Dissertation second: exhibiting a general view of the progress of mathematical and physical science, since the revival of letters in Europe

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DISSERTATION SECOND:

EXHIBITING A GENERAL VIEW UF THE

ele of MWathematical and JOhpsical
Science, i

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SINCE THE REVIVAL OF LETTERS IN EUROPE.
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BY JOHN PLAYFAIR,

Professor of Natural Philosophy in the University of Edinburgh, Fellow of the Royal Society
of London, and Secretary of the Royal Society of Edinburgh.

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DISSERTATION SECOND.

PART I.

Iw conformity to the plan which has been traced and exe-
cuted with so much ability in the First Dissertation, I am
now to present the reader with an historical sketch of the
principal discoveries made in Natural Philosophy, from the
revival of letters down to the present time. In entering
on this task, and on looking at the instructive but formida-
ble model already set before me, I should experience no

- small solicitude, did I not trust that the subject of which I

am to speak, in order to be interesting, needs only to be
treated with clearness and precision. These two requi-
sites I will endeavour to keep steadily in view.

In the order which I am to follow, I shall be guided
solely by a regard to the subserviency of one science to
the progress of another, and to the consequent priority of
the former in the order of regular study. For this reason,
the history of the pure Mathematicks will be first consider-
ed, as that science has been one of the two principal instru-
ments applied by the moderns to the advancement of natu-

_ tal knowledge. The other instrument is Experience ; and,

therefore, the principles of the inductive method, or-of the

branch of Logick which teaches the application of ex-

periment and observation to the interpretation of nature,
: | 1

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6 DISSERTATION SECOND. [PART 1.

must be the second object of inquiry ; and in this article I
shall give an account of Bacon’s Philosophy, as applied to
Physical investigation. After these two sections, which
may in some measure be considered as introductory, I am
to treat of Natural-Philosophy, under the divisions of
Mechanicks, Astronomy, and Opticks. Under the general
denomination of Mechanicks I include the Theory of Mo-
tion, as applied not only to solids, but to fluids, both incom-
pressible and elastick. Opticks I have placed after Astro-
nomy, because the discoveries in Mechanicks have much
less affected the progress of the former of these sciences
than of the latter. To these will succeed a sixth division,
containing the laws of the three unknown substances, if,
indeed, they may be called substances,—Heat, Electricity
and Magnetism. These, though very different, agree in
some general characters. They permeate all substances,
though not with the same facility ; and, if other bodies had
been formed in the same manner with them, the idea of im-
penetrability would never have been suggested to the
mind. "They seem to receive motion, without taking any
away from the body which communicates it; so that they
can hardly be considered as inert. Two of them, Heat
and Electricity, are perceived by the sense of touch; but
the impression which they make does not convey an idea
of resistance. The third is not perceived by toueh ; and,
therefore, all the three might be denominated impalpable
substances. If they have any gravity, it cannot be appre-
ciated ; and, for these reasons, had it not too paradoxical an
appearance, we might class them together as material, but
incorporeal substances. We know, indeed, nothing of
them but as powers transferable from one body to another ;
and it is in consequence of this last circumstance alone that
they are entitled to the name of substances.

Though the general design of this historical sketch ex-
tends from the revival of letters to the beginning of the

. | DISSERTATION SECOND. 7
nineteenth century, I shall, in the present Part, confine
myself entirely, as has been done in the first Discourse, to
the period preceding the end of the seventeenth century,
or, more precisely, to that preceding the invention of the
fluxionary calculus, and the discovery of the principle of
gravitation ;—one of the most remarkable epochas, without
doubt, in the history of human knowledge.

. SECTION TI.

MATHEMATICKS.
1. GEOMETRY.

Tue great inheritance of mathematical knowledge, which
the ancients bequeathed to posterity could not, on the re-
vival of learning, be immediately taken possession of, nor
could even its existence be discovered, but by degrees.
Though the study of the Mathematicks had never been en-

~ tirely abandoned, it had been reduced to matters of very sim-

pleand easy comprehension, suchas were merely subservient
to practice. There had been men who could compute the
area of a triangle, draw a meridian line, or even construct a
sun-dial, in the worst of times; but between such skill, and
the capacity to understand, or the taste to relish, the de-
monstrations of Euclid, Apollonius, or Archimedes, there
was a great interval; and many difficulties were to be over-
come, for which much time, and much subsidiary know-
ledge, were necessary. The repositories of the ancient
treasures were to be opened, and made accessible ; the
knowledge of the languages was to be acquired ; the manu-
scripts were to be deciphered ; and the skill of the gram-

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ra) DISSERTATION SECOND. [PART 1

‘marian and the critick were to precede, in a certain degree,
that of the geometrician or the astronomer. The obliga-
tions which we have to those who undertook this laborious
and irksome task, and who rescued the ancient books from
the prisons to which ignorance and barbarism had con-
demned them, and from the final destruction by which
they must soon have been overtaken, are such as we can
never sufficiently acknowledge; and, indeed, we shall
never know even the names of many of the benefactors to
whom our thanks are due. Inthe midst of the wars, the
confusion, and bloodshed, which overwhelmed Europe
during the middle ages, the religious houses and monas-
teries afforded to the remains of ancient learning an asylum,
which a salutary prejudice forced even the most lawless to
respect; and the authors who have given the best account
of the revival of letters, agree, that it is in a great measure
to those establishments, that we owe the safety of the books
which have kept alive the scientifick and literary attain-
ments of Greece and Rome.

The study of the remains of antiquity gradually produc-
ed men of taste and intelligence, who were able to correct
the faults of the manuscripts they copied, and to explain
the difficulties of the authors they translated. Such were
Purbach, Regiomontanus, Commandine, Maurolycus, and
many others. By their means, the writings of Euclid,
Archimedes, Apollonius, Ptolemy, and, Pappus, became
known and accessible to men of science. Arabia contri-
buted its share towards this great renovation, and from the
language of that country was derived the knowledge of
many Greek books, of the originals of which, some were
not found till long afterwards, and others have never yet
been discovered.

In nothing, perhaps, is the inventive and elegant genius
of the Greeks better exemplified than in their geometry.

sec. t.] DISSERTATION SECOND. 9

The elementary truths of that science were connected by
Euclid into one great chain, beginning from the axioms,
and extending to the properties of the five regular solids ;
the whole digested into such admirable order, and explain-
ed with such clearness and precision, that no similar work
of superiour excellence has appeared, even in the present
advanced state of mathematical science.

Archimedes had assailed the more difficult problems of
geometry, and by means of the method of Exhaustions,
had demonstrated many curious and important theorems,
with regard to the lengths and areas of curves, and the
contents of solids. The same great geometer had given a
beginning to physico-mathematical science, by investigat-
ing several propositions, and resolving several problems in
Mechanicks and Hydrostaticks.

Apollonius had treated of the Conick Sections,—the .
Curves which, after the circle, are the most simple and
important in geometry ; and, by his elaborate and profound
researches, had laid the foundation of discoveries which
were to illustrate very distant ages.

Another great invention, the Geometrical Analysis, as-
cribed very generally to. the Platonick school, but most
successfully cultivated by the geometer just named, is one
of the most ingenious and beautiful contrivances in the
mathematicks. It is a method of discovering truth by
reasoning concerning things unknown, or propositions
merely supposed, as if the one were given, or the other
were really true. A quantity that is unknown, is only to
be found from the relations which it bears to quantities that
are known. By reasoning on these reiations, we come at
last to some one so simple; that the thing sought is thereby
" determined. By this analytical process, therefore, the
- thing required is discovered, and we are at the same time
put in possession of an instrument by which new fruths |

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10 DISSERTATION SECOND. [parr 1.

may be found out, and which, when skill in using it has
been acquired by practice, may be applied to an unlimited
extent.

A similar process enables us to discover the demonstra-
tions of propositions, supposed to be true, or, if not true,
to discover that they are false. ,

This method, to the consideration of which we shall
again have an epportunity of returning, was perhaps the
most valuable part of the ancient mathematicks, inasmuch
as a method of discovering truth is more valuable than the
truths it bas already discovered. Unfortunately, how-
ever, the fragments containing this precious remnant had
suffered more from the injuries of time than almost any
other.

In the fifteenth century, Regiomontanus, already men-
tioned, is the mathematician who holds the highest rank.
To him we owe many translations and commentaries, to-
gether with several original and yaluable works of his own.
Trigonometry, which had never been known to the Greeks
as a separate science, and which took that form in Arabia,
advanced, in the hands of Regiomontanus, to a great degree
of perfection, and approached very near to the condition
which it has attained at the present day. He also intro-
duced the use of decimal fractions into arithmetick, and
thereby gave to that scale its full extent, and to numerical -
computation the utmost degree of simplicity and enlarge-
ment which-it seems capable of attaining.

This eminent man was cut off in the prime of life; and
his untimely death, says Mr. Smith, amidst innumerable
projects for the advancement of, science, is even at this
day a matter of regret.’ He was buried in the Pantheon
at Rome; and the honours paid to him at his death prove

' History of Astronomy, p. 90. Regiomontanus was born in
. 1456, and died in 1496.

sec. 1.] DISSERTATION SECOND. il

that science had now become a distinction which the great
were disposed to recognise.

Werner, who lived in the end of this century, is the first
among the moderns who appears to have been acquainted
with the geometrical analysis. His writings are very rare,

and I have never had an opportunity of examining them.
What I here assert is on the authority of Montucla,
whose judgment in this matter may be safely relied on, as
he has shown, by many instances, that he was well ac-
quainted with the nature of the analysis referred to. _ It is
not a little remarkable that Werner should have understood
this subject, when we find many eminent mathematicians,
long after his time, entirely unacquainted with it, and con-
tinually expressing their astonishment how the ancient:
geometers found out those simple and elegant constructions
and demonstrations, of which they have given so many ex-
amples. In the days of Werner, there was no ancient
book known, except the Data of Buclid, from which any
information concerning the geometrical analysis could be
collected ; and it is highly to his credit, that, without any
other help, he should have come to the knowledge of a
method, not a little recondite in its principles, and among
the finest inventions either of ancient or of modern science.
Werner resolved, by means of it, Archimedes’s problem
of cutting a sphere into two segments, having a given ratio
to one another. He proposed also to translate, from the
Arabick, the work of Apollonius, entitled Sectio Rationis,
rightly judging it to be an elementary work in that analy-
sis, and to come next after the Data of Euclid. '

Benedetto, an Italian mathematician, appears also to have
been very early acquainted with the principles of the same
ingenious method, as he published-a book on the ceometri-
cal analysis at Turin in 1585.

See Montuela, vol. I, p. 581.

12 DISSERTATION SECOND. [rant 1.

Maurolycus of Messina flourished in the middle of the
sixteenth century, and is justly regarded as the first geo-
meter of that age. Beside furnishing many valuable trans-
lations and commentaries, he wrote a treatise on the conick
sections, which is highly esteemed. He endeavoured also
io restore the fifth book of the conicks of Apollonius, in
which that geometer treated of the maxima and minima
of the conick sections. His writings all indicate a man of
clear conceptions, and of a strong understanding ; though
he is taxed with having dealt in astrological prediction.

In the early part of the seventeenth century, Cavalleri
was particularly distinguished, and made an advance in the
higher geometry, which occupies the middle place be-
tween the discoveries of Archimedes and those of New-
ton.

For the purpose of determining the lengths and areas of
curves, and the contents of solids contained within curve
superficies, the ancients had invented a method, to which
the name of Exhaustions has been given; and in nothing,
perbaps, have they more displayed their powers of mathe-
matical invention. :

Whenever it is required to measure the space bounded
by curve ‘lines, the length of a curve, or the solid contain-
ed within a curve superficies, the investigation does not
fall within the range of elementary geometry. Rectilineal
figures are compared, on the principle of superposition, by
help of the notion of equality which is derived from the
coincidence of magnitudes both similar and equal. Two
rectangles of equal bases and equal altitudes are held to be
equal, because they can perfectly coincide. A rectangle
and an oblique angled parallelogram, having equal bases
‘and altitudes, are shown to be equal, because the same
triangle, taken from the rectangle on one side, and added
fo it on the other, converts it into the parallelogram; and

sxcr. 1.] DISSERTATION SECOND. 13

thus two magnitudes which are not similar, are shown to
have equal areas. In like manner, if a triangle and a paral-
lelogram have the same base and altitude, the triangle is
shown to be half the parallelogram; because, if to the
triangle there be added another, similar and equal to it-
self, but in the reverse position, the two together will com-
pose a parallelogram, having the same base and altitude
with the given triangle. The same is true of the com-
parison of all other rectilineal figures ; and if the reasoning
be carefully analyzed, it will always be found to be redu-
cible to the primitive and original idea of equality, derived
from things that coincide or occupy the same space ; that
‘is to say, the areas which are proved equal, are always
such as, by the addition or subtraction of equal and simi-
lar parts, may be rendered capable of coinciding with one
another.

_This principle, which is quite general with respect to
rectilineal figures, must fail, when we would compare cur-
vilineal and rectilineal spaces with one another, and make
the latter serve as measures of the former, because no ad-
dition or subtraction of rectilineal figures can ever produce
a figure which is curvilineal. It is possible, indeed, to com-
bine curvilineal figures, so as to produce one that is rectili-
neal; but this principle is of very limited extent; it led
to the quadrature of the lunulae of Hippocrates, but has
hardly furnished any other result which can be considered
as valuable in science.

In the difficulty to which geometers were thus reduced,
it might occur, that, by inscribing a rectilineal figure within
a curve, and circumscribing another round it, two limits
could be obtained, one greater and the other less than the
area required. It was also evident, that, by increasing the
number, and diminishing the sides of those figures, the two
limits might be brought continually nearer to one another,

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14 DISSERTATION SECOND. [parr 3,

and of course nearer to the curvilinear area, which was
always intermediate between them. In prosecuting this

sort of approximation, a result was at length found out, -

which must have occasioned no less surprise than delight
to the mathematician who first encountered it. The result
I mean is, that, when the series of inscribed figures was
continually increased, by multiplying the number of the
sides, and diminishing their size, there was an assignable
rectilineal area, to which they continually approached, so
as to come nearer it than any difference that could be sup-
posed. The same limit would also be observed to belong
to the circumscribed figures, and therefore it could be no
other than the curvilineal area required.

It appears to have been to Archimedes that a truth of
this sort first occurred, when he found that two-thirds of
the rectangle, under the ordinate and abscissa of a parabo-
la, was a limit always greater than the inscribed rectilineal
figure, and less than the circumscribed. In some other
curves, a similar conclusion was found, and Archimedes
contrived to show that it was impossible to suppose that
the area of the curve could differ from the said limit, with-
out admitting that the circumscribed figure might become
less, or the inscribed figure greater than the curve itself.
The method of Exhaustions was the name given to the in-
direct demonstrations thus formed. ‘Though few things
more ingenious than this method have been devised, and
though nothing could be more conclusive than the demon-
strations resulting from it, yet it laboured under two very
considerable defects. In the first place, the process by
which the demonstration was obtained was long and diffi-
cult; and, in the second place, it was indirect, giving no
insight into the principle on which the investigation was
founded. Of consequence, it did not enable one to find
out similar demonstrations, nor increase one’s power of

“ur

skcr, 1.] DISSERTATION SECOND. 15

making more discoveries of the same kind. It was a de-
monstration purely synthetical, and required, as all indi-
rect reasoning must do, that the conclusion should be known
before the reasoning is begun. A more compendious, and
a more analytical method, was therefore much to be wished
for, and was an improvement, which, at a moment when the
field of mathematical science was enlarging so fast, seemed
particularly to be required.

Cavalleri, bern at Milan in the ycar 1598, is the person
by whom this great improvement was made. The princi-
ple on which he proceeded was, that areas may be consi-
dered as made up of an infinite number of parallel lines ;
solids of an infinite number of parallel planes; and even
lines themselves, whether curve or straight, of an infinite
number of points. The cubature of a solid being thus re-
duced to the summation of a series of planes, and the quad-
rature of a curve to the summation of a series of ordinates,
each of the investigations was reduced to something more
simple. Ii added to this simplicity not a little, that the
sums of series are often more easily found, when the num-
ber of terms is infinitely great, than when it is finite, and
actually assigned.

It appears that a tract on stereometry, written by Kep-
ler, whose name will hereafter be often mentioned, first
Jed Cavalleri to take this view of geometrical magnitudes.
In that tract, which was published in 1615, the measure-
ment of many solids was proposed, which had not before
fallen under the consideration of mathematicians. Such,
for example, was that of the solids generated by the re-
volution of a curve, not about its axis, but about any line
whatsoever. Solids of that kind, on account of their af-
finity with the figure of casks, and vessels actually employ-
ed for containing liquids, appeared to Kepler to offer both
curious and useful subjects of investigation. There were

16 DISSERTATION SECOND. [parte

no less than eighty-four such solids, which he proposed
for the consideration of mathematicians. He was, how-
ever, himself unequal to the task of resolving any but a
small number of the simplest of these problems. In these
solutions, he was bold enough to introduce into yeometry,
for the first time, the idea of infinitely great and infinitely
small quantities, and by this apparent departure from the
rigour of the science, he rendered it in fact a most essen-
tial service. Kepler conceived a circle to be composed
of an infinite number of triangles, having their common
vertex in the centre of the circle, and their infinitely small
bases in the circumference. It is to be remarked, that
Galileo had also introduced the notion of infinitely small
quantities, in his first dialogue, De Mechanica, where he
treats of a cylinder cut out of a hemisphere; and he has
done the same in treating of the acceleration of falling
bodies. Cavalleri was the friend and disciple of Galileo,
but much more profound in the mathematicks. In his
hands the idea took a more regular and systematick form,
and was explained in his work on indivisibles, published
in 1635.

The rule for summing an infinite series of terms in arith-
metical progression had been long known, and the appli-
cation of it to find the area of a triangle, according to the
method of indivisibles, was a matter of no difficulty. The
next step was, supposing a series of lines in arithmetical
progression, and squares to be described on each of them,
fo find what ratio the sum of all these squares bears to the
greatest square, taken as often as there are terms in the
progression. Cavalleri showed, that when the number of
terms is infinitely great, the first of these sums is just one
third of the second. ‘This evidently led to the cubature
of many solids,

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sor. 1] DISSERTATION SECOND. 17

Proceeding one step farther, he sought for the sum of
the cubes of the same lines, and found it to be one fourth
of the greatest, taken as often as there are terms ; and,
continuing this investigation, he was able to assign the sum
of the nth powers of a series in arithmetical progression,
supposing always the difference of the terms to be infinite-
ly small, and their number to be infinitely great. The
number of curious results obtained from these investiga-
tions may be easily conceived. It gave, over geometrical
problems of the higher class, the same power which the
integral calculus, or the inverse method of fluxions does,
in the case when the exponent of the variable quantity is
an integer. The method of indivisibles, however, was not
without difficulties, and could not but be liable to objection,
with those accustomed to the rigorous exactness of the an-
cient geometry. In strictness, lines, however multiplied,
can never make an area, or any thing but a line; nor can
areas, however they may be added together, compose a
solid, or any thing but an area. This is certainly true,
and yet the conclusions of Cavalleri, deduced ona con-
trary supposition, are true also. This happened, because,
though the suppositions that a certain series of lines, in-
finite in number, and contiguous {o one another, may com-
pose a certain area, and that another series may compose
another area, are neither of them true; yet is it strictly
true, that the one of these areas must have to the other
the same ratio, which the sum of the one series of lines has
to the sum of the other series. Thus, it isthe ratios of
the areas, and not the areas absolutely considered, which
are determined by the reasonings of Cavalleri ; and that
this determination of their ratios is quite accurale, can
very readily be demonstrated by the method of exhaus»
tions.

18 DISSERTATION SECOND. [paer 1.

The method of indivisibles, from the great facility with
which it could be managed, furnished a most ready method
of ascertaining the ratios of areas and solids to one ano-
ther, and, therefore, scarcely seems to deserve the epithet
which Newton himself bestows upon it, of involving in its
conceptions something harsh, (durwm,) and not easy to be
admitted. It was the doctrine of infinitely small quanti-
ties carried to the extreme, and gave at once the result of
an infinite series of successive approximations. Nothing,
perhaps, more ingenious, and certainly nothing more hap-
py, ever was contrived, than to arrive at the conclusion of
ail these approximations, without going through the ap-
proximations themselves. This is the purpose served by
introducing into mathematicks the consideration of quanti-
ties infinitely small in size, and infinitely great in number ;
ideas which, however inaccurate they may seem, yet, when
carefully and analogically reasoned upon, have never led
into errour.

Geometry owes to Cavalleri, not only the general method
just described, but many particular theorems, which that
method was the instrument of discovering. Among these
is the very remarkable proposition, that as four right an-
gles, to the excess of the three angles of any spherical
triangle, above two right angles, so is the superficies of
the hemisphere to the area of the triangle. At that time,
however, science was advancing so fast, and the human
mind was every where expanding itself with so much ener-
gy, that the same discovery was likely to be made by
more individuals than one at the same time. It was not
known in Italy in 1632, when this determination of the area
of a spherical triangle was given by Cavalleri, that it had
been published three years before by Albert Girard, a
mathematician of the Low Countries, of whose inventive
powers we shail soon have more occasion to speak.

axon, 1.) DISSERTATION SECOND. , 19

The Cycloid afforded a number of problems, well cal-
culated to exercise the proficients in the geometry of in-
divisibles, or of infinites. It is the curve described by a
point in the circumference of a circle, while the circle it-
self rolls in a straight line along a plane. It is not quite
certain when this curve, so remarkable for its curious pro-
perties, and for the place which it occupies in the history
of geometry, first drew the attention of mathematicians.
In the year 1639, Galileo informed his friend Torricelli,
that, forty years before that time, he had thought of this
curve, on account of its shape, and the graceful form it
would give to arches in architecture. The same philoso-
pher had endeavoured to find the area of the cycloid ; but
though he was one of those who first introduced the con-
sideration of infinites into geometry, he -was not expert
enough in the use of that doctrine, to be able to resolve
this problem. It is still more extraordinary, that the same
problem proved too difficult for Cavalleri, though he cer-
tainly was in complete possession of the principles by
which it was to be resolved. it is, however, not easy to
determine whether it be to Torricelli, the scholar of Caval-
leri, and his successor in genius and talents, or to Roberval,
a French mathematician of the same period, and a man
also of great originality and invention, that science is in-
debted for the first quadrature of the cycloid, or the proof
that its area is three times that of its generating circle.
Both these mathematicians laid claim to it. The French
and Italians each took the part of their own countryman ;
and in their zeal have so perplexed the question, that it is
hard to say on which side the truth is to be found. 'Tor-
ricelli, however, was a man of a mild, amiable, and can-
did disposition ; Roberval of a temper irritable, violent,
and envious; so that, in as far as the testimony of the in-
dividuals themselyes is concerned, there is no doubt which

20 DISSERTATION SECOND. (parr 1.

ought to preponderate. They had both the skill and talent
which fitted them for this, or even for more difficult re-
searches.

The other properties of this curve, those that respect
its tangents, its length, its curvature, &c. exercised the
ingenuity, not only of the geometers just mentioned, but of
Wren, Wallis, Huygens, and, even after the invention of
the integral calculus, of Newton, Leibnitz, and Bernoulli.

Roberval also improved the method of quadratures in-
vented by Cavalleri, and extended his solutions to the
case, when the powers of the terms in the arithmetical
progression of which the sum was to be found were frac-
tional ; and Wallis added the case when they were nega-
tive. Fermat, who, in his inventive resources, as well as
in the correctness of his mathematical taste,' yielded to
none of his contemporaries, applied the consideration of
infinitely small quantities to determine the maxima and
minima of the ordinates of curves, as also their tangents.
Barrow, somewhat later, did the same in England. After-
wards the geometry of infinites fell into the hands of Leib-
nitz and Newton, and acquired that new character which
marks so distinguished an era in the mathematical sciences.

2. ALGEBRA. )

It was not from Greece alone that the light proceeded
which dispelled the darkness of the middle ages ; for, with
the first dawn of that light, a mathematical science, of a
name and character unknown to the geometers of antiquity,

' He also was very skilful in the geometrick analysis, and
seems to have more thoroughly imbibed the spirit of that inge-
nious invention than any of the moderns before Halley.

agcr, 1.) DISSERTATION SECOND. 21

was received in Europe from Arabia. As early as the be-
ginning of the thirteenth century, Leonardo, a merchant of
Pisa, having made frequent visits to the East, in the course
of commercial adventure, returned to Italy enriched by the
traflick, and instructed by the science of those countries.
He brought with him the knowledge of Algebra; anda
late writer quotes a manuscript of his, bearing the date of
1202, and another that of 1228." The importation of Al-
gebra into Europe is thus carried back nearly 200 years
farther than has generally been supposed, for Iieonardo has
been represented as flourishing in the end of the fourteenth
century, instead of the very beginning of the thirteenth.
It appears by an extract from his manuscript, published by
the above author, that his knowledge of Algebra extended
as far as quairatick equations. The language was very
imperfect, corresponding to the infancy of the science ; the
quantities and the operations being expressed in words,
with the help only of a few abbreviations. The rule for
resolving quadraticks by completing the squaye, is demon-
strated geometrically.

Though Algebra was brought into Europe from Arabia,
it is by no means certain that this last is its native coun-
try. There is, indeed, reason to think that its invention
must be sought for much farther to-the East, and probably
not nearer than Indostan. We are assured by the Arabian
writers, that Mahomet Ben Musa of Chorasan, distinguish-
ed for his ‘mathematical knowledge, travelled, about the
year 959, into India, for the purpose of receiving farther
instruction in the science which he cultivated. It is like-
wise certain, that some books, which have lately been
brought from India into this country, treat of Algebra in a
manner that has every appearance of originality, or at least

1

M. Cossali of Pisa, in a Tract on the Origin of Algebra,
1797.

3

22 DISSERTATION SECOND. {pAR? 2.

of being derived from no source with which we are at all
acquainted.

Before the time of Leonardo of Pisa, an important ac-
quisition, also from the East, had greatly improved the
science of arithmetick. This was the use of the Arabick
notation, and the contrivance of making the same charac-
ter change its signification, according to a fixed rule, when
it changed its position, being increased tenfold for every
place that it advanced towards the left. The knowledge
of this simple but refined artifice was learned from the
Moors by Gerbert, a monk of the Low Countries, in the
tenth century, and by him made known in Europe. Ger-
bert was afterwards Pope, by the name of Silvester the
Second ; but from that high dignity derived much less glo-
ry, than from having instructed his countrymen in the de-
_ cimal notation.

The writings of Leonardo, above mentioned, have re-
mained in manuscript; and the first printed book in Alge-
bra is that of Lucas de Burgo, a Franciscan, who, towards
ihe end of the fifteenth century, travelled, like Leonardo,
into the East, and was there instructed in the principles of
Algebra. The characters employed in his work, as in those
of Leonardo, are mere abbreviations of words. The let-
' ters p and m denote plus and minus; and the rule is laid
down, that, in multiplication, plus into minus gives mi-
nus, but minus into minus gives plus. Thus the first
appearance of Algebra is merely that of a system of short-
hand writing, or an abbreviation of common language, ap-
‘plied to the solution of arithmetical problems. It was a
contrivance merely to save trouble; and yet to this con-
trivance we are indebted for the most philosophical and
refined art which men have yet employed for the ex-
pression of their thoughts. This scientifick language,
therefore, like those in common use, has grown up slowly,

i]

sxcr. 1.] DISSERTATION SECOND. 23

from a very weak and imperfect state, till it has reached
the condition in which it is now found.

