign of that quantity is -+-
or — .

All loci of the 2d degree are conic sections, viz, either
the parabola, the circle, ellipsis, or hyperbola. There-
fore when an equation is given, whose locus is of the 2d
degree, and it is required to draw that locus, or, which is
the same thing, to construct the equation generally ; bring
over all the terms of the equation to one side, so that the
other side be. 0; then to know which of the conic sec-
tions it denotes, there will be two general cases, viz, either
when the rectangle xy is in the equation, or when it is not
in it.

Case 1 . When the term xy is not in the proposed equa-
tion. Then, 1st, if only one of the squares x'2,y~ be found
in it, the locus will be a parabola. 2d, If both the squares
be in it, and if they have the same sign, the locus will be a
circle or an ellipse. 3d, But if the signs of the squares
x'l,yL be different, the locus will be an hyperbola, or the
opposite hyperbolas.

Case 2. When the rectangle xy is in the proposed equa-
tion ; then 1st, If neither of the squares x~,y\ or only one
of them be in the equation, the locus will bean hyperbola
between the asymptotes. 2d, If both xr and y" be in it,
having different signs, the locus will be an hyperbola,
having the abscisses on its diameter. 3d, If both the
squares be in it, and with the same sign, then if the coef-
ficient of x2 be greater than the square of half the coeffi-
cient ofxy, the locus will be an ellipse; if equal, a para-
bola ; and if less, an hyperbola.

This method of determining geometric loci, by reducing
them to the most compound or general equations, was
first published by Mr. Craig, in his Treatise on the Qua-
drature of Curves, in 1693. It is explained at large in the
7th and Sth books of I Hospital's Conic Sections. This
subject is also particularly illustrated in Maclaurin's Al-
gebra. The method of Descartes for finding the loci of
equations of the 2d order, is by extracting the root of the
equation. Sec his Geometry ; as also Stirling's llluslratio
Linearum Tertii Ordinis. The doctrine of these loci is
likewise well treated by De Witt in his Elementa Curva-
rura. And Bartholomrcus Intieri, in his Aditus ad Nova
Arcana Geometrica delegenda, lias shown how to find the
loci of equations of the higher orders. Mr. Stirling too,
in his treatise above-mentioned, has given an example or
two of finding the loci of equations of 3 dimensions. Eu-
clid, Apollonius, Aristreus, Fermat, Viviani, have also
written on the subject of loci.

LOG, in Navigation, is a piece of thin board, of a sec-
toral or quadtantal form, loaded in the circular side with
lead sufficient to make it swim upright in the water; to
which is fastened a line of about 150 fathoms, or 300 yards
long, called the log-line, divided into certain spaces,
called knots, and wound on a reel turning very freely, for
the line to wind easily off.

3

LOG

The use of the log, or log-line, is to measure the velo-
city of the ship, or rate at which she runs, which is done
from time to time, as the foundation on which the ship's
reckoning, or finding her place, is kept ; and the practice
is, to heave the log into the sea, with the line tied to it,
and by observing how much of the line is run off the reel,
while the ship sails, during the space of half a minute,
which time is measured by a. sand-glass made to run
that time very exactly. About 10 fathoms of stray or
waste line is left next the log before the knotting or count-
ing commence, that space being usually allowed to carry
the log out of the eddy of the ship's wake.

The using of the log for finding the velocity of the ship,
is called heaving the log, and is thus performed : One
man holds the reel, and another the half-minute glass ; an
officer of the watch throws the log over the ship's stern,
on the lee-side, and when he observes the stray line, and
the first mark is going off, he cries Turn ! when the glass-
holder instantly turns the glass crying out Done ! then
watching the glass, the moment it is run out he says Slop !
upon which the reel being quickly stopt, the last mark
run off shows the number of knots, and the distance of
that mark from the reel is estimated in fathoms : then the
knots and fathoms together show the distance run in half
a minute, or the distance per hour nearly, by considering
the knots as miles, and the fathoms as decimals of a mile :
Thus, if 7 knots and 4 fathoms be observed, then the ship
runs at the rate of 7'4 miles an hour.

It follows, therefore, that the length of each knot, or
division of the line, ought to be the same part of a sea;
mile, as half a minute is of an hour, that is T4-oth part.
Now it is found that a degree of the meridian contains
nearly 366,000 feet, therefore -^ of this, or a nautical
mile, will be 6l00 feet ; the T^th of which, or 51 feet
nearly, should be the length of each knot, or division of
the log-line. But, because it is safer to have the reckon-
ing rather before the ship, than after it, it is therefore usual
now to make each knot equal to S fathoms or 48 feet. But
the knots are made sometimes to contain only 42 feet;
which method of dividing the log-line is founded on the
supposition, that 60 miles, of 5000 feet each, make a de-
gree; for -rl-^th of 5000 is 41|, or in round numbers 42
feet. And though many mariners find by experience that
this length of the knot is too short, yet rather than quit
the old way, they use sand-glasses lor half-minute ones
that run only 24 or 25 seconds. The sand, or hall-minute
glass, may be tried by a pendulum vibrating seconds, in
the following manner: On a hook or peg, hang a thread
or fine string that has a muskel-ball fixed to one end,
carefully measuring between the centre of the ball and the
string's loop over the nail 3Q£ inches, being the length of
a second pendulum ; then make it swing or vibrate very
small arches, and count one for every time it passes un-
der the nail, beginning at the second time it passes ; and
the number of swings made during the time the glass is
running out, shows the seconds in the glass.

It is not known who was the inventor of this method of
measuring the ship's way, or her rale of sailing ; but no
mention qf it occurs lill the year l6"07, in an East-India
voyage, published by Purchas ; and from that time its
name occurs in other -voyages in his collections; after
which it became more known, being noticed both by our
own authors, and by foreigners ; as by Gunter in 16'23;
Snellius, in l6'24; Melius, in lfiSl ; Oughtred, in IG'33 ;
llerigone, in 16'34; Saltonstall, in lo"36' ; Norwood, iu

LOG

[ 745 ]

LOG

1637 ; Fournicr, in 1643; and almost atl the succeeding
writers on navigation of every country. Various improve-
ments have lately been made on this instrument by differ-
ent persons. See Robertson's Navigation, the Preface,
pa. v. Also Hist. Math. torn, iv, 525 — 537.

Loa-Board, in Navigation, a board or table usually di-
vided into 5 columns : in the first of which is entered the
hour of the day ; in the 2d, the course steered ; in the 3d,
the number of knots run off the reel each time of heaving
the log; in the 4th, the direction of the wind; and in the
5th, observations on the weather, variation of the com-
pass, &c.

~LoG-Book, a book ruled in columns like the log-board,
into which the account on this is transcribed every day at
noon ; from whence, after it is corrected, &c, it is entered
into the journal.

LOGARITHM, from the Greek kayos ratio, and
xptSao; number; q. d. ratio of numbers, or perhaps rather
number of ratios ; the indices of the ratios of numbers to
one another; or a series of numbers in arithmetical pro-
portion, corresponding to as many others in geometrical
proportion, in such sort that 0 corresponds to, or is the
index of \, in the geometricals. They have been devised
for the ease of large arithmetical calculations.

Thus,

0, 1, 2, 3, 4, &c,indicesorlogarithms.
f 1,2, 4,' S , 16 , &c,\
\ot 2°, 21, 2°, 23, 24, &c, the geometrical

f 1,3, 9, 27, SI, ccc, \ progressions, or
"^or 3', 31, 3", 33, 34, &c, I common num-

f 1 , 10, 100, 1000, 10000, &c, j bers.
|orlO°, 10", 10-, 103, 104, &c,J

Where the same indices, or logarithms, serve equally for
any geometric series ; and from which it is evident, that
there may be an endless variety of sets of logarithms to
the same common numbers, by varying the 2d term 2, or
3, or 10; &c of the geometric series ; as this will change
the original series of terms whose indices are the numbers
1, 2, 3, &c ; and, by interpolation, the whole system
of numbers may be made to enter the geometrical series,
and receive their proportional logarithms, whether inte-
gers or decimals.

Or the logarithm of any given number, is the index of
such a power of some other number, as is equal to the
given one. So if n be = r", then the logarithm of n is n,
which may be either positive or negative, and r any num-
ber whatever, according to the different systems of loga-
rithms. When k is 1, then n is = 0, whatever the value
of r is ; and consequently the logarithm of 1 is always 0
in every system of logarithms. When n = 1, then N =r;
consequently the root r is always the number whose loga-
rithm is 1, in every system. When r = 2-718281S2S459
&c, the indices are the hyperbolic logarithms ; so that n
is always the hyperbolic logarithm of (2'71S &c)". But
in the common logarithms r = 10; so that the common
logarithm of any number, is the index of that power of 10
which is equal to the said number; so the common loga-
rithm of n = 10", is n the index of the power of 10 ;
example, 1000, being the 3d power of 10, has 3 for its lo-
garithm ; and if 50 be = lO1'6^, then is 1-69S97 the com-
mon logarithm of 50. And hence it follows that this de-
i imal series of terms

Vol. I.

1000, 100, .10 , 1 , •], '01, -001,

or 103, 10-, 10', 10°, 10~', lo"\ 10~3,
have 3, 2, 1, 0, —1, —2, —3,
respectively for the logarithms of those terms.

The logarithm of a number contained between any two
terms of the first series, is included between the two tor-
responding terms of the latter; and therefore that loga-
rithm will consist of the same index, whether positive or
negative, as the smaller of those two terms, to°ether with
a decimal fraction, which will always be positive. So the
number 50 falling between 10 and 100, its logarithm will
fall between 1 and 2, being indeed equal to 1 "69897 near-
ly : also the number -05 falling between the terms -1 and
•01, its logarithm will fall between —1 and —2, and is
indeed =-2 + -69S97, the index of the less term toge-
ther with the decimal -69S9T. The index is also called
the characteristic of the logarithms, and is always an in-
teger, either positive or negative, or else = 0 ; and it
shows what place is occupied by the first significant figure
of the given number, either above or. below the place of
units, being in the former case -+- or positive; in the
latter — or negative.

When the index of a logarithm is negative, the sign —
is commonly set over it, to distinguish it from the decimal
part, which, being the logarithm found in the tables, is
always positive : so — 2 •+■ -698fl7, or the logarithm of
•05, is written thus 2'69S97. But on some occasions it
is convenient to reduce the whole expression^ to a negative
torm ; which is done by making the characteristic less by
1, and taking the arithmetical complement of the decimal,
that is, beginning, at the left hand, subtract each figure
from 9, except the last significant figure, which is sub-
tracted from 10; so shall the remainders form the loga-
rithm wholly negative: thus the logarithm of -05, which
is 2-69897 or — 2 -+- "69897, is also expressed by
— 1"30103, which is all negative. It is also sometimes
thought more convenient to express such logarithms en-
tirely as positive, namely by only joining to the tabular
decimal the complement of the index to 10; and in this
way the above logarithm is expressed by S'69897; which
is only increasing the indices in the scale by 10.

The Properties of Logarithms. — From the definition of
logarithms, either as being the indices of a series of geo-
metricals, or as the indices of the power of the same root,
it follows, that the multiplication of the numbers will
answer to the addition of their logarithms ; the division of
numbers, to the subtraction of their logarithms; the
raising of powers, to the multiplying the logarithm of the
root by the index of the power; and the extracting of
roots, to the dividing the logarithm of the given number
by the index of the root required to be extracted.

So, 1st,
Log. ab or of a x b is = log. a -+- log. b,
Log. 1 b or of 3 x 6 is = log. 3 ■+■ log. 6,
Log. 5 x 9 x 73 is = log. 5 -f- log. 9 + log. 73.

Secondly,
Log. a ■— b is = log. a — log. h,

Log. 18 -r- 6 is = log. IS - log. 6,

Log. 79 x 5 -f- 9 is = log. 79 ■+■ log. 5 — log. 9,
Log. \ or 1 -7- 2 is = 1. 1 — 1. 2 = 0 - 1. 2 = - 1. 2,

Log": - or 1 -7- n is = 1.1 — l.n = — l.n.

LOG

[ m ]

LOG

Thirdly,
Log. r" is = n 1. r; log. r» or of »/r is = - 1. r;
Lot. r» is = - 1- r; log. 26 is =61. 2; log. 9J or

of^2 is = ■§■!. 2; and log. 2* is = £ 1.2.
So that any number and its reciprocal have the same lo-
garithm, but with contrary signs ; and the sum of the
logarithms of any number and its reciprocal, or comple-
ment, is equal to 0. For other particulars, see the Intro-
duction to my Logarithms.