Though in ali this the moderns received none of their
information from the Greeks, yet a work in the Greek
language, treating of arithmetical questions, in a manner
that may be accounted algebraick, was discovered in the
course of the next century, and given to the world, in a
Latin translation, by Xylander, in 1575. This is the work
of Diophantus of Alexandria, who had composed thirteen
books of Arithmetical Questions, and is supposed to have
flourished about 150 years after the Christian era. The
questions he resolves are often of considerable difficulty ;
and a great deal of address is displayed in stating them, so
as to bring out equations of such a form, as to involve
only one power of the unknown quantity. The expression
is that of common Janguage, abbreviated and assisted by a
few symbols. The investigations do not extend beyond
quadratick equations; they are, however, extremely in-
genious, and prove the author to have been a man of talent,
though the instrument he worked with was weak and im-
perfect.

The name of Cardan is famous in the history of Algebra.
He was born at Milan in 1501, and was a man in whose
character good and ill, strength and weakness, were mixed
up in singular profusion. With great talents and industry, -
he was capricious, insincere, and vainglorious to excess.
Though a man of real science, he professed divination, and
was such a believer in the influence of the stars, that he
died to accomplish an astrological prediction. He remains,
accordingly, a melancholy proof, that there is no folly or
weakness too great to be united to high intellectual attain-
ments.

Before his time very little advance had been made in
the solution of any equations higher than the second de-

24 DISSERTATION SECOND. [pant 1.

gree ; except that, as we are told, about the year 1508,
Scipio Ferrei, professor of mathematicks at Bologna, had
found out a rule for resolving one of the cases of cubick
equations, which, however, he concealed, or communicated
only to a few of his scholars. One of these, Florido, on
the strength of the secret he possessed, agreeably to a
practice then common among mathematicians, challenged
Tartalea of Brescia, to contend with him in the solution of
algebraick problems. Florido had at first the advantage ;
but Tartalea, being a man of ingenuity, soon discovered his
rule, and also another much more general, in consequence
of which, he came off at last victorious. By the report of
this victory, the curiosity of Cardan was strongly excited ;
for, though he was himself much versed in the mathema-
ticks, he had not been able to discover a method of resolv-
ing equations higher than the second degree. By the
most earnest and importunate solicitation, he wrung from
Tartalea the secret of his rules, but not till he had bound
himself, by promises and oaths, never to divulge them.
Tartalea did not communicate the demonstrations, which,
however, Cardan soon found out, and extended, in a very
ingenious and systematick manner, to all cubick equations
whatsoever. Thus possessed of an important discovery,
which was at least ina great part his own, he soon forgot
his promises to Tartalea, and published the whole in 1545,
not concealing, however, what he owed to the latter.
Though a proceeding, so directly contrary to an express
stipulation, cannot be defended, one does not much regret
the disappointment of any man, who would make a mysiery
of knowledge, or keep his discoveries a secret, for pur-
poses merely selfish.

Thus was first published the rule which still bears the
name of Cardan, and which, at this day, marks a point in
the progress of algebraick investigation, which all the ef-

skcr. 1.] DISSERTATION SECOND. 25

forts of succeeding analysts have hardly been able to go
beyond. As to the general doctrine of equations, it ap-
pears that Cardan was acquainted both with the negative
and positive roots, the former of which he called by the
name of false roots. He also knew that the number of
positive, or, as he called them, true roots, is equal to the
number of the changes of the signs of the terms; and that
the coefficient of the second term is the difference between
the sum of the true and -the false roots. He also had per-
ceived thé difficulty of that case of cubick equations, which
cannot be reduced to his own rule. He was not abie to
overcome the difficulty, but showed how, in all cases, an
approximation to the roots might be obtained.

There is the more merit in these discoveries, that the
Janguage: of Algebra still remained very imperfect, and
consisted merely of abbreviations of words. Mathema-
ticians were then in the practice of putting their rules into
verse. Cardan has given his a poetical dress, in which, as
may be supposed, they are very awkward and obscure ;
for whatever assistance in this way is given to the memory,
must be entirely at the expense of the understanding. It
is, at the same time, a proof that the language of Algebra
was very imperfect. Nobody now thinks of translating an
algebraick formula into verse ; because, if one has acquired
any familiarity with the language of the science, the formu-
la will be more easily remembered than any thing that can
be substituted in ifs room.

Italy was not the only country into which the algebraick
analysis had by this time found its way; in Germany it
had also made considerable progress, and Stiphelius, in a
' book of Algebra, published at Nuremberg in 1544, em-
ployed the same numeral exponents of powers, both posi-
tive and negative, which we now use, as far as integer
numbers are concerned ; but he did not carry the solution
of equations farther than the second degree. He introdu’

296 DISSERTATION SECOND. [PART 1.

ced the same characters for plus and minus which are at
present employed.

Robert Recorde, an English mathematician, published
about this time, or a few years later, the first English trea-
tise on Algebra, and he there introduced the same sign of
equality which is now in use.

The properties of algebraick equations were discovered,
however, very slowly. Pelitarius, a French mathemati-
cian, ina treatise which bears the date of 1558, is the first
who observed that the root of an equation is a divisor of
the last term; and he remarked also this curious property
of numbers, that the sum of the cubes of the natural
numbers is the square of the sum of the numbers them-

selves.

The knowledge of the solution of cubick equations was
still confined to Italy. Bombelli, a mathematician of that
country, gave a regular treatise on Algebra, and consi-
dered, with very particular attention, the irreducible case
of Cardan’s rule. He was the first who made the remark,
that the problems belonging to that case can always be re-
solved by the trisection of an arch.’

1 A passage in Bombelli’s book, relative to the Algebra of
India, has become more interesting, from the information con-
cerning the science of that country, which has reached Europe
within the last-tWwenty years. He tells us, that he had seen in
the Vatican library, a. manuscript of a certain Diophantus, a
Greek author, which he admired so much, that he had formed
the design of translating it. He adds, that in this manuscript he
had found the Indian authors often quoted ; from which it ap-
peared, that Algebra was known to the Indians before it was
known to the Arabians. Nothing, however, of all this is to be
found in the work of Diophantus, which was pub! shed about
three years after the tine when Bombelli wrote. As it is, at
the same time, impossible that he could be so much mistaken
about a manuscript which he had particularly examined, this
passage remains a mystery, which those who are curious about
the ancient history of science would be very eth to have un-
ravelled. See Hutton’s History of Algebra.

secr. 1.] DISSERTATION SECOND. 27

Vieta was a very learned man, and an excellent mathe-
matician, remarkable both for industry and invention. He
was the first who employed letters to denote the known as
well as the unknown quantities, so that it was with him
that the language of algebra first became capable of express-
ing general truths, and attained to that extension which
has since rendered it such a powerful instrument of in-
vestigation. He has also given new demonstrations of the
rule for resolving cubick, and even biquadratick equations.
He also discovered the relation between the roots of an
equation of any degree, and the coefficients of its terms,
though only in the case where none of the terms are want-
ing, and where all the roots are real or positive. It is, in-
deed, extremely curious to remark, how gradually the
truths of this sort came in sight. This proposition belong-
ed toa general truth, the greater part of which remained
yet to be discovered. Vieta’s treatises were originally
published about the year 1600, and were afterwards col-
lected into one volume by Schooten, in 1646.

In speaking of this illustrious man, Vieta, we must not
omit his improvements in trigonometry, and stil! less his
treatise on angular sections, which was a most important
application of Algebra to investigate the theorems, and re-
solve the problems of geometry. He alse restored some
of the books of Apollonius, in a manner highly*creditable
to his own ingenuity, but not perfectly in the taste of the
Greek geometry ; because, though the constructions are
elegant, the demonstrations are all synthetical.

About the same period, Algebra became greatly indebt-
ed to Albert Girard, a Flemish mathematician, whose prin-
cipal work, Invention Nouvelle en Algebre, was printed
in 1669. This ingenious author perceived a greater ex-
tent, but not yet the whole of the truth, partially discov-
ered by Vieta, viz. the successive formation of the coeffi-

28 DISSERTATION SECOND. [paRY 1.

cients of an equation from the sum of the roots; the sum
of their products taken two and two; the same taken three
and three, &c. whether the roots be positive or negative.
He appears also to have been the first who understood the
use of negative roots in the solution of geometrical pro-
blems, and is the author of the figurative expression, which
gives to negative quantities the name of quantities less
than nothing ; a phrase that has been severely censured
by those who forget that there are correct ideas, which
correct language can hardly be made to express. The
same mathematician conceived the notion of imaginary
roots, and showed that the number of the roots of an equa-
tion could not exceed the exponent of the highest power of
the unknown quantity. He was also in possession of the
very refined and difficult rule, which forms the sums of the
powers of the roots of an equation from the coefficients of
its terms. This is the greatest list of discoveries which
the history of any algebraist could yet furnish.

The person next in order, as an inventor in Algebra, is
Thomas Harriot, an English mathematician, whose book,
Artis Analyticae Praxis, was published after his death, in
1631. This book contains the genesis of all equations, by
the continued multiplication of simple equations ; that is to
say, it explains the truth in its full extent, to which Vieta
and Girard had been approximating. By Harriot also, the
method of extracting the roots of equations was greatly im-
proved; the smaller letters of the alphabet, instead of the
capital letters employed by Vieta, were introduced ; and
by this improvement, trifling, indeed, compared with the
rest, the form and exteriour of algebraick expression were
brought nearer to those which are now in use.

I have been the more careful to note very particularly
the degrees by which the properties of equations were
thus unfolded, because J think it forms an instance hardly

seer. 1.) DISSERTATION SECOND. | 29

paralleled in science, where asuccession of able men, with-
out going wrong, advanced, nevertheless, so slowly in the
discovery of a truth which, when known, does not seem to
be of a very hidden and absiruse nature. ‘Their slow prov
gress arose from this, that they worked with an instrument,
the use of which they did not fully comprehend, and em- |
ployed a language which expressed more than they were
prepared to understand ;—a language which, under the no-
tion, first of negative and then of imaginary quantities, seem-
ed to involve such mysteries as the accuracy of mathemati-
cal science must necessarily refuse to admit.

The distinguished author of whom I have just been
speaking was born at Oxford in 1560. He was employed
in the second expedition sent out by Sir Walter Raleigh, to
Virginia, and on his return published an account of that
country. He afterwards devoted himself entirely to the
study of the mathematicks; and it appears from some of
his manuscripts, lately discovered, that he observed the
spots of the sun as early as December 1610, not more than
a month later than Galileo. He also made observations
on Jupiter’s satellites, and on the comets of 1607, and of
1618."

The succession of discoveries, above related, brought
the algebraick analysis, abstractly considered, into a stale
of perfection, little short of that which it has attained at
the present moment. It was thus prepared for the step
which was about to be taken by Descartes, and which forms
one of the most important epochas in the history of the

'The manuscripts which contain these observations, and pro-
bably many other things of great interest, are preserved in the
collection of the Earl of Egremont, having come into the posses-
sion of his family from Henry Percy, Earl of Northumberland, a
most liberal patron of science, with whem Harriot appears to
have chiefly lived after his return from Virginia.

4

,

30 DISSERTATION SECOND. {rant

-

mathematical sciences. This was the application of the alge-
braick analysis, to define the nature, and investigate the
properties, of curve lines, and, consequently, to represent
the notion of variable quantity. It is often said, that Des-
cartes was the first who applied algebra to geometry ; but
this is inaccurate; for such applications had been made
before, particularly by Vieta, in his treatise on angular sec-
tions. The invention just mentioned is the undisputed
property of Descartes, and opened vast fields of disco-
very for those who were to come after him.

The work in which this was contained is a tract of no

more than 106 quarto pages; and there is probably no
book of the same size which has conferred so much and so
just celebrity on its author. It was first published in
1637. - -
In the first of the three books into which the tract just
mentioned is divided, the author begins with the considera-
tion of such geometrical problems as may be resolved by
circles and straight lines ; and explains the method of con-
structing algebraick formulas, or of translating a truth from
the language of algebra into that of geometry. He then
proceeds to the consideration of the problem, known among
the ancients by the name of the locus ad quatuor rectas,.
and treated of by Apollonius and Pappus. The algebraick
analysis afforded a method of resolving this problem in its
full extent ; and the consideration of it is again resumed in
the second book. The thing required is, to find the locus
of a point, from which, if perpendiculars be drawn to four
lines given in position, a given function of these perpendicu-
lars in which the variable quantities are only of two dimen-
sions, shall be always of the same magnitude.’ Descartes

' It will easily be perceived, that the word function is not con-
tained in the original enunciation of the problem. It is a term
but fately introduced into mathematica! language, and affords

suer. 1.) DISSERTATION SECOND. 3i

shows the locus, on this hypothesis, to be always a conick
section ; and he distinguishes the cases in which it is a cir-
cle, an ellipsis, a parabola, or a hyperbola. It was an in-
stance of the most extensive investigation which had yet
been undertaken in geometry, though, to render it a com-
plete solution of the problem, much more detail was doubt-
less necessary. The investigation is extended to the cases
where the function, which remains the same, is of three,
four, or five dimensions, and where the locus is a line of a
higher order, though it may, in certain circumstances, be-
come a conick section. The lines given in position may
be more than four, or than any given number ; and the lines
drawn to them may either be perpendiculars, or lines mak-
ing given angles with them. The same analysis applies to
all the cases ; and this problem, therefore, afforded an ex-
cellent example of the use of algebra in the investigation of
geometrical propositions. ‘The author takes notice of the
unwillingness of the ancients to transfer the language of
arithmetick into geometry, so that they were forced to
have recourse to very circuitous methods of expressing
those relations of quantity in which powers beyond the third
are introduced. Indeed, to deliver investigation from those
modes of expression which involve the composition of ra-
tios, and to substitute in their room the multiplication of the
numerical measures, is of itself a very great advantage,
arising from the introduction of algebra into geometry.

In this book also, an ingenious method of drawing tan-
gents to curves is proposed by Descartes, as following from
his general principles, and it is an invention with which he
appears to have been particularly pleased. He says, “‘ Nec
verebor dicere problema hoc non modo eorum, quae scio;

here, as on many other occasions, a more general and more con-
eise expression than could be otherwise obtained.

32 DISSERTATION SECOND. [rant i.

utilissimum et generalissimum esse, sed etiam eorum quae in
geometria scire unquam desideraverim.”' This passage is
not a little characteristick of Descartes, who was very much
disposed to think well of what he had done himself, and
even to suppose that it could not easily be rendered more
perfect. The truth, however, is, that his method of draw-
ing tangents is extremely operose, and is one of those hasty
views which, though ingenious and even profound, require
to be vastly simplified, before they.can be reduced to
practice. Fermat, the rival and sometimes the superiour
of Descartes, was far more fortunate with regard to this
problem, and his method of drawing tangents to curves, is
the same, in effect, that has been followed by all the geome-
ters since his time,—while that of Descartes, which could
only be valued when the other was unknown, has been long
since entirely abandoned. 'The remainder of the second
book is occupied with the consideration of the curves, which
have been called the ovals of Descartes, and with some in-
vestigations concerning the centres of lenses ; the whole in-
dicating the hand of'a great master, and deserving the most
diligent study of those, who would become acquainted with
this great enlargement of mathematical science. _—

The third book of the geometry treats of the construc-
tion of equations by geometrick curves, and it also contains
a new method of resolving biquadratick equations.

The leading principles of algebra were now unfolded,
and the notation was brought, from a mere contrivance for
abridging common language, to a system of symbolical
writing, admirably fitted to assist the mind in the exercise
of thought.

The happy idea, indeed, of expressing quantity, and the
operations on quantity, by conventional symbols, instead of

* Cartesii Geometria, p. 40.

sucr. 1.) DISSERTATION SECOND. 33

representing the first by real magnitudes, and enunciating
the second in words, could not but make a great change on
the nature of mathematical investigation. The language of
mathematicks, whatever may be its form, must always con-
sist of two parts ; the one denoting quantities simply, and
the other denoting the manner in which the quantities are
combined, or the operations understood to be performed
on them. Geometry expresses the first of these by real
magnitudes, or by what may be called natural signs ; a line
by a line, an angle by an angle, an area by an area, &c. ;
and it describes the latter by words. Algebra, on the oth-
er hand, denotes both quantity, and the operations on quan-
tity, by the same system of conventional symbols. Thus,
in the expression «°—a x? 4 b? = 0, the letters a, b, x,
denote quantities, but the terms 2°, a 2°, &c. denote cer-
tain operations performed on those quantities, as well as.
the’ quantities themselves ; x° is the quantity x raised to the
cube; and ax* the same quantity x raised to the square,
and then multiplied into a, &c.; the combination, by addi.
tion or subtraction, being also expressed by the signs +
and —.

Now, it is when applied to this latter purpose that the
algebraick language possesses such exclusive excellence.
The mere magnitudes themselves might be represented by
figures, as in geometry, as well as in any way whatever ;
but the operations they are to be subjected to, if described
in words, must be set before the mind slowly, and in suc-
cession, so that the impression is weakened, and the clear
apprehension rendered difficult. In the algebraick expres-
sion, on the other hand, so much meaning is concentrated
into a narrow space, and the impression made by all the
parts is so simultaneous, that nothing can be more favour-
able to the exertion of the reasoning powers, to the continu-
ance of their action, and their security against errour.

34 DISSERTATION SECOND. {eaux .

Another advantage resulting from the use of the same nota-
tion, consists in the reduction of all the different relations
among quantities to the simplest of those relations, that of
equality, and the expression of it by equations. This
gives a great facility of generalization, and of comparing
quantities with one another. A third arises from the sub-
stitution of the arithmetical operations of multiplication and
division, for the geometrical method of the composition ©
and resolution of ratios. Of the first of these, the idea is
so clear, and the work so simple ; of the second, the idea is
comparatively so obscure, and the process so complex, that
the substitution of the former for the latter could not but
be accompanied with great advantage. This is, indeed,
what constitutes the great difference in practice between
the algebraick and the geometrick method of treating quan-
tity. When the quantities are of. a complex nature, so as
to go beyond what in algebra is called the third power, the
geometrical expression is so circuitous and involved, that
it renders the reasoning most Jaborious and intricate. The
great facility of generalization in algebra, of deducing one
thing from another, and of adapting the analysis to every
kind of research, whether the quantities be constant or vari-
able, finite or infinite, depeads on this principle more than
any other. Few of the early algebraists seem to have been
aware of these advantages. :

The use of the signs plus and minus has given rise to
some dispute. These signs were at first used, the one to
denote addition, the other subtraction, and for a long time
were applied to no other purpose. But as, in the multipli-
cation of a quantity, consisting of parts connected by those
signs, into another quantity similarly composed, it was al-
ways found, and could be universally demonstrated, that,
in uniting the particular products of which the total was
made up, those of which both the factors had the sign mz:

saer. 1.] DISSERTATION SECOND. 33

nus before them, must be added into one sum with those of
which all the factors had the sign plus; while those of
which one of the factors had the sign plus, and the other
the sign minus, must be subtracted from the same,—this
general rule came to be more simply expressed by saying,
that in multiplication like signs gave plus, and that unlike
signs gave minus. :
Hence the signs plus and minus were considered, not as
merely denoting the relation of one quantity (o another
placed before it, but, by a kind of ficlton, they were con-
sidered as denoting qualities inherent in the quantities to
the names‘of which they were prefixed. This fiction was
found extremely useful, and it was evident that no errour
could arise from it. It was necessary to have a rule for
determining the sign belonging to a product, from the signs
of the factors composing that product, independently of
every other consideration ; and this was precisely the pur-
pose for which the above fiction was introduced. So ne-
cessary is this rule in the generalizations of algebra, that
we meet with it in Diophantus, notwithstanding the imper-
fection of the language he employed; for he states, that
Awfis into Acnlis gives Yragkc, &c. The reduction, there-
fore, of the operations on quantity fo an arithmetical form,
_ necessarily involves this use of the signs plus or minus ;
- that is, their application to denole something like absolute
qualities in the objects they collect together. The at-
tempts to free algebra from this use of the signs have of
course failed, and must ever do so, if we would preserve
to that science the extent and facility of its operations.
Even the most scrupulous purist in mathematical lan-
guage must admit, that no real errour is ever introduced by
employing the signs in this most abstract sense. If the
equation 2° + pa? -- qa + r=o, be said to have one posi-
tive and two negative roots, this is certainly as exception-

36 DISSERTATION SECOND. [Parr 2.

able an application of the term negative, as any that can
be proposed ; yet, in reality, it means nothing but this in-
telligible and simple truth, that 2° 4 pa? + gx + r=(«—a)
(x + b) (w 4+ c;) or that the former of these quantities is
produced by the multiplication of the three binomial fac-
tors, x—a, x 4+ b, a +c. We might say the same nearly
as to imaginary roots; they show thatthe simple factors
cannot be found, but that the quadratick factors may be
found ; and they also point out the means of discovering
them.

_ ‘The aptitude of these same signs to denote contrariety
of position among geometrick magnitudes, makes the fore-
going application of them infinitely more extensive and
more indispensable.

From the same source arises the great simplicity intro-
duced into many of the theorems and rules of the mathe-
matical sciences. Thus, the rule for finding the latitude
of a place from the sun’s meridian altitude, if we employ
the signs plus and minus for indicating the position of the
sun and of the place relatively to the equator, is enunciat-
ed in one simple proposition, which includes every case,
without any thing either complex or ambiguous. But if
this is not done,—if the signs plus and minus are not em-
ployed, there must be at least two rules, one when the sun
and place are on the same side of the equator, and another
when they are on different sides. In the more complicated
calculations of spherical trigonometry, this holds still more
remarkably. When one would accommodate such rules
to those who are unacquainted with the use of the algebra-
ick signs, they are perhaps not to be expressed in less than
four, or even six different propositions ; whereas, if the use
of these signs is supposed, the whole is comprehended in
asingle sentence. In such cases, it is obvious that both
the memory and understanding derive great advantage from

ia

sKor. 1.] DISSERTATION SECOND. 37

the use of the signs, and profit by a simplification, which
is the work entirely of the algebraick language, and cannot
be imitated by any other.

‘That I might not interrupt the view of improvements so
closely connected with one another, I have passed over
one of the discoveries, which does the greatest honour to
the seventeenth century, and which took place near the
beginning of it. ¢

As the accuracy of astronomical observation had been

continually advancing, it was necessary that the correct-
ness of trigonometrical calculation, and of course its diffi-
culty, should advance in the same proportion, ‘The signs
and tangents of angles could not be expressed with. suffi-

cient correctness without decimal fractions, extending to five

or six places below unity, and when to three such numbers

a fourth proportional was to be found, the work of multipli-
cation and division became extremely laborious. Accord-

ingly, in the end of the sixteenth century, the time and
labour consumed in such calculations had become exces-

sive, and were felt as extremely burdensome by the mathe-
maticians and astronomers all over Europe. Napier of
-Merchiston, whose mind seems to have been peculiarly

turned to arithmetical researches, and who was also devoted
to the study of astronomy, had early sought for the means of
relieving himself and others from this difficulty. He had.
viewed the subject in a variety of lights, and a number of
ingenious devices had occurred to him, by which the te-

diousness of arithmetical operations might, more or less com-

pletely, be avoided. In the course of these attempts, he

did not fail to observe, that whenever the numbers to be

multiplied or divided were terms of a geometrical progres-

sion, the product or the quotient must also be a term of
that progression, and must occupy a place in it pointed

out by the places of the given numbers, so that it might be

5

38 DISSERTATION SECOND. - feanr a.

found from mere inspection, if the progression were far
enough continued. If, for instance, the third term of the
progression were to be multiplied by the seventh, the pro-
duct must be the tenth, and if the twelfth were to be divid-
ed by the fourth, the quotient must be the eighth; so that
the multiplication and division of such terms was reduced
to the addition and subtraction of the numbers which indi-
cated their places in the progression.

This observation, or one very similar to it, was made by
Archimedes, and was employed by that great geometer to
_convey an idea of a number too vast to be correctly express-
ed by the arithmetical notation of the Greeks. Thus far,
however, there was no difficulty, and the- discovery might
certainly have been made by men much inferiour either to
Napier or Archimedes. What remained to. be done, what
Archimedes did not attempt, and what Napier completely
performed, involved two great difficulties. It is plain, that
the resource of the geometrical progression was sufficient,
when the given numbers were terms of that progression ; but
if they were not, it did not seem that any advantage could
be derived from it. Napier, however, perceived, and it
was by no means obvious, that all numbers whatsoever
might be inserted in the progression, and have their places
assigned init. After conceiving the possibility of this, the
next difficulty was, to discover the principle, and to execute
the arithmetical process, by which these places were to be
ascertained. It is in these two points that the peculiar merit
of his invention consists; and at a period when the nature
of series, and when every other resource of which be could
avail himself were so little known, his success argues a depth
and originality of thought which, I am persuaded, have
rarely been surpassed.

The way in which he satisfied himself that all numbers
might be intercalated between the terms of the given pro-

Te ern ee A TE

axcr, 1.] DISSERTATION SECOND. 39

_ gression, and by which he found the places they must oc-

cupy, was founded on a most ingenious supposition,—that
of two points describing two different lines, the one with a
constant velocity, and the other with a velocity always in-
creasing in the ratio of the space the point had already
gone over: the first of these would generate magnitudes in
arithmetical, and the second magnitudes in geometrical pro-
gression. It is plain, that all numbers whatsoever would
find their places among the magnitudes so generated ; and,
indeed, this view of the subject is as simple and profound

as any which, after two hundred years, has yet presented

itself to mathematicians. ~The mode of deducing the results
has been simplified ; but it can hardly be said that the prin-
ciple has been more clearly developed.

I need not observe, that the numbers which indicate the
places of the terms of the geometrical progression are call-
ed by Napier the logarithms of those terms.

Various systems of logarithms, it is evident, may be con-
structed according to the geometrical progression assumed ;
and of these, that which was first contrived by Napier,

though the simplest, and the foundation of the rest, was

not so convenient for the purposes of calculation, as one
which soon afterwards occurred, both to himself and his
friend Briggs, by whom the actual calculation was per-
formed. The new system of logarithms was an improve-
ment, practically considered ; but in as far as it was con-
nected with the principle of the invention, it is only of se-
condary consideration. The original tables had been also
somewhat embarrassed by too close a connexion between
them and trigonometry. The new tables were free from
this inconvenience.

It is probable, however, that the greatest inventor in sci-
ence was never able to do more than to accelerate the pro-
gress of discovery, and to anticipate what time, “ the au-

4O DISSERTATION SECOND. 2) ofeammen

thor of authors,’ would have gradually brought to light.
Though logarithms had not been invented by Napier, they —
would have been discovered in the progress of the alge-
braick analysis, when the arithmetick of powers and expo-
nents, both integral and fractional, came to be fully under-
stood. The idea of considering all numbers, as powers of
one given number, would then have readily occurred, and
the doctrine of series would have greatly facilitated the cal-
culations which it was necessary to undertake. Napier
had none of these advantages, and they were all supplied
by the resources of his own mind. Indeed, as there never
was any invention for which the state of knowledge had less
prepared the way, there never was any where more merit
fell to the share of the inventor.
His good fortune, also, not less than his great sagacity, |
may be remarked. Had the invention of logarithms been
delayed to the end of the seventeenth century, it would
have come about without effort, and would not have con-
ferred on the author the high celebrity which Napier so
justly derives from it. In another respect he has also been
fortunate. Many inventions have been eclipsed or obscur-
ed by new discoveries; or they have been so altered by
subsequent improvements, that their original form can hard-
ly be recognised, and, in some instances, has been entirely
forgotten. This has almost always happened to the disco-
veries made at an early period in. the progress of science,
and before their principles were fully unfolded. It has been
quite otherwise with the invention of logarithms, which
came out of the hands of the author so perfect, that it has
never received but one material improvement, that which
it derived, as has just been said, from the ingenuity of his
friend in conjunction with his own. Subsequent improve-
ments in science, instead of offering any thing that could
supplant this invention, have only enlarged the circle to

ster. a.] DISSERTATION SECOND. 41

which its utility extended. Logarithms have been ap-
plied to numberless purposes, which were not thought of at
the time of their first construction. Even the sagacity of
their author did not see the immense fertility of the prin-
ciple he had discovered ; he calculated his tables merely to
facilitate arithmetical, and chiefly trigonometrical computa-
tion, and little imagined that he was at the same time con-
structing a scale whereon to measure the density of the
strata of the atmosphere, and the heights of mountains ; that
he was actually computing the areas and the lengths of in-
numerable curves, and was preparing for a calculus which
was yet to be discovered, many of the most refined and most
valuable of its resources. Of Napier, therefore, if of any
man, it may safely be pronounced, that his name will never
be eclipsed by any one more conspicuous, or his invention
superseded by any thing more valuable.

As a geometrician, Napier has left behind him a_ noble
monument in’ the two trigonometrical theorems, which are
known by his name, and which appear first to have been
communicated in writing to Cavalleri, who has mentioned
them with great eulogy... They are theorems not a little
difficult, and of much use, as being particularly adapted to
logarithmick calculation. They were published in the
Canon Mirificus Logarithmorum, at Edinburgh, in 1614.°

* Wallis, Opera Math. Tom. IT. p. 875.

* A reprint of the Canon Mirificus, from the original edition,
is given in the 6th volume of the great Thesaurus, in which Ba-
ron Maseres, with his usual zeal and intelligence, has collected
and illustrated every thing of importance that has been written on
the subject of logarithms. See Scripleres Logarithmici, Ato. vol.
VI, p. 470.

42 DISSERTATION SECOND. [ranr 2.
tare 2 ee, ae yi | rid, eae 7

ures cet! BY "a0
SECTION II. wet dita

t MAREE Hedy 4

EXPERIMENTAL INVESTIGATION.

In this section I shall begin with a short view of the state
of Physical Knowledge before the introduction of the In-
ductive Method ; I shall next endeavour to explain that me-
thod by an analysis of the Novum Organum ; and shall
then inquire how far the principles established in that work
have actually contributed to the advancement of Natural
Philosophy.

+e.) eb -

1. Ancient Puysicks.

Though the phenomena of the material world ¢ould not
but early excite the curiosity of a being who, like man, re-
ceives his strongest impressions from without, yet an accu-
rate knowledge of those phenomena, and their laws, was
not to be speedily acquired. The mere extent and variety
of the objects were, indeed, such obstacles to that acquisi-
tion, as could not be surmounted but in the course of many
ages. Man could not at first perceive from what point he —
must begin his inquiries, in what direction he must carry
them on, or by what rules he must be guided. He was
like a traveller going forth to explore a vast and unknown
wilderness, in which a multitude of great and interesting
objects presented themselves on every side, while there
was no path for him to follow, no rule to direct his survey,
and where the art of observing, and the instruments of ob-
servation, must equally be the work of his own invention.
In these circumstances, the selection of the objects to be
studied was the effect of instinct rather than of reason, or
of the passions and emotions, more than of the understand-
ing. When things new and unlike those which occurred in

excr. 11.] Dida su anponD 43

“a:

the course of every day’s experience presented themselves,
they excited wonder or surprise, and created an anxiety to
discover some principle which might connect them with the
appearances commonly observed. About these last, men
felt no desire to be farther informed ; but when the common
’ order of things was violated, and something new or singu-
lar was produced, they began to examine into the fact, and
attempted to inquire into the cause. Nobody sought to
know why a stone fell to the ground, why smoke ascended,
or why the stars revolved round the earth. But if a fiery
meteor shot across the heavens,—if the flames of a volcano
burst forth,—or if an earthquake shook the foundations of
the world, terrour and curiosity were bothawakened; and
when the former emotion had subsided, the latter was sure
to become active. Thus, to,trace a resemblance between
the events with which the observer was most familiar, and
those to which he was less accustomed, and which had ex-
cited his wonder, was the first object of inquiry, and pro-
duced the first advances towards generalization and philo-
sophy.'

This principle, which it were easy to trace, from tribes
the most rude and barbarous, to nations the most highly
refined, was what yielded the first attempts towards classi-
fication and arrangement, and enabled man, out of individuals,
subject to perpetual change, to form certain fixed and per-
manent objects of knowledge,—the species, genera, orders,
and classes, into which he has distributed these individuals.
By this effort of mental abstraction, he has created to him-
self a new and intellectual world, free from those changes
and vicissitudes to which all material things are destined,

' La maraviglia
Dell ‘ignoranza é la figlia,
E del sapere
La madre.

Ad DISSERTATION SECOND. [part 1.

This, too, is a work not peculiar to the philosopher, but, in
a certain degree, is performed by every man who compares
one thing with another, and who employs the terms of or-
dinary language. ,

Another great branch of knowledge is occupied, not
about the mere arrangement and classification of objects,
but about events or changes, the laws which those changes
observe, and the causes by which they are produced. In
a science, which treated of events and of change, the nature
and properties of motion came of course to be studied, and
the ancient philosophers naturally enough began their in-
quiries with the definition of motion, or the determination
of that in which it consists. Aristotle’s definition is highly
eharacteristical of the vagueness and obscurity of his phy-
sical speculations. He calls motion “the act of a being in
power, as far as in power,’’—words to which it is impossi-
ble that any distinct idea can ever have been annexed.

The truth is, however, that the best definition of motion
can be of very little service in physicks. Epicurus defined
it to be the ‘‘ change of place,’ which is, no doubt, the
simplest and best definition that can be given; but it must,

_at the same time, be confessed, that neither he nor the
moderns who have retained his definition, have derived the
least advantage from it in their subsequent researches.
The properties, or, as they are called, the laws of motion,
cannot be derived from mere definition; they must be
sought for in experience and observation, and are not to be
found without a diligent comparison, and scrupulous exami-
nation of facts. Of such an examination, neither Aristotle,
nor any other of the ancients, ever conceived the necessity,
and hence those laws remained quite unknown throughout
all antiquity.

When the laws of motion were unknown, the other parts
of natural philosophy could make no great advances. In-

skCY. 11.) - DISSERTATION SECOND. 49

stead of conceiving that there resides in body a natural and
universal tendency to persevere in the same state, whether
of rest, or of motion, they believed that terrestrial bodies
tended naturally either to fall to the ground, or to ascend
from it, till they attain their own place ; but that, if they
were impelled by an oblique force, then their motion be-
came unnatural or violent, and tended continually to decay.
With the heavenly bodies, again, the natural motion was
circular and uniform, eternal in its course, but perpetually
varying in its direction. Thus, by the distinction between
natural and violent motion among the bodies of the earth,
and the distinction between what we may call the laws of
motion in terrestrial and celestial bodies, the ancients threw
into all their reasonings upon this fundamental subject a
confusion and perplexity, from which their philosophy never
was-delivered.

There was, however, one part of physical knowledge in
which their endeavours were attended with much better
success, and in which they made important discoveries.
This was in the branch of Mechanicks, which treats of the
action of forces in equilibrio, and producing not motion but
rest ;—a subject which may be understood, though the laws
of motion are unknown.

The first writer on this subject is Archimedes. He
treated of the lever, and of the centre of gravity, and has
shown that there will be an equilibrium between two heavy
bodies connected by an inflexible rod or lever, when the
point in which the lever is supported is so placed between
the bodies, that their distances from it are inversely as their
weights. Great ingenuity is displayed in this demonstra-
tion; and it is remarkable, that the author borrows no
principle from experiment, but establishes his conclusion
entirely by reasoning a priori. He assumes, indeed, that
equal bodies, at the ends of the equal arms of a lever, will

6

46 DISSERTATION SECOND. [parr 1.

balance one another ; and also, that a cylinder, or parallelo-
piped of homogeneous matter, will be balanced about its
centre of magnitude. These, however, are not inferences
from experience ; they are, properly speaking, conclusions
deduced from the principle of the sufficient reason.

The same great geometer gave a beginning to the science
of Hydrostaticks, and discovered the law which determines
the loss of weight sustained by a body on being immersed
in water, or in any other fluid. His demonstration rests on
a principle, which he lays down as a postulatum, that, in
water, the parts which are less pressed are always ready to
yield in any direction to those that are more pressed, and
from this, by the application of mathematical reasoning, the
whole theory of floating bodies is derived. The above is
the same principle on which the modern writers on hydros-
taticks proceed ; they give it not as a postulatum, but as
constituting the definition of a fluid.

Archimedes, therefore, is the person who first made the
application of mathematicks to natural philosophy. No in-
dividual, perhaps, ever laid the foundation of more great
discoveries than that geometer, of whom Wallis has said
with so much truth, ‘ Vir stupendae sagacitatis, qui pri-
ma fundamenta posuit inventionum feré omnium in quibus
promovendis aetas nostra gloriatur.”’

The mechanical inquiries, begun by the geometer of Sy-
racuse, were extended by Ctesibius and Hero; by Anthe-
mius of Tralles; and, lastly, by Pappus Alexandrinus.—
Ctesibius and Hero were the first who analyzed mechanical
engines, reducing them all to combinations of five simple
mechanical contrivances, to which they gave the name of
Avrapes, or Powers, the same which they retain at the pre-
sent moment.

Even in mechanicks, however, the success of these in-
vestigations was limited; and failed in those cases where

szcr. 11.) DISSERTATION SECOND. 47

the resolution of forces is necessary, that principle being
then entirely unknown. Hence the force necessary to sus-

tain a body on an inclined plane, is incorrectly determined
~ by Pappus, and serves to mark a point to which the me-
chanical theories of antiquity did not extend.

In another department of physical knowledge, Astrono-
my, the endeavours of the ancients were also accompanied
with success. I do not here speak of their astronomical
theories, which were, indeed, very defective, but of their
discovery of the apparent motions of the heavenly bodies,
from the observations begun by Hipparchus, and continued
by Ptolemy. In this their success was great; and while
the earth was supposed to be at rest, and while the instru-
ments of observation had but a very limited degree of ac-
curacy, a nearer approach to the truth was probably not
within the power of human ingenuity. Mathematical rea-
soning was very skilfully applied, and no men whatever, in
the same circumstances, are likely to have performed more
than the ancient astronomers. They succeeded, because
they were observers, and examined carefully the motions
which they treated of. The philosophers, again, who stu-
died the motion of terrestrial bodies, either did not observe
at all, or observed so slightly, that they could obtain no ac-
curate knowledge, and, in general, they knew just enough of
the facts to be misled by them.

The opposite ways which the ancients thus took to study
the Heavens and the Earth, observing the one, and dream-
ing, as one may say, over the other, though a striking incon-
sistency, is not difficult to be explained.

No information at all could be obtained in astronomy,
without regular and assiduous observation, and without in-
struments capable of measuring angles, and of measuring
time, either directly or indirectly. The steadiness and re-
gularity of the celestial motions seemed to invite the most

48 DISSERTATION SECOND. [parr i.

scrupulous attention. On the other hand, as terrestrial ob-
jects were always at hand, and spontaneously falling under
men’s view, it seemed unnecessary to take much trouble to
become acquainted with them, and as for applying mea-
sures, their irregularity appeared to render every idea of
such preceeding nugatory. The Aristotelian philosophy
particularly favoured this prejudice, by representing the
earth, and all things on its surface, as full of irregularity and
confusion, while the principles of heat and cold, dryness
and moisture, were in a state of perpetual warfare. The
unfortunate division of motion into natural and violent, and
the distinction, still more unfortunate, between the proper-
ties of motion and of body, in the heavens and on the earth,
prevented all intercourse between the astronomer and the
naturalist, and all transference of the maxims of the one to
the speculations of the other. ;

Though, on account of this inattention to experiment, no-
thing like the true system of natural philosophy was known
to the ancients, there are, nevertheless, to be found in their
writings many brilliant conceptions, several fortunate conjec-
tures, and gleams of the light which was afterwards to be so
generally diffused.

Anaxagoras and Empedocles, for example, taught that
the moon shines by light borrowed from the sun, and were
Jed to that opinion, not only from the phases of the moon,
but from its light being weak, and unaccompanied by heat.
That it was a habitable body, like the earth, appears tobea
- doctrine as old as Orpheus ; some lines, ascribed to that po-
et, representing the moon as an earth, with mountains and
cities on its surface.

Democritus supposed the spots on the face of the moon
to arise from the inequalities of the surface, and from the
shadows of the more elevated parts projected on the plains.
Every one knows how conformable this is to the discoveries
made by the telescope.

szer. 11] DISSERTATION SECOND. 49

Plutarch considers the velocity of the moon’s motion as
the cause which prevents that body from falling to the
earth, just as the motion of a stone in a sling prevents it
from falling to the ground. The comparison is, in a cer-
tain degree, just, and clearly implies the notion of centri-
fugal force ; and gravity may also be considered as pointed
at for the cause which gives the moon a tendency to the
earth. Here, therefore, a foundation was laid for the true
philosophy of the celestial motions ; but it was laid without
effect. It was merely the conjecture of an ingenious mind,
wandering through the regions of possibility, guided by no
evidence, and having no principle which could give stabili-
ty to its opinions. Democritus, and the authors of that
physical system which Lucretius has so beautifully illus-
trated, were still more fortunate in some of their conjec-
tures. ‘They taught that the Milky Way is the light of a
great number of smal! stars, very close to one another; a
magnificent conception, which the Jatest improvements of
the telescope have fully verified. Yet, as if to convince
us that they derived this knowledge from no pure or cer-
tain source, the same philosophers maintained, that the sun
and the moon are bodies no larger than they appear to us to
be.

Very just notions concerning comets were entertained by
some of the ancients. The Chaldeans considered those be-
dies as belonging to the same order with the planets; and
this was also the opinion of Anaxagoras, Pythagoras, and
Democritus. The remark of Seneca on this subject is tru-
ly philosophical, and contains a prediction which has been

fully accomplished: “Why do we wonder that comets,
_ which are so rare a spectacle in the world, observe laws
which to us are yet unknown, and that the beginning and
end of motions, so seldom observed, are not yet fully un-
derstood ?”— Veniet tempus, quo ista quae nunc latent,

50 DISSERTATION SECOND. [pant 1.

in lucem dies extrahat, et longioris aevi diligentia: ad
inquisitionem tantorum aetas una non sufficit. Veniet
tempus, quo posterit nostri tam aperta nos nescisse mi-
rentur.*

It was, however, often the fate of such truths to give way
to errour. The comets, which these ancient philosophers
had ranked so justly with the stars, were degraded by
Aristotle into meteors floating in the earth’s atmosphere ;
~ and this was the opinion concerning them which ultimately
prevailed.

But, notwithstanding the above, and a few other splendid
conceptions which shine through the obscurity of the an-
cient physicks, the system, taken on the whole, was full of
errour and inconsistency. ‘Truth and falsehood met almost
on terms of equality ; the former separated from its root,
experience, found no preference above the latter ; to the lat-
ter, in fact, it was generally forced to give way, and the do-
minion of errour was finally established.

One ought to listen, therefore, with caution to the enco-
miums sometimes bestowed on the philosophy of those early
ages. If these encomiums respected only the talents, the
genius, the taste of the great masters of antiquity, we would
subscribe to them without any apprehension of going be-
yond the truth. But if they extend to the methods of phi-
losophizing, and the discoveries actually made, we must be
excused for entering our dissent, and exchanging the lan-
guage of panegyrick for that of apology. The infancy of
science could not be the time when its attainments were
the highest ; and, before we suffer ourselves to be guided by
ihe veneration of antiquity, we ought to consider in what
real antiquity consists. With regard to the progress of
knowledge and improvement, “ we are more ancient than

; Nat. Quaest. Lib. Vii. Cc. Boe

sacr. 11.J DISSERTATION SECOND. 51

those who went before us.”’* The human race has now
more experience than in the generations that are past, and
of course may be expected to have made higher attainments
in science and philosophy. Compared with natural phi-
losophy, as it now exists, the ancient physicks are rude and
‘imperfect. The speculations contained in them are vague
and unsatisfactory, and of little value, but as they eluci-
date the history of the errours and illusions to which the
human mind is subject. Science was not merely stationa-
ry, but often retrograde ; the earliest opinions were fre-
quently the best; and the reasonings of Democritus and
Anaxagoras were in many instances more solid than those
of Plato and Aristotle. Extreme credulity disgraced the
speculations of men who, however ingenious, were little ac-
quainted with the laws of nature, and unprovided with the
great criterion by which the evidence of testimony can alone
be examined. Though observations were sometimes made,
experiments were never instituted ; and philosophers, who
were little attentive to the facts which spontaneously offer-
ed, did not seek to increase their number by artificial com-
binations. Experience, in those ages, was a light which
darted a few tremulous and uncertain rays on some small
portions of the field of science, but men had not acquired
the power over that light which now enables them to con-
centrate its beams, and to fix them steadily on whatever ob-
ject they wish to examine. This power is what distin-
guishes the modern physicks, and is the cause why later
philosophers, without being more ingenious than their pre-
decessors, have been infinitely more successful in the study
of nature. .

' Bacon.

52 DISSERTATION SECOND. [PART 1.

2. Novum Organum.

The defects which have been ascribed to the ancient phy-
sicks were fot likely tobe corrected in the course of the
middie ages. It is true, that, during these ages, a science
of pure experiment had made its appearance in the world,
and might have been expected to remedy the greatest of
these defects, by turning the attention of philosophers to
experience and obsefvation. This effect, however, was far
from being immediately produced ; and none who professed
to be in search of truth ever wandered over the regions of
fancy, in paths more devious and eccentrick, than the first
experimenters in chemistry. They had become acquainted
with a series of facts so unlike to any thing already known,
that the ordinary principles of belief were shaken or sub-
verted, and the mind laid open to a degree of credulity far
beyond any with which the philosophers of antiquity could

be reproached. An anlooked for extension of human pow-
er had taken place; itslimits were yet unknown; and the
boundary between the possible and the impossible was no lon-
ger to be distinguished. The adventurers in an unexplored
country, given up tothe guidance of imagination, pursued
objects which the kindness, no less than the wisdom of na-
ture, have rendered uuattainable by man; and in their spe-
culations peopled the air, the earth, and all the elements,
with spirits and genii, the invisible agents destined to con-
nect together all the facts which they knew, and all those
which they hoped to discover. Chemistry, in this state,
might be said to have an elective attraction for all that was
most absurd and extravagant in the other parts of knowledge ;
alchemy was its immediate offspring, and it allied itself in
succession with the dreams of the Cabbalists, the Rosicru-
cians, and the Theosophers. Thus a seience, founded in
experiment, and destined one day to afford such noble

sEcr. 11.} DISSERTATION SECOND. 53

examples of its use, exhibited for several ages little else than
a series of illusory pursuits, or visionary speculations, while
now and then a fact was accidentally discovered.

Under the influence of these circumstances arose Para-
celsus, Van Helmont, Fiudde, Cardan, and several others,
conspicuous no less for the weakness than the force of their
understandings: men who united extreme credulity, the
most extravagant pretensions, and the most excessive vani-
ty, with considerable powers of invention, a complete con-
tempt for authority, and a desire to consult. experience ;
but destitute of the judgment, patience, and comprehensive
views, without which the responses of that oracle are never
to be understood. Though they appealed to experience,
and disclaimed subjection to the old legislators of science,
they were in too great haste to become legislators themselves,
and to deduce an explanation of the whole phenomena of
nature from a few facts, observed without accuracy, arrang-
ed without skill, and never compared. or confronted with
one another. Fortunately, however, from the turn which
their inquiries had taken, the ill done by them has passed
away, and the good has become permanent. The reveries
of Paracelsus have disappeared, but his application of che-
mistry to pharmacy has conferred a lasting benefit on the
world. The Archeus of Van Helmont, and the army of
spiritual agents with which the discovery of elastick fluids
had filled the imagination of that celebrated empirick, are
laughed at, or forgotten; but the fluids which he had the
sagacity to distinguish, form, at the present moment, the
connecting principles of the new chemistry.

_ Earlier than any of the authors just named, but in a great

measure under the influence of the same delusions, Roger

Bacon appears to have been more fully aware than any of

them of the use of experiment, and of mathematical reason-

ing, in physical and mechanical inquiries. But, in the thir-
7

54 DISSERTATION SECOND. [pane 2.

teenth century, an appeal from the authority of the schools,
even to nature herself, could not be made with impunity.

» Bacon, accordingly, incurred the displeasure both of the
University and of the Church, and this forms one of his
claims to the respect of posterity, as it is but fair to consi-
der persecution inflicted by the ignorant and bigoted as
equivalent to praise bestowed by the liberal and enlighten-
ed.

Much more recently, Gilbert, in his treatise on the Mag-
net, had given an example of an experimental inquiry, car-
ried on with more correctness, and more enlarged views,
than had been done by any of his predecessors. Neverthe-
less, in the end of the sixteenth century, it might still be
affirmed, that the situation of the great avenue to knowledge
was fully understood by none, and that its existence, to the
bulk of philosophers, was utterly unknown.

It was about this time that Francis Bacon (Lord Veru-
lam) began to turn his powerful and creative mind to con-
template the state of human knowledge, to mark its imper-
fections, and to plan its improvement. One of the consi-
derations which appears to have impressed his mind most
forcibly, was the vagueness and uncertainty of all the phy-
sical speculations then existing, and the entire want of con-
nexion between the sciences and the arts.

Though these two things are in their nature so closely
united, that the same truth which is a principle in science,
becomes a rule in art, yet there was at that time bardly any
practical improvement which had arisen from a theoretical
discovery. The natural alliance between the knowledge
and the power of man seemed entirely interrupted ; nothing
was to be seen of the mutual support which they ought to
afford to one another; the improvement of art was left to
the slow and precarious operation of chance, and that of
science to the collision of opposite opinions.

¥
secr. 11.] DISSERTATION SECOND. 58

* But whence,” said Bacon, “can arise such vagueness
and sterility in all the. physical systems which have bither-
to existed inthe world? [tis not certainly from any thing in —
nature itself; for the steadiness.and regularity of the laws
by which it is governed clearly mark them out as objects
of certain and precise knowledge. Neither can it arise from
any want of ability in those who have pursued such inqui-
ries, many of whom have been men of the highest talent
and genius of the ages in which they lived; and it cans
therefore, arise from nothing else but the perverseness and
insufficiency of the methods that have been pursued.
Men have sought to make a world from their own con-
ceptions, and to draw from their own minds all the mate-
rials which they employed; but if, instead of doing so,
they had consulted experience and observation, they would
have had facts, and not opinions, to reason about, and
might have ultimately arrived at the knowledge of the
Jaws which govern the material world.”

** As things are at present conducted,”’ he adds, “a sud-
. den transition is made from sensible objects and particular
facts to general propositions, which are accounted princi-

ples, and round which, as round so many fixed poles, dis-
putation and argument continually revolve. From the pro-
positions thus hastily assumed, all things are derived, by a
process compendious and precipitate, ill suited to discovery,
but wonderfully accommodated to debate. The way that
promises success is the reverse of this. It requires that
we should generalize slowly, going from particular things
to those that are but one step more general ; from those to
others of still greater extent, and so on to such as are uni-
-versal. By such means, we may hope to arrive at princi-
ples, not vague and obscure, but luminous and well defined,
such as nature herself will not refuse to acknowledge.”

%
56 DISSERTATION SECOND. {parr y.

Before laying down the rules to be observed in this in-
ductive process, Bacon proceeds to enumerate the causes
of errour,—the Idols, as he terms them, in his figurative
language, or false divinities to which the mind had so long
been accustomed to bow. He considered this enumeration
as the more necessary, that the same idols were likely to
return, even after the reformation of science, and to avail
themselves of the real discoveries that might have been
made, for giving a colour to their deceptions.

These idols he divides into four classes, to which he
gives names, fantastical, no doubt, but, at the same time,
abundantly significant.

Idola Tribus, Idols of the Tribe,
Specus, of the Den,
Fori, of the Forum,

of the Theatre.

—— Theatri,

1. The idols of the tribe, or of the race, are the causes
of errour founded on human nature in general, or on princi-
ples common to all mankind. “The mind,” he observes,
‘‘ig not like a plain mirror, which reflects the images of
things exactly as they are; it is like a mirror of an uneven
surface, which combines its own figure with the figures of
the objects it represents.’ *

Among the idols of this class, we may reckon the propen-
sity which there is in all men to find in nature a greater de-
gree of order, simplicity, and regularity, than is actually in-
dicated by observation. ‘Thus, as soon as men perceived
the orbits of the planets to return into themselves, they im-
mediately supposed them to be perfect circles, and the mo-
tion in those circles to be uniform; and to these hypothe-
ses, so rashly and gratuitously assumed, the astronomers

* Novum Organum, Lib. i. Aph. 41.

eEcr. 11.] DISSERTATION SECOND. 57

and mathematicians of all antiquity laboured incessantly to
reconcile their observations.

The propensity which Bacon has here characterized so
well, is the same that has been, since his time, known by
the name of the spirit of system. The prediction, that the
sources of errour, would return, and were likely to infest
science in its most flourishing condition, has been fully ve-
rified with respect to this illusion, and in the case of scien-
ces which had no existence at the time when Bacon wrote.
When it was ascertained, by observation, that a considera-
ble part of the earth’s surface consists of minerals, dispos-
ed in horizontal strata, it was immediately concluded, that
the whole exteriour crust of the earth is composed, or has
been composed, of such strata, continued all around without
interruption; and on this, as on a certain and general
fact, entire theories of the earth have been constructed.

There is no greater enemy which science has to struggle
with than this propensity of the mind; and it is a struggle
from which science is never likely to be entirely relieved ;
because, unfortunately, the illusion is founded on the same
principle from which our love of knowledge takes its rise.

2. The idols of the den are those that spring from the
peculiar character of the individual. Besides the causes
of errour which are common to alb mankind, each indivi-
dual, according to Bacon, has his own dark cavern or den,
into which the light is imperfectly admitted, and in the ob-
scurity of which a tutelary idol lurks, at whose shrine the
truth is often sacrificed.

One great and radical distinction in the capacities of men
is derived from this, that some minds are best adapted to
mark the differences, others to catch the resemblances, of
things. Steady and profound understandings are dispos-
ed to attend carefully, to proceed slowly, and to examine
the most minute differences ; while those that are sublime

58 DISSERTATION SECOND. [arr 1.

and active are ready to lay hold of the slightest resemblan-
ces. Hach of these easily runs into excess; the one by
catching continually at distinctions, the other at affinities.