History and Construction of Logarithms. — The properties
of logarithms hitherto mentioned, or of arithmetical in-
dicesato powers or geometricals, with their various uses
and properties, as above-mentioned, are taken notice of
by Stifelius, in his Arithmetic; and indeed they were not
unknown to the ancients ; but they come all far short of
the use of logarithms in trigonometry, as first discovered
by John Napier, baron of Merchiston in Scotland, arid
published at Edinburgh in 1614, in his Mirifici Logarith-
morum Canonis Descriptio; which contained a large canon
of logarithms, with the description and uses of them ; but
theirDconstruction Was reserved till the sense of the learned
concerning his invention should be known. This work
was translated into English by the celebrated Mr. Edward
Wright, and published by his son in l6l6. In the year
1619, Robert Napier, son of the inventor of logarithms,
published a new edition of his late father's work, together
with the promised construction of the logarithms, with
other miscellaneous pieces written by his father and Mr.
Briggs. And in the same year, 1619, Mr. John Speidell
published his New Logarithms, being an improved form of
Napier's.

All these tables were of the kind that have since been
called hyperbolical, because the numbers express the areas
between the asymptote and curve of the hyperbola. And
logarithms of this kind were also soon after published by
several other persons; as by Ursinus in 1619, Kepler in
1624, and some others.

On the first publication of Napier's logarithms, Henry
Briggs, then professor of geometry in Gresham-college in
London, immediately applied himself to the study and im-
provement of them, and soon published the logarithms of
the first 1000 numbers, but on a new scale, which he had
invented, viz, in which the logarithm of the ratio of 10 to
1 is 1, the logarithm of the same ratio in Napier's system
bcina 2-3025S &c ; and in l624, Briggs published his
Arithmetica Logarithmica, containing the logarithms of
30,000 natural numbers, to 14 places of figures besides
the index, in a form that Napier and he had agreed upon
together, being the present form of logarithms; also in
]633 was published, to the same extent of figures; his Tri-
gonometria Britannica, containing the natural and logarith-
mic sines, tangents, &c.

With various and gradual improvements, logarithms
were also published successively, by Gunter in l620, Win-
gate in 1624, Henrion in 16*26, Miller and Norwood in
1631, Cavalerius in l6"32 and 1643, Vlacq and Rows in
16'33, Frobenius in 1634, Newton in 16'58, Caramuel in
]6'70, Sherwin in 1706', Gardiner in 1742, and Dodson's
Aatilogarithmic Canon in the same year, Taylor in 1793,
besides many others of lesser note; not to mention the ac-
curate and comprehensive! tables in the Tables Portative
in 1793, &c, those of Borda and Delambre in 1801, and

in my own Logarithms lately published, where a complete
history of this science may be seen, with the various ways
of constructing them, that have been invented by different
authors.

In Napier's construction of logarithms, the natural
numbers, aud their logarithms, as he sometimes called
them, or at other times the artificial numbers, are sup-
posed to arise, or to be generated, by the motions of points,
describing two lines, of which the one is the natural num-
ber, and the other its logarithm, or artificial. Thus, he
conceived the line or length of the radius to be described,
or run over, by a point moving along it in such a manner,
that in equal portions of time it generated, or cut off, parts
in a decreasing geometrical progression, leaving the several
remainders, or sines, in geometrical progression also;
while another point described equal parts of an indefinite
line, in the same equal portions of time ; so that the re-
spective sums of these, or the whole line generated, were
always the arithmeticals or logarithms of the aforesaid
natural sines. In this idea of the generation of the lo-
garithms and numbers, Napier assumed 0 as the logarithm
of the greatest sine or radius ; and next he limited his
system, not by assuming a particular value to some as-
signed number, or part of the radius, but by supposing
that the two generating points, which, by their motions
along the two lines, described the natural numbers and
logarithms, should have their velocities equal at the be-
ginning of those lines. And this is the reason that, in his
table, the natural sines and their logarithms, at the com-
plete quadrant, have equal differences or increments ; and
this is also the reason why his scale of logarithms happens
accidentally to agree with what have since been called the
hyperbolical logarithms, which have likewise numeral dif-
ferences equal to those of their natural numbers at the
beginning ; except only that these latter increase with the
natural numbers, while his on the contrary decrease ; the
logarithm of the ratio of 10 to 1 being the same in both,
namely 2-30258509 &c.

Having thus limited his system, Napier proceeds, in
the posthumous work of l6l9> to explain his construction
of the logarithmic canon. This he effects in various ways,
but chiefly by generating, in a very easy manner, a scries
of proportional numbers, and their arithmeticals or loga-
rithms; and then finding, by proportion, the logarithms
to the natural sines from those of the natural numbers,
among the original proportionals ; a particular account of
which may be seen in my book of logarithms above men-
tioned.

The methods above alluded to, relate to Napier's or the
hyperbolical system of logarithms, and indeed are in a
manner peculiar to that kind of them. But in an appen-
dix to the posthumous work, mention is made of other
methods, by which the common logarithms, agreed on by
him and Briggs, may be constructed, and which it appears
were written aflcr that agreement. One of these methods
is as follows: Having assumed 0 for the logarithm of 1,
and 10000 &c, for the logarithm of 10; this logarithm
of 10, and the successive quotients, are to be divided ten
times by 5, by which divisions there will be obtained these
other ten logarithms, namely 2000000000, 400000000,
S0O000O0, 16000000, 3200000, 640000, 12S0£>0, 'J56'00,
5120, 1024; then this last logarithm, and ifs'qiioticnts,
being divided ten times by 2, will give these other ten lo-
garithms,

LOG [ 747 J

viz, 512, 256, 128, 64, 32, 16, S, 4, 2, 1.
Also the numbers answering to these twenty logarithms
are to be found in this manner, viz, Extract the 5th root
of 10 (with ciphers), then the 5th root of that root, and
so on for ten continual extractions of the 5th root: so
shall these ten roots be the natural numbers belonging to
the first ten logarithms above found, in dividing conti-
nually by 5. Next, out of the last 5th root is to be ex-
tracted the square root, then the square root of this last
root, and so on for ten successive extractions of the square
root: so shall these last ten roots be the natural numbers
corresponding to the logarithms or quotients arising from
the last ten divisions by the number 2. And from these
twenty logarithms, 1, 2, 4, 8, &c, and their natural
numbers, the author observes that other logarithms and
their numbers may be formed, namely by adding the lo-
garithms, and multiplying their corresponding numbers.
But, besides the immense labour of this method, it is evident
that this process would generate rather an antilogarithmic
canon, such as Dodson's, than the table of Briggs.

Napier next mentions another method of deriving a
few of the primitive numbers and their logarithms,
namely, by taking continually geometrical means, first be-
tween 10 and 1, then between 10 and this mean, and again
between 10 and the last mean, and so on; and then taking
the arithmetical means between their corresponding lo-
garithms.

He then lays down various relations between numbers
mid their logarithms, such as, that the products and quo-
tients of numbers, answer to the sums and differences of
their logarithms; and that the powers and roots of num-
bers, answer to the products and quotients of the loga-
rithms when multiplied or divided by the index of the
power or root, &c; as also that, of any two numbers,
whose logarithms are given, if each number be raised to the
power denoted by the logarithm of the other, the two re-
mits will be equal; thus, if x be the logarithm of any
number x, and y the logarithm of y, then is x>" = yx.
Napier then adverts to another method of making the
.logarithms to a few of the prime integer numbers, which
is well adapted for the construction of the common table
of logarithms: this method easily follows from what has
been said above, depending on this property, that the lo-
garithm of any number in this scale, is one less than the
number of places or figures contained in that power of the
given number whose exponent is 10000000000, or the lo-
garithm of 10, at least as to integer numbers, for they really
differ by a fraction, as is shown by Mr. Briggs in his illus-
trations of these properties; printed at the end of the Ap-
pendix to the Construction of Logarithms.

Kepler gave, a construction of logarithms somewhat
varied from Napier's. His work is divided into two parts :
In the first, he raises a regular and purely mathematical
system of proportions, and the measures of them, demon-
strating both the nature and principles of the construction
of logarithms, which he calls the Measures of Ratios:
and in the second part, he applies those principles in the
actual construction of his table, which contains only 1000
numbers and their logarithms. The fundamental prin-
ciples are briefly these: That at the beginning of the lo-
garithms, their increments or differences are equal to those
of the natural numbers: that the natural numbers may
be considered as the decreasing cosines of increasing arcs :
and that the secants of those arcs at the beginning have
the same differences as the cosines, and therefore the

LOG

same differences as the logarithms. Then, since the se-
cants are the reciprocals of the cosines of the same arcs,
from the foregoing principles, he establishes the following
method of raising the first 100 logarithms, to the num-
bers 1000, 999, <;9S, &c, to yOO ; viz, in this manner:
Divide the radius 1000, increased with seven ciphers, by
each of these numbers separately, and the quotients will be
the secants of those arcs which have the divisors for their
cosines; continuing the division to the Sth figure, as it is
in that place only that the arithmetical and geometrical
means differ. Then by adding continually the arith-
metical means between every two successive secants, the
sums will be the series of logarithms. Or by adding
continually every two secants, the successive sums
wjll be the series of the double logarithms. He then de-
rives all the other logarithms from these first 100, by com-
mon principles.

Briggs first adverts to the methods mentioned above, in
the Appendix to Napier's Construction, which methods
were common to both these authors, and had doubtless
beenjointiy agreed on by them. He then gives an example
of computing a logarithm by the property, that the lo-
garithm is one less than the number of places or figures
contained in that power of the given number whose ex-
ponent is the logarithm of 10 with ciphers. Briggs next
treats of the other general method of finding the loga-
rithms of prime numbers, which he thinks is an easier way
than the former, at least when many figures are required.
This method consists in taking a great number of conti-
nued geometrical means between 1 and the given number
whose logarithm is required; that is, first extracting the
square root of the given number, then the root of the 1st
root, the root of the 2d root, the root of the 3d root, and
so on, till the last root shall exceed 1 by a very small de-
cimal, greater or less according to the intended number of
places to be in the logarithm sought: then finding the lo-
garithm of this small number, by easy methods described
afterwards, he doubles it as often as he made extractions of
the square root, or, which is the same thing, he multiplies
it by suGh power of 2 as is denoted by the said number
of extractions, and the result is the required logarithm of
the given number; as is evident from the nature of loga-
rithms.

But, as the extraction of so many roots is a very trou-
blesome operation, our author devises some ingenious
methods of abridging that labour, chiefly by a proper ap-
plication of the several orders of the differences of num-
bers, forming the first instance of what may be called The
Differential Method; for a particular description of which,
see my Logarithms, above quoted, pa. 65, &c.

Mr. James Gregory, in his Vera Circuli Hyperbolae
Quadrature, printed at Padua in 1667, having approxi-
mated to the hyperbolic asymptotic spaces by means of a
series of inscribed and circumscribed polygons, from thence
shows how to compute the logarithms, which are analo-
gous to the areas of those spaces : and thus the quadrature
of the hyperbolic spaces became the same thing as the com-
putation of the logarithms. He here also lays down various
methods to abridge the computation, with the assistance of
some properties of numbers themselves, by which the loga-
rithms of all prime numbers under 1000 may be computed,
each by one multiplication, two divisions, and the extrac-
tion of the square root. And the same subject is further
pursued in his Exercitationes Geometricaj. In this latter
place, he first finds an algebraic expression, in an infinite
5 C 2.

LOG

t 748 ]

LOG

series, for the logarithm of , and then likewise for the

logarithm of ; and as the one series has all its terms

° 1 —a

positive, while those of the other are alternately positive

and negative, by adding the two together, every 2d term

is cancelled, and the double of the other terms gives the

logarithm of the product of

and ,or the logarithm oi

1 l — a °

ratio of 1 — a to 1 + a :
Thus, he finds,
first a —

and a -+-

-, that is, of the

theref. 2a

Which may be accounted Mr. James Gregory's method of
making logarithms.

In l66'S, Nicholas Mercator published his Logarith-
niotechnia, sive Methodus Construendi Logarithmos, nova,
accurata, et facilis ; in which he delivers a new and in-
genious method for computing the logarithms on princi-
ples purely arithmetical; and here, in his modes of think-
ing and expression, he closely follows the celebrated
Kepler, in his writings on the same subject; accounting
logarithm^ as the measures of ratios, or as the number of
ratiunculoe contained in the ratio which any number bears
to unity. Purely from these principles then, the number
of the equal ratiunculoe contained in some one ratio, as of
10 to 1, being supposed given, our author shows how the
logarithm, or measure, of any other ratio may be found.
But this, however, only by-the-bye, as not being the prin-
cipal method he intends to teach, as his last and best. Hav-
ing shown then, that these logarithms, or numbers of small
ratios, or measures of ratios, may be all properly repre-
sented by numbers, and that of 1, or the ratio of equality,
the logarithm or measure being always 0, the logarithm of
10, or the measure of the ratio of 10 to 1, is most conve-
niently represented by 1 with any number of ciphers; he
then proceeds to show how the measures of all other ratios
may be found from this last supposition: and he explains
these principles by some examples in numbers.

In the latter part of the work, Mercator treats of his
other method, given by an infinite series of algebraic terms,
which are. collected in numbers by common addition only.
He here squares the hyperbola, and finally finds that the
hyperbolic logarithm of 1 ■+■ a, is equal to the infinite
scries a — \az ■+■ \u? — £a' &c ; which may be consi-
dered as Mercator's quadrature of the hyperbola, or his
general expression of an hyperbolic logarithm, in an in-
finite series.

And this method was further improved by Dr. Wallis,
in the Philos. Trans, for the year l668 ; by showing that
the hyp. log. of the ratio of 1 to 1 — a is equal to the in-
finite series a •+- \ai -+- ^a3 -+- Ja* -t- &c. The celebrated
Newton invented also the same series for the quadrature
of the hyperbola, and the construction of logarithms, and
that before they were given by Gregory and Mercator,
though unknown to one another, as appears by his letter
to Mr. Oldenburg, dated October 24, 1676- The expla-
nation and construction of the logarithms are also further
pursued in his Fluxions, published in 1736' by Mr. Colson.

Or. Halley, in the Philos. Trans, for the year l6y5,

gave a very ingenious essay on the construction of loga-
rithms, entitled, " A most compendious and facile method
for constructing the logarithms, and exemplified and de-
monstrated from the nature of numbers, without any re-
gard to the hyperbola, with a speedy method for rinding
the number from the given logarithm."

Instead of the more ordinary definition of logarithms,
viz, 'numerorumproportionaliumasquidifferentescomites,*
the learned author adopts this other, ' numeri rationum
exponentes/ as better adapted to the principle on which
logarithms are here constructed, considering them as the
number of ratiunculae contained in the given ratios whose
logarithms are in question. In this way he first arrives at
the logarithmic series before given by Newton and others,
and afterwards, by various combinations and sections of
the ratios, he derives others, converging still faster than
the former. Thus he found the logarithms of several
ratios, as below, viz, when multiplied by the modulus pe-
culiar to the scale of logarithms,

q — ±q" -t- -Ij3 — ^q* &c, the log. of 1 to 1 •+- q,

q -+- iq'2 '•+■ -Jo3 -+• ^q* &c, the log. of 1 to 1 — q,

— s- h r — — t &c, the log. of a to b, or

a 2a" 3a 4a °

—rr &c, the same log. of a to b, or

Ox7

— -■ &c, the same log. of a to b,

7z7 °

a*
— 5- &c, the log. of /Jab to iz, or

— -. &c, the same log. of \/ab to iz ;
72/" ' ° v *

where a, b, q, are any quantities, and the values of x, y, z,
are thus, viz, x = 6 — a, z = b -+- a, y= ab -+- £z*.

Dr. Halley also, the first who performed the reverse of
the problem, by assigning the number to a given loga-
rithm ; viz,

*</

ui3

Bar1

IF

^10

= 1 -h l-t- il°-

■I* &c, or

-P &c. where I is the

0 T '2.3 2.3.4

logarithm of the ratio of a the less, to b the greater of any
two terms.

Mr. Abraham Sharp of Yorkshire made many calcula-
tions and improvements in logarithms, &c. The most re-
markable of these were, his quadrature of the circle to 72
places of figures, and his computation of logarithms to 6l
figures, viz, for all numbers to 100, and for all prime num-
bers to 1 100.

The celebrated Mr. Roger Cotes gave to the world a
learned tract on the nature and construction of loga-
rithms: this was first printed in the Philos. Trans. No. 338,
and afterwards with his Harmonia Mensurarum in 1722,
under the title Logometria. This tract however has justly
been complained of, as very obscure and intricate, and the
principle is something between that of Kepler and the
method of fluxions. He invented the terms modulus and
modular ratio, this being the ratio

ecc to 1 or

of 1

of 1 to 1 h

&c;

that is the ratio of 2'71 8281828459 &c to 1,
or the ratio of 1 to 0'S(>7879*4117J &c;

LOG

[ 749 ]

LOG

the modulus of any system being the measure or logarithm
of that ratio, which in the hyp. logarithms is 1, which in
Briggs's or the common logarithms isO'4342944S 1903 &c.

The learned Dr. Brook Taylor gave another method of
computing logarithms in the Philos. Trans. No. 352, which
is founded on these three principles, viz, 1st, That the sum
of the logarithms of any two numbers is the logarithm of
the product of those numbers; 2d, That the logarithm of
1 is 0, and consequently that the nearer any number is to
1, the nearer will its logarithm be to 0; 3d, That the pro-
duct of two numbers or factors, of which the one is greater,
and the other less than 1, is nearer to 1, than that factor
is which is on the same side of 1 with itself; so of the two
numbers § and |, the product f- is less than 1, but yet
rearer to it than \ is, which is also less than 1. — And on
these principles he founds an ingenious, though not very
obvious, approximation to the logarithms of given numbers.

In the Philos. Trans, a Mr. John Long gave a method of
constructing logarithms, by means of a small table, some-
thing in the manner of one of Briggs's methods for the same
purpose.

Also in the Philos. Trans, vol. 6l, a tract on the con-
struction of logarithms is given by the ingenious Mr. Wil-
liam Jones. In this method, all numbers are considered
as some certain powers of a constant determined root:
thus, any number x 4s considered as the r power of any
root r, or x = rz is taken as a general expression for all
numbers in terms of the constant root r and a variable ex-
ponent z. Now the index z being the logarithm of the
number x, therefore to find this logarithm, is the same
thing as to find what power of the radix r is equal to the
number x.

An elegant tract on logarithms, as a comment on Dr.
Halley's method, was also given by Mr. Jones in his Sy-
nopsis Palmariorum Matheseos, published in the year 1706.

In the year 1742, Mr. James Dodson published his Anti-
logarithmic Canon, containing all logarithms under
100,000, and their corresponding natural numbers to
eleven places of figures, with all their differences and the
proportional parts ; the whole arranged in the order con-
trary to that used in the common tables of numbers and
logarithms, the exact logarithms being here placed first,
and their corresponding nearest numbers in the columns
opposite to them.

And in 176'7, Mr. Andrew Reid published an "Essay
on logarithms," in which he shows the computation of lo-
garithms from principles depending on the binomial theo-
rem, and on the nature of the exponents of powers, the
logarithms of numbers being here considered as the expo-
nents of the powers of 10. In this way he brings out the
usual series for logarithms, and exemplifies Dr. Halley's
construction of them. But for the particulars of this, and
the methods given by the other authors, we must refer to
the historical preface to my treatise on logarithms.

Besides the authors above-mentioned, many others have
treated on the subject of logarithms ; among the principal
of whom are, Leibnitz, Euler, Maclaurin, Wolfius, Keill.
Lagrange, and professor Simson in an ingenious geometri-
cal tract on logarithms, contained in his posthumous works,
elegantly printed at Glasgow in the year 1776, at the ex-
pense of the learned Earl Stanhope, and by his lordship
disposed of in presents among gentlemen most eminent for
mathematical learning.

For the description and uses of logarithms in numeral
calculations, with the various methods of constructing

them, see the historical introduction to my Logarithms
pa. 124 et seq.

Briggs's or Common Logarithms, are those that have
1 for the logarithm of 10, or which have 0-4342944319
&c for the modulus ; as has been explained above.

Hyperbolic Logarithms, are those that were computed
by the inventor Napier, and called also sometimes natural
logarithms, having 1 for their modulus, or 2*302585092994
&c for the logarithm of 10. These have since been called
hyperbolical logarithms, because they are analogous to
the areas of a right-angled hyperbola between the asym-
ptotes and the curve. See Logarithms, also Hyperbola
and Asymptotic Space.

Logistic Logarithms, are certain logarithms of sexa-
gesimal numbers or fractions, useful in astronomical cal-
culations. The logistic logarithm of any number of se-
conds, is the difference between the common logarithm of
that number and the logarithm of 3600, the seconds in 1
degree.

The chief use of the table of logistic logarithms, is for
the ready computing a proportional part in minutes and
seconds, when two terms of the proportion are minutes
and seconds, or hours and minutes, or other such sexage-
simal numbers. See the introd. to my Logarithms, pa. 144.

Imaginary Logarithm, a term used in the Log. of
imaginary and negative quantities; such as — a,or^/ — a2
or a \/ — 1. The fluents of certain imaginary expres-
sions are also imaginary logarithms; as of — -. , or of

*v- i

— — — , &c. See Euler Analys. Infin. vol. i. pa. 72, 74.

It is well known that the expression - represents the
fluxion of the. logarithm of x, and therefore the fluent of
- is the logarithm of x ; and hence the fluent of — ;

x a V"1

is the imaginary logarithm of x.

However, when these imaginary logarithms occur in
the solutions of problems, they may be transformed into
circular arcs or sectors; that is, the imaginary logarithm,
or imaginary hyperbolic sector, becomes a real circular
sector. See Bernoulli Oper. torn, i, pa. 400, and pa. 512.
Maclaurin's Fluxions, art. 762. Cotes's Harmon. Mens,
pa. 45. Walmesley, Anal, des Mes. pa. 63.

LOGARITHMIC, or Logistic Curve, a curve so
called from its properties and uses, in explaining and con-
structing the logarithms, because its ordinates are in geo-
metrical progression, while the abscisses are in arithmetical
progression ; so that the abscisses are as the logarithms of
the corresponding ordinates. And hence the curve will
be constructed in this manner : Upon any right line, as an
axis, take the equal parts ab, bc, cd, &c, or the arith-
metical progression ab, ac, ad, &c; and at the points
A, b, c, n, &c, erect the perpendicular ordinates ap, bq,
cr, ds, &c, in a geometrical progression ; so is the curve
line drawn through all the points p, q, r, s, &e, the lo-
garithmic, or logistic curve; so called, because any absciss
ab, is as the logarithm of its ordinate bq. So that the
axis abc &c is an asymptote to the curve.

LOG

[ 750 ]

L O N

Hence, if any absciss an = x, its ordinate no =3/,
ap = 1, and a = a certain constant quantity, or the mo-
dulus of the logarithms ; then the equation of the curve
is x = a x log. of y = \og.y.

And if the fluxion of this equation be taken, it will be

x= — ; which gives this proportion,

j : x : :y : a
but in any curve j : x : : y : the subtangent at ; there-
fore the subtangent of this curve is every-where equal to the
constant quantity a, or the modulus of the logarithms.
To find the Area contained between two ordinates. Here

the ftuxion of the area a oryx is y x 2- = aj ; and the

correct fluent is a = a x (ap — y) = a x (ap — wo)
= a x pv = at x pv. That is, the area apon be-
tween any two ordinates, is equal to the rectangle of the
constant subtangent and the difference of the ordinates.
And hence, when the absciss is infinitely long, or the far-
ther ordinate equal to nothing, then the infinitely long
area apz is equal at x ap, or doublethe triangle apt.
To find the Solid formed by the curve revolved about its
xis az. The fluxion of the solid is s = py2x =

py2 x — = payj, where p is = 3" 1416 ; and the correct

fluent is s ■= ^pa x (apj- y1) = ip x at x (ap2— no-),
which is half the difference between two cylinders of the
common altitude a or at, and the radii of their bases ap,
no. And hence, supposing the solid infinitely long to-
wards z, where y or the ordinate is nothing, the infinitely
long solid will be equal to \j)a x ap" = \p x at x ap3,
or half the cylinder on the same base and its altitude at.
It has been said that Gunter gave the first idea of a
curve whose abscisses are in arithmetical progression,
while the corresponding ordinates are in geometrical pro-
gression, or whose abscisses are the logarithms of their or-
dinates; but I do not find it noticed in any part of his wri-
tings. This curve was afterwards considered by others,
and named the logarithmic or logistic curve, by Huygens,
in his Dissertatio de Causa Gravitatis, where he enume-
rates all the principal properties of it, showing its analogy
to logarithms. Many other learned men have also treated
of its properties ; particularly Ja. Gregory, in the preface
to his Geometrioe Pars Universalis &c ; Le Seur and Jac-
quier, in their Comment on 'Newton's Principia; Dr. John
Keill, in the elegant little Tract on Logarithms subjoined
to his edition of Euclid's Elements ; and Francis Maseres
Esq. Cursitor Baron of the Exchequer, in his ingenious
Treatise on Trigonometry : see also Bernoulli's Discourse
in the Acta Eruditorum for the year l6Q6\ pa.2l6; Guido
Grando's Demonstratio Thcorcmatum Huygeneanorum
circa Logisticam seu Logarithmicam Lineam ; and Emerson
on Curve Lines, pa. 1 p. — It is indeed rather extraordinary
that this curve was not sooner announced to the public,
since it results immediately from Napier's manner of con-
ceiving the generation of logarithms, by only supposing
the lines which represent the natural numbers as placed at
right angles to that upon which the logarithms are taken.
This curve greatly facilitates the conception of loga-
rithms, and affords an almost intuitive proof of the very
important property of their fluxions, or very small incre-
ments, namely, that the fluxion of the number is to the
fluxion of the logarithm, as the number is to the subtan-
gent; as also this property, that if three numbers be taken
very nearly equal, so that their ratios may differ but a

little from a ratio of equality, as the three numbers
10000000, 10000001, 10000002, their differences will
be very nearly proportional to the logarithms of the ratios
of those numbers to each other : all which follows from
the logarithmic arcs differing very little from their chords,
when they are taken very small. And the constant sub-
tangent of this.curve is what was afterwards by Cotes called
the modulus of the system of logarithms.