The studies, also, to which a man is addicted, have a
great effect in influencing his opinions. Bacon complains,
that the chemists of his time, from a few experiments with
the furnace and the crucible, thought that they were fur-
nished with principles sufficient to explain the structure of
the universe; and he censures Aristotle for having deprav-
ed his physicks so much with his dialecticks, as to render
the former entirely a science of words and controversy. In
like manner, he blames a philosopher of his own age, Gil-
bert, who had studied magnetism to good purpose, for hav-
ing proceeded to form out of it a general system of philoso-
phy. Such things have occurred in every period of sci-
ence. ‘Thus electricity has been applied to explain the
motion of the heavenly bodies ; and, of late, galvanism and
electricity together have been held out as explaining, not
only the aifinities of chemistry, but the phenomena of gravi-
tation, and the laws of vegetable and animal life. It were
a good caution for a man who studies nature, to distrust
those things with which he is particularly conversant, and
which he is accustomed to contemplate with pleasure.

3. The idols of the forum are those that arise out of the
commerce or intercourse of society, and especially from lan-
guage, or the means by which men communicate their
thoughts to one another.

Men believe that their thoughts govern their words ; but
it also happens, by a certain kind of reaction, that their
words frequently govern their thoughts. This is the more
pernicious, that words, being generally the work of the
multitude, divide things according to the lines most con-
picuous to vulgar apprehensions. Hence, when words are
examined, few instances are found in which, if at all ab-

szcr. 11,] DISSERTATION SECOND. 59

stract, they convey ideas tolerably precise and well defin-
ed. For such imperfections there seems to be no remedy,
but by having recourse to particular instances, and diligent-
ly comparing the meanings of words with the external arche-
types from which they are derived.

4. The idols of the theatre are the last, and are the de-
ceptions which have taken their rise from the systems or
dogmas of the different schools of philosophy. In the opi-
nion of Bacon, as many of these systems as had been invent-
ed, so many representations of imaginary worlds had been
brought upon the stage. Hence the name of idola theatri.
They do not enter the mind imperceptibly like the oth-
er three ; a man must labour to acquire them, and they are
often the result of great learning and study.

“ Philosophy,” said he, ‘as hitherto pursued, has taken
much from a few things, or a little from a great many ; and,
in both cases, has too narrow a basis to be of much duration
or utility.”” The Aristotelian philosopiy is of the latter
kind ; it has taken its principles from common experience,
but without due attention to the evidence or the precise
nature of the facts; the philosopher is left to work out the
rest from his own invention. Of this kind, called by Ba-
con the sophistical, were almost all the physical systems of
antiquity.

When philosophy takes all its principles from a few
facts, he calls it empirical,—such as was that of Gilbert,
and of the chemists.

It should be observed, that Bacon does not charge the
physicks of antiquity with being absolutely regardless of
experiment. No system, indeed, however fantastical, has
ever existed, to which that reproach could be applied in
its full extent; because, without some regard to fact, no
theory can ever become in the least degree plausible.
The fault lies not, therefore, in the absolute rejection of ex-

60 DISSERTATION SECOND. {parr 1.

perience, but in the unskilful use of it ; in taking up princi-
ples lightly from an inaccurate and careless observation of
many things; or, if the observations have been more accu-
rate, from those made on a few facts, unwarrantably gene- -
ralized.

Bacon proceeds to point out the circumstances, in the
history of the world, which had hitherto favoured these
perverse modes of philosophizing. He observes, that the
periods during which science had been cultivated were not
many, nor of long duration. They might be reduced to
three; the first with the Greeks; the second with the Ro-
mans; and the third with the western nations, after the re-
vival of letters. In none of all these periods had much at-
tention been paid to natural philosophy, the great parent of
the sciences.

With the Greeks, the time was very short during which
physical science flourished in any degree. The seven
Sages, with the exception of Thales, applied themselves
entirely to morals and politicks ; and in later times, after
Socrates had brought down philosophy from the heavens
to the earth, the study of nature was generally abandon-
ed. In the Roman republick, the knowledge most cul-
tivated, as might be expected among a martial and ambi-
tious people, was such as had a direct reference to war and
politicks. During the empire, the introduction and estab-
lishment of the Christian religion drew the attention of men
to theological studies, and the important interests which
were then at stake left but a smali share of talent and ability
to be occupied in inferiour pursuits. The corruptions
which followed, and the vast hierarchy which assumed the
command both of the sword and the sceptre, while it oc-
cupied and enslaved the minds of men, looked with suspi-
cion on sciences which could not easily be subjected to its
control.

sucr. 11.] DISSERTATION SECOND. 61

At the time, therefore, when Bacon wrote, it might truly
be said, that a small portion even of the learned ages, and
of the abilities of learned men, had been dedicated to the
study of Natural Philosophy. This served, in his opi-
nion, to account for the imperfect state in which he found
human knowledge in general; for he thought it certain,
that no part of knowledge could attain much excellence
without having its foundation laid in physical science.

He goes on to observe, that the end and object of know-
ledge had been very generally mistaken; that many, instead
of seeking through it to improve the condition of human
life, by new inventions and new resources, had aimed only
at popular applause, and had satisfied themselves with the
knowledge of words more than of things: while others, who
were exceptions to this rule, had gone still farther wrong,
by directing their pursuits to objects imaginary and unat-
tainable. The alchemists, for example, alternately the
dupes of their own credulity and of their own imposture,
had amazed and tormented the world with hopes which
were never (o be realized. Others, if possible more visionary,
had promised to prolong life, to extinguish disease and in-
firmity, and to give man a command over the world of spir-
its, by means of mystick incantations. <All this,’ says
he, ‘‘is the mere boasting of ignorance; for, when the
knowledge of nature shall be rightly pursued, it will lead to
discoveries that will as far excel the pretended powers of
magick, as the real exploits of Cesar and Alexander exceed
the fabulous adventures of Arihur of Britian or Amadis of
Gaul.” '

Again, the reverence for antiquity, and the authority of
great names, have contributed much to retard the progress
of science. Indeed, the notion of antiquity which men

"Nov. Org. Lib. i. Aph. $7.
g

62 DISSERTATION SECOND. [pan 1.

have taken up seems to be erroneous and inconsistent. It
is the duration of the world, or of the human race, as reck-
oned from the extremity that is past, and not from the point
of time which is present, that constitutes the true antiquity to
which the advancement of science may be conceived to bear
some proportion; and just as we expect more wisdom and
experience in an old than in a young man, we may expect
more knowledge of nature from the present than from any
of the ages that are past.

“If is not to be esteemed a small matter in this estimate,
that, by the voyages and travels of these later times, so
much more of nature has been discovered than was known
at any former period. It would, indeed, be disgraceful to
mankind, if, after such tracts of the material world have
been laid open, which were unknown in former times,—so
many seas traversed,—so many countries explored,—so
many stars discovered,—that philosophy, or the intelligible
world, should be circumscribed by the same boundaries as
before.”

Another cause has greatly obstructed the progress of
philosophy, viz. that men inquire only into the causes of
rare, extraordinary, and great phenomena, without troubling
themselves about the explanation of such as are common,
and make a part of the general course of nature.' It is,
however, certain, that no judgment can be formed concern-
ing the extraordinary and singular phenomena of nature,
without comparing them with those that are ordinary and
frequent.

The laws which are every day in action, are those which
it is most important for us to understand ; and this is well
illustrated by what has happened in the scientifick world
since the time when Bacon wrote. The simple falling

"Nov. Org. Lib. i. Aph. 119.

exer. 11.) DISSERTATION SECOND. 63

of a stone to the ground has been found to involve princi-
ples which are the basis of all we know in mechanical phi-
losophy. Without accurate experiments on the descent
of bodies at the surface of the earth,-the objections against
the earth’s motion could not have been answered, the iner-
tia of body would have remained unknown, and the nature
of the force which retains the planets in their orbits could
never have been investigated. Nothing, therefore, can be
more out of its place than the fastidiousness of those philo-
sophers, who suppose things to be unworthy of study, be -
cause, with respect to ordinary life, they are trivial and un-
important. It is an errour of the same sort which leads
men to consider experiment, and the actual application of
the hands, as unworthy of them, and unbecoming of the dig-
nity of science. ‘‘ There are some,” says Bacon, ‘ who,
delighting in mere contemplation, are offended with our fre-
quent reference to experiments and operations to be per-
formed by the hand, things which appear to them mean and
mechanical ; but these men do in fact oppose the attainment
of the object they profess to pursue, since the exercise of
contemplation, and the construction and invention of experi-
ments, are supported on the same principles, and perfected

by the same means.’’*

After these preliminary discussions, the great restorer
of philosophy proceeds, in the second book of the Novum
Organum, to describe and exemplify the nature of the in-
duction, which he deems essential to the right interpretation
of nature.

.The first object must be to prepare a history of the phe-
nomena to be explained, in all their modifications and varie-
ties. This history is to comprehend not only all such facts
as spontaneously offer themselves, but all the experiments

‘Impetus Phil. p. 681. Note C.

64 DISSERTATION SECOND. jrarr 2.

instituted for the sake of discovery, or for any of the pur-
poses of the useful arts. It ought to be composed with
great care; the facts accurately related, and distinctly ar-
ranged ; their authenticity diligently examined ; those that
rest on doubtful evidence, though not rejected, being noted
as uncertain, with the grounds of the judgment so formed.
This last is very necessary ; for facts often appear incredi-
ble, only because we are ill informed, and cease to appear
marvellous, when our knowledge is farther extended.

All such facts, however, as appear contrary to the ordi-
nary course of our experience, though thus noted down and
preserved, must have no weight allowed them in the first
steps of investigation, and are to be used only when the
general principle, as it emerges from the inductive pro-
cess, serves to increase their probability.

This record of facts is what Bacon calls natural history,
and it is material to take notice of the comprehensive sense
in which that term is understood through all bis writings.
According to the arrangement of the sciences, which he has
explained in his treatise on the advancement of knowledge,
all learning is classed relatively to the three intellectual fa-
culties of Memory, Reason, and Imagination. Under the
first of these divisions is contained all that is merely Narra-
tion or History, of whatever kind it may be. Under the
second are contained the different sciences, whether they
respect the Intellectual or the Material world. Under the
third are comprehended Poetry and the Fine Arts. It is
with the first of these classes only that we are at present .
concerned. The two first divisions of it are Sacred and
Civil History, the meaning of which is sufficiently under-
stood. The third division is Natural History, which com-
prehends the description of the facts relative to inanimate
matter, and to all animals, except man. Natural history is
again subdivided into three parts: 1. The history of the

sucr. 11.] DISSERTATION SECOND. 65

phenomena of nature, which are uniform; 2. Of the facts
which are anomalous or extraordinary ; 3. Of the processes
in the different arts.

We are not to wonder at finding the processes of the arts
thus enrolled among the materials of natural history. The
powers which act in the processes of nature and in those
of art are precisely the same, and are only directed, in the
latter case, by the intention of man, toward particular ob-
jects. In art,/as Bacon elsewhere observes, man does noth-
ing more than bring things nearer to one another, or carry
them farther off; the rest is performed by nature, and, on
most occasions, by means of which we are quite igno-
rant.

Thus, when a man fires a pistol, he does nothing but
make a piece of flint approach a plate of hardened steel, with
a certain velocity. It is nature that does the rest ;—that
makes the small red hot and fluid globules of steel, which
the flint had struck off, communicate their fire to the gun-
powder, and, by a process but little understood, set loose
the elastick fluid contained in it; so that an explosion is
produced, and the ball propelled with astonishing velocity.
It is obvious that, in this instance, art only gives certain
powers of nature a particular direction.

To the rules which have been given from Bacon, for the
composition of natural history, I may be permitted to add
this other,—that theoretical language should, as much as
possible, be avoided. Appearances ought to be described
in terms which involve no opinion with respect to their
causes: These last are the objects of separate examination,
and will be best understood if the facts are given fairly,
without any dependence on what should yet be considered
as unknown. This rule is very essential where the facts
are in a certain degree complicated ; for it is then much
easier to describe with a reference to theory than without it.

66 DISSERTATION SECOND. [parr 1

It is only from a skilful physician that you can expect a
description of a disease which is not full of opinions con-
cerning its cause. A similar observation might be made
with respect to agriculture ; and with respect to no science
more than geology.

The natural history of any phenomenon, or class of phe-
nomena, being thus prepared, the next object is, by a com-
parison of the different facts, to find out the cause of the
phenomenon, its form, in the language of Bacon, or its es-
sence. ‘The form of any quality in body is something
‘convertible with that quality ; that is, where it exists, the
quality is present, and where the quality is present, the
form must be so likewise. Thus, if transparency in bodies
be the thing inquired after, the form of it is something that,
wherever it is found, there is transparency ; and, vice ver-
sa, wherever there is. transparency, that which we have
called the form is likewise present.

The form, then, differs in nothing from the cause ; only
we apply the word cause were it is event or change that
is the effect. When the effect or result isa permanent
quality, we speak of the form or essence.

Two other objects, subordinate to forms, but often es-
sential to the knowledge of thenf, are also occasionally sub-
jects of investigation. These are the latent process, and
the latent schematism ; latens processus, et latens schema-
f&ésmus. ‘The former is the secret and invisible progress
by which sensible changes are brought about, and seems,
in Bacon’s accepfation, to involve the principle, since call-
ed the law of continuity, according to which, no change,
however small, can be effected but in time. 'To know the
relation between the time and the change effected in it,
would be to have a perfect knowledge of the latent process.
In the firing of a cannon, for example, the succession of
events during the short interval between the application of

szcr. 12,] DISSERTATION SECOND. 67

the match and the expulsion of the ball, constitutes a latent
process of a very remarkable and complicated nature,
which, however, we can now trace with some degree of
accuracy. In mechanical operations, we can often follow
this process still more completely. When motion is com-
municated from any body to another, it is distributed
through all the parts of that other, by a law quite beyond
the reach of sense to perceive directly, but yet subject to
investigation, and determined by a principle, which, though
late of being discovered, is now generally recognised.—
The applications of this mechanical principle are perhaps
the instances in which a latent, and, indeed, a very recon-
dite process, has been most completely analyzed.

The latent schematism is that invisible structure of bo-
dies, on which so many of their properties depend. When
we inquire into the constitution of crystals, or into the in-
ternal structure of plants, &c. we are examining into the
latent schematism. We do the same when we attempt to
explain elasticity, magnetism, gravitation, &c. by any pe-
culiar structure of bodies, or any arrangement of the par-
ticles of matter. '

In order to inquire into the form or cause of any thing
by induction, having brought together the facts, we are to
begin with considering what things are thereby excluded
from the number-of possible forms. This exclusion is the
first part of the process of induction: it confines the field
of hypothesis, and brings the true explanation within nar-
rower limits. Thus, if we were inquiring into the quality
which is the cause of transparency in bodies ; from the
fact that the diamond is transparent, we immediately ex-
clude rarity or porosity as well as fluidity from those caus-
es, the diamond being a very solid and dense body.

\ ? Nov. Org. Lib. ii. Aph. 5, 6, &e.

68 DISSERTATION SECOND. {Part 1.

Negative instances, or those where the given form is
wanting, are also to be collected.

That glass, when pounded, is not transparent, is a nega-
tive fact, and of considerable importance when the form of
transparency is inquired into; also, that collections of va-
pour, such as clouds and fogs, have not transparency, are
negative facts of the same kind. The facts thus collected,
both affirmative and negative, may, for the sake of refe-
rence, be reduced into tables.

Bacon exemplifies his method on the subject of Heat ;
and, though his collection of facts be imperfect, his method
of treating them is extremely judicious, and the whole
disquisition highly interesting." He here proposes, as an
experiment, to try the reflection of the heat of opaque
bodies.” He mentions also the vitrum calendare, or ther-
mometer, which was just then coming into use. His re-
flections, after finishing his enumeration of facts, show
how sensible he was of the imperfect state of his own
knowledge. * 7

After a great number of exclusions have left but a few
principles, common to every case, one of these is to be
assumed as the cause; and, by reasoning from it synthe-
tically, we are to try if it will account for the phenomena.

So necessary did this exclusive process appear to Ba-
con, that he says, “‘ It may perhaps be competent to angels,
or superiour intelligences, to determine the form or essence
directly, by affirmations from the first consideration of the
subject ; but it is certainly beyond the power of man, to
whom if is only given to proceed at first by negatives, and,
in the last place, to end in an affirmative, after the exclu-
sion of every thing else.’ 4

* Nov. Org. Lib. ii. Aph. 18, 20, &c. * Tbid. Aph. 11.
* Thid. Aph. 14. * Tbid. Apb. 15.

suer. 11.] DISSERTATION SECOND. 69

The method of induction, as laid down here, is to be
considered as applicable to all investigations where expe-
rience is the guide, whether in the moral or natural world.
“Some may doubt whether we propose to apply our me-
thod of investigation to natural philosophy only, or to
other sciences, such as logick, ethicks, politicks. We an-
swer, that we mean it to be so applied. And as the com-
mon logick, which proceeds by the syllogism, belongs
not only to natural philosophy, but to all the sciences, so
our logick, which proceeds by induction, embraces every
thing.”’!

Though this process had been pursued by a person of
much inferiour penetration and sagacity to Bacon, he could
not but have discovered that all facts, even supposing them
truly and accurately recorded, are not of equal value in the
discovery of truth. Some of them show the thing sought
for in its highest degree, some in its lowest; some exhibit
it simple and uncombined, in others it appears confused
with a variety of circumstances. Some facts are easily
interpreted, others are very obscure, and are understood
only in consequence of the light thrown on them by the
former. Thisled our author to consider what he calls Pre-
rogativae Instantiarum, the comparative value of facts as
means of discovery, or as instruments of investigation. He
enumerates twenty-seven different species, and enters at
some length into the peculiar properties of each. I must
gontent myself, in this sketch, with describing a few of the
most important, subjoining, as illustrations, sometimes the
examples which the author himself has given, but more
frequently such as have been furnished by later discove-
ries in science.

' Noy. Org. Lib. i. Aph. 127.
9

70 DISSERTATION SECOND. [rant 1,

I. The first place in this classification is assigned to
what are called instantiae solitariae, which are either ex-
amples of the same quality existing in two bodies, which
have nothing in common but that quality, or of a quality
differing in two bodies, which are in all other respects the
same. In the first instance, the bodies differ in all things
but one; in the second, they agree in all but one. The
hypotheses that in either case can be entertained, concern-
ing the cause or form of the said quality, are reduced toa
small number ; for, in the first, they can involve none of
the things in which the bodies differ; and, in the second,
none of those in which they agree.

Thus, of the cause or form of colour now inquired into,
instantiae solitariae are found in crystals, prisms of glass,
drops of dew, which occasionally exhibit colour, and yet
have nothing in common with the stones, flowers, and me-
tals, which possess colour permanently, except the colour
itself. Hence Bacon concludes, that colour is nothing
else than a modification of the rays of light, produced, in
the first case, by the different degrees of incidence ; and, in
the second, by the texture or constitution of the surfaces
of bodies. He may be considered as very fortunate in
fixing on these examples, for it was by means of them that
Newton afterwards found out the composition of light.

Of the second kind of instantiae solitariae, Bacon men-
tions the white or coloured veins which occur in limestone
or marble, and yet hardly differ in substance or in struc-
ture from the ground of the stone. He concludes, very
justly, from this, that colour has not much to do with the
essential properties of body. .

If. The instantiae migrantes exhibit some nature or
property of body, passing from one condition to another,
either from less to greater, or from greater to less; arriving

sucr. 11.] DISSERTATION SECOND. 71

nearer perfection in the first case, or verging towards ex-
tinction in the second.

Suppose the thing inquired into were the cause ef white -
ness in bodies; an instantia migrans is found in glass,
which, when entire, is without colour, but, when pulveriz-
ed, becomes white. The same is the case with water
unbroken, and water dasbed into foam. In both cases,
the separation into particles produces whiteness. So also
the communication of fluidity to metals by the application
of heat; and the destruction of that fluidity by the ab-’
straction of heat, are examples of both kinds of the instan-
tia migrans. Instances of this kind are very powerful
for reducing the cause inquired after into a narrow space,
and ‘for removing all the accidental circumstances. It is_
necessary, however, as Bacon’ very justly remarks, that
we should consider not merely the case when a certain
quality is lost, and another produced, but the gradual
changes made in those qualities during their migration, viz.
the increase of the one, and the corresponding diminution
of the other. The quantity which changes proportionally
to another, is connected with it either as cause and effect,
or as a Collateral effect of the same cause. When, again,
we find two qualities which do not increase proportionally,
they afford a negative instance, and assure us that the twe
are not connected simply as cause and effect.

The mineral kingdom is the great theatre of the instan-
tiae migrentes, where the same nature is seen in all gra-
dations, from the most perfect state, till it become entire-
ly evanescent. Such are the shells which we see so perfect
in figure and structure in limestone, and gradually losing
themselves in the finer marbles, till they can no longer be
distinguished. The use, also, of one such fact to explain

* Nov. Org. Lib. ii. Aph. 23.

72 DISSERTATION SECOND. [pant 1,

or interpret another, is nowhere so well seen as in the his-
tory of the mineral kingdom. |

III. In the third place are the instantiae ostensivae,
which Bacon also calls elucescentiae and predominantes.
They are the facts which show some particular nature in
its highest state of power and energy, when it is either
freed from the impediments which usually counteract it,
or is itself of such force as entirely to repress those im-
pediments. For as every body is susceptible of many
different conditions, and has many different forms combin-
ed in it, one of them often confines, depresses, and hides
another entirely, so that it is not easily distinguished.
There are found, however, some subjects in which the
nature inquired into is completely displayed, either by
the absence of impediments, or by the predominance of its
own power.

Bacon instances the thermometer, or vitrum calendare,
as exhibiting the expansive power of heat, in a manner
more distinct and measurable than in common cases. To
this example, which is well chosen, the present state of
science enables us to add many others.

If the weight of the air were inquired into, the Torri-
cellian experiment or the barometer affords an ostensive
instance, where the circumstance which conceals the
weight of the atmosphere in common cases, namely, the
pressure of it in all directions, being entirely removed,
that weight produces its full effect, and sustains the whole
column of mercury in the tube. The barometer: affords
also an example of the instantia migrans, when the change
is not total, but only partial, or progressive. If it be the
weight of the air which supports the mercury in the tube
of the barometer, when that weight is diminished, the
mercury ought to stand lower. On going to the top of
a mountain, the weight of the incumbent air is diminish-

sxcr. 11] DISSERTATION SECOND. 73

ed, because a shorter column of air is to be sustained ;
the mercury in the barometer ought therefore to sink,
and it is found to do so accordingly.

These are instances in which the action of certain
principles is rendered visible by the removal of all the
opposing forces. One may be given where it is the dis-
tinct and decisive nature of the fact which leads to the
result.

Suppose it were inquired, whether the present land
had ever been covered by the sea. If we look at the
stratified form of so large a portion of the earth’s surface,
we cannot but conclude it to be very probable that such
land was formed at the bottom of the sea. But the deci-
sive proof is afforded by the shells and corals, or bodies
having the perfect shape of shells and corals, and of other
marine exuviae, which are found imbedded in masses of
the most solid rock, and often on the tops of very high
mountains. This leaves no doubt of the formation of the
land under the sea, though it does not determine whe-
ther the land, since its formation, has been elevated to
its present height, or the sea depressed to its present
level. The decision of that question requires other facts
to be consulted.

IV. The instantia clandestina, which is, as it were,
opposed to the preceding, and shows some power or
quality just as it is beginning to exist, and in its weakest
state, ts often very useful in the generalization of facts.
Bacon also gave to this the fanciful name of inslantia
crepusculi.

An example of this may be given from hydrostaticks.
If the suspension of water in capillary tubes be inquired
into, it becomes very useful to view that effect when it is
least, or when the tube ceases to be capillary, and be-
comes a vessel of a large diameter. The column is then

74 DISSERTATION SECOND. {parr i.

reduced te a slender ring of water which goes all round
the vessel, and this, though now so inconsiderable, has.
the property of being independent of the size of the vessel,
so as to be in all cases the same when the materials are
the same. As there can be no doubt that this ring pro-
ceeds from the attraction of the sides, and of the part im-
mediately above the water, so there can be no doubt that
the capillary suspension, in part at least, is derived from
the same cause. An effect of the opposite kind takes place
when a glass vessel is filled with mercury.

V. Next to these may be placed what are called instan-
tiae manipulares, or collective instances, that is, general
facts, or such as comprehend a great number of particular
cases. As human knowledge can but seldom reach the
mos‘ general cause or form, such collective instances are
often the utmost extent to which our generalization can
be carried. They have great value on this account, as
they likewise have on account of the assistance which
they give to farther generalization.

Of this we have a remarkable instance in one of the most
important steps ever taken in any part of human know-
ledge. The laws of Kepler are facts of the kind now
treated of, and consist of three general truths, each belong-
ing to the whole planetary system, and it was by means
of them that Newton discovered the principle of gravita-
tion. The first is, that the planets all move in elliptical
orbits, having the sun for their common focus; the next,
that about this focus the radius vector of each planet de-
scribes equal areas in equal times. The third and last, that
the squares of the periodick times of the planets are as the
cubes of their mean distances from the sun. The know-
ledge of each of these was the result of much research, and
of the comparison of a vast multitude of observations, in-
somuch that it may be doubted if ever three truths in

a a i lt i ee

excr. 11.] DISSERTATION SECOND. 75

science were discovered at the expense of so much labour
and patience, or with the exertion of more ingenuity and
‘invention in imagining and combining observations. These
discoveries were all made before Bacon wrote, but he is
silent concerning them ; for the want of mathematical know-
ledge concealed from his view some of the most splendid
and interesting parts of science.

Astronomy is full of such collective instances, and af-
fords them, indeed, of the second and third order, that is
to say, two or three times generalized. ‘The astronomer
observes nothing but that a certain luminous disk, or per-
haps merely a luminous point, is in a certain position, in
respect of the planes of the meridian and the horizon, at a
certain moment of time. By comparing a number of such
observations, he finds that this luminous point moves ina
certain plane, with a certain velocity, and performs a revo-
lution in a certain time. ‘Thus, the periodick time of a
planet is itself a collective. fact, ora single fact expressing
the result of many hundred observations. This holds
with respect to each planet, and with respect to each ele-
ment, as it is called, of the planet’s orbit, every one of
which is a general fact, expressing the result of an indefi-
nite number of particulars. This holds stil! more remark-
ably of the inferences which extend to the distances of the
planet from the earth, or from the sun. The laws of Kep-
ler are therefore collective facts of the second, or evena
higher order; or such as comprehend a great number of
general facts, each of which is itself a general fact, includ-
ing many particulars. It is much to the credit of astrono-
my, that, in all this process, no degree of truth or certain-
ty is sacrificed ; and that the same demonstrative evidence
is preserved from the lowest to the highest point. No-
thing but the use of mathematical reasoning could secure
this advantage to any of the sciences.

76 DISSERTATION SECOND. [part 1

VI. In the next place may be ranked the instances
which Bacon calls analogous, or parallel. These consist
of facts, between which an analogy or resemblance is visi-
ble in some particulars, notwithstanding great diversity in all
the rest. Such are the telescope and microscope, in the
works of art, compared with the eye in the works of na-
ture. This, indeed, is an analogy which goes much be-
yond the mere exterior; it extends to the internal struc-
ture, and to the principle of action, which is the same in
the eye and in the telescope,—to the latent schemutism,
in the language of Bacon, as far as material substance is
concerned, It was the experiment of the camera obscura
which led to the discovery of the formation of the images
of external objects in the bottom of the eye by the action
of the crystalline lens, and the other humours of which the
eye is formed. ;

Among the instances of conformity, those are the most
useful which enable us to compare productions of an un-
known formation, with similar productions of which the
formation is well understood. Such are basalt, and the
other trap rocks, compared with the lava thrown out from
volcanoes. They have a structure so exactly similar, that
it is hardly possible to doubt that their origin is the same,
and that they are both produced by the action of subter-
raneous fire. There are, however, amid their similarity, |
somé very remarkable differences in the substances which
they contain, the trap rocks containing calcareous spar,
and the lava never containing any. On the supposition
that they are both of igneous origin, is there any circum-
stance, in the conditions in which heat may have been ap-
plied to them, which can account for this difference? Sir
James Hall, in a train of most philosophical and happily
contrived experiments, has explained the nature of those
conditions, and has shown that the presence of calcareous

aEcr. 11.} DISSERTATION SECOND. V7

spar, or the want of it, may arise from the greater or less
compression under which the fusion of the basalt was per-
formed. This has served to explain a great difficulty in
the history of the mineral kingdom.