Logarithmic, or Logistic Spiral, is a curve constructed
as follows. Divide the arch of a circle into any number
of equal parts ab, bd, de, &c;
and upon the radii drawn to the
points of division take cb, cd, ce, &c,
in a geometrical progression; so is
the curve Abde &c the logarithmic
spiral; being thus called, because it
is evident that ab, ad, ae, &c, being
arithmeticals, are as the logarithms
of ca, cb, ed, ce, &c which are geo-
metrical ; and a spiral, because it winds continually about
the centre c, coming continually nearer, but without ever
really falling into it. In the Philos. Trans. Dr. Halley has
happily applied this curve to the division of the meridian
line in Mercator's chart. See alo Cotes's Harmonia Mens.,
Guido Grando's Demonst. Theor. Huygcn., the Acta
Erudit. 16.01, and Emerson's Curves, &c.

LOGISTICS, or LOGISTICAL Arithmetic, a name
sometimes employed for the arithmetic of sexagesimal
fractions, used in astronomical computations. This name
was perhaps taken from a Greek treatise of Barla;mus, a
monk, who wrote a book on Sexagesimal Multiplication,
which he called logistic. Vossius places this author about
the year 1350, but he mistakes the work for a treatise on
algebra. The same term however has been used for the
rules of computations in algebra, and in other species of
arithmetic: witness the logistics of Vieta and other writer?.
Shakerly, in his Tabular Britannica?, has a table of
logarithms adapted to sexagesimal fractions, which he,
calls logistical logarithms ; and the expeditious arith-
metic, obtained by means of them, he calls logistical
arithmetic.

Logistical Curve, Line, or Spiral, the same as the Lo-
garithmic ; which see.

LONG (Roger), d. d. master of Pembroke-hall in
Cambridge, Lowndes's professor of astronomy in that uni-
versity, &c. He was author of a well-known and much-
approved treatise of astronomy, and the inventor of a re-
markably curious astronomical machine. This was a hol-
low sphere, of IS feet diameter, in which more than 30
persons might sit conveniently. Within side the surface,
which represented the heavens, was painted the stars and
constellations, with the zodiac, meridians, and axis parallel
to the axis of the world, upon which it was easily turned
round by a winch. He died, December 16", 1770, at 01
years of age.

A few years before his death, Mr. Jones gave some anec-
dotes of Dr. Long, as follows: " He is now in the SSth
year of his age, and for his years vegcte and active. He
was lately put in nomination for the office of vice-chan-
cellor: he executed that trust once before, I think in the
year 1737. He is a very ingenious person, and sometimes
very facetious. At the public comrm ..cement, in the year
1713, Dr. Greene (master of Bennet- college, and after-
wards bishop of Ely) being then vice-chancellor, Mr.
Long was pitched upon for the tripos performance; it was

L O N

I 751 ]

L O N

witty and humourous, and has passed through divers edi-
tions. Some that remembered the delivery of it, told me,
that in addressing the vice-chancellor (whom the univer-
sity wags usually styled Miss Greene), the tripos-orator,
being a native of Norfolk, and assuming the Norfolk dia-
lect, instead of saying Domine Vice-Cancellarie, archly
pronounced the words thus, Domina Vice-Cancellaria;
which occasioned a general smile in that great auditory.
His friend the late Mr. Bonfoy of Ripton related to me
the following little incident: ' That he and Dr. Long
walking together in Cambridge in a dusky evening, and
coming to a short post fixed in the pavement, which Mr.
Bonfoy in the midst of chat and inattention took to be a
boy standing in his way, he said in a hurry, ' Get out of
my way, boy !' ' That boy, Sir,' said the doctor very
calmly and slily, ' is a post-boy, who turns out of his way
for nobody.' I could recollect several other ingenious re-
partees if there were occasion. One thing is remarkable,
lie never was a hale and hearty man, always of a tender and
delicate constitution, yet took great care of it: his common
drink water; he always dines with the fallows in the hall.
Of late years he left off eating flesh ; in the room thereof,
puddings, vegetables, &c; sometimes a glass or two of wine."

LONGIMETRY, the art of measuring lengths or
distances, both accessible and inaccessible, forming a part
of what is called heights and distances, being an applica-
tion of geometry and trigonometry to such measurements.
As to accessible lengths, they are easily measured by the
actual application of a rod, a chain, or wheel, or some
other measure of length.

But inaccessible lengths require the practice and pro-
perties of geometry and trigonometry, either in the mea-
surement and construction, or in the computation. For
example, Suppose it were required to know the length or
distance between the two places a and b, to which places
there is free access, but not to the intermediate parts, on
account of water or some other impediment ; measure
therefore, from a and b, the distances to any convenient
place c, which suppose to be thus, viz. ac = 735, and
bc = 840 links ; and let the angle at c, taken with a the-
odolite or other instrument, be 55° 40'. From these mea-
sures the length or distance ab may be determined, either
by geometrical construction, or by trigonometrical com-
putation. Thus, first lay down an angle c = 55° 40', and
upon its legs set off, from any convenient scale of equal
parts, ca = 735, and cb = 840; then measure the di-
stance between the points a and B by the same scale of
equal parts, which will be found to be 740 nearly. Or

thus by calculation,

840 180° - 55° 40' = 124° 20', its half 62° 10',

735
Sum 1575 - MS7280G

Dif. 105 0.02118<)3

102773793
p-1012880
9-9711092
9-9168593
0-9242793

Tang. 62° 10'
Tang. MljJ, - . -

s. Sum or £. a =99° 21|$
to s. /I c — 55 40 -
So bc = 840
To ab = 741-2

For a 2d Example — Suppose it were required to find
the distance between two inaccessible objects, as between
the house and mill, h and m; first measure any conveni-
ent line on the ground, as ab, 300 yards; then at the
station a take the angles bam = 58° 20', and mah =
37 ; also at the station b take the angles abh =
53° 30', and hbm = 45° 15'; hence the distance or length
mh may be found, either by geometrical construction, or
by trigonometrical calculation, thus :

First draw a line ab of the given length of 300, by a
convenient scale of equal parts ; then at the point a lay
down the angles bam and mah of the magnitudes above
given; and also at the point b the given angles abh and
hbm ; then by applying the length hm to the same scale of
equal parts, it is found to be nearly 480 yards.

Otherwise, by calculation. First, by adding and sub-
tracting the angles, there is found as below :

37°

00'

58° 20' 53° 30'

58

20

53 30 45 15

53

30

45 15 suiiiq8 45 ^ ABM

sums

148

50

157 05

from

180

00

180 00

/.AHB

31

10

22 55 ^.ame

Then, as sin.

AHB

: sin. abh : : ab : ah = 4659776",

and, as

sin.

AMB

: sin. abm : : ab ; am = 76l-4655,

their sum is 1227-4431
and their diff. 295-4879
Then as sum am 4- ah : to dif. am — ah : :

tang. I AHM -+- \ AMII = 71° 30',

to tang. J ahm — \ amh = 35 44

the dif. of which is a mh = 35 46.
Lastly, as s. amh : s. mah : : ah : hm = 479'7933, the
distance sought.

LONGITUDE of the Earth, is sometimes used to de-
note its extent from west to east, according to the
direction of the equator : by which it stands contra-dis-
tinguished from the latitude of the earth, which denotes
its extent from one pole to the other.

Longitude of a Place, in Geography, is its longitudi-
nal distance from some first meridian, which is assumed
arbitrarily by different nations : the English call the first
meridian, that which passes over the Royal Observatory
at Greenwich ; the French take the meridian of Paris for
the first meridian ; the Spaniards, Madrid ; and so most
other nations reckon that the first meridian which passes
over their metropolis.

/ The longitude of any place is therefore an arch of the
equator intercepted between the meridian of that place,
and the first meridian.

Longitude in the Heavens, as of a star, &c, is an arch
of the ecliptic, counted from the beginning of Aries, to
the place where it is cut by a circle perpendicular to it,
and passing through the place of the star.

Longitude of the Sun or Star from the next equinoctial
point, is the degrees they are distant from the beginning
of Aries or Libra, either before or after them ; which can
never exceed 180 degrees.

LOS

L 762 ]

LON

Longitude, Geocentric, Heliocentric, &c, the longitude
of a planet as seen from the earth, or from the sun. See
the respective terms.

Longitude, in Navigation, is the distance of a ship, or
place, east or west, from some other place or meridian,
counted in degrees of the equator. When this distance is
counted in leagues, or miles, or in degrees of the meridian,
and not in those proper to the parallel of latitude, it is
usually called departure.

An easy practicable method of finding the longitude at
sea, is the only thing wanted to render the art of naviga-
tion perfect, and is a problem that has greatly perplexed
mathematicians for the last two centuries : accordingly
most of the commercial nations of Europe have offered
great rewards for the discovery of it ; and in consequence
very considerable advances have been made towards a per-
fect solution of the problem, especially by the English.

In the year 1598, the government of Spain offered a
reward of 1000 crowns for the solution of this problem;
and soon after the States of Holland offered 10 thousand
florins for the same. Encouraged by such offers, in l635,
M. John Morin, professor of mathematics at Paris, pro-
posed to cardinal Richlieu, a method of resolving it ; and
though the commissioners, who were appointed to examine
this method, on account of the imperfect state of the lunar
tables, judged it insufficient, cardinal Mazarin, in 1645,
procured for the author a pension of 2000 livres.

In 1714 an act was passed in the British parliament,
allowing 2000/. towards making experiments; and also
offering a reward to the person who should discover the
longitude at sea, proportioned to the degree of accuracy
that might be attained by such discovery ; viz, a reward
of 10,000/. if it determines the longitude to one degree of
a great circle, or 60 geographical miles; 15,000/. if it de-
termines the same to two-thirds of that distance; and
20,000/.' if it determines it to half that distance; with
other regulations and encouragements. 12 Ann. cap. 15.
See also stat. 14 Geo.' 2, cap. 3°, and 26 Geo. 2, cap. 25.
But, by stat. Geo. 3, all former acts concerning the
longitude at sea are repealed, except so much of them as
relates to the appointment and authority of the commis-
sioners, and such clauses as relate to the publishing of
nautical almanacs, and other useful tables ; and it enacts,
that any person who shall discover a method for finding
the longitude by means of a time-keeper, the principles of
which not having hitherto been made public, shall be en-
titled to the reward of 5000/. if it shall enable a ship to
keep her longitude, during a voyage of 6 months, within
60 geographical miles, or one degree of a great circle; to
7500/. if within 40 geographical miles, or two-thirds of a
degree of a great circle; or to a reward of 10,000/. if
within 30 geographical miles, or half a degree of a great
circle. But if the method shall be by means of improved
solar and lunar tables, the author of them shall be entitled
to a reward of 5000/. if they show the distance of the
moon from the sun and stars within 15" of a degree, an-
swering to about 7' of longitude, after making an allow-
ance of half a degree for the errors of observation, and
after comparison with astronomical observations for a pe-
riod of 18^ years, or during the period of the irregularities
of the lunar motions. Or that in case any other method
shall be proposed for finding the longitude at sea, besides
those before-mentioned, the author shall be entitled to
5000/. il it shall determine the longitude within one degree
of a great circle, or 60 geographical miles; to 7500/. if

within two-thirds of that distance; and to 1?),000/. if
within half the said distance.

Accordingly, many attempts have been made for such
discovery, and several ways proposed, with various degrees
of success. These however have been chiefly directed to
methods of determining the difference of time between any
two points on the earth ; for the longitude of any place
being an arc of the equator intercepted between two me-
ridians, and this arc being proportional to the time re-
quired by the sun to move from the one meridian to the
other, at the rate of 4 minutes of time to one degree of the
arc, it follows that the difference of time being known,
and turned into degrees according to that proportion, it
will give the longitude.

This measurement of time has been attempted by some
persons by means of clocks, watches, and other automata :
for if a clock or watch were contrived to go uniformly at
all seasons, and in all places and situations ; such a ma-
chine being regulated, for instance, to London or Green-
wich time, would always show the time of the day at Lon-
don or Greenwich, wherever it should be carried to; then
the time of the day at this place being found by observa-
tions, the difference between these two times would give
the difference, of longitude, according to the proportion
of one degree to 4 minutes of time.

Gemma Frisius, in his tract De Principiis Astronomias
et Geographic, printed at Antwerp in 1530, it seems, first
suggested the method of finding the longitude at sea by
means of watches, or time-keepers; which machines, he
says, were then but lately invented. And soon after, the
same was attempted by Metius, and some others ; but the
state of watch-making was then too imperfect for that
purpose. Dr. Hooke and Mr. Huygens also, about the
year 1664, applied the invention of the pendulum-spring
to watches; and employed it for the purpose of discover-
ing the longitude at sea. Some disputes however between
Dr. Hooke and the English ministry prevented any expe-
riments from being made with watches constructed by
him; but many experiments were made with some con-
structed by Huygens; particularly Majoi Holmes, in a
vo3'age from the coast of Guinea in 1665, by one of these
watches predicted the longitude of the island of Fuego to
a great degree of accuracy. This success encouraged
Huygens to improve the structure of his watches, (see
Philos. Trans, for May l66o) ; but experience soon con-
vinced him, that unless methods could be discovered for
preserving the regular motion of such machines, and pre-
venting the effects of heat and cold, and other disturbing
causes, they could never answer the intention of disco-
vering the longitude, and on this account his attempts
failed.

The first person who turned his thoughts this way, after
the public encouragement held out by the act of 1714,
was Henry Sully, an Englishman; who, in the same year,
printed at Vienna, a small tract on the subject of watch-
making; and afterwards removing to Paris, he employed
himself there in improving time-keepers for the discovery
of the longitude. It is said he greatly diminished the
friction in the machine, and rendered uniform that which
remained : and to him is principally to be attributed what
Is yet known of watch-making in France: for the cele-
brated Julier le Roy was his pupil, and to him owed most
of his inventions, which he afterwards perfected and exe-
cuted : and this gentleman, with his son, and M. Ber-
thoud, are the principal persons in France who have

L O N

[ 753 ]

L O N

turned their thoughts this way since the time of Sully.
Several watches made by these last two artists, have been
tried at sea, it is said with good success, and many ac-
counts have been published of these trials, at considerable
length.

In the year 1726 our countryman, Mr. John Harrison,
produced a time-keeper of his own construction, which
did not err above one second in a month, for 1.0 years to-
gether : and in the year 1736 he had a machine tried in
a voyage to and from Lisbon ; which was the means of
correcting an error of almost a degree and a half in the
computation of the ship's reckoning. In consequence of
this success, Mr. Harrison received public encourage-
ment to proceed, and he made three other time-keepers,
each more accurate than the former, which were finished
successively in the years 1739, 1758, and 176l ; the last
of which proved so much to his own satisfaction, that he
applied to the commissioners of longitude to have this
instrument tried in a voyage to some port in the West-
Indies, according to the directions of the statute of the
12th of Anne above cited. Accordingly, Mr. William
Harrison, son of the inventor, embarked in November
176l, on a voyage for Jamaica, with this 4th time-keeper
or watch ; and on his arrival there, the longitude, as
shown by the time-keeper, differed but one geographical
mile and a quarter from the true longitude, deduced from
astronomical observations. The same gentleman returned
to England, with the time-keeper, in March 1762; when
he found that it had erred, in the 4 months, no more than
1' 54"| in time, or 28| minutes of longitude ; whereas the
act requires no greater exactness than 30 geographical
miles, or minutes of a great circle, in such a voyage. Mr.
Harrison now claimed the whole reward of 20,000<, offered
by the said act: but some doubts arising in the minds of
the commissioners, concerning the true situation of the
island of Jamaica, and the manner in which the time at
that place had been found, as well as at Portsmouth ;
and it being farther suggested by some, that though the
time-keeper happened to be right at Jamaica, and after its
return to England, it was by no means a proof that it had
been always so in the intermediate times; another trial
was therefore proposed, in a voyage to the island of Bar-
badoes, in which precautions were taken to obviate as
many of these objections as possible. Accordingly, the
commissioners previously sent out proper persons to make
astronomical observations at that island, which, when
compared with other corresponding ones made in England,
would determine, beyond a doubt, its true situation: and
Mr. William Harrison again set out with his father's time-
keeper, in March 1764, the watch having been compared
with equal altitudes at Portsmouth, before hp set out, and
he arrived at Barbadoes about the middle of May; where,
on comparing it again by equal altitudes of the sun, it
was found to show the difference of longitude, between
Portsmouth and Barbadoes, to be 3h 55™ 3s ; the true dif-
ference of longitude between these places, by astronomical
observations, being 3h 54m 20'; so that the error of the
watch was 43*, or 10' 45" of longitude. In consequence
of this, and the former trials, Mr. Harrison received one
moiety of the reward offered by the 12th of Queen Anne,
after explaining the principles on which his watch was
constructed, and delivering this as well as the three former
to the commissioners of the longitude, for the use of the
public: and he was promised the other moiety of the
reward, when Other time-keepers should be made, on the •

Vol. I.

same principles, either by himself or others, performing
equally well with that which he had last made. In the
mean time, this last time-keeper was sent down to the
Royal Observatory at Greenwich, to be tried there under
the direction of the'Rev. Dr. Maskelyne, the astronomer-
royal. But it did not appear, during this trial, that the
watch went with the regularity that was expected; from
which it was apprehended, that the performance even of the
same watch, was not at all times equal ; and consequently
that little certainty could be expected in the performance
of different ones. Moreover, the watch was now found to
go faster than during the voyage to and from Barbadoes,
by 18 or 19 seconds in 24 hours: but this circumstance
was accounted for by Mr. Harrison ; who informs us that
he had altered the rate of its going by trying some expe-
riments, which he had not time to finish before he was
ordered to deliver up the watch to the Board. Soon
after this trial, the commissioners of longitude agreed with
Mr. Kendal, one of the watch-makers appointed by them
to receive Mr. Harrison's discoveries, to make another
watch on the same construction with this, to determine
whether such watches could be made from the account
which Mr. Harrison nad given, by other persons, as well
as himself. The event proved the affirmative ; for the
watch produced by Mr. Kendal, in consequence of this
agreement, went considerably better than Mr Harrison's
did. Mr. Kendal's watch was sent out with C;ipt. Cook,
in his 2d voyage towards the south pole and round the
globe, in the years 1772, 1773, 1774, and 1775; when
the only fault found in the watch was, that its rate of
going was continually accelerated ; though in this trial,
of 3 years and a half, it never amounted to 14"| a day.
The consequence was, that the house of commons in 1774,
to whom an appeal had been made, were pleased to order
the 2d moiety of the reward to be given to Mr. Harrison,
and to pass the act above-mentioned. Mr. Harrison had
also at different times received some other sums of money,
as encouragements to him to continue his endeavours,
from the Board of Longitude, and from the India Com-
pany, as well as from many individuals. Mr. Arnold,
Mr. Mudge, Mr. Ernshaw, Mr. Hardy, and some other
persons have since also made several very good watches
for the same purpose.

Others have proposed various astronomical methods for
finding the longitude, which chiefly depend on having an
ephemeris or almanac suited to the meridian of some
place, as Greenwich for instance, to which the nautical
almanac is adapted, that shall contain for every day com-
putations of the times of all remarkable celestial motions
and appearances, as adapted to that meridian. So that,
if the hour and minute be known when any of the same
phenomena are observed in any other place, whose longi-
tude is desired, the difference between this time and that
to which the time of the said phenomenon was calculated
and set down in the almanac, will be known, and conse-
quently the difference of longitude also becomes known,
between that place and Greenwich, allowing at the rate
of 15 degrees to an hour.

Now it is easy to find the time at any place, by means
of the altitude or azimuth of the sun or stars ; which time
it is necessary to find by the same means, both in these
astronomical modes of determining the longitude, and in
the former by a time-keeper; and it is the difference be-
tween that time, so determined, and the time at Green-
wich, known cither by the time-keeper or by the astrono-
5D

LON

[ 754 ]

LON

mical observations of celestial phenomena, which gives
the difference of longitude, at the rate above-mentioned.
But here the great difficulty is, that there are only a
few phenomena proper, or capable of being thus
observed ; for all slow motions, such as belong to the
planet Saturn for instance, are quite excluded, as afford-
ing too small a difference, in a considerable space of time,
to be properly observed ; indeed there are no phenomena
in the heavens proper for this purpose, except the eclipses
or motions of Jupiter's satellites, and the eclipses or mo-
tions of the moon, viz, such as her distance from the sun
or certain fixed stars lying near her path, or her longitude
or place in the zodiac, &c.

1st, That by the eclipses of the moon is very easy, and
sufficiently accurate, if they did but happen often, as every
night. • For at the moment when the beginning, or middle,
or end of an eclipse is observed by a telescope, there is
no more to be done but to determine the time by observing
the altitude or azimuth of some known star; which time
oeing compared with that in the tables, set down for the
happening of the same phenomenon at Greenwich, gives
the difference in time, and consequently of longitude
sought. But as the beginning or end of an eclipse of the
moon cannot generally be observed nearer than one mi-
nute, and sometimes 2 or 3 minutes of time, the longitude
■cannot certainly be determined by this method, from a
single observation, nearer than one degree of longitude.
However, by two or more observations, as of the begin-
ning and end &c, a much greater degree of exactness may
be attained.

2d, The moon's place in the zodiac is a phenomenon
more frequent than that of her eclipses ; but then the ob-
servation of it is difficult, and the calculus perplexed and
intricate, by reason of two parallaxes; so that it is hardly
practicable, to any tolerable degree of accuracy.

3d, But the moon's distances from the sun, or certain
fixed stars, are phenomena to be observed many times in
almost every night, and afford a good practical method of
determining the longitude of a ship at almost any time;
either by computing, from thence, the moon's true place,
to compare with the same in the almanac; or by com-
paring her observed distance itself with the same as there
set down.

It is said that the first person who recommended the
finding the longitude from this observed distance between
the moon and some star, was John Werner, of Nuremberg,
who printed his annotations on the first book of Ptolemy's
Geography in 1514. And the same thing was recom-
mended in 1524, by Peter Apian, professor of mathema-
tics at Ingolstadt; also about 1530, by Oronce Fin6, of
Briancon ; and the same year by the celebrated Kepler,
and by Gemma Frisius, at Antwerp; and in 1560, by
Nonius or Pedro Nunez.

Nor were the English mathematicians behind hand on
this head. In 1665 Sir Jonas Moore prevailed on king
Charles the 2d to erect the Royal Observatory at Green-
wich, and to appoint Mr. Flamstced his astronomical ob-
server, with this express command, that he should apply
himself with the utmost care and diligence to the rectify-
ing the table of the motions of the heavens, and the places
of the fixed stars, in order to find out the longitude at sea,
lor perfecting the art of navigation. And to the fidelity
and industry with which Mr. Flamsteed executed his
commission, it is that we arc chielly indebted for that cu-
rious theory of the moon, which was afterwards formed by

the immortal Newton. This incomparable . philosopher
made the best possible use of the observations with which
he was furnished ; but as these were interrupted and im-
perfect, his theory would sometimes differ from the hea-
vens by 5 minutes or more.