Comparative anatomy is full of analogies of this kind,
which are most instructive, and useful guides to discovery.
It was by remarking in the blood-vessels a contrivance si-
milar to the valves used in hydraulick engines, for prevent-
ing the counter current ofa fluid, that Harvey was led to
the discovery of the circulation of the blood. The analo-
gies between natural and artificial productions are always
highly deserving of notice.

The facts of this class, however, unless the analogy be
very close, are apt to mislead, by representing accidental
regularity as if it were constant. Of this we have an ex-
ample in the supposed analogy between the colours in the
prismatick spectrum and the divisions of the monochord.
The colours in the prismatick spectrum do not occupy the
same proportion of it in all cases: the analogy depends on
the particular kind of glass, not on any thing that is com-
mon toall refraction. The tendency of man to find more
order in things than there actually exists, is here to be
cautiously watched over. ;

VII. Monodick, or singular facts, are the next in order.
They comprehend the instances which are particularly
distinguished from all those of the genius or species to
which they belong. Such is the sun among the stars, the
magnet among stones, mercury among metals, boiling foun-
tains among springs, the elephant among quadrupeds. So
also among the planets, saturn is singular from his ring, the
new ‘planets are so likewise from their small size, from be-
ing extrazodiacal, &c.

Connected with these are the irregular and deviating in-
stances, in which nature seems to depart from her ordinary
10

78 DISSERTATION SECOND. [pant 3

course. Earthquakes, extraordinary tempests, years of
great scarcity, winters of singular severity, &c. are of this
number. All such facts ought to be carefully colleeted ;
and there should be added an account of all monstrous
productions, and of every thing remarkable for its novelty
and its rareness. Here, however, the most severe criti-
cism must be applied; every thing connected with su-
perstifion is suspicious, as well as whatever relates to al-
chemy or magick.

A set of facts, which belongs to this class, consists of
the instances in which stones have so often of late years
been observed to fall from the heavens. Those stones are
so unlike other atmospherical productions, and their origin
must be so different from that of other minerals, that it is
scarcely possible to imagine any thing more anomalous,
and wore inconsistent with the ordinary course of our ex-
perience. Yet the existence of this phenomenon is so
well authenticated by testimony, and by the evidence
‘arising from certain physical considerations, that no doubt
with respect to it can be entertained, and it must therefore
be received, as making a part of the natural history of me-
teors. But as every fact, or class of facts, which is per-
fectly singular, must be incapable of explanation, and can
only be understood when its resemblance to other things
has been discovered, so at present we are unable to assign
the cause of these phenomena, and have no right to offer
any theory of their origin.

VIII. Another class of facts is composed of what Ba-
con calls instantiae comitatus, or examples of certain
’ qualities which always accompany one another. Such are
flame and heat,—flame being always accompanied by heat,
and the same degree of heat in a given substance being al-
ways accompanied with flame. So also heat and expan-
sion,—an increase of heat being accompanied with an in-

emer. 11.] DISSERTATION SECOND. 79

crease of volume, except in a very few cases, and in Cir-
‘cumstances very particular.

The most perfect instantia comitatus known, as being
without any negative, is that of bedy and weight. What-
ever is impenetrable and inert, is also heavy ina degree
proportional to its inertia. To this there is no exception,
though we do not perceive the connexion as necessary.

Hostile instances, or those of perpetual separation, are
the reverse of the former.

Examples of this are found in air, and the other elastick
fluids, which cannot have.a solid form induced on them by
any known means, when not combined with other substan-
ces. So also in solids, transparency and malleability are
never joined, and appear to be incompatible, though it is
not obvious for what reason.

IX. Passing over several classes which seem of infe-
riour importance, we come to the instantia crucis, the
division of this experimental logick which is most frequent-
ly resorted to in the practice of inductive investigation.
When, in such an investigation, the understanding is plac-
ed in equilibrio, as it were, between two or more causes,
each of which accounts equally well for the appearances,
as far as they are known, nothing remains to be done, but
to look out for a fact which can be explained by the one
of these causes, and not by the other; if such a one can be
found, the uncertainty is removed, and the true cause is
determined. Such facts perform the office of a cross,
erected at the separation of two roads, to direct the tra-
veller which he is to take, and, on this account, Bacon
gave them the name of inslantiae crucis.

Suppose that the subject inquired into were the motion
of the planets, and that the phenomena which first present
themselves, or the motion of these bodies in longitude,
could be explained equally on the Ptolemaick and the Co»

80 DISSERTATION SECOND. [rant a.

pernican system, that is, either on the system which makes
the Earth, or that which makes the Sun, the centre of the
planetary motions, a cautious philosopher would hesitate
about which of the two he should adopt, and notwithstand-
ing that one of them was recommended by its superiour
simplicity, he might not think himself authorized to give
to it a decided preference above the other. If, however,
he consider the motion of these bodies in latitude, that 1s to
say, their digressions from the plane of the ecliptick, he
will find a set of phenomena which cannot be reconciled
with the supposition that the earth is the centre of the pla-
netary motions, but which receive the most simple and
satisfactory explanation from supposing that the sun is at
rest, and is the centre of those motions. The laiter phe-
nomena would therefore serve as instantiae crucis, by
which the superiour credibility of the Copernican system
was fully evinced.

Another example which I shall give of an instantia cru-
cis is taken from chemistry, and is, indeed, one of the most
remarkable experiments which has been made in that
science.

It is a general fact observed in chemistry, that metals
are always rendered heavier by calcination. Whena mass
of tin or lead, for instance, is calcined in the fire, though
every precaution is taken to prevent any addition from the
adhesion of ashes, coals, &c. the absolute weight of the
mass is always found to be increased. It was long before
the cause of this phenomenon was understood. There
might be some heavy substance added, though what it was
could not easily be imagined; or some substance might
have escaped, which was in its nature light, and possessed
a tendency upwards. Other phenomena, into the nature of
which it is at present unnecessary to inquire, induced
chemists to suppose, that in calcination a certain substance

ancr. 11.] DISSERTATION SECOND. 81

actually escapes, being present in the regulus, but not in
the calx of the metal.. This substance, to which they gave
the name of phlogiston, was probably that which, by its
escape, rendered the metal heavier, and must, therefore,
be itself endued with absolute levity.

The instantia crucis which extricated philosophers from
this difficulty, was furnished by an experiment of the cele-
brated Lavoisier. That excellent chemist included a
quantity of tin ina glass retort, hermetically sealed, and ac-
curately weighed together. wiih its contents; he then ap-
plied the necessary heat, and when the calcination of the
tin was finished, he found the weight of the whole precise-
ly the same as before. This proved, that no substance,
which was either light or heavy, ina sensible degree, had
made its way through the glass. The experiment went
still farther. When the retort was cooled and opened, the
air rushed in, so that it was evident that a part of the air
haddisappeared, or had lost its elasticity. On weighing
the whole apparatus, it was now found that its weight was
increased by ten grains ; so that ten grains of air had enter-
ed into the retort when it was opened. The calx was
next taken out, and weighed separately, and it was found
to have become heavier by ten grains precisely. The ten
grains of air then which had disappeared, and which had
made way for the ten grains that rushed into the retort,
had combined with the metal during the process of cal-
cination. The farther prosecution of this very decisive
experiment led to the knowledge of that species of air
which combines with metals when they are calcined.
The doctrine of phlogiston was of course exploded and
a creature of the imagination replaced by areal exist-

ence.
The principle which conducts to the contrivance of an

expertmentum crucis is not difficult to be understood.

“+>

82 DISSERTATION SECOND. (Pann t.

Taking either of the hypotheses, its consequences must

be attempted to be traced, supposing a different experi-

ment to be made. This must be done with respect to
the other hypothesis, and a case will probably at last oc-
cur, where the two hypotheses would give different results.
The experiment made in those circumstances will furnish
an instantia crucis. ,

Thus, if the experiment of calcination be performed in
a close vessel, and if phlogiston be the cause of the in-
crease of weight, it must either escape through the ves-
sel, or it mast remain in the vessel after separation from
the calx. Ifthe former be the case, the apparatus will be
increased in weight ; if the latter, the phlogiston must make
its escape .on opening the vessel. If neither of these be
the case, it is plain that the theory of phlogiston is insuf-
ficient to explain the facts.

The experimentum crucis is of such weight in matters of
induction, that in all those branches of science where it
cannot easily be resorted to (the circumstances of an ex-
periment being out of our power, and incapable of being
varied at pleasure,) there is often a great want of conclu-
sive evidence. ‘This holds of agriculture, medicine, poli-
tical economy, &c. To make one experiment, similar to
another in all respects but one, is what the experimentum
cructs, and, in general, the process of induction, principally
requires; but it is what, in the sciences just named, can
seldom be accomplished. Hence the great difficulty of
separating the causes, and allotting to each its due propor-
tion of the effect. Men deceive themselves in consequence
of this continually, and think they are reasoning from fact
and experience, when, in reality, they are only reasoning
from a mixture of truth and falsehood. ‘The only end an-
swered by facts so incorrectly apprehended, is that of mak-
ing errour more incorrigible.

szcr. 11.] DISSERTATION SECOND. 83

Of the twenty-seven classes into which instantiae are
arranged by the author of the Novum Organum, fifteen
immediately address themselves to the Understanding ; five
serve to correct or to inform the Senses; and seven to
direct the hand in raising the superstructure of Art on the
foundation of Science. The examples given above are
from the first of these divisions, and wil! suffice for a sum-
mary. ‘To the five that follow next, the general name of
instantiae lampadis is given, from their office of assisting
or informing the senses.

Of these the instantiae januae assist the immediate ac-
tion of the senses, and especially of sight. The examples
quoted by Bacon are the microscope and telescope (which
last he mentions as the invention of Galileo,) and he speaks
of them with great admiration, but with some doubt of
their reality. '

The instantiae citantes enable us to perceive things
which are in themselves insensible, or not at all the objects
of perception. They cite or place things, as it were, be-
fore the bar of the senses, and from this analogy to judicial
proceedings is derived the name of instantiae citantes.
Such, to employ examples which the progress of science
has unfolded since the time of Bacon, are the airpump and
the barometer for manifesting the weight and elasticity of
air; the measurement of the velocity of light, by means of
the eclipses of the satellites of jupiter, and the aberration
of the fixed stars ; the experiments in electricity and gal-
vanism, and in the greater part of pneumatick chemistry. ,
In all these instances things are made known, which before
had entirely escaped the senses.

The instantiae viae are facts which manifest the con-
tinuous progress of nature in her operations. here is a
propensity in men to view nature as it were at intervals, or
at the ends of fixed periods, without regarding her gradual

84 DISSERTATION SECOND. [part t.

and unceasing action.' The desire of making observation
easy is the great source of this propensity. Men wish for
knowledge, but would obtain it at the least expense of
time and labour. As there is no time, however, at which
the hand of nature ceases to work, there ought to be none
at which observation ceases to be made.

The instantiae persecantes, or vellicantes, are those
which force us to attend to things which, frem their sub-
tilty and minuteness, escape common observation.

Some of-Bacon’s remarks on this subtilty are such as
. would do credit to the most advanced state of science, and
show how much his mind was fitted for distinguishing and
observing the great and admirable in the works of nature.

The last division contains seven classes, of which I men-
tion only two. The experiments of this division are those
most immediately tending to produce the improvement of
art from the extension of science. ‘ Now there are,’’ says
Bacon, “ two ways in which knowledge, even when sound
in itself, may fail of becoming a safe guide to the artist, and
these are either when it is not sufficiently precise, or when
it leads to more complicated means of producing an effect
than can be employed in practice. There are therefore
two kinds of experiments which are of great value in pro-
moting the alliance between knowledge and power ;—those
which tend to give accurate and exact measures of objects,
and those which disencumber the processes deduced from
scientifick principles of all unnecessary operations.”

In the inslantiae radii we measure objects by lines and
angles ; in the instantiae curriculi by time or by motion.

To the former of these classes are to be referred a num-
ber of instruments which now constitute the greater part of
the apparatus of natural philosophy. Though Bacon had

* Nov. Org. I]. Aph. 41:

J
!

akCP. 11.] DISSERTATION SECOND. 85

a just idea of their utility in general, he was unacquainted
with most of them. The most remarkable at present are
those that follow :

1. Astronomical instruments, or, more generally, all in-
struments for measuring lines and angles.

2. Instruments for measuring weight or force; such are
the common balance, the hydrostatick balance, the baro-
meter, the instruments used in England by Cavendish, and
in France by Coulomb, which measure smal! and almost in-
sensible actions by the force of torsion.

These last rather belong to the class of the instantiae
luctae, where force is applied as the measure of force, than
to the instantiae radii. a

3. The thermometer, newly invented in the time of
Bacon, and mentioned by him under the name of Vitrum
Calendare, an instrument to which we owe nearly all the
knowledge we have of one of the most powerful agents in
nature, viz. Heat. ¥

4. The hygrometer, an instrument for measuring the
quantity of humidity contained in the air; and in the con-
struction of which, after repeated failures by the most
skilful experimenters, the invention of Professor Leslie now
promises success. Almost every one of these instruments,
to which several more might be added, has brought in sight
a new country, and has enriched science not only with new
facts, but with new principles. .

Among the remarks of Bacon on the experimenta radii,
some are very remarkable for the extent of view which
they display even in the infancy of physical science. He
mentions the forces by which bodies act on one another at
a distance, and throws out some hints at the attraction
which the heavenly bodies exert on one another.

*‘ Inquirendum est,” says he, “si sit vis aliqua magneti-
ca quae operetur per consensum inter globum terrae et

11

86 DISSERTATION SECOND. [part &

ponderosa, aut inter globum lunae et aquas maris, aut inter
coelum stellatum, et planetas per quam avocentur et atol-
lantur ad sua apogaea; haee omnia operantur ad distantias
admodum longinquas.”’ *

Under the head of the instantia curriculi, or the mea-
suring of things by time; after remarking that every change
and every motion requires time, and illustrating this by a
variety of instances, he has the following very curious an-
ticipation of facts, which appeared then doubtful, but which
subsequent discovery has ascertained :

“The consideration of those things produced in mea
doubt altogether astonishing, viz. Whether the face of
the serene and starry heavens be seen at the instant it really
exists, or not till some time later; and whether there be
not, with respect to the heavenly bodies, a true time and
an apparent time, no less than a true place, and an appa-
rent place, as astronomers say, on account of parallax. For
it seems incredible that the species or rays of the celestial
bodies can pass through the immense interval between them
and us in an instant, or that they do not even require some
considerable portion of time.” ?

The measurement of the velocity of light, and the won-
derful consequences arising from it, are the best commen-
taries on this passage, and the highest eulogy on its au-
thor.

Such were the speculations of Bacon, and the rules he
laid down for the conduct of experimental inquiries, before
any such inquiries had yet been instituted. The power
and compass of a mind which could form such a plan be-
forehand, and trace not merely the outline, but many of the
most minute ramifications of sciences which did not yet

1 Nov. Org. Il. Apb. 45. ? Ibid. Aph. 46.

szor. 11] DISSERTATION SECOND. 87

exist, must be an object of admiration to all succeeding

ages. He is destined, if, indeed, any thing in the world
be so destined, to remain an instantia singularis among
men, and as he has had no rival in the times which are
past, so is he likely to have none in those which are to
come. Before any parallel to him can be found, not only
must a man of the same talents be produced, but he must
be placed’in the same circumstances ; the memory of his
predecessor must be effaced, and the light of science, after
being entirely extinguished, must be again beginning to re-
vive. If asecond Bacon is ever to arise, he must be ig-
norant of the first.

Bacon is eften compared with two great men who lived
nearly about the same time with himself, and who were
both eminent reformers of philosophy, Descartes and Ga-
lileo.

Descartes flourished about forty years later than Bacon,
but does not seem to have been acquainted with his writ-
ings. Like him, however, he was forcibly struck with
the defects of the ancient philosophy, and the total inapti-
tude of the methods which it followed, for all the purposes
of physical investigation. Like him, too, he felt himself
strongly impelled to undertake the reformation of this er-
roneous system; but the resemblance between them goes
no farther; for it is impossible that two men could pur-
sue the same end by methods more diametrically oppo-
site.

Descartes never proposed to himself any thing which
had the least resemblance to induction. He began with
establishing principles, and from the existence of the Deity
and his perfections, he proposed to deduce the explanation
of all the phenomena of the world, by reasoning @ prior.
Instead of proceeding upward from the effect to the cause,
he proceeded continually downward from the cause to the

a”
‘+ >

88 DISSERTATION SECOND. [parr 3.

effect. It was in this maaner that he sought to determine
the laws of motion, and of the collision of bodies, in which
last all his conclusions were erroneous. From the same
source he deduced the existence of a plenum, and the con-
tinual preservation of the same quantity of motion in the
universe ; a proposition which, in a certain sense, is true,
but in the sense in which he understood it, is altogether
false. Reasonings of the kind which he employed may
possibly suit, as Bacon observed, with intelligences of a
higher order than man, but to his case they are quite inap-
plicable. Of the fruit of this tree nature has forbidden him
to eat, and has ordained, that, with the sweat of his brow,
and the labour of his hands, he should earn his knowledge
as well as his subsistence.

Descartes, however, did not reject experiment altogether,
though he assigned it a very subordinate place in his phi-
losophy. By reasoning down from first principles, he tells
us that he was always able to discover the effects; but the
number of different shapes which those effects might as-
sume was so great, that he could not determine, without
having recourse to experiment, which of them nature had
preferred to the rest. ‘* We employ experiment,” says
he, “not as a reason by which any thing is proved, for
we wish to deduce effects from their causes, and not con-
versely causes from their effects. We appeal to experi-
ence only, that out of innumerable effects which may be
produced from the same cause, we may direct our atten-
tion to one rather than another.’’ It is wonderful, that
Descartes did not see what a severe censure he was here
passing on himself; of how little value the speculations
must be that led to conclusions so vague and indefinite ;
and how much more philosophy is disgraced by affording
an explanation of things which are not, than by not afford-
ing an explanation of things which are,

sect, H1.] DISSERTATION SECOND. 89

Asa system of philosophy and philosophick investiga-
tion, the method of Descartes can, therefore, stand in no
comparison with that of Bacon. Yet his physicks contri-
buted to the advancement of science, but did so, much
more by that which they demolished, than by that which
they built up. In some particular branches the French
philosopher far excelled the English. He greatly improv-
ed the science of opticks, and in the pure matheuaticks, as
has been already shown, he left behind him many marks of
a great and original genius. He will, therefore, be always
numbered among those who have essentially contributed to
the advancement of knowledge, though nothing could be
more perverse than his method of philosophizing, and no-
thing more likely to impede the progress of science, had
not an impulse been at that time given to the human mind
which nothing could resist.

Galileo, the other rival and contemporary of Bacon, is,
in truth, one of those to whom human knowledge is under
the greatest obligations. His discoveries in the theory of
motion, in the laws of the descent of heavy bodies, and in
the motion of projectiles, laid the foundation of all the great
improvements which have since been made by the appli-
cation of mathematicks to natural philosophy. If to this
we add the invention of the telescope, the discoveries made
by that instrument, the confirmations of the Copernican
system which these discoveries afforded, and, lastly, the
wit and argument with which he combated and exposed
the prejudice and presumption of the schools, we must
admit that the history of human knowledge contains few
greater names than that of Galileo. On comparing him
with Bacon, however, I have no hesitation in saying, that
the latter has given indications of a genius of a still higher
order. In this I know that I differ from a historian, who

90 DISSERTATION SECOND. [parr 1.

was himself a philosopher, and whose suffrage, of conse-
quence, is here of more than ordinary weight. _

“The great glory of literature,” says Hume, “in this
island, during the reign of James, was Lord Bacon. If we
consider the variety of talents displayed by this man, as
a publick speaker, a man of business, a wit, a courtier, a
companion, an author, a philosopher, he is justly entitled
to great admiration. If we consider him merely as an
, author and a philosopher, the light in which we view him
at present, though very estimable, he was yet inferiour to
his contemporary Galileo, perhaps even to Kepler. Ba-
con pointed out, at a distance, the road to philosophy ;

Galileo both pointed it out to others, and made himself —

considerable advances init. The Englishman was igno-
rant of geometry ; the Florentine revived that science, ex-
celled in it, and was the first who applied it, together with
experiment, to natural philosophy. The former reject-
ed, with the most positive disdain, the system of Coperni-
cus; the latter fortified it with new proofs, derived both
from reason and the senses. Bacon’s style is stiff and
rigid; his wit, though often brilliant, is also often unnatu-
ral and far-fetched. Galileo is a lively and agreeable,
though somewhat a prolix writer.”’

Though it cannot be denied that there is considerable
truth in these remarks, yet it seems to me that the com-
parison is not made with the justness and discrimination
which might have been expected from Hume, who appears
studiously to have contrasted what is most excellent in
Galileo, with what is most defective in Bacon. It is true
that Galileo showed the way in the application of mathema-
ticks and of geometry to physical investigation, and that
the immediate utility of his performance was greater than

* Hist. of England, vol. VI. Appendix.

sEcr. 11.] DISSERTATION SECOND. 9}

that of Bacon’s; as it impressed more movement on the
age in which he lived, example being always so much more
powerful than precept. Bacon, indeed, wrote for an age
more enlightened than his own, and it was long before the
full merit of his work was understood. But though Galileo
was a geometer, and Bacon unacquainted with the mathe-
maticks,—though Galileo added new proofs to the system
of the earth’s motion, which Bacon rejected altogether,—
yet is it certain, [ think, that the former has more fellows
or equals in the world of science than the latter, and that
his excellence, though so high, is less unrivalled. The
range which Bacon’s speculations embraced was altogether
immense. He cast a penetrating eye on the whole of
science, from its feeblest and most infantine state, to that
strength and perfection from which it was then so remote,
and which it is perhaps destined to approach to continually,
but never to attain. More substitutes might be found for
Galileo than for Bacon. More than one could be mention-
ed, who, inthe place of the former, would probably have
done what he did; but the history of human knowledge
points out nobody of whom it can be said, that, placed in
the situation of Bacon, he would have done what Bacon
did ;—no man whose prophetick genius would have enabled
him to delineate a system of science which had not yet
begun to exist!—who could have derived the knowledge of
what owght to be from what was not, and who could have
become so rich in wisdom, though he received from his
predecessors no inheritance but their errours. Iam in-
clined, therefore, to agree with D’Alembert, “that when
one considers the sound and enlarged views of this great
man, the multitude of the objects to which his mind was
turned, and the boldness of his style, which unites the most
sublime images with the most rigorous precision, one is

92 DISSERTATION SECOND. [rant 1.

disposed to regard him as the greatest, the most universal,
and the most eloquent of philosophers.’”'

ey Remarks, &c.

It will hardly be doubted by any one who attentively
considers the method explained in the Novum Organum,
which we have now attempted to sketch, that it contains
a most comprehensive and rigorous plan of inductive inves-
tigation. A questich, however, may occur, how far has
this method been really carried into practice by those who
have made the great discoveries in natural philosophy, and
who have raised physical science to its present height in
the scale of human knowledge? Is the whole method ne-
cessary, or have not circumstances occurred, which have
rendered experimental investigation easier in practice than
it appears to be in theory? ‘To answer these questions
completely, would require more discussion than is consis-
tent with the limits of this Dissertation; I shall, therefore,
attempt no more than to point out the principles on which
such an answer may be founded.

In°a very extensive department of physical science, it
cannot be doubted that investigation has been carried on,
not perhaps more easily, but with a less frequent appeal to
experience, than the rules of the Novum Organum would
‘seem to require. In all the physical inquiries where ma-
thematical reasoning has been employed, after a few princi-
ples have been established by experience, a vast multitude
of truths, equally certain with the principles themselves,
have been deduced from them by the were application of
geometry and algebra.

In mechanicks, for example, after the laws of motion were
discovered, which was done by experiment, the rest of the

* Discours Préliminaire de PEncyclopédie.

‘

ext. 11.) DISSERTATION SECOND. 93

science, fo a great extent, was carried on by reasoning from
those laws, in the same manner that the geometer makes his
discoveries by reasoning on the definitions, by help of a
few axioms, or self-evident propositions. The only differ-
ence is, that, in the one case, the definitions and axioms are
supplied solely from the mind itself, while, in the other, all
the definitions and axioms, which are not those of pure
' geometry, are furnished by experience.!

Bacon certainly was not fully aware of the advantages
that were thus to accrue to the physical sciences. He was
not ignorant, that the introduction of mathematical reason-
ing into those sciences is not only possible, but that, under
certain conditions, it may be attended with the greatest ad-
vantage. He knew also in what manner this application had
been abused by the Platonists, who had attempted, by
means of geometry, to establish the first principles of phy-
sicks, or had used thém, in axiomalis constituendis, which
is exactly the province belonging exclusively to experience.
At the same time, he pointed out, with great precision, the
place which the mathematicks may legitimately occupy, as
serving to measure and compare the objects of physical in-
quiry. He did not, however, perceive beforehand, nor was
it possible that he should, the vast extent to which the ap-
plication of that science was capable of being carried. In
the book, De Augmentis, he has made many excellent re-
marks on this subject, full of the sagacity which penetrated
so far into futurity, but, nevertheless, could only perceive
a small part of the scene which the genius of Newton was
afterwards to unfold.

? The part of mechanicks which involves only statical considera-
tions, or the equilibrium of forces, is capable of being treated by
reasoning @ priori entirely, without any appeal to experience.
This will appear, when the subject of Mechanicks is. more parti-
eularly treated of.

ft 12

94 DISSERTATION SECOND. , {ean

Hence, the route which leads to many of the richest and
most fertile fields of science, is not precisely that which
Bacon pointed out; it is safer and easier, so that the voyager
finds he can trust to his chart and compass alone, with-
out constantly looking out, or having the sounding-line per-
petually in his hand.

Another remark I must make on Bacon’s method is, that
it does not give sufficient importance to the instantiae radii,
or those which furnish us with accurate measures of physi-
cal quantities. The experiments of this class are introduced
as only subservient to practice ; they are, however, of in-
finite value in the theoretical part of induction, or for ascer-
taining the causes and essences of the things inquired into.
We have an instance of this in the discovery of that important
truth in physical astronomy, that the moon is retained in
her orbit by the force of gravity, or the same which, at the
earth’s surface, makes a stone fall to the ground. This
proposition, however it might have been suspected to be
true, could never have been demonstrated but by such ob-
servations and experiments as assigned accurate geometri-
cal measures to the quantities compared. The semidiame-
ter of the earth ; the velocity of falling bodies at the earth’s
surface ; the distance of the moon, and her velocity in her
orbit ;—all these four elements must be determined with
great precision, and afterwards compared together by cer-
tain theorems deduced from the laws of motion, before the
relation between the force which retains the moon in her or-
bit, and that which draws a stone to the ground, could pos-
sibly be discovered. The discovery also, when made, car-
ried with it the evidence of demonsfration, so that here, as
in many other cases, the instantiae radii are of the utmost
importance in the theoretical part of physicks.