Dr. Halley bestowed much time on the same object •
and a starry zodiac was published under his direction
containing all the stars to which the moon's appulse can
be observed; but for want of correct tables, and proper
instruments, he could not proceed in making the necessary
observations. In a paper on this subject, in the Philos.
Trans. No. 421, he expresses his hope, that the instrument
just invented by*Mr. Hadley might be applied to takin"
angles at sea with the desired accuracy. This oreat astro-
nomer, and after him Lacaille, and others, 'have reckoned
the best astronomical method for finding the longitude at
sea, to be that in which the distance of the moon from the
sun or from a star is used; for the moon's daily motion
being about 13 degrees, her hourly mean motion is above
half a degree, or one minute of a degree in two minutes of
time; so that an error of one minute of a degree in posi-
tion will produce an error of 2 minutes in time, or half a
degree in longitude. Now from the great improvements
made by Newton in the theory of the moon, and more
lately by Euler and others on his principles, Professor
Mayer of Gottingen, was enabled to calculate lunar ta-
bles more correct than any former ones ; havinor so far
succeeded as to give the moon's place within one minute
of the truth, which has been proved by a comparison of
the tables with the observations made at the Greenwich
observatory by the late Dr. Bradley, and by Dr. Maske-
lyne, the astronomer royal ; and the same have been
still further improved under his direction, by the late
Mr. Charles Mason, by several new equations, and the
whole computed to tenths of a second. These new tables,
when compared with the above-mentioned series of obser-
vations, a proper allowance being made for the unavoida-
ble error of observation, seem to give always the moon's
longitude in the heavens correctly within 30 seconds of a
degree; which greatest error, added to a possible error of
one minute in taking the moon's distance from the sun or
a star at sea, will at a medium only produce an error of 42
minutes of longitude. Further successive improvements
in them have since been made, by the labours of Messieurs
Lagrange, Laplace, Lalande, Delambre, Burg, and Vince;
by which means it is thought the astronomical tables have
been brought to the most correct form that the present
state of astronomical knowledge will permit.

To facilitate the use of the tables, Dr. Maskelyne pro-
posed a nautical ephemeris, the scheme of which was
adopted by the commissioners of longitude, and first exe-
cuted in the year 1707; since which time it has been re-
gularly continued, and published as far as for the year
1816. But as the rules that were given in the appendix
to one of those publications, for correcting the effects of
refraction and parallax, were thought too difficult for ge-
neral use, they have been reduced to tables. So that, by
the help of the ephemeris, these tables, and others that are
■also provided by the Board of Longitude, the calculations
relating to the longitude, which could not be performed
by the most expert mathematician in less than 4 hours,
may now be completed with great ease and accuracy in
halfan hour.

As this method of determining the longitude depends on
the use of the tables annually published for this purpose, .

LON

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Lax

those who wish for further information, are referred to the
instructions that accompany them, and particularly to
those that are annexed to the Tables requisite to be vised
with the Astronomical and Nautical Ephemeris, 3d edit.
1802. See Lunar Method, below.

4th, The phenomena of Jupiter's satellites have com-
monly been preferred to those of the moon, for finding the
longitude; because they are less liable to parallaxes than
these are, and besides they afford a very commodious ob-
servation whenever the planet is above the horizon. Their
motion is very swift, and must be calculated for every
hour. These satellites of Jupiter were no sooner an-
nounced by Galileo, in his Syderius Nuncius, first printed
at Venicein l6l0, than the frequency of their eclipses re-
commended them for this purpose; and among those who
treated on this subject, none was more successful than
Cassini. This great astronomer published, at Bologna, in
l68S, tables for calculating the appearances of their
eclipses, with directions for finding the longitudes of places
by them ; and being invited to France by Louis the 14th,
he there, in the year 1693, published more correct tables
of the same. But the mutual attractions of the satellites
rendering their motions very irregular, those tables soon
became useless for this purpose; insomuch that they re-
quire to be renewed from time to time; a service which
has been performed by several ingenious astronomers, as
Dr. Pound, Dr. Bradley, M. Cassini the son, and more
especially by Mr. Wargentin, whose tables are much
esteemed, and have been published in several places, as
well as in the Nautical Almanacs for 1771 and 1779.

Now, to find the longitude by these satellites; with a
good telescope observe some of their phenomena, as the
conjunction of two of them, or of one of them with Ju-
piter, &c ; and at the same time find the hour and minute,
from the altitudes of the stars, or by means of a clock or
watch, previously regulated for the place of observation ;
then, consulting tables of the satellites, observe the time
when the same appearance happens in the meridian of the
place for which the tables are calculated ; and the differ-
ence of time, as before, will give the longitude.

The eclipses of the first and second of Jupiter's satellites
are the most proper for this purpose; and as they happen
almost daily, they afford a ready means of determining the
longitude of placesat land, having indeed contributed much
to the modern improvements in geography; and if it were
possible to observe them with proper telescopes, in a ship
under sail, they would be of great service in ascertaining
its longitude from time to time. To obviate the inconve-
nience to which these observations are liable from the mo-
tions of the ship, a Mr. Irwin invented what he called a
marine chair; this was tried by Dr. Maskclyne, in his
voyage to Barbadoes, when it was not found that any be-
nefit could be derived from the use of it. And indeed,
considering the great power requisite in a telescope proper
for these observations, and the violence, as well as irregu-
larities in the motion of a ship, it is to be feared that the
complete management of a telescope on ship-board, will
always remain among the desiderata in this part of nauti-
cal science. And further, since all methods that depend
on the phenomena of the heavens, have also this other de-
fect, that they cannot be observed at all times, this renders
the improvement of time-keepers an object of the more
importance.

Many other schemes and proposals have been made by
different persons, but most of them of very little or no

use ; such as by the space between the flash and report of
a great gun, proposed by Messrs. Whiston and Ditton;
and another proposed by Mr. Whiston, by means of the in-
clinatory or dipping needle ; besides a method by the va-
riation of the magnetic-needle, &c, &c.

For approximate .constructions for clearing the lunar
distances from the effects of parallax and refraction, the
reader may consult Dr. Kelly's Spherics and Nautical
Astronomy, &c. And for more detailed information re-
specting the above, and other methods of ascertaining the
longitude, see Dr. Mackay's and Mr. Mqndoza's valuable
works ; also Vince's Astronomy, vol.2, pa. 515, &c, and
O. Gregory's Astronomy, pa. 446 to 470.

Longitude of Motion, is a term used by Dr. Wallis
for the measure-of motion, estimated according to its line
of direction ; or it is the distance or length gone through
by the centre of any moving body, as it moves on in a right
line. — The same author calls the measure of any motion,
estimated according to the line of direction of the vis mo-
trix, the altitude of it.

LONGOMONTANUS (Christian), a learned astro-
nomer, born in Denmark in 1562, in the village of Lon-
gomontum, whence he took his name. Vossius, by mistake,
calls him Christopher. He was the son of a poor plough-
man,and-he was obliged to suffer, during his studies, all the
hardships to which he could be exposed, dividing his time,
like the philosopher Cleanthes, between the cultivation of
the earth and the lessons he received from the minister of
the place. At length, at 15 years of age he stole away
from his family, and went to Wiburg, where there was a
college, in which he spent 1 1 years ; and though he was
obliged to earn his livelihood as he could, his close appli-
cation to study, enabled him to make a great progress in
learning, particularly in the mathematical sciences.

Hence he went to Copenhagen ; where thes professors of
that university soon conceived a very high opinion of him,
and recommended him to the celebrated Tycho Brahe,
with whom Longomontanus lived 8 years, and was of
great service to him in his observations and calculations.
At length, being very desirous of obtaining a professor's
chair in Denmark, Tycho Brahe consented, with some dif-
ficult}', to his leaving him, giving him a discharge filled
with the highest testimonies of his esteem, and furnishing
him with money for the expense of his long journey from
Germany, whither Tycho had retired. He accordingly
obtained a professorship of mathematics in the university
of Copenhagen in 1605, the duty of which he discharged
very worthily till his death, which happened in l647> at
85 years of age.

Longomontanus was author of several works, which
show great talents in mathematics and astronomy. The
most distinguished of which, is his Astronomica Danica,
first printed in 4to, 1621, and afterwards in folio in 1640,
with augmentations. He amused himself with endeavour-
ing to square the circle, and pretended that he had made
the discovery of it; but our countryman Dr. John Pell
attacked him warmly on that subject, and proved that he
was mistaken. — It is remarkable that, obscure as his villago
and father were, he contrived to dignify and eternize them
both ; for he took his name from his village, and in the
title page to some of his works he wrote himself Christianus
Longomontanus Severini filius, his father's name being Se-
verin or Severinus.

LOXODROMIC Curve, or Spiral, is the same as the
rhumb line, or path of a ship sailing always on the same
5 D 2

LUN

[ 756 ]

LUN

course in an oblique direction, or making always the same of finding the longitude at sea by celestial observations,
angle with every meridian. It is a species of logarithmic the moon is the chief instrument; for the quickness of her

spiral, described on the surface of the sphere, having the
meridians for its radii.

LOXODROMICS, the art or method of oblique sail-
ing, by the loxodromic or rhumb line.

LOZENGE, an oblique-angled parallelogram ; being
otherwise called a rhombus, or a rhomboides.

LUBIENIETSKI (Stanislaus), a Polish gentleman,
born at Cracow, in 1623, and educated with great care by
his father. He was learned in astronomy, and became a
celebrated Socinian minister. He took great pains to ob-
tain a toleration from the German princes for his Socinian
brethren. His endeavours however were all in vain ;
being himself persecuted by the Lutheran ministers, and
banished from place to place ; till at length he was banished
out of the world, with his two daughters, by poison, in
1675, his wife narrowly escaping.

We have, of his writing, A History of the Reformation
in Poland ; and a Treatise on Comets, entitled Theatrum
Cometicum, printed at Amsterdam in 2 volumes folio ;
which is a most elaborate work, containing a minute histo-
rical account of every single comet that had been seen or
recorded.

LUCIDA Coronje, a fixed star of the 2d magnitude,
in the northern crown. See Corona Borealis.

Lucid a Hydrje. See Cor Hydra.

Lucida Ltrjs, a bright star of the first magnitude in
the constellation Lyra.

LUCIFER, a name given to the planet Venus, when she
appears in the morning before sunrise.

LUDOLPH van Ceulen. See Ceulen.

LUMINARIES, a term used for the sun and moon,
by way of eminence, for their extraordinary lustre, and
the great quantity of light they give us.

LUNA, the Moon ; which see.

LUNAR, something relating to the moon.

Lunar Cycle, or Cycle of the Moon. See Cycle.

Lunar Distance, in nautical astronomy, a term de-
noting the distance of the moon from the sun, or from a
fixed star lying nearly in the line of its path. The mea-
surement of the apparent lunar distances, and the deter-
mination of the true lunar distances from thence, are of
great use in determining the longitude at sea, or on land.

Lunar Method for the Longitude, a method of keepinjg
or finding the longitude by means of \ the moon's motions,
particularly by her observed distances from the sun and
stars; for which, see the article Longitude.

This method consists in taking and reducing the distance
between the moon and the sun or a fixed star, in order to
find the longitude at sea; which has been already ex-
plained generally under the article longitude ; but the
great utility and importance of the subject require here a
further and more minute explanation of its principles and
operations, with an account of the different methods that
have been devised for obtaining the solution; and I shall
conclude the article with a very simple projection, in-
vented by Dr. Kelly, which is found sufficiently correct
for the common purposes of nautical practice, and even
where the greatest accuracy is required, it is found useful
as a guide or check to calculation.

The lunar method of finding the longitude at sea, is the
greatest modern improvement in navigation. In order to
form a clear and comprehensive idea of the subject, it
should be first considered, that in the most practical methods

motion renders her peculiarly well adapted for measuring
small portions of correspondent time. — Now, as she is seen
in the same part of the heavens nearly, at the same instant
of absolute time, from all parts of the earth where she is
visible, and as she is continually and sensibly changing her
place, it is evident that if two correspondent observers
were to note the precise moment of their respective times,
when she is seen in any particular part of the heavens the
difference between those times would show the difference
of longitude.

In every method of finding the longitude by the moon,
the first object is to be able to ascertain the part of the
heavens where she is : — this is easily seen at the time of
her eclipses, or at the occultation of a fixed star, and these'
were naturally the first methods resorted to ; but they oc-
cur too seldom for general use : the moon's place how-
ever may be marked with equal precision, by taking her
distance from some fixed object, whose latitude and lon-
gitude are known ; the stars in or near the zodiac are pre-
ferred, as the nearer such objects are to the moon's path,
the quicker will be her motion with respect to them : and
though her motion is not uniform, yet, during the short
space of time that she is near any star, it may be consi-
dered as such.

It may be observed, that if two persons under different
meridians were to mark the moon's place, and also their
relative times of observation, they might thence tell their
difference of longitude; but they could not communicate
their observations sufficiently soon for practical purposes;
but even admitting the possibility of this, it were neces-
sary that the longitude of one place should be known, in
order to determine that of the other. Now the Nautical
Almanac is calculated to supply all these wants. This
work may be considered a perpetual observer, that com-
municates universally and instantaneously certain celestial
appearances, as they take place at Greenwich observatory.
Here the distances are given between the moon and the
sun, and certain remarkable stars in or near the zodiac,
for every three hours ; and any intermediate distance, or
time, may be thence found by the rule of proportion, with
sufficient accuracy. If therefore, under any meridian, a
lunar distance be observed, the difference between the
time of observation and the time in the almanac, when
the same distance was to take place at Greenwich, will
show the longitude. For example, if the observed dis-
tance between the sun and moon be 50° at 8 o'clock, but
by the almanac the same distance of 50° will take place at
Greenwich at 6, it is evident that the difference between
the observed and computed time is 2 hours, and therefore
the longitude is 30°; and it is also clear that the longi-
tude is east, the time being most advanced at the place of
observation.

A method so apparently simple must have been long
since adopted, but for two difficulties which occurred :
one the want of proper instruments ; which now has been
happily supplied by the invention and subsequent im-
provement of Hadley's quadrant; and the other, correct
lunar tables; for the moon, though so near and so con-
spicuous to the earth, has always perplexed astronomers
more than any other celestial body. The various in-
equalities of her motions were never properly understood,
until Sir Isaac Newton discovered the physical laws which
govern them ; and from his theory professor Mayer formed

L U N

[ 707 ]

L U N

Junar tables, from which those tables, in the Nautical
Almanac, of the lunar distances were calculated, under
the direction of Dr. Maskelyne for many years. In 1806
the French Board of Longitude published new Lunar Ta-
bles, calculated by Du Burg, from the theory of Laplace
and the observations of Dr. Maskelyne ; and from these
tables the lunar distances in the Nautical Almanac of
1813, and the following years, are computed.

The above two difficulties having been obviated, a third
seems still to remain ; and though this is in some measure
removed by the application of the Nautical Almanac and
requisite tables, yet still the calculation is more tedious
than might be wished ; nor is it possible to render it much
shorter, as the problem necessarHy comprehends the solu-
tions of two spheric triangles : this arises from the circum-
stance of the observed distances between the heavenly bo-
dies not being the true distances ; for the altitudes of those
bodies are more or less affected both by refraction and
parallax ; and though these effects only operate in a ver-
tical direction, yet that which changes the altitude of two
bodies, must also change the distance between them. This
is evident from the consideration, that the altitude of a ce-
lestial object is an arc of an azimuth circle intercepted
between the object and the horizon, and as all azimuth
circles incline gradually to each other, from the horizon
to the zenith, where they meet, it is plain that the more
two bodies are apparently raised, the less will be their
apparent distance asunder, and the contrary.

It has been already stated that the heavenly bodies are
raised by refraction, and depressed by parallax; and that
these effects are greatest in the horizon, and gradually di-
minish to the zenith, where they become nothing. Re-
fraction depends on altitude alone, but parallax depends
on both altitude and distance from the earth. All celes-
tial objects, except the moon, are more affected by re-
fraction than by parallax, and therefore appear above
their true places; but the moon is always seen, excepting
in the zenith, below her true place, being more affected
by parallax than by refraction, on account of her prox-
imity to the earth.

These effects of parallax and refraction, though coun-
teracting each other, seldom do it so equally as to render
all correction unnecessary. Sometimes the apparent di-
stance is nearly a whole degree more or less than the true
distance; and the principal cause of so great a difference
is the moon's parallax; for this body, which is the chief
guide to the longitude, is also the great cause of error in
the distances, and is therefore the principal object of cor-
rection.

In making a lunar observation, four persons are usually
employed, one of whom takes the distance, two the alti-
tudes, and the fourth notes the time. These things should
be performed at the same instant; and if the observation
be repeated several times, and a mean taken, the work is
likely to be the more correct; and great care is here ne-
cessary, for an error in this part of the operation, parti-
cularly in taking the distance, will pervade the subsequent
parts of the work, and will of course produce a wrong solu-
tion. The manner of adjusting the instruments, and of
making the observations, is best taught by practice. Those
who wish for written instructions on the subject, are re-
ferred to the British Mariner's Guide by Dr. Maskelyne,
to Dr. Mackay's book on the Longitude, or to professor
Vinee's Practical Astronomy.

Of correcting the Altitudes of the observed Objects. — When
a lunar observation is made, the first object is to clear the
altitudes from semidiameter, dip, refraction, and parallax,
according to usual practice. The moon's parallax in al-
titude must next be calculated : by saying, " As radius is
to the sine of her zenith distance, so is the sine of her ho-
rizontal' parallax (as given in the Nautical Almanac) to
the sine of her parallax in altitude.

In correcting the moon's altitude, an allowance should
be made for the augmentation of her semidiameter, which
gradually takes place from the horizon to the zenith.
This increase is given, in the 4th of the Requisite Tables,
for every 5 degrees of altitude, which correction is to be
added to her horizontal semidiameter given in the Nau-
tical Almanac.

The augmentation of the moon's semidiameter is caused
by her being nearer to the spectator in the zenith than in
the horizon, by a semidiameter of the earth — for the ap-
parent magnitude of a body is in the inverse ratio of its
distance from the observer ; and as the earth's semidia-
meter bears a very sensible proportion to the moon's di-
stance, she is seen under the greatest angle in the zenith,
which angle gradually diminishes to the horizon.

There are other corrections of the altitudes, which may
be necessary in cases of peculiar nicety, but which are
seldom noticed at sea. These are, an allowance for the
contraction of the vertical semidiameters of the sun and
moon by refraction ; a correction of the moon's parallax,
supposing the earth an oblate spheroid ; a correction for
the refraction according to the actual state of the atmo-
sphere, as shown by a thermometer and barometer, and not
according to the mean astronomical refraction which is
commonly used. These corrections, though perhaps ne-
cessary towards the perfection of this problem, being very
small, and frequently counteracting each other, are ge-
nerally considered of little consequence in nautical prac-
tice, where greater errors arc unavoidable.

From the corrected Altitudes to find the true Distance. — It
is easy to conceive that by a lunar observation, the three
sides of a spheric triangle are measured in the heavens,
which sides are the apparent co-altitudes of the observed
bodies, and their apparent distance asunder.

The co-altitudes or zenith distances being corrected, the
question is, to find the true distance between the observed
bodies; but here only two things are given, and therefore
it cannot be performed until the angle at the zenith is
known, which is determined from the three given sides of
the triangle, by the rules of spheric trigonometry. See
Kelly's Spherics, pa. 182.

As the effects of parallax, refraction, Sec, operate only
in a vertical direction, it is evident that the corrections
of the zenith distances, or containing sides, will not change
the included angle at the zenith ; and therefore three things
are now known, namely, the corrected zenith distances
and the included angle, whence the other side is deter-
mined by spherics, and this side is the true distance
sought.

A general View of the different Methods of working the
lunar Observations. — Few problems have ever been more in-
vestigated and studied, than that of clearing the lunar di-
stances, and many ingenious methods have been devised for
contracting the operation. These methods are founded on
some of the following general principles. The first is
spheric trigonometry, as before explained ; the second is

L U N

C 758 ]

L U N

the doctrine of proportional errors, by which the effects
that the errors in the altitudes produce on the distance,
are solved by fluxions, or by the differential calculus ; and
a third principle has been lately discovered, which is
founded on the properties of a quadrangle inscribed in a
circle, as explained and exemplified by the inventor, Dr.
Andrew, in his Astronomical and Nautical Tables.

Various methods of working the lunar observations
have been devised, chiefly by Halley, Euler, Mayer, Mas-
kelyne,Dunthorn, Lyons, Witchell, Burrow, Borda, Wales,
Mackav, Kelly, Gerrard, Andrew, and Mendoza. The
methods of the two last authors appear to be the most
concise, but all are sufficiently correct, and seamen gene-
rally prefer such as they have first learnt. It may indeed
be observed, that operations which appear the most con-
cise, are not always the most expeditiously performed ;
as much depends on the number and variety of tables re-
quired, and the manner of applying them. No method
has hitherto obtained an exclusive preference over the rest,
nor does it appear possible to reduce the calculation to
a conciseness to answer the general wants or wishes of sea-
men; and hence, other modes have been devised of obtain-
ing approximate solutions by projection or graphic ope-
ration.

Graphic Operations. — The first graphic method for clear-
ing the distances, was that published by Lacaille, called
the Chassis de Reduction, which has since been copied
by Lalande, Mackay, and others. It is an orthographic
projection, consisting of a great number of lines accurately
drawn, and various scales for obtaining the different cor-
rections.

On similar principles, and for the same purpose, the
late Mr. James Fergusson, teacher of navigation and exa-
miner in that science to the East-India Company, con-
structed a Rotula, or longitude instrument, which did
great credit to him as a man of science.

Another graphic operation, of a different description,
was executed by Mr. George Margetts, and published in
1790- It consists of 70 large plates, containing numerous
lines drawn from the solution of lunar distances in Dr.
Shepherd's Tables. By Margett's Longitude Tables, the
solution of a lunar observation may be obtained in about
one-fourth of the time required by calculation; and the
answer, though not given as perfectly accurate, is suffi-
ciently so for the general purposes of navigation.

Dr. Kelly has devised an orthographic projection
founded on the fluxional analogies of spheric triangles,
which is published in his Introduction to Spherics and
Nautical Astronomy, where an investigation of its princi-
ples is given (pa. 195. edit. 2, 3, and 4), with a demon-
stration, showing that, in proper altitudes, it cannot es-
sentially err, but must give the true distance within a few
seconds. The simplicity of this projection is extremely
curious, as giving an approximate solution of a complicated
problem, by drawing four right lines only, from the scale
of chords, and it must therefore be very useful, especially
where expedition is required.

Since the first appearance of this projection, which was
in 1795, several mathematicians have turned their, at-
tention to the subject, and various ingenious plans have
been proposed for shortening and simplifying the opera-
tion, among which may be mentioned a Rotula by Mr. B.
Donne of Bristol, which solves the problem with great
accuracy.

The following is an abstract of Kelly's Projection, in-
sertedhere by permission of the author.

Exam. 1. Given the apparent distance between the
moon, and the sun, or a star
The moon's altitude -

The star's altitude -
The moon's horizontal parallax
To find the true distance.

30"

20

40

1

With the sweep of 60°, from a scale of chords, describe a
semicircle, and bisect it by the perpendicular ab. On the
right hand side of this line, from b to c, lay off 30°, the
apparent distance, from the same scale. Also the moon's
altitude, 20°, from e to a, and from p to J; and draw ab.
Then lay off the sun's co-altitude, 50°, both ways from c
to m and n, and draw mn, intersecting ab in d. — Then is
d (J the mean correction, to be measured on the scale of
chords, calling every degree on the scale a minute; which
must be reduced to the true correction by the following
general rule : first observing that if this line d (J falls on
the right-hand side, as in fig. 1, the correction is negative
or subtractive ; but if on the left, as in fig. 2, the correc-
tion is positive or additive.

Rule. Multiply the mean correction, d<j, by the
given horizontal parallax, and divide the product by 62
when the correction is subtractive, but by 53 when it is
additive. Then the quotient will be the true correction.
Thus, in the first case D $ measures 41J, or 4l' 15", and
being subtractive, it is multiplied by 60', and divided by
62, which gives the true correction 39' 55", being only one
second more than by calculation ; hence the true distance
is 29° 20' 5".

Examp. 2. Given the apparent distance 30°
The moon's altitude 40

The sun or star's altitude 20

The moon's horizontal parallax 1
To find the true distance.

Describe the semicircle as before (or a greater arc to m
if wanted, as in fig. 2), and lay off the apparent distance
BC = 30°. Make En and f6 = 40° the moon's altitude,
and draw ab. Lay off the star's co-altitude 70°, from c
to m and n, and drawn mn. Then is d ([ the mean cor-
rection, which here measures 24' 10"; and being multi-
plied by 60, and divided by 53, it gives 27' 21" for the
true correction, being only 3" more than the result by cal-
culation ; and the correction being additive, it gives
30° 27' 21" for the true distance.