Another thing to be observed is, that, in many cases, the
result of a number of particular facts, or the collective in-

ee ee Ee ee ee ee eee

SECT. 11.] _ DISSERTATION SECOND. 95

stance arising from them, can only be found out by geome-
try, which, therefore, becomes a necessary instrument in
completing the work of induction. An example, which the
science of opticks furnishes, will make this clearer than
any general description. When light passes from one
transparent medium to another it is refracted, that is, it
ceases to go en in a straight line, and the angle which the
incident ray makes with the superficies which bounds the
two media, determines that which the refracted ray makes
with the same superficies. Now, if we would learn any
thing about the relation which these angles bear to one ano-,
ther, we must have recourse to experiment, and all that ex-
periment can do is, for any particular angle of incidence, to
determine the corresponding angle of refraction. This
may be done in innumerable cases ; but, with respect to the
general rule which, in every possible case, determines the
one of those angles from the other, or expresses the con-
stant and invariable relation which subsists between them,—
with respect to it, experiment gives no direct information.
The methods of geometry must therefore be called in to
our assistance, which, when a constant though unknown re-
lation subsists between two angles, or two variable quantities
of any kind, and when an indefinite number of values of
these quantities are given, furnishes, infallible means of dis-
covering that unknown relation, either accurately, or at
least by approximation. [In this way it has been found,
that, when the two media remain the same, the cosines of
the angles above mentioned have a constant ratio to one
another. Thus it appears, that, after experiment has done
its utmost, geometry must be applied before the business
of induction can be completed. This can oaly happen
when the experiments afford accurate measures of the quan-
tities concerned, like the instantiae radii, curriculi, &c.
and this advantage of admitting generalization with so much

96 DISSERTATION SECOND. (parr 1.

certainty is one of their properties, of which it does not ap-
pear that even Bacon himself was aware.

Again, from the intimate connexion which prevails
among the principles of science, the success of one investi-

gation must often contribute to the success of another, in ©

such a degree as to make it unnecessary to employ the
complete apparatus of induction from the beginning.—
When certain leading principles have been once established,
they serve, in new investigations, to narrow the limits with-
in which the thing sought for is contained, and enable the
inquirer to arrive more speedily at the truth.

Thus, suppose that, after the nature of the reflection and
refraction of light, and particularly of the colours produc-
ed by the latter, had been discovered by experiment, the
cause of the rainbow were to be inquired into. It would,
after a little consideration, appear probable, that the phe-
nomenon to be explained depends on the reflection and re-
fraction of light by the rain falling from a cloud opposite
to the sun. Now, since the nature of reflection and re-
fraction are supposed known, we have the principles pre-
viously ascertained which are likely to assist in the ex-
planation of the rainbow. We hare no occasion, there-
fore, to enter on the inquiry, as if the powers to be inves-
tigated were wholly unknown. It is the combination of
them only which is unknown, and our business is to seek
so to combine them, that the result may correspond with
the appearances. 'This last is precisely what Newton ac-
complished, when, by deducing from the known laws of
refraction and reflection the breadth of the coloured arch,
the diameter of the circle of which it is a part, and the re-
lation of the latter to the place of the spectator and of the
gun, he found all these to come out from his calculus,
just as they are observed in nature. Thus he proved the

OT

sxcr. 11.) DISSERTATION SECOND. 97

truth of his solution by the most clear and irresistible evi-
dence.

The strict method of Bacon is therefore only necessary
where the thing to be explained is new, and where we have
no knowledge, or next to none, of the powers employed.
This is but rarely the case, at least in some of the branches
of Physicks; and, therefore, it occurs most commonly in
actual investigation, that the inquirer finds himself limited,
almost from the first outset, to two or three hypotheses, all
other suppositions involving inéonsistencies which cannot
for a moment be admitted. His business, therefore, is to
compare the results of these hypotheses, and to consider
what consequences may in any case arise from the one that
would not arise from the other. If any such difference can
be found, and if the matter is a subject of experiment, we
have then an instantia crucis which must decide the ques-
tion.

Thus, the instantia crucis comes in real practice to be
the experiment most frequently appealed to, and that from
which the most valuable information is derived.

In executing the method here referred to, the application
of much reasoning, and frequently of much mathematical
reasoning, is necessary, before any appeal to the experi-
ment can be made, in order to deduce from each of the
hypotheses an exact estimate of the consequences to which
it leads. Suppose, for instance, that the law by which the
magnetick virtue decreases in ifs intensity, as we recede
from its poles, were to be inquired into. It is obvious that
the number of hypotheses is here indefinite; and that we
have hardly any choice but to begin with the simplest, or
with that which is most analogous to the law of other forces
propagated from a centre. Whatever law we assume, we
must enter into a good deal of geometrick reasoning, before
a conclusion can be obtained, capable of being brought to

98 DISSERTATION SECOND. jrarrr.

the test of experience. The force itself, like all other
forces, is not directly perceived, and its effects are not the
result of its mere intensity, but of that intensity combined
with the figure and magnitude of the body on which it
acts; and, therefore, the calculus must be employed to ex-
press the measure of the effect, in terms of the intensity
and the distance only. This being done, the hypothesis
which gives results most nearly corresponding to the facts
observed, when the magnet acts on the same body, at dif-
ferent distances, must be taken as the nearest approxima-
tion to the truth. We have here an instance of the use of
hypothesis in inductive investigation, and, indeed, of the
only legitimate use to which it can ever be applied.

It also appears that Bacon placed the ultimate object of
philosophy too high, and too much out of the reach of
man, even when his exertions are most skilfully conducted.
He seems to have thought, that, by giving a proper direc-
tion to our researches, and carrying them on according to
the inductive method, we should arrive at the knowledge
of the essences of the powers and qualities residing in
bodies ; that we should, for instance, become acquainted
with the essence of heat, of cold, of colour, of transparen-
cy. The fact, however, is, that, in as far as science has
yet advanced, no one essence has been discovered, either
as to matter in general, or as to any of its more extensive
modifications. We are yet in doubt, whether heat isa
peculiar motion of the minute parts of bodies, as Bacon
himself conceived it to be ; or something emitted or radiat-
ed from their surfaces; or lastly, the vibrations of an
elastick medium, by which they are penetrated and sur-
rounded. Yet whatever be the form or essence of heat,
we have discovered a great number of its properties and
its laws ; and have done so, by pursuing with more or less
accuracy the method of induction. We haye also this con-

i i ti tia a a

Oe See eee ee ee

secT. 11.] DISSERTATION SECOND. 99

solation for the imperfection of our theoretical knowledge,
that, in as much as art is concerned, or the possession of
power over heat, we have perhaps all the advantages that

could be obtained from a complete knowledge of its es-
sence.

An equal degree of mystery hangs over the other pro-
perties and modifications of body ; light, electricity, mag-
netism, elasticity, gravity, are all in the same circumstan-
ces; and the only advance that philosophy has made _to-

ward the discovery of the essences of these qualities or -

substances is, by exploding some theories, rather than by
establishing any,—so true is Bacon’s maxim, that the first
steps in philosophy necessarily consist in negative propo-
sitions. Besides this, in all the above instances the laws
of action have been ascertained ; the phenomena have been
reduced to a few general facts, and in some cases, as in
that of gravity, to one only; and for ought that yet ap-
pears, this is the highest point which our science is des-
tined to reach. |

In consequence of supposing a greater perfection in
knowledge than is ever likely to be attained, Bacon ap-
pears, in some respects, to have misapprehended the way
in which it is ultimately to become applicable to art. He
conceives that, if the form of any quality were known, we
should be able, by inducing that form on any body, to
communicate to it the said quality. It is not probable,
however, that this would often lead to a more easy and
simple process than that which art has already invented.
In the case of colour, for example, though ignorant of its
form, or of the construction of surface which enables bo-
dies to reflect only light of a particular species, yet we
know how to communicate that power from one body to
another. Nor is it likely, though this structure were known
with ever so great precision, that we should be able to im-

EE

100 DISSERTATION SECOND. [Pant 1.

part it to bodies by any means so simple and easy, as by
the common process of immersing them ina liquid of a
given colour. 4 ae
In some instances, however, the theories of chemistry
have led to improvements of art very conformable to the
anticipations of the Novum Organum. A remarkable in-
‘stance of this occurs in the process for bleaching, invented
by Berthollet. It had been for some time known, that the
combination of the chemical principle of oxygen with the
colouring matter in bodies, destroyed, or discharged, the
colour; and that, in the common process of bleaching, it
was chiefly by the union of the oxygen of the air with the
colouring matter in the cloth that this effect was produced.
The excellent chemist just named conceived, therefore,
that if the oxygen could be presented to the cloth ina
dense state, and, at the same time, feebly combined with
any other body, it might unite itself to the colouring
matter so readily, that the process of bleaching would by
that means be greatly accelerated. His skill in chemistry
suggested to him a way in which this might easily be done,
by immersing the cloth in a liquid containing much oxygen
in a loose state, or one in which it was slightly combined
with other substances, and the effect followed so exactly,
that he was able to perform in a few hours what required
weeks, and even months, according to the common protess.
This improvement, therefore, was a real gift from the
sciences to the arts; and came nearly, though not alto-
gether, up to the ideas of Bacon. I suspect not altogether,
because the manner in which oxygen destroys the colour
of bodies, or alters the structure of their surfaces, remains
quite unknown.
It was natural, however, that Bacon, who studied these
subjects theoretically, and saw nowhere any practical re-
sult in which he could confide, should listen to the inspira-

seer. 111.] * DISSERTATION SECOND. 101

tions of his own genius, and ascribe.to philosophy a per-
fection which it may be destined never to attain. He
knew, that from what it had not yet done, he could con-
clude nothing against what it might hereafter accomplish.
Bat after his method has been followed, as it has now been,
with greater or less accuracy, for more than two hundred
years, circumstances are greatly changed ; and the impe-
diments, which, during all that time, have not yielded in
the least to any effort, are perhaps never likely to be re-
moved. This may, however, be a rash inference ; Bacon,
after all, may be in the right; and we may be judging un-
der the influence of the vulgar prejudice, which has con-
vinced men, in every age, that they had nearly reached
the farthest verge of human knowledge. This must be left
for the decision of posterity; and we should rejoice to
think, that judgment will hereafter be given against the
opinion which at this moment appears most probable.

SECTION Ii.

MECHANICKS.

1. Turory or Morion.

Berore the end of the sixteenth century, mechanical
science had never gone beyond the problems which treat
of the equilibrium of bodies, and had been able to resolve
these accurately, only in the. cases which can be easily
reduced to the lever. Guido Ubaldi, an Italian mathema-
tician, was among the first who attempted to go farther
than Archimedes and the ancients had done in such in-

18

102 DISSERTATION SECOND. ® [parr 2

quiries. In a treatise which bears the date of 1577, he re-
duced the pulley to the lever; but with respect to the
inclined plane, he continued in the same errour with Pap-
pus Alexandrinus, supposing that a certain force must be
applied to sustain a body, even on a plane which has no
inclination.

Stevinus, an engineer of the Low Countries, is the first
who can be said to have passed beyond the point at which
the ancients had stopped, by determining accurately the
force necessary to sustain a body on a plane inclined at
any angle to the horizon. He resolved also a great num-
ber of other problems connected with the preceding, but,
nevertheless, did not discover the general principle of the
composition of forces, though he became acquainted with
this particular case, immediately applicable to the inclined
plane.

The remark, that a chain laid on an inclined plane, with
a part of it hanging over at top, in a perpendicular line,
will be in equilibrio, if the two ends of the chain reach
down exactly to the same level, led him to the conclusion,
that a body may be supported on such a plane by a force
which draws in a direction parallel to it, and has to the
weight of the body the same ratio that the height of the
plane has to its length.

Though it was probably from experiencetthat Stevinus
derived the knowledge of this proposition, he attempted to
prove the truth of it by reasoning a priori. He suppos-
ed the two extremities of the chain, when disposed as
above, to be connected by a part similar to the rest, which,
therefore, must hang down, and form an arch. If in this
state, says he, the chain were to move at all, it would con-
tinue to move for ever, because ifs sifuation, on the whole,
never changing, if it were determined to move at one in-
stant, it must be so determined at every other instant.

sacT, 111.) . DISSERTATION SECOND. 103

Now, such perpetual motion, he adds, is impossible, and
therefore the chain, as here supposed, with the arch hang-
ing below, does not move. But the force of the arch be-
low draws down the ends of the chain equally, because the
arch is divided in the middle or lowest point into two parts
similar and equal. Take away these two equal forces, and
the remaining forces will also be equal, that is, the tenden-
cy of the chain to descend along the inclined plane, and
the opposite tendency of the part hanging perpendicularly
down, are equal, or are in equilibrio with one another.
Such is the reasoning of Stevinus, which, whether per-
fectly satisfactory or not, must be acknowledged to be
extremely ingenious, and highly deserving of attention, as
having furnished the first solution of a problem, by which
the progress of mechanical science had been long arrested.
The first appearance of his solution is said to have had
the date of 1585; but his works, as we now see them,
were collected after his death, by his countryman AI-
bert Girard, and published at Leyden in 1634.1 Some
discoveries of Stevinus in hydrostaticks will be hereafter
mentioned.

The person who comes next in the history of mecha-
nicks made a great revolution in the physical sciences,
Galileo was born at Pisa in the year 1564. He early ap-
plied himself to the study of mathematicks and natural
philosophy ; and it is from the period of his discoveries
that we are to date the joint application of experimental

* The edition of Albert Girard is entitled Oeuvres Mathemati-
ques de Stevins, in folio. See Livre I. De la Stalique, theorem
1l. Stevinus also wrote a treatise on navigation, which was
published in Flemish in 1586, and was afterwards honoured with
a translation into Latin, by Grotius. The merit of Stevinus has
been particularly noticed by La Grange. Mecanique Analytique,
Tom. I. Sect. 1. 6 5.

104 DISSERTATION SECOND. [pant 1.

and geometrical reasoning to explain the phenomena of ©

nature. ;

As early as 1592 he published a treatise, della Sciensa
Mechanica, in which he has given the theory, not of the
lever only, but of the inclined plane and the screw; and
has also laid down this general proposition, that mechani-
cal engines make a small force equivalent to a great one,
by making the former move over a greater space in the
same time than the latter, just in proportion as it is less.
No contrivance can make a small weight put a great one in
motion, but such a one as gives to the small weight a velo-
city which is as much greater than that of the large weight,
as this last weight is greater than the first. These gene-
ral propositions, and their influence on the action of ma-
chinery, Galileo proceeded to illustrate with that clearness,
simplicity, and extent of view, in which he was quite un-
rivalled ; and hence, I think, it is fair to consider him as
the first person to whom the mechanical principle, since
denominated that of the virtual velocities, had occurred
in its full extent. The object of his consideration was the
action of machines in motion, and not merely of machines
in equilibrio, or at rest; and he showed, that, if the effect
of a force be estimated by the weight it can raise to a
ziven height ina given time, this effect can never be m-
creased by any mechanical contrivance whatsoever.

In the same treatise, he lays it down as a postulate (sup-
posisione,) that the effect of one heavy body to turn an-
other round a centre of motion, is proportional to the per-
‘pendicular drawn from that centre to the vertical passing
‘through the body, or in general to the direction of the
force. This proposition he states without a demonstration,
and passes by means of it to the oblique lever, and thence
io the inclined plane. To speak strictly, however, the
demonstrations with respect to both these last are incom-

sxcr. 11.] DISSERTATION SECOND. 105

plete, the preceding proposition being assumed in them
without proof. Itis probable that he satisfied himself of
the truth of it, on the principle, that the distances of
forces from the centre of motion must always be measured
by lines making the same angles with their directions, and
that of such lines the simplest are the perpendiculars.
His demonstration is regarded by La Grange as quite sa-
tisfactory. '

Galileo extended the theory of motion still farther. He
had begun, while pursuing his studies at the university of
Pisa, to make experiments on the descent of falling bodies,
and discovered the fact, that heavy and light bodies fall to
the ground from, the same height in the same time, or in
times so nearly the same, that the difference can only be
ascribed to the resistance of the air. From observing the
vibrations of the lamps in the cathedral, he also arrived at
this very important conclusion in mechanicks, that the
great and the small. vibrations of the same pendulum are
performed in the same time, and that this time depends only
on the length of the pendulum. The date of these observa-
tions goes back as far as the year 1583.

These experiments drew upon him the displeasure of
his masters, who considered it as unbecoming of their pupil
to seek for truth in the Book of Nature, rather than in the
writings of Aristotle, when elucidated by iheir commenta-
ries, and, from that moment began the persecutions with
which the prejudice, the jealousy, and bigotry of his con-
temporaries continued to harass or afflict this great man
throughout his whole life.

That the acceleration of falling bodies is uniform, or,
that they receive equal increments of velocity in equa!
times, he appears first to have assumed as the law which

* Mecanique Analytique, Tom. I. Sect. 1. 4 6.

-

7

106 DISSERTATION SECOND. [parr 1.

they follow, merely on account of its simplicity. Having
once assumed this principle, he showed, by mathematical
reasoning, that the spaces descended through must be as
the squares of the times, and that the space fallen through
in one second is just the half of that which the body would
have described in the same time with the velocity last ac-
quired.

The knowledge which he already had of the properties
of the inclined plane enabled him very readily to per-
ceive, that a body descending on such a plane must be
uniformly accelerated, though more slowly than when it
falls directly, and is aecelerated by its whole weight. By
means of the inclined plane, therefore, he was able to bring
the whole theory of falling bodies to the test of experiment,
and to prove the truth of his original assumption, the uni-
formity of their acceleration.

His next step was to determine the path of a heavy bo-
dy, when obliquely projected. He showed this path to
be a parabola; and here, for the first time, occurs the use
of a principle which is the same with the composition of
motion in its full extent. Galileo, however, gave no name
to this principle; he did not enunciate it generally, nor
did he give any demonstration of it, though he employed
it in his reasonings. The inertia of body was assumed in
the same manner; it was, indeed, involved in the uniform
acceleration of falling bodies, for these bodies did not lose
in one minute the motion acquired in the preceding, but,
retaining it, went on continually receiving more.

The theory of the inclined plane had led to the know-
ledge of this proposition, that, if a circle be placed verti-
cally, the chords of different arches terminating in the
lowest point of the circle, are all descended through in the
same space of time. This seemed to explain why, in a
circle, the great and the small vibrations are of equal du-

tecr. mt.) DISSERTATION SECOND. 107

ration. Here, however, Galileo was under a mistake, as
the motions in the chord and in the arch are very dissimi-
lar. The accelerating force in the chord remains the same
from the beginning to the end, but, in the arch, it varies
continually, and becomes, at the lowest point, equal to no-
thing. The times in the chords, and in the arches, are
therefore different, so that here we have a point marking
the greatest distance in this quarter, to which the me-
chanical discoveries of Galileo extended. The first per-
son who investigated the exact time of a vibration in an
arch of a circle was Huygens, a very profound mathema-
tician. :

To this list of mechanical discoveries, already so im-
portant and extensive, we must add, that Galileo was the
first who maintained the existence of the law of continuity,
and who made use of it as a principle in his reasonings on
the phenomena of motion. *

The vibrations of the pendulum having suggested to
Galileo the means of measuring time accurately, it ap-
pears certain that the idea of applying it to the clock had
also occurred to him, and of using the chronometer so
formed for finding the longitude, by means of observations
made on the eclipses of the satellites of Jupiter. How
far he had actually proceeded in an invention which re-
quired great practical knowledge, and which afterwards
did so much credit to Huygens, appears to be uncertain,
and not now easy to be ascertained. But that the project
had occurred to him, and that he had taken some steps to-
wards realizing it, is sufficiently established.

One forms, however, a very imperfect idea of this phi-
losopher, from considering the discoveries and inventions,
numerous and splendid as they are, of which he was the

* Opere de Galileo, Tom. [V. Dial. 1. p. 32, Florence edi-
tion, and in many other parts.

ll ” iy acu Ta ae Ren a
P :

108 DISSERTATION SECOND. parr 1.

undisputed author. It is by following his reasonings, and
by pursuing the train of his thoughts in his own elegant,
though somewhat diffuse exposition of them, that we be-
come acquainted with the fertility of his genius, with the
sagacity, penetration, and comprehensiveness of his mind.
The service which he rendered to real knowledge, is to
be estimated not only from the truths which he discovered,
but from the errours which he detected,—not merely from
the sound principles which he established, but from the
pernicious idols which he overthrew. His acuteness was
strongly displayed in the address with which he exposed
thé errours of his adversaries, and refuted their opinions,
by comparing one part of them with another, and proving
their extreme inconsistency. Of all the writers who have
lived in an age, which was yet only emerging from igno-
rance and barbarism, Galileo has most entirely the tone
of true philosophy, and is most free from any contamina-
tion of the times in taste, sentiment, and opinion.

The discoveries of this great man concerning motion
drew the attention of philosophers more readily, from the
circumstance that the astronomical theories of Copernicus
had directed their attention to the same subject. It had
become evident, that the great point in dispute between °
his system and the Ptolemaick must be finally decided by
an appeal to the nature of motion and its laws. 'The great
argument to which the friends of the latter system naturally
had recourse was the impossibility, as it seemed to them
to be, of the swift motion of the earth being able to exist,
without the perception, nay, even without the destruction,
of its inhabitants. It was natural for the followers of Co- ,
pernicus to reply, that it was not certain that these two
things were incompatible; that there were many cases in
which it appeared, that the motion common to a whole
system of bodies did not affect the motion of those bodies

gxcr. 111.) DISSERTATION SECOND. 109

relatively to one another ; that the question must be more
deeply inquired into; and that, without this, the evidence
on opposite sides could not be fairly and accurately com-
pared. Thus it was, at a very fortunate moment, that
Galileo made his discoveries in Mechanicks, as they were
rendered more interesting by those, which, at that very
time, he himself was making in Astronomy. The system
of Copernicus had, in this manner, an influence on the
theory of motion, and, of course, on all the parts of na-
tural philosophy. The inerlia of matter, or, the tenden-
cy of body, when put in motion, to preserve the quantity
and direction of that motion, after the cause which impress-
ed it has ceased to act, is a principle which might still
have been unknown, if it had not been forced upon us by
the discovery of the motion of the earth.

The first addition which was made to the mechanical
discoveries of Galileo, was by Torricelli, in a treatise De
Motu Gravium naturaliter descendeniium et projectorum. }
To this ingenious man we are indebted for the discovery
of a remarkable property of the centre of gravity, anda
general principle with respect to the equilibrium of bodies.
It is this: If there be any number of heavy bodies con-
nected together, and so circumstanced, that by their mo-
tion their centre of gravity can neither ascend nor descend,
these bodies will remain at rest. This proposition often
furnishes the means of resolving very difficult questions in
mechanicks.

Descartes, whose name is so great in philosophy and
mathematicks, has also a place in the history of mechani-
cal discovery. With regard to the action of machines,
he laid down the same principle which Galileo had estab-
lished,—that an equal effort is necessary to give to a weight.

* Vitae Italorum Illustrium, vol. TI. p. 347.
14

110 DISSERTATION SECOND. [PARP 1.

a certain velocity, as to give to double the weight the
half of that velocity, and so on in proportion, the effect
being always measured by the weight multiplied into the
velocity which it receives. He could hardly be ignorant
that this proposition had been already stated by Galileo,
but he has made no mention of it. He, indeed, always
affected a disrespect for the reasonings and opinions of the
Italian philosopher, which has done him no credit in the
eyes of posterity.

The theory of motion, however, has in some points been
considerably indebted to Descartes. Though the reason-
ings of Galileo certainly involve the knowledge of the dis-
position which matter has to preserve its condition either of
rest or of rectilineal and uniform motion, the first distinct
enunciation of this law is found in the writings of the
French philosopher. It is, however, there represented,
not as mere inactivity, or indifference, but as a real force,
which bodies exert in order to preserve their state of rest
or of motion, and this inaccuracy affects some of the rea-
sonings concerning their action on one another.

Descartes, however, argued very justly, that all motion
being naturally rectilineal, when a body moves in a curve,
this must arise from some constraint, or some force urging
it in a direction different from that of the first impulse, and
that if this cause were removed at any time, the motion
would become rectilineal, and would be in the direction of
a tangent to the curve at the point where the deflecting
force ceased to act.

Lastly, He taught that the quantity of motion in the uni-
verse remains always the same.

The reasoning by which he supported the first and se-
cond of these propositions is not very convincing, and
though he might have appealed to experience for the truth
of both, it was not in the spirit of his philosophy to take

ener. 111.) DISSERTATION SECOND. 11i

that method of demonstrating its principles. His argument
was, that motion is a state of body, and that body or matter
cannot change its own state. This was his demonstration of
the first proposition, from which the second followed ne-
cessarily.

The evidence produced for the third, or the preserva-
tion of the same quantity of motion in the universe, is
founded on the immutability of the Divine nature, and is
an instance of the intolerable presumption which so often
distinguished the reasonings of this philosopher. Though
the immutability of the Divine nature will readily be ad-
mitted, it remains to be shown, that the continuance of the
Same quantity of motion in the universe is a consequence
of it. This, indeed, cannot be shown, for that quantity,
in the sense in which Descartes understood it, is so far
from being preserved uniform, that it varies continually
from one instant to another. It is nevertheless true, that
the quantity of motion in the universe, when rightly esti-
mated, is invariable, that is, when reduced to the direc-
tion of three axes at right angles to one another, and
when opposite motions are supposed to have opposite
signs. This is a truth now perfectly understood, and is a
corollary to the equality of action and reaction, in conse-
quence of which, whatever motion is communicated in one
direction, is either lost in that direction, or generated in the
opposite. This, however, is quite different from the pro-
position of Descartes, and if expressed in his language,
would assert, not that the sum, but that the difference of
the opposite motions in the universe remains constantly
the same. When he proceeds, by help of the principle
which he had thus mistaken, to determine the laws of the
collision of bodies, his conclusions are almost all false, and
have, indeed, such a want of consistency and analogy with
one another, as ought, in the eyes of a mathematician, to

‘

112 DISSERTATION SECOND. [parr 3.

have appeared the most decisive indications of errour.

How this escaped the penetration of a man well acquainted —

with the harmony of geometrical truths, and the gradual
transitions by which they always pass into one another, is
not easily explained, and perhaps, of all his errours, is the
least consistent with the powerful and systematick genius
which he is so well known to have possessed.

Thus, the obligation which the theory of motion has to
this philosopher, consists in his having pointed out the
nature of centrifugal force, and ascribed that force to the
true cause, the inertia of body, or its tendency to uniform
and rectilineal motion.

The laws which actually regulate the collision of bodies
remained unknown till some years later, when they were
recommended by the Royal Society of London to the
particular attention of its members. Three papers soon
appeared, in which these laws were all correctly laid down,
though no one of the authors had any knowledge of the
conclusions obtained by the other two. The first of these
was read to the Society, in November, 1668, by Dr. Wal-
fis of Oxford; the next by Sir Christopher Wren in the
month following, and the third by Huygens, in January,
i669. The equality of action and reaction, and the max-
im, that the same force communicates to different bodies
velocities which are inversely as their masses, are the
principles on which these investigations are founded.

The ingenious and profound mathematician last men-
tioned, is also the first who explained the true relation
between the length of a pendulum, and the time of its least
vibrations, and gave a rule by which the time of the rec-
tilineal descent, through a line equal in length to the pen-
dulum, might from thence be deduced. ~ He next applied
the pendulum to regulate the motion of a clock, and gave
an account of his constraction, and the principles of if, in

a

suc, 111.) DISSERTATION SECOND. 4313

his Horologium Oscillatorium, about the year 1670,
though the date of the invention goes as far back as 1656."
Lastly, He taught how to correct the imperfection of a
pendulum, by making it vibrate between cycloidal cheeks,
in consequence of which its vibrations, whether great or
small, became, not approximately, but precisely, of equal
duration.

Robert Hooke, a very celebrated English mechanician,
laid claim to the same application of the pendulum to the
clock, and the same use of the cycloidal cheeks. There
is, however, no dispute as to the priority of Huygens’s
claim, the invention of Hooke being as late as 1670. Of
the cycloidal cheeks, he is not likely te have been even
the second inventor. Experiment could hardly lead any
one to this discovery, and he was not sufficiently skilled
in the mathematicks to have found it out by mere reason-
ing. The fact is, that though very original and inventive,
Hooke was jealous and illiberal in the extreme; he ap-
propriated to himself the inventions of all the world, and
accused all the world of appropriating his.

It has already been observed, that Galileo conceived the
application of the pendulum to the clock earlier, by seve-
ral years, than either of the periods just referred to. The
invention did great honour to him and to- his two rivals;
but that which argues the most profound thinker, and the
most skilful mathematician of the three, is the discovery
of the relation between the length of the pendulum and
the time of its vibration, and this discovery belongs ex-
clusively to Huygens. ‘The method which he followed in
his investigation, availing himself of the properties of the
cycloid, though it be circuitous, is ingenious, and highly
instructive.

* Montucla, Tom. iI. p. 418, 2d edit.

114 DISSERTATION SECOND, [pant 1.

An invention, in which Hooke has certainly the priority
to any one, is the application of a spiral spring to regulate
the balance of a watch. It is well known of what practi-
cal utility this invention has been found, and how much it
has contributed to the solution of the problem of finding the
longitude at sea, to which not only he, but Galileo and
Huygens, appear all to have had an eye.

In what respects the theory of motion, Huygens has
still another strong claim to our notice. This arises from
his solution of the problem of finding the centre of oscilla-
tion of a compound pendulum, or the length of the simple
pendulum vibrating in the same time with it. Without
the solution of this problem, the conclusions respecting the
pendulum were inapplicable to the construction of clocks,
in which the pendulums used are of necessity compound.
The problem was by no means easy, and Huygens was
obliged to introduce a principle which had not before been
recognised, that if the compound pendulum, after descend-
ing to its lowest point, was to be separated into particles
distinct and unconnected with one another, and each left
at liberty to continue its own vibration, the common centre
of gravity of all those detached weights would ascend to
the same height to which it would have ascended had they
continued to constitute one body. The above principle
Jed him to the true solution, and his investigation, though
less satisfactory than those which have been since given,
does great credit to his ingenuity. This was the most
difficult mechanical inquiry which preceded the invention
of the differential or fluxionary calculus.

2. HyprostatTicks.

While the theory of motion, as applied to solids, was
thus extended, in what related to fluids, it was making

seer. 111.] DISSERTATION SECOND. 115

equal progress. ‘The laws which determine the weight of
bodies immersed in fluids, and also the position of bodies
floating on them, had been discovered by Archimedes, and
were farther illustrated by Galileo. It had also been dis-
covered by Stevinus, that the pressure of fluids is in pro-
portion to their depth, and thus the two leading principles
ef hydrostaticks were established. Hydraulicks, or the
motion of fluids, was a matter of more difficulty, and here
the first step is to be ascribed to Torricelli, who, though
younger than Galileo, was for some time his contemporary.
He proved that water issues from a hole in the side or bot-
tom of a vessel, with the velocity which a body would ac-
quire, by falling from the level of the surface to the level
of the orifice. This proposition, now so well known as the
basis of the whole doctrine of Hydraulicks, was first pub-
lished by Torricelli at the end of his book, De Motu Gra-
vium et Projectorum ; but it is not the greatest discovery
which science owes to the friend and disciple of Galileo.
The latter had failed in assigning the reason why water
cannot be raised in pumps higher than thirty-three feet,
but he had remarked, that if a pump is more than thirty-
three feet in length, a vacuum will be left in it. Torricelli,
reflecting on this, conceived, that if a heavier fluid than
water were used,'a vacuum might be produced, in a way
far shorter, and more compendious. He tried mercury,
therefore, and made use of a glass tube about three feet
long, open at one end, and close at the other, where it ter-
minated in a globe. He filled this tube, shut it with his
finger, and inverted it in a basin of mercury. The result
is well known ;--he found that a column of mercury was
suspended in the tube, an effect which he immediately as-
cribed to the pressure of the atmosphere. So disinterest-
ed was this philosopher, however, that he is said to have
lamented that Galileo, when inquiring into the cause why

116 DISSERTATION SECOND. [panr x.

. water does not ascend in pumps above a certain height, had
not discovered the true cause of the phenomenon. The
generosity of Torricelli was perhaps rarer than his ge-
nius ;—there are more who might have discovered the sus-
pension of mercury in the barometer, than who would have
been willing to part with the honour of the discovery toa
master or a friend.

This experiment opened the door to a multitude of new
discoveries, and demolished a formidable idol, the horrour
of a vacuum, to which so much power had _ been long attri-
buted, and before which even Galileo himself had conde-
scended to bow.

The objections which were made to the explanation of
the suspension of the mercury in the tube of the barometer,
were overthrown by carrying that instrument to the top of
Puy de Dome, an experiment suggested by Pascal. The
descent of the mercury showed, that the pressure which
supported it was less there than at the bottom ; and it was
afterwards found, that the fall of the mercury corresponded
exactly to the diminution of the length of the pressing co-
jumn, so that it afforded a measure of that diminution, and,
consequently, of the heights of mountains. The invention
of the airpump by Otto Guericke, burgomaster of Magde-
burg, quickly followed that of the barometer by Torricelli,
though it does not appear that the invention of the Italian
philosopher was known to the German. In order to obtain
a space entirely void of air, Otto Guericke filled a barrel
with water, and having closed it exactly on all sides, began
to draw out the water by a sucking pump applied to the
lower part of the vessel. He had proceeded but a very

little way, when the air burst into the barrel with a loud _

noise, and its weight was proved by the failure of the ex-
periment, as effectually as it could have been by its suc-
cess. After some other trials, which also failed, he

ager, 111.] DISSERTATION SECOND. 117

thought of employing a sphere of glass, when the experi-
ment succeeded, and a vacuum was obtained. This was
about the year 1654.

The elasticity of the air, as well as its weight, now be-
came known; its necessity to combustion, and the absorp-
tion of a certain proportion of it, during that process; its
necessity for conveying sound ;—all these things were
clearly demonstrated. ‘The necessity of air to the respi-
ration of animals required no proof from experiment, but
the sudden extinction of life, by immersion in a vacuum,
was a new illustration of the fact.

The first considerable improvements made on the air-
pump are due to Mr. Boyle. He substituted for the glass
globe of Otto Guericke a receiver of a more commodious
form, and constructed his pump so as to be worked with
much more facility. His experiments were farther extend-
ed,—they placed the weight and elasticity of the air ina
variety of new lights,—they made known the power of air
to dissolve water, &c. Boyle had great skill in contriving,
and great dexterity in performing experiments. He had,
indeed, very early applied himself to the prosecution of ex-
perimental science, and was one of the members of the
small but distinguished body, who, during the civil wars,
held private meetings for cultivating natural knowledge, on
the plan of Bacon. They met first in London, as early as
1645, afterwards at Oxford, taking the name of the Philoso-
phick College. Of them, when Charles the Second as-
cended the throne, was formed the Royal Society of Lon-
don, incorporated by letters patent in 1662. No one was
more useful than Boyle in communicating activity and vi-
gour to the new institution. A real lover of knowledge, he
was most zealous in the pursuit of it; and having thorough-
ly imbibed the spirit of Bacon, was an avowed enemy to the
philosophy of Aristotle.

118 DISSERTATION SECOND. [pant i.

SECTION IV.
ASTRONOMY.

1. Ancient AsTrronomy.

Ir has already been remarked, that the ancients made
more considerable advances in astronomy than in almost any
other of the physical sciences. They applied themselves
diligently to observe the heavens, and employed mathema-
tical reasoning to connect together the insulated facts, which
are the only objects of direct observation. ‘The astrono-
mer discovers nothing by help of his instruments, but that,
at a given instant, a certain luminous point has a particular
position in the heavens: The application of mathematicks,
and particularly of spherical trigonometry, enables him to
trace out the precise tract of this luminous spot; to disco-
ver the rate of its motion, whether varied or uniform, and
thus to resolve the first great problem which the science of
astronomy involves, viz. to express the positions of the
heavenly bodies, relatively to a given plane in functions of
the time. The problem thus generally enunciated, compre-
hends all that is usually called by the name of descriptive
or mathematical astronomy.

The explanation of the celestial motions, which natural-
ly occurred to those who began the study of the heavens,
was, that the stars are so many luminous points fixed in the
surface of a sphere, having the earth in its centre, and re-
volying on an axis passing through that centre in the space
of twenty-four hours. When it was. observed that all the
stars did not partake of this diurnal motion in the same de-
gree, but that some were carried slowly towards the east,
and that their paths estimated in that direction, after cer-
tain intervals of time, returned into themselves, it was be-
lieved that they were fixed in the surfaces of spheres,

’
szer. 1.] DISSERTATION SECOND. 119

which revolved westward, more slowly than the sphere of
the fixed stars. These spheres must be transparent, or
made of some crystaliine substance, and hence the name of
the crystalline spheres, by which they were distinguish-
ed. This system, though it grew more complicated in
proportion to the number and variety of the phenomena
observed, was the system of Aristotle and Eudoxus, and,
with a few exceptions, of all the philosophers of anti-
quity.

But when the business of observation came to be regu-
larly pursued ; when Timocharis and Aristillus, and their
successors in the Alexandrian school, began to study the
phenomena of the heavens, little was said of these orbs ; and
astronomers seemed only desirous of ascertaining the laws
or the general facts concerning the planetary motions.

To do this, however, without the introduction of hypo-
thesis, was certainly difficult, and probably was then impossi-
ble. The simplest and most natural hypothesis was, that
the planets moved eastward in circles, and at a uniform rate.
But when it was found that, instead of moving uniformly
to the eastward, every one of them was subject to great
irregularity, the motion eastward becoming at certain pe-
riods slower, and at length vanishing altogether, so that the
planet became stationary, and afterwards acquiring a mo-
tion in (he contrary direction, proceeded for a time toward
the west, it was far from obvious how all these appear-
ances could be reconciled with the idea of a uniform circu-
lar motion.

The solution of this difficulty is ascribed to Apollonius
Pergeus, one of the greatest geometers of antiquity. He
conceived that, in the circumference of a circle, having the
earth for its centre, there moved the centre of another
circle, in the circumference of which the planet ac-
tually revolved. The first of these circles was called

120 DISSERTATION SECOND. [PART 1.

the deferenf, and the second the epicycle, and the mo-
tion in the circumference of each was supposed uni-
form. Lastly, it was conceived that the motion of the
centre of the epicycle in the circumference of the deferent, —
and of the planet in that of the epicycle, were in opposite
directions, the first being towards the east, and the second
towards the west. In this way, the alterations from pro-
gressive to retrograde, with the intermediate stationary
points, were readily explained, and Apollonius carried his
investigation so far as to determine the ratio between the ra-
dius of the deferent, and that of epicycle, from knowing the

atations and retrogradations of any particular planet.

An object, which was then considered as of great impor-
tance to astronomy, was thus accomplished, viz. the pro-
duction of a variable motion, or one which was continually
changing both its rate and its direction from two uniform
circular motions, each of which preserved always the same
quantity and the same direction.

It was not long before another application was made of
the method of epicycles. Hipparchus, the greatest, astro-
nomer of antiquity, and one of the inventors in science most
justly entitled to admiration, discovered the inequality of
the sun’s apparent motion round the earth. To explain or
to express this irregularity, the same observer imagined
an epicycle of a small radius with its centre moving uni-
* formly in the circumference of a large circle, of which the
earth was the centre, while the sun revolved in the circum-
ference of the small circle with the same angular velocity
as this last, but in a contrary direction,

As other irregularities in the motions of the moon and of
the planets were observed, other epicycles were introduc-
ed, and -Piclemy, in his Almagest, enumerated all which
then appeared necessary, and assigned to them such di-
mensions as enabled them to express the phenomena with
accuracy. It is not to be denied that the system of the

aucr. 1V.] DISSERTATION SECOND. 121

heavens becamein this way extremely complicated ; though,
when fairly examined, it will appear to be a work of great
ingenuity and research. The ancients, indeed, may be re-
garded as very fortunate in the contrivance of epicycles,
because, by means of them, every inequality which can
exist in the angular motion of a planet may be at least
nearly represented. This I call fortunate, because, at the
time when Apollonius introduced the epicycle, he had no
idea of the extent to which his contrivance would go, as he
could have none of the conclusions which the author of the
Mécanique Céleste was to deduce from the principle of
gravitation.

The same contrivance had another great advantage ; it
subjected the motions of the sun, the moon, and the plan-
ets, very readily to a geometrical construction, or an arith-
metical calculation, neither of them difficult. By this means
the predictions of astronomical phenomena, the calculation
of tables, and the comparison of those tables with observa-
tion, became matters of great facility, on which facility,
in a great measure, the progress of the science depended.
It was on these circumstances, much more than on the sim-
plicity with which it amused or deceived the imagination,
that the popularity of this theory was founded ; the ascend-
ant which it gained over the minds of astronomers, and the
resistance which, in spite of facts and observations, it was
ao lorig able to make to the true system of the world.

It does not appear that the ancient astronomers ever con-
sidered the epicycles and deferents which they employed
in their system as having a physical existence, or as serving
to explain the causes of the celestial motions. They seem
to have considered them merely as mathematical diagrams,
serving to eapress or to represent those motions as geo-
metrical expressions of certain general facts, which readily
furnished the rules of astronomical calculation.

eS

122 DISSERTATION SECOND. [parr 1:

The language in which Ptolemy speaks of the epicycles
is not a little curious, and very conformable to the notion,
that he considered them as merely the means of expressing
a general law. After laying down the hypothesis of certain
epicycles, and their dimensions, it is usual with him to add,
‘these suppositions will save the phenomena.’’ Save is
the literal translation of the Greek word, which is always a
part of the verb Zafw, or some of its compounds. Thus,
in treating of certain phenomena in the moon’s motion, he
lays down two hypotheses, by either of which they may
be expressed; and he concludes, “in this way the simili-
tude of the ratios, and the proportionality of the times, will
be saved (dsacwfowre) on both suppositions.”' It is plain
from these words, that the astronomer did not here consid-
er himself as describing any thing which actually existed,
but as explaining two artifices, by either of which, certain
irregularities in the moon’s motion may be represented, in
consistence with the principle of uniform velocity. The

* Mathematica Syntaxis, Lib. 1V. p. 223 of the Paris edi-
tion. Milton, the extent and accuracy of whose erudition
can never be too much admired, had probably in view this phra-
seology of Ptolemy, when he wrote the following lines :—

‘“‘ He his fabrick of the Heavens

Hath left to their disputes, perhaps to move
His laughter at their quaint opinions wide
Hereafter, when they come to model Heaven
And calculate the stars, how they will wield
The mighty frame, how build, unbuild, contrive
To save appearances, how gird the sphere

With centrick and eccentrick scribbled o’er,
Cycle and epicycle, orb in orb.”

he obsolete verb to salve is employed by Bacon, and many
other of the old English writers in the same sense with Zoey in
the work of Ptolemy here referred to. ‘ The schoolmeu were
like the astronomers, who, to salve phenomena, framed to their
conceit eccentricks and epicycles; so they. to salve the practice
of the church, had devised a great number of strange positions.”
Bacon.

azcr. 1V.] DISSERTATION SECOND. 123

hypothesis does not relate to the explanation, but merely
to the expression of the fact; it is first assumed, and its
merit is then judged of synthetically, by its power to save,
to reconcile, or to represent appearances. At atime when
the mathematical sciences extended little beyond the ele-
ments, and when problems which could not be resolved by
circles and straight lines, could hardly be resolved at
all, such artifices as the preceding were of the greatest va-
lue. They were even more valuable than the truth itself
would have been in such circumstances; and nothing is
more certain than that the real elliptical orbits of the plan-
ets, and the uniform description of areas, would have been
very unseasonable discoveries at the period we are now
treating of. The hypotheses of epicycles, and of centres
of uniform motion, were well accommodated to the state of
science, and are instances of a false system which has
materially contributed to the establishment of truth.

2. CorprerRNicus anv Tycuo.

On the revival of learning in Europe, astronomy was the
first of the sciences which was regenerated. Such, indeed,
is the beauty and usefulness of this branch of knowledge,
that, in the thickest darkness of the middle ages, the study
of it was never entirely abandoned. In those times of ig-
norance, it also derived additional credit from the assistance
which it seemed to give to an imaginary and illusive science.
Astrology, which has exercised so durable and extensive a
dominion over the human mind, is coéval with the first ob-
servations of astronomy. In the middle ages, remarkable
for the mixture of a few fragments of knowledge and truth
in a vast mass of ignorance and errour, it was assiduously
cultivated, and, in conjunction with alchemy and magick,
shared the favour of the people, and the patronage of the

124 DISSERTATION SECOND. [Pann 1.

great. During the thirteenth and fourteenth centuries, it
was taught in the universities of Italy, and professors were
appointed, at Padua and Bologna, to instruct their pupils in
the influence of the stars. Every where through Europe
the greatest respect was shown for this system of imposture,
and they who saw the deceit most clearly, could not always
avoid the disgrace of being the instruments of it. Astro-
nomy, however, profited by the illusion, and was protected
for the great assistance which it seemed to afford to a sci-
ence more important than itself.

Of those who cultivated astronomy, many were infected
by this weakness, though some were completely superiour
to it. Alphonso, the King of Castile, was among the lat-
ter. He flourished about the middle of the thirteenth
century, and was remarkable for such freedom of thought,
and such boldness of language, as it required his royal
dignity to protect. He applied himself diligently to the
study of astronomy ; he perceived the inaccuracy of Ptole-
my’s tables, and endeavoured to correct their errours by
new tables of his own. ‘These, in the course of the next
age, were found to have receded from the heavens, and it
became more and more evident that astronomers had not
yet discovered the secret of the celestial motions.

Two of the men who, in the fifteenth century, contri-
buted the most to the advancement of astronomical science,
Purbach and Regiomontanus, were distinguished also for
their general knowledge of the mathematicks. Purbach
was fixed at Vienna by the patronage of the Emperour
Frederick the Third, and devoted himself to astronomical
observation. He published’a new edition of the Almagest,
and, though he neither understood Greek nor Arabick, his
knowledge of the subject enabled him to make it much
more perfect than any of the former translations. He is
said to have been the first who applied the plummet to

‘gaer. 1v.] DISSERTATION SECOND. 125

astrenomical instruments ; but this must not be understood
strictly, for some of Ptolemy’s instruments, the paral-
lactick for instance, were placed perpendicularly by the
plumb-line.

Regiomontanus was the disciple of Purbach, and is still
more celebrated than his master. He was a man of great
learning and genius, most ardent for the advancement of
knowledge, and particularly devoted io astronomy. To
him we owe the introduction of decimal fractions, which
completed our arithmetical notation, and formed the se-
cond of the three steps by which, in modern times, the
science of numbers has been so greatly improved.

In the list of distinguished astronomers, the name of
Copernicus comes next, and stands at the head of those
men, who, bursting the fetters of prejudice and authority,
have established truth on the basis of experience and ob-
servation. He was born at Thorn in Prussia, in 1473 ;
he studied at the university of Cracow, being intended at
first for a physician, though he afterwards entered into the
church. A decided taste for astronomy led him early to
the study of the science in which he was destined to make
such an entfre revolution, and as soon as he found himself
fixed and independent, he became a diligent and careful
observer. ,

It would be in the highest degree interesting to know by
what steps he was led to conceive the bold system which
removes the earth from the centre of the world, and as-
cribes to it a twofold motion. It is probable that the
complication of so many epicycles and deferents as were
necessary, merely to express the laws of the planetary
motions, had induced him to think of all the possible sup-
positions which could be employed for the same purpose,
in order to discover which of them was the simplest, -

16

126 - DISSERTATION SECOND. {pant 3°

{t appears extraordinary, that so natural a thought should
have occurred, at so late a period, for the first, or nearly
for the first time. We are assured, by Copernicus him-
self, that one of the first considerations which offered. itself
to his mind, was the effect produced by the motion of a
Spectator, in transferring that motion to the objects ob-
served, but ascribing to it an opposite direction.' From
this principle it immediately followed, that the rotation of
the earth on an axis, from west to east, would produce the
apparent motion of the heavens in the direction from east
to west.

In considering some of the objections which might be
made to the system of the earth’s motion, Copernicus rea-
sons with great soundness, though he is not aware of the
full force of his own argument. Ptolemy had alleged, that,
if the earth were to revolve on its axis, the violence of the
motion would be sufficient to tear it in pieces, and to dis-
sipate the parts. This argument, it is evident, proceeds
on a confused notion of a centrifugal force, the effect of
which the Egyptian astronomer overrated, as much as he
undervalued the firmness and solidity of the earth. Why,
says Copernicus, was he not more alarmed for the safety
of the heavens, if the diurnal revolution be ascribed to
them, as iheir motion must be more rapid, in proportion as
their magnitude is greater ? The argument here suggested,
now that we know how to measure centrifugal force, and to
compare it with others, carries demonstrative evidence
with it, because that force, if the diurnal revolution were
really performed by the heavens, would be such, as the
forces which hold together the frame of the material world
would be wholly unable to resist.

+ Astronomia Instaurata, Lib. 1. cap. 5.

ager. 1v.] DISSERTATION SECOND. 127

There are, however, in the reasonings of Copernicus,
some unsound parts, which show, that the power of his
genius was not able to dispel all the clouds which in that
age hung over the human mind, and that the unfounded
distinctions of the Aristotelian physicks sometimes afford-
ed arguments equally fallacious to him and !o his adversa-
ries. One of his most remarkable physical mistakes was
his misconception with respect to the parallelism of the
earth’s axis; to account for which, he thought it necessary
to assume, in addition to the earth’s rotation on an axis,
and revolution round the sun, the existence of a third mo-
tion altogether distinct from either of the others. In this
he was mistaken. ; the axis naturally retains its parallelism,
and it would require the action of a force to make it do
otherwise. This, as Kepler afterwards remarked, is a con- |
sequence of the inertia of matter ; and, for that reason, he
very justly accused Copernicus of not being fully acquaint-
ed with his own riches.

The first edition of the Astronomia Instaurata, the
publication of which was solicited by Cardinal Schoen-
berg, and the book itself dedicated to the Pope, appeared
in- 1543, a few days before the death of the author.
Throughout the whole book, the new doctrine was ad-
vanced with great caution, as if from a presentiment of the
opposition and injustice which it was one day to expe-
rience. At first, however, the system attracted little no-.
tice, and was rejected by the greater part even of astrone-
mers. It lay fermenting in secret with other new dis-
coveries for more than fifty years, till, by the exertions of
Galileo, it was kindled into so bright a flame as to consume
the philosophy of Aristotle, to alarm the hierarchy of
Rome, and to threaten the existence of every opinion not
founded on experience and observation.

os

128 DISSERTATION SECOND. [PART 1.

After Copernicus, Tycho Brahé was the most distin-
guished astronomer of the sixteenth century. An eclipse
of the sun which he witnessed in 1560, when he was yet a
very young man, by the exactness with which it answered to
the prediction, impressed him with the greatest reverence
for a science which could see so far and so distinctly into
the future, and from that moment he was seized with the
strongest desire of becoming acquainted with it. Here,
indeed, was called into action a propensity nearly allied
both to the strength and the weakness of the mind of this
extraordinary man, the same that attached him, on one
hand, to the calculations of astronomy, and, on the otber,
to the predictions of judicial astrology.

In yielding hiniself up, however, to his love of astrono-
my, he found that he had several difficulties to overcome.
He belonged to a class in society, elevated, in the opinion
of that age, above the pursuit of knowledge, and jealous
of the privilege of remaining ignorant with impunity. Ty-
cho was of a noble family in Denmark, so that it required
all the enthusiasm and firmness inspired by the love of
knowledge, to set him above the prejudices of hereditary
rank, and the opposition of his ‘relations. He succeeded,
however, in these objects, and also in obtaining the pa-
tronage of the King of Denmark, by which he was enabled"
to erect an observatory, and form an establishment in the
island of Huena, such as had never yet been dedicated to
astronomy. ‘The instruments: were of far greater size,
more skilfully contrived, and more nicely divided, than any
that had yet been directed to the heavens. By means of
them, Tycho could measure angles to ten seconds, which
may be accounted sixty times the accuracy of the instru-
ments of Ptolemy, or of any that had belonged to the
school of Alexandria.

sxcr. 19.] DISSERTATION SECOND. 129

Among the improvements which he made in the art of
astronomical observation, was that of verifying the instru-
ments, or determining their errours by actual observation,
instead of trusting, as had been hitherto done, to the sup-
posed infallibility of the original construction.

One of the first objects to which the Danish astronomer
applied himself was the formation of a new catalogue of the
fixed stars. That which was begun by Hipparchus, and
continued by Ptolemy, did not give the places of the
stars with an accuracy nearly equal to that which the new
instruments were capable of reaching ; and it was besides
desirable to know, whether the lapse of twelve centuries
had produced any unforeseen changes in the heavens.

The great difficulty in the execution of this work arose
from the want of a direct and easy method of ascertaining
the distance of one heavenly body due east or west of an-
other. ‘The distance north or south, either from one
another or from a ‘fixed plane, that of the equator, was
easily determined by the common method of meridian al-
titudes, the equator being a plane which, for any given
place on the earth’s surface, retains always the same posi-
tion. But no plane extending from north to south, or pass-
ing through the poles, retains a fixed position with respect
to an observer, and, therefore, the same way of measuring
distances from such a plane cannot be applied. 'The na-
tural substitute is the measure of time; the interval
between the passage of two stars over the meridian, bear-
ing the same proportion to twenty-four hours, that the arch
which measures their distance perpendicular to the meri-
dian, or their difference of right ascension, does to four

right angles.
- ° An accurate measure of time, therefore, would answer
the purpose, but such a measure no more existed in the
age of Tycho, than it had done in the days of Hipparchus

130 DISSERTATION SECOND. [ean x.

or Ptolemy. ‘These ancient astronomers determined the
longitude of the fixed stars by referring their places to
those of the moon, the longitude of which, for a given
time, was known from the theory of her motions. ‘Thus
they were forced to depend on the most irregular of all
the bodies in the heavens, for ascertaining the positions of
the most fixed, those which ought to have been the basis
of the former, and of so many other determinations. Tycho
made use of the planet Venus instead of the moon, and
his method, though more tedious, was more accurate than
that of the Greek astronomers. His catalogue contained
the places of 777 fixed stars. »

The irregularities of the moon’s motions were his next
subjects of inquiry. The ancients had discovered the in-
equality of that planet depending on the eccentricity of
the orbit, the same which is now called the equation of the
centre.' Ptolemy had added the knowledge of another
inequality in the moon’s motion, to which the name of the
evection has been given, amounting to an increase of the
former equation at the quarters, and a diminution of it at
the times of new and full moon. Tycho discovered another
inequality, which is greatest at the octants, and depends
on the difference between the longitude of the moon and
that of the sun. A fourth irregularity to which the moon’s
motion is subject, depending wholly on the sun’s place, was
known to Tycho, but included among the sun’s equations.
Besides, these observations made him acquainted with the
changes in the inclination of the plane of the moon’s orbit ;
and, lastly, with the irregular motion of the nodes, which,
instead of being always retrograde at the same rate, are
subject to change that rate, and even to become progres-

* The allowance made for any such equality, when the place
of a planet is to be computed for a ,given time, is called an
equation in the language of astronomy.

sxcr. 1v,] DISSERTATION SECOND. 131

sive according to their situation in respect of the sun.
These are the only inequalities of the moon’s motion known
before the theory of Gravitation, and, except the two first,
are all the discoveries of Tycho.