This projection will appear still more simple by draw-
ing the triangle only, and laying off the altitudes from the
line of sines. Thus, in fig. 1 , project the angle bac = 30°,
and make a ([ = sine of 20°, also a * = 40°. Then per-
pendicular lines from d and # will meet in d as before.

From the foregoing projection, the following general
rules are obvious, for estimating the correction of a lunar
observation without any operation. — 1. When the moon is
lower than the star, the correction is always subtractive;
but when it is the higher body, beyond a certain extent,

L U N

[ 759 ]

L U N

and within a certain limit as to distance, the correction is
additive. — 2. When the sine of the moon's altitude is to
that of the star's, as radius is to the cosine of the distance,
then the true and apparent distances are nearly equal :
but when the moon's altitude is greater than the star's,
beyond that proportion, the correction is additive; except
when the distance is about *)0°, or more, and then it is
subtractive. — 3. Though the correction additive does not
occur so frequently as the subtractive, nor is it in general
so great, yet it admits of a wider variation, being sometimes
above 1° ; but the subtractive correction is never quite so
much.

by the moon, even till the time of Julius Caesar. The
Jews too had their lunar month and year.

Lunar Dial, Eclipse, Horoscope, and Ruinboiv. See the
several substantives.

LUNATION, the period or time between one new moon
and another; it is also called thesynodical month, consist-
ing of 29 days 12 hrs. 44 m. 3 sec. 11 thirds; exceeding
the periodical month by 2 ds. 5 hrs. 0 m. 55 sec.

LUNE, or Lunula, or little moon, is a geometrical
figure, in form of a crescent, terminated by the arcs of two
circles that intersect each other within.

Though the quadrature of the whole circle has never been

pressly says it was found out by
Oenopidas of the same place. See Heinius in Mem. de
l'Acad. de Berlin, torn. ii. pa. 410, where he gives a disser-
tation concerning this Oenopidas. See also Circle, and
Quadrature.

Thelune'of Hippocrates is this: Let abc be a semi-
circle, having its centre e, and adc a quadrant, having its.
centre f; then the figure abcda, contained betweenthe
arcs of the semicircle and quadrant, is his lune ; and it is
equal to the right-angled triangle- acf, as is thus easily
proved. Since af*= 2ae% that is,, the square of the ra-
dius of the quadrant equal to double the square of the
radius of the semicircle; therefore the quad rantal area
adcfa is = the semicircle a bcea; from each of these take
away the common space adcea, and there remains the
triangle acf = the lune abcda.

The author of this construction further observes that, effected, yet many of its parts have been squared. The first
" In solving the lunar problem by this projection, a slate of these partial quadratures was that of the lunula, given
will perhaps be found more convenient than paper; and a by Hippocrates of Scion, or Chios ; who, from beinaa ship-
large scale of chords is, of course, preferable to a small wrecked merchant, commenced geometrician. But'though
one.— That, in the foregoing examples, round numbers - the quadrature of the lune be generally ascribed to Hip-
have been chosen for the sake of simplicity ; but frac- pocrates, yet Proclus expressly says it was found out bv

tional quantities might have been adopted with equal ac-
curacy. It is not however presumed that the results of all

lunar problems will be alike correct by this method, but

they will be found seldom to vary above half a minute

from the true distance ; which cannot be deemed of

much importance, when it is considered that greater inac-
curacies are perhaps unavoidable in the observations. It

is extremely difficult, for instance, to ascertain within a

minute the true point of contact between the moon and

the sun or star. Their altitudes too are often doubtful,

on account of the changes of refraction, and the haziness

of the horizon : to which may be added the imperfection

of instruments, the incompetency of observers, and even

the inaccuracy of the lunar tables themselves, which have

been hitherto only in a progressive state of improvement.
It is well known that, from various unavoidable causes,

all astronomical computations are but approximations to

perfect accuracy; and that in this science difficulties and

obstructions are much greater at sea than at land. It is

also allowed, that of all the operations in nautical astro-
nomy, the lunar problem is the most embarrassing, and

the most liable to error; and it is likewise agreed that the

best mode of diminishing such uncertainties, is by multi-
plying the observations, and taking a mean of many trials.

Hence an approximate method, like the present, which

cannot essentially err, and which is performed with so

much expedition, must be highly useful, as affording the

most time for repeating the observations. Even where no

such repetitions are required, this projection may be ad-
vantageously used as a substitute for troublesome compu-
tations, particularly in the common course of nautical
practice. It may indeed be observed, that the more ur-
gent duties of seamen seldom allow them sufficient leisure
for tedious numerical operations. Besides, it seems rather
a vain pursuit to sacrifice so much time in search of a few
seconds, where there are probably errors of whole minutes
in the data. It may even be observed, that long and la-
borious calculations contain within themselves many
sources of error, to which projection is not exposed."

Lunar Month, is either periodical, synodical, or illu-
minative: which see; also Month.

Lunar Year, consists of 354 days,, or 12 synodical
months, of 2£)£ days each. See Year. In the early
ages, the lunar year was used by all nations; the variety
of course being more frequent and conspicuous in this
planet, and consequently better known to men, than those
of any other. The Romans regulated their year, in part,

Another property of this lune is, that if fg be any line
drawn from the point f, and ah perpendicular to it; then
is the intercepted part of the June agia = the triangle
agw, cut off by the chord line ag; or in general, that the
small segment akga is equal to the trilineal aiha. For,,
the angle afg being at the centre of the one circle, and at
the circumference of the other, the arcs cut off ag, al
are similar to the wholes abc, adc; therefore the small,
seg. akga is to the semisegment aih, as the whole semi-
circle abca, to the semisegment or quadrant adcf, that
is in a ratio of equality.

Again, if abc (fig. 2) be a triangle, right-angled at c,,
and if semicircles be described on the three sides as dia-
meters ; then the triangle t (abc) is equal to the sum of
the two lunes l 1 , L2. For, the greatest semicircle is equal
to the sum of both the other two; from the greatest semi-
circle take away the segments si and s2, and there remains-
the triangle t; also from the two less semicircles take away
the same two segments si and s2, and there remains the
two lunes hi and l2 ; therefore the triangle t = Ll -t- l2 .
the two lunes.

LUNETTE, in Fortification, an inveloped counterguard,
or mound of earth, made beyond the second ditch, op.po-

L Y O

[ 7«0 ]

l y a

site to the place of arms; differing from the ravelins only
in their situation. Lunettes are usually made in wet ditches,
and serve the same purpose as fausse-brays, to defend the
passage of the ditch.

LUPUS, the Wolf, a southern constellation, joined to the
Centaur, containing together 19 stars in Ptolemy's cata-
logue, but 24 in the Britannic catalogue.

LYNX, a constellation of the northern hemisphere,
composed by Hevelius out of the unformed stars. Pn his
catalogue it consists of 19 stars, but in the Britannic of 44.

LYONS (Israel), a respectable mathematician and
botanist, was the son of a Polish Jew silversmith, and
teacher of Hebrew at Cambridge in England, where he had
come to settle, and where young Lyons was born, 1739-
He was a very extraordinary young man for parts and
ingenuity ; and showed very early in life a great inclina-
tion to learning, particularly mathematics, on which ac-
count he was much patronized by Dr. Smith, master of
Trinity-college. About 1753 he began to study botany,
which he continued occasionally till his death; in which
he made a considerable progress, and could remember not
only the Linnasan names of almost all the English plants,
but even the synonyma of the old botanists; and he had
prepared large materials for a Flora Cantabrigiensis, de-
scribing fully every part of each plant from the specimen,
without being obliged to consult, or being liable to be misled
by, former authors.

In 1758, he obtained much celebrity by publishing A
Treatise on Fluxions, dedicated to his. patron, Dr. Smith ;
andinl7o"3, FasciculusPlantarumcircaCantabrigiam,&c.
In the same year, or the year before, he read lectures on
botany at Oxford with great applause, to at least 6"0 pu-
pils; but he could not be prevailed on to make a long ab-
sence from Cambridge.

Mr. Lyons was some time employed as one of the com-
puters of the Nautical Almanac; besides which, he re-
ceived frequent presents from the Board of Longitude for
his own inventions. — He had studied the English history;

and could quote whole passages from the monkish writers
verbatim. He could read Latin and French with ease,
but wrote the former indifferently. He was appointed by
the Board of Longitude to sail with Capt. Fhipps, in his.
voyage towards the north pole, in 1773, as astronomical
observator ; which office he discharged much to the satis-
faction of lis employers. After his return from this voyage,
he married, and settled in London, where he died of the
meazles about two years after.

At the time of his death he was engaged in preparing
for the press, a complete edition of all the works of the
learned Dr. Halley ; a work very much wanted. — His
Calculations in Spherical Trigonometry abridged, were
printed in the Philos. Trans, vol. 65, for the year 1775,
pa. 470. — After his death, his name appeared in the title-
page of A Geographical Dictionary, the astronomical
parts of which were said to be " taken from the papers of
the late Mr. Israel Lyons of Cambridge, author of several
valuable mathematical productions, and astronomer in
Lord Mulgrave's voyage to the northern hemisphere." —
The astronomical and other mathematical calculations,
printed in the account of Captain Phipps's voyage towards
the north pole, mentioned above, were made by Mr. Lyons.
This appeared afterwards, by the acknowledgment of
Captain Phipps, when Dr. Horsley detected a material
error in some part of them, in his Remarks on the Obser-
vations made in the late Voyage, &c, 1774.

The Scholar's Instructor, or Hebrew Grammar, by
Israel Lyons, teacher of the Hebrew tongue in the Uni-
versity of Cambridge, the 2d edit. &c, 1757, 8vo, was the
production of his father; as was also another treatise
printed at the Cambridge press, under the title of Obser-
vations and Enquiries relating to various parts of Scripture
History, 176"l.

LYRA, the Harp, a constellation in the northern hemi-
sphere; containing 10 stars in Ptolemy's catalogue, 1 1 in
Tycho's, 17 in Hevelius's, and 21 in the Britannic cata-
logue.

ADDENDUM.
After line 17, col. .2, page 341, on the nature of comets, add: — Dr. Herschel, in several volumes of the Philos.
Trans, states his opinions of the nature and constitution of comets. His hypothesis is, that they are formed of the
condensed nebulous matter diffused through the universal space, and tying within the sphere of the sun's activity;
which fchus gives them a revolutionary motion in an elliptic orbit. Of the same opinion also is Count Laplace, as
appears by a learned memoir on the subject, in the Connoissance des Temps for the year 1816.

ERRATA.

Pa. 135, col. 2, end of Ifcie 6 from the bottom, for M% read 2N.

— 269, — 1, lit). 24, 25, for Chichester, read Chester.

— 480, — 2, end of line 28, for it, read ll.

— 484, — 1, line 20, for p'ate viii, read plate x.

— 484, — 2, line 5, for fc, read K.

— 698, — 1, line 28, for f, read Jj.

END OF THE FIRST VOLUME.

Printed bj S. Hamilton, Weybridgc, Surrey.

Hat& Z.

AIR - BALLOON s

Montgolfter's .
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Monto-oUier's .
Fazucbowg of S'iOcrnwin

Chaiies Sc Robert's.

C/iamp de Mars .

1

Plate H.

Fiq.l. ^4im gt/jv.

Fig. 2. « The lock.

Fit/. 3. Ijiiprcn'ed ^im gu^

Fig. 4. ^4lr Cane

Fig. S. Jlac/azJJte ^4ir GirJ\r .

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Fig.S.

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^iKEMOMETERS .

Lind's .

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Fig. 8.

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Tiff. 4

Fig. 3

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Plate (7.

Perspective View ^ part of a CastAl ivith locks . Fuji.

FlateVU.

Plan Sc Elevution of a Dome, cotutructcd uillwut Centring .

Fui. 1. ^

Fig. 2.

Compass .

Fig.l

D!" Knight's Compasses .

Azimvt/i Compass.

Fig. 4

Proportional Compulses .

Fig. 3

Compass Card.

F{9-5

Elliptic Compafte-? .

Fig. 6

FOl/^TAII^S .

^■9

Fig. 8

Vol.L

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Plate X.

Fig. 4

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Fig. i

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Fig. Q

Vol.1.

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Plate MIL.

Fig. 4

roi.i.

Artificial Globe

Fw. 2

Bate jar.

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Wtie* sc.SiifiM 1

Ilate XVI.

Fu}. 5.

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Vol.I.

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Knots of different kinds .
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BOSTON COLLEGE

3 9031 01629881 2

DATE DUE