The atmospherical refraction, by which the heavenly
bodies are made to appear more elevated above the hori-
zon than they really are, was suspected before the time
of this astronomer, but not known with certainty to exist.
He first became acquainted with it by finding that the
latitude of his observatory, as determined from observa-
tions at the solstices, and from observations of the greatest
and least altitudes of the circumpolar stars, always differed
about four minutes. The effect of refraction he supposed
to be 34’ at the horizon, and to diminish from thence up-
ward, till at 45° it ceased altogether. This last supposi-
tion is erroneous, but at 45° the refraction is less than 1%,
and probably was not sensible in the altitudes measured
with his instruments, or not distinguishable from the errours
of observation. An instrument which he contrived on
purpose to make the refraction distinctly visible, shows
the scale on which his observatory was furnished. It was
an equatorial circle of ten feet diameter, turning on an axis
parallel to that of the earth. ‘With the sights of this equa-
torial he followed the sun on the day of the summer sol-
stice, and found, that, as it descended towards the horizon,
it rose above the plane of the instrument. At its setting,
the sun was raised above the re by more than its own
diameter.

The comet of 1570 was carefully observed by Tycho,
and gave rise to a new theory of those bodies. He found
the horizontal parallax to be 20’, so that the comet was
nearly three times as far off as the moon. He considered
comets, therefore, as bodies placed far beyond the reach
of our atmosphere, and moving round the sun.. This wae

mt

132 DISSERTATION SECOND. [part 1.

a severe blow to the physicks of Aristotle, which regarded
comets as meteors generated in the atmosphere. His ob-
servation of the new star in 1572, was no less hostile to
the argument of the same philosopher, which maintained,
that the heavens are a region in which there is neither
generation nor corruption, and in which existence has
neither a beginning nor an end.

Yet Tycho, with this knowledge of astronomy, and
after having made ebservations more numerous and accu-
rate than all the astronomers who went before him, con-
tinued to reject the system of Copernicus, and to deny
the motion of the earth. He was, however, convinced .
that the earth is not the centre about which the planets
revolve, for he had himself observed Mars, when in oppo-
sition, to be nearer to the earth than the earth was to the
sun, so that, if the planets were ranged as in the Ptole-
maick system, the orbit of Mars must have been within
the orbit of the sun. He therefore imagined the system
still known by his name, according to which the sun moves
round the earth, and is at the same time the centre of the
planetary motions. It cannot be denied, that the phe-
nomena purely astronomical may be accounted for on this
hypothesis, and that the objections to it are rather derived
from physical and mechanical considerations, than from the
appearances themselves. It is simpler than the Ptole-
maick system, and free from its inconsistencies ; but it is
more complex than the Copernican, and, in no respect,
affords a better explanation of the phenomena. The true
place of the Tychonick system is between the two former ;
an advance beyond the one, and a step short of the other ;
and such, if the progress of discovery were always perfectly
regular, is the place which it would have occupied in the
history of the science. If Tycho had lived before Co-
pernicus, his sysiem would have been a step in the ad-

sxer. 1v.] DISSERTATION SECOND. 135

vancement of knowledge ; coming after him, it was a step
backward.

It is not to his credit as a philosopher to have made this
retrograde movement, yet he is not aliegether without apo-
logy. The physical arguments in favour of the Coperni-
can system, founded on the incongruity of supposing the
greater body to move round the smaller, might not be sup-
posed to have much weight, in an age when the equality of
action and reaction was unknown, and when it. was not
clearly understood that the sun and the planets act at all
on one another. The arguments, which seem, in the judg-
ment of Tycho, to have. balanced the simplicity of the
Copernican system, were founded on certain texts of
Scripture, and on the difficulty of reconciling the motion
of the earth with the sensations which we experience at its
surface, or the phenomena which we observe, the same, in

_all respects, as if the earth were at rest. The experiments
and reasonings of Galileo had not yet instructed men in
the inertia of matter, or in the composition of motion ; and
the followers of Copernicus reasoned on principles which
they he!d in common with their adversaries. A ball, it
was said by the latter, dropt from the mast-head of a ship
under sail, does not fall at the foot of the mast, but some-
what behind it; and, in the same manner, a stone, dropt
from a high tower, would not fall, on the supposition of the
earth’s motion, at the bottom of the tower, but to the west
of it, the earth, during its fall, having gone eastward from
under it. The followers of Copernicus were not yet pro-
vided with the true answer to this objection, viz. that the’
ball does actually fall at the bottom of the mast. It was
admitted that it must fall behind it, because the ball was
no part of the ship, and that the motion forward was not
natural, either to the ship or to the ball. The stone, on
the other hand, let fall from the top of the tower, was a part

7

134 DISSERTATION SECOND. [parr 2.

of the earth; and, thecitafaets the diurnal and annual revo-
lutions which were natural to the earth, were also natural
to the stone; the stone would, therefore, retain the same
motion with the tower, and strike the ground precisely at
the bottom of it.

It must be confessed, that neither of these logicians had
yet thoroughly awakened from the dreams of the Aristote-
lian metaphysicks, but men were now in possession of the
truth, which was finally to break the spell, and set the
mind free from the fetters of prejudice and authority. An-
other charge, against which it is more difficult to defend
Tycho, is his belief in the predictions of astrology. He
even wrote a treatise in defence of this imaginary art, and
regulated his conduct continually by its precepts. Credu-
lity, so unworthy of a man deeply versed in real science,
is certainly to be set down less to his own account than

to that of the age in which he lived.

3. Kerxter Aanp GALILEO.

Kepler followed Tycho, and in his hands astronomy
underwent a change only second to that which it had un-
dergone in the hands of Copernicus. He was born in
1571. He early applied himself to study and observe the
heavens, and was soon distinguished as an inventor. He
began with taking a more accurate view of astronomical
refraction than had yet been done, and he appears to have
been the first who conceived that there must be a certain
fixed law which determined the quantity of it, correspond-
ing to every altitude, from the horizon to the zenith. The
application of the principles of opticks to astronomy, and
the accurate distinction between the optical and real in-
equalities of the planets, are the work of the same astrono-

sitc?. 1v.] DISSERTATION SECOND. 135

mer. It was by the views thus presented that he was led
to the method of constructing and calculating eclipses, by
means of projections, without taking into consideration the
diurnal parallax. These are valuable improvements, but
they were, however, obscured by the greatness of his fu-
ture discoveries.

The planes of the orbits of the planets were naturally,
in the Ptolemaick system, supposed to pass through the
earth, and the reformation of Copernicus did not go so far
as to change the notions on that subject which had gene-
rally been adopted. Kepler observed that the orbits of the
planets are in planes passing through the sun, and that, of
consequence, the lines of their nodes all intersect in the
centre of that luminary. This discovery contributed es-
sentially to those which followed.

~The oppositions of the planets, or their places when
they pass the meridian at midnight, offer the most fayour-
able opportunities for observing them, both because they
are at that time nearest to the earth, and because their
places seen from thence are the same as if they were seen
from the sun. The true time of the opposition had, how-
ever, been till now mistaken by astronomers, who held it to
be at the moment when the apparent place of the planet
was opposite to the mean place of the sun. It ought, how-
ever, to have been, when the apparent places of both were
opposed to one another. This reformation was proposed
by Kepler, and, though strenuously resisted by Tycho,
was finally received.

‘Having undertaken to examine the orbit of Mars, i in which
the irregularities are most considerable, Kepler discovered,
by comparing together seven oppositions of that planet,
that its orbit is elliptical ; that the sun is placed in one of
the foci; and that there is no point round which the angu-
lar motion is uniform. In the pursuit of this inquiry he

136 DISSERTATION SECOND. / [pant a.

found that the same thing is true of the earth’s orbit round
the sun; hence by analogy it was reasonable to think,
that all the planetary orbits are elliptical, having the sun
in their common focus.

The industry and patience of Kepler, in this investiga-
tion, were not less remarkable than his ingenuity and in-
vention. Logarithms were not yet known, so that arith-
metical computation, when pushed to great accuracy, was
carried on at a vast expense of time and labour. In the
calculation of every opposition of Mars, the work filled ten
folio pages, and Kepler repeated each calculation ten
times, so that the whole work for each opposition extended
to one hundred such pages; seven oppositions thus calcu-
lated produced a large folio volume.

In these calculations the introduction of hypotheses was
unavoidable, and Kepler’s candour in rejecting them,
whenever they appeared erroneous, without any other re-
eret than for the time which they had cost him, cannot be
sufficiently admired. He began with hypothesis, and ended
with rejecting every thing hypothetical. In this great as-
tronomer we find genius, industry, and candour, all uniting
together as instruments of investigation.

Though the angular motion'of the planet was not found
to be uniform, it was discovered that a very simple law
connected that motion with the rectilineal distance from
the sun, the former being every .where inversely as the
square of the latter; and hence it was easy to prove, that
the area, described by the line drawn from the planet to
ithe sun, increased at a uniform rate, and, therefore, that
any two such areas were proportional to the times in which
they were described. The picture presented of the hea-
vens was thus, for the first time, cleared of every thing
hypothetical.

Ker. 1V.) DISSERTATION SECOND. 137

The same astronomer was perhaps the first person who
conceived that there must be always a law capable of
being expressed by arithmetick or geometry, which con-
nects such phenomena as have a physical dependence on
one another. His conviction of this truth, and the delight
which he appears to have experienced in the contempla-
tion of such laws, led him ‘to seek, with great eagerness,
for the relation between the periodical times of the planets,
and their distances from the sun. He seems, indeed, to
have looked towards this object with such earnestness,
that, while it was not attained, he regarded all his other
discoveries as incomplete. He at last found, infinitely to
his satisfaction, that in any two planets, the squares of the
times of the revolution are as the cubes of their mean dis-
tances from the sun. This beautiful and simple law had a
value beyond what Kepler could possibly conceive ; yet a
sort of scientifick instinct instructed him in its great im-
portance. He has marked the year and the day when it
became known to him; it was on the 8th of May, 1618;
and perhaps philosophers will: agree, that there are few
days in the scientifick history of the world which deserve
so well to be remembered.

These great discoveries, however, were not much at-
tended to by the astronomers of that period, or by those
who immediately followed. They were but little con-
sidered by Gassendi,—they were undervalued by Riccioli,
—and were never mentioned by Descartes. It was an
honour reserved for Newton to estimate them at their true
value.

- Indeed, the discoveries of Kepler were at first so far
from being duly appreciated, that they were objected to,
not for being false, but for offering to astronomers, in the
calculation of the place of a planet in its orbit, a problem
too difficult to be resolved by elementary geometry. To

138 DISSERTATION SECOND. [parr 1.

cut the area ofa semi-ellipsis. in a given ratio by a line
drawn through the focus, is the geometrical problem into
which he showed that the above inquiry ultimately re-
solved. . As if he had been answerable for the proceedings
of nature, the difficulty of this question was considered as
an argument against his theory, and he himself seems
somewhat to have felt it as an objection, especially when
he found that the best solution he could obtain was no
more than an approximation. With all his power of in-
vention, Kepler was a mathematician inferiour to many of
that period; and though he displayed great ability in the |
management of this difficult investigation, his solution fell
very far short of the simplicity which it was afterwards
found capable of attaining.

In addition to all this, he rendered another very impor-
tant service to the science of astronomy and to the system
of Copernicus. Copernicus, it has been already mention-
ed, had supposed that a force was necessary to enable
the earth to preserve the parallelism of its axis during its
revolution round the sun. He imagined, therefore, that a
third motion belonged to the earth, and that, besides turn-
ing on its axis and revolving round the sun, it had another
movenient by which its axis was preserved always equally
inclined to the ecliptick. Kepler was the first to observe
that this third motion was quite superfluous, and that the
parallelism of the earth’s axis, in order to be preserved,
required nothing but the absence of all force, as it neces-
sarily proceeded from the inertia of matter, and its tenden-
cy to persevere ina state of uniform motion. Kepler had
a clear idea of the inertia of body; he was the first who
employed the term ; and, considering all motion as’ natu-
rally rectilineal, he concluded that when a body moves in
a curve, it is drawn or forced out of the straight line by
the action of some cause, not residing in itself. Thus he

txcr. 1v.] DISSERTATION SECOND. 139

prepared the way for physical astronomy, and in these
ideas he was earlier than Descartes.

The discoveries of Kepler were secrets extorted from
nature by the most profound and laborious research. The
astronomical discoveries of Galileo, more brilliant and im-
posing, were made at a far less expense of intellectual !a-
bour. By this it is not meant to say that Galileo did not
possess, and did not exert intellectual powers of the very
highest order, but it was less in his astronomical discove-
ries that he had occasion to exert them, than in those
which concerned the theory of motion. The telescope
turned to the heavens for the first time, in the hands of a
man far inferiour to the Italian philosopher, must have un-
folded a series of wonders to astonish and delight the
world.

It was in the year 1609 that the news of a discovery,
made in Holland, reached Galileo, viz. that two glasses
had been so combined, as greatly to magnify the objects
seen through them. More was not told, and more was not
necessary to awaken a mind abundantly alive to all that
interested the progress either of science or of art. Galileo
applied himself to try various combinations of lenses, and
he quickly fell on one which made objects appear greater
than when seen by the naked eye, in the proportion of
three to one. He soon improved on this construction, and
found one which magnified thirty-two times, nearly as
much as the kind of telescope he used is capable of.
That telescope was formed of two lenses ; “the lens next
the object convex, the other concave; the objects were
presented upright, and magnified in their lineal dimensions
in the proportion just assigned.

Having tried the effect of this combination on terres-
trial objects, he next directed it to the moon. What the
telescope discovers on the ever-varying, face of that lumi.

140 DISSERTATION SECOND. [pan 1.

nary, is now well known, and needs not to be described ;
but the sensations which the view must have communicat-
ed to the philosopher who first beheld it, may be con-
ceived more easily than expressed. To the immediate
impression which they made upon the sense, to the won-
der they excited in all who saw them, was added the
proof, which, on reflection, they afforded, of the close re-
semblance between the earth and the celestial bodies,
whose divine nature had been so long and so erroneously
contrasted with the ponderous and opaque substance of
our globe. The earth and the planets were now proved
to be bodies of the same kind, and views were entertained
of the universe, more suitable to the simplicity, and the
magnificence of nature. _

When the same philosopher directed his telescope to
the fixed stars, if he was disappointed at finding their magni-
tudes not increased, he was astonished and delighted to
find them multiplied in so great a degree, and such num-
bers brought into view, which were invisible to the naked
eye. In Jupiter he perceived a large disk, approaching
in size to the moon. Near it, as he saw it for the first
time, were three luminous points ranged in a straight line,
two of them on one side of the planet, and ene on the
other. This occasioned no surprise, for they might be
small stars not visible to the naked eye, such as he had
already discovered in great numbers. By observing them,
however, night after night, he found these small stars to be
four in number, and to be moons or satellites, accempany-
ing Jupiter, and revolving round him, as the moon revolves
round the earth.

The eclipses of these satellites, their conjunctions with
the planet, their disappearance behind his disk, their
periodical revolutions, and the very problem of distinguish-

axcr. 1v.] DISSERTATION SECOND. 141

ing them from one another, offered, to an astronomer, a
series of new and interesting observations.

In Saturn he saw one large disk, with two smaller ones
very near if, and diametrically oppusite, and always seen
in the same places; but more powerful telescopes were
required before these appearances could be interpreted.

The horned figure of Venus, and the gibbosity of Mars,
added to the evidence of the Copernican system, and veri-
fied the conjectures of its author, who had ventured to say,
that, if the sense of sight were sufficiently powerful, we
should see Mercury and Venus exhibiting phases similar
to those of the moon.

The spots of the sun derived an inferest from their con-
trast with the luminous disk over which they seemed to
pass. They were found to have such regular periods of
return, as could be derived only from the motion of the
disk itself; and thus the sun’s revolution on his axis, and
the time of that revolution, were clearly ascertained.

This succession of noble discoveries, the most splendid,
probably, which it ever fell to the lot of one individual to
make, in a better age would have entitled its author to the
admiration and gratitude of the whole scientifick world,
but was now viewed from several quarters with suspicion
and jealousy. The ability and success with which Galileo
had laboured to overturn the doctrines of Aristotle and the
schoolmen, as well as to establish the motion of the earth,
and the immobility of the sun, had excited many enemies.
There are always great numbers who, from habit, indo-
lence, or fear, are the determined supporters of what is es-
. tablished, whether in practice or in opinion. To these
the constitution of the universities of Europe, so entirely
subjected to the church, had added a numerous and learn-
ed phalanx, interested to preserve the old systems, and to
resist all innovations which could endanger their authority

18

142 DISSERTATION SECOND. © [pant r.

or their repose. ‘The church itself was roused to action,
by reflecting that it had staked the infallibility of its judg-
ments on the truth of the very opinions which were now in
danger of being overthrown. Thus was formed a vast
combination of men, not very scrupulous about the means
which they used to annoy their adversaries; the power
was entirely in their hands, and there was nothing but
truth and reason to be opposed to it.

The system of Copernicus, however, while it remained
obscure, and known only to astronomers, created no alarm
in the church. It had even been ushered into the world
at the solicitation of a cardinal, and under the patronage of
the Pope; but when it became more popular, when the
ability and acuteness of Galileo were enlisted on its side,
the consequences became alarming ; and it was determined
to silence by force an adversary who could not be put
down by argument. His dialogues contained a full exposi-
tion of the evidence of the earth’s motion, and set forth
the errours of the old, as well as the discoveries of the
new philosophy, with great force of reasoning, and with the
charms of the most lively eloquence. They are written,
indeed, with such singular felicity, that one reads them at
the present day, when the truths contained in them are
known and admitted, with all the delight of novelty,
and feels one’s self carried back to the period when the
telescope was first directed to the heavens, and when the
earth’s motion, with all its train of consequences, was
proved for the first time. The author of such a work
could not be forgiven. Galileo, accordingly, was twice
brought before the Inquisition. The first time a council
of seven cardinals pronounced a sentence which, for the
sake of those disposed to believe that power can subdue
truth, ought never to be forgotten: ‘* That to maintain the
sun to be immovable, and without local motion, in the

sxcr. 1v.] DISSERTATION SECOND. 143

centre of the world, is an absurd proposition, false in phi-
losophy, heretical in religion, and contrary to the testimony
of Scripture. That it is equally absurd and false in phi-
losophy to assert that the earth is not immovable in the
centre of the world, and, considered theologically, equally
erroneous and heretical.”

These seven theologians might think themselves official-
ly entitled to decide on what was heretical or orthodox in
faith, but that they should determine what was true or
false in philosophy, was an insolent invasion of a territory
inte which they had no right to enter, and is a proof how
ready men are to suppose themselves wise, merely be-
cause they happen te be powerful. At this time a promise
was extorted from Galileo, that he would not teach the
doctrine of the earth’s motion, either by speaking or by
writing. ‘To this promise he did not conform. His third
dialogue, published, though not till long afterwards, con-
tained such a full display of the beauty and simplicity of
the new system, and such an exposure of the inconsisten-
cies of Ptolemy and Tycho, as completed the triumph of
Copernicus.

In the year 1663, Galileo, now seventy years old, being
brought before the Inquisition, was forced solemnly to
disavow his belief in the earth’s motion; and condemned
to perpetual imprisonment, though the sentence was after-
wards mitigated, and he was allowed to return to Flo-
rence. The Court of Rome was very careful to publish
this second recantation all over Europe, thinking, no doubt,
that it was administering a complete antidote to the be-

* He was thrown into prison previously to his trial, and at-
tempts were made to render him obnoxious to the people. From
the text of a priest who preached against him, we may judge
of the wit and the sense with which this persecution was cen-
ducted. Virt Galilaci quid statis in caelum suspicientes ?

144 DISSERTATION SECOND. [PART 1.

lief of the Copernican system. The sentence, indeed, ap-
pears to have pressed very heavily on Galileo’s mind, and
he never afterwards either talked or wrote on the subject
of astronomy. Such was the triumph of his enemies, on
whom ample vengeance would have long ago been execut-
ed, if the indignation and contempt of posterity could
reach the mansions of the dead.

Conduct like this, in men professing to be the ministers
of religion and the guardians of truth, can give rise to none
but the most painful reflections. That an aged philoso-
pher should be forced, laying his hand on the sacred
Scriptures, to disavow opinions which he could not cease
io hold without ceasing to think, was as much a _profana-
tion of religion, as a violation of truth and justice. Was it
the act of hypocrites, who considered religion as ‘a state.
engine, or of bigots, long trained in the art of believing
without evidence, or even in opposition to it? These
questions it were unnecessary to resolve ; but one conclu-
sion cannot be denied, that the indiscreet defenders of re-
ligion have often proved its worst enemies.

At length, however, by the improvements, the discove-
ries, and the reasonings, first of Kepler, and then of Gali-
leo, the evidence of the Copernican system was fully
developed, and nothing was wanting to its complete es-
tablishment, but time sufficient to allow opinion to come
gradually round, and to give men an opportunity of study-
ing the arguments placed before them. Of the adherents
of the old system, many had been too long habituated to it
to change their views; but as they disappeared from the
scene, they were replaced by young astronomers, not under
the influence of the same prejudices, and eager to follow
doctrines which seemed to offer so many new subjects of
investigation. In the next generation the systems of Ptole-
my and Tycho had no followers.

sBcr. 1¥.] DISSERTATION SECOND. 145

It was not astronomy alone which was benefited by this
revolution, and the discussions to which it had given rise.
A new light, as already remarked, was thrown on the phy-
sical world, and the curtain was drawn aside which had so
long concealed the great experiment, by which nature her-
self manifests, at every instant, the inertia of body, and the
composition of forces. To reconcile the real motion of the
earth with its appearance of rest, and with our feeling of its
immobility, required such an examination of the nature of
motion, as discovered, if not its essence, at least its most
general and fundamental properties. The whole science of
rational mechanics profited, therefore, essentially by the
discovery of the earth’s motion. 5

A great barrier to philosophick improvement had arisen
from the separation so early made, and so strenuously sup-
ported in the ancient systems, between terrestrial and celes-
tial substances, and between the laws which regulate motion
on the earth, and in the heavens. This barrier was now
entirely removed ; the earth was elevated to the rank of a
planet; the planets were reduced to the condition of earths,
and by this mutual approach, the same rules of interpreta-
tion became applicable to the phenomena of both. Princi-
ples derived from experiments on the earth, became guides
for the analysis of the heavens, and men were now in a
situation to undertake investigations, which the mest hardy
adventurer in science could not before have dared to ima-
gine. Philosophers had ascended to the knowledge of the
affinities which pervade all nature, and which mark so
strongly both the wisdom and unity of its author.

_ The light thus struck out darted its rays into regions the
most remote from physical inquiry. When men saw opi-
nions entirely disproved, which were sanctioned by all anti-
quity, and by the authority of the greatest names, they be-
gan to have different notions of the rules of evidence, of the

146 _ DISSERTATION SECOND. [rant i.

principles of philosophick inquiry, and of the nature of the
mind itself. It appeared that science was destined to be
continually progressive ; provided it was taken for an invio-
lable maxim, that all opinion must be ultimately amenable to
experience and observation.

It was no slight addition to all these advantages, that, in
consequence of the discussions from which Galileo had un-
happily been so great a sufferer, the line was at length
definitely drawn, which was to separate the provinces of faith
and philosophy from one another. It became a principle,
recognised on all hands, that revelation, not being intended
to inform men of those things which the unassisted powers
of their own understanding would in time be able to discover,
had, in speaking of such matters, employed the language
and adopted the opinions of the times ; and thus the magick
circle by which the priest had endeavoured to circumscribe
the inquiries of the philosopher entirely disappeared.—
The reformation in religion which was taking place about
the same time, and giving such energy to the human mind,
contributed to render this emancipation more complete, and.
to reduce the exorbitant pretensions of the Romish church.
The prohibition against believing in the true system of the
world either ceased altogether, or was reduced to an empty
form, by which the affectation of infallibility still BregEye?
the memory of its errours.?

? The learned fathers who have, with so much ability, com-
mented on the Principia of Newton, have prefixed to the third
book this remarkable declaration :—“ Newtonus in hoc tertio
libro telluris motae hypothesin assumit. Auctoris propositiones
aliter explicari non poterant, nisi eadem fact& hypothesi. Hinc
alienam coacti sumus gerere personam. Ceterum latis a summis
Pontificibus contra telluris motum Decretis nos obsequi profitemur.”
There is an archness in the last sentence, that looks as if the
authors wanted to convey meanings that would differ according
to the orthodoxy of the readers.

sxcr. 1y.] DISSERTATION SECOND. 147

4. Descartes, Huygens, &c.

Descartes flourished about this period, and has the merit
of being the first who undertook to give an explanation of
the celestial motions, or who formed the great and philoso-
phick conception of reducing all the phenomena of the

“universe to the same law. The time was now arrived
when, from the acknowledged assimilation of the planets to
the earth, this might be undertaken with some reasonable
prospect of success. No such attempt had hitherto been
made, unless the crystalline spheres or homocentrick orbs
of the ancients are to be considered in that light. The
conjectures of Kepler about a kind of animation, and of or-
ganick structure, which pervaded the planetary regions, were
too vague and indefinite, and too little analogous to any thing
known on the earth, to be entitled to the name of a theory.
To Descartes, therefore, belongs the honour of being the
first who ventured on the solution of the most arduous pro-
blem which the material world offers to the consideration of
philosophy. For this solution he sought no other data than
matter and motion, and with them alone proposed to explain
the structure and constitution of the universe. The matter
which he required, too, was of the simplest kind, possessing
no properties but extension, impenetrability, and inertia.
It was matter in the abstract, without any of its peculiar or
distinguishing characters. To explain these characters,
was indeed a part of the task which he proposed to himself,
and thus, by the simplicity of his assumptions, he added
infinitely to the difficulty of the problem which he under-
took to resolve.

The matter thus constituted was supposed to fill all
space, and its parts, both great and small, to be endued
with motion in an infinite variety of directions. From the

148 DISSERTATION SECOND. [PART 1.

combination of these, the rectilineal motion of the parts
became impossible ; the atoms or particles of matter were
continually diverted from the lines in which they had be-

gun to move; so that circular motion and centrifugal force

originated from their action on one another. ‘Thus matter
came to be formed into a multitude of vortices, differing
in extent, in velocity, and in density ; the more subtile
parts constituting the real vortex, in which the denser bo-
dies float, and by which they are pressed, though not equal-
ly, on all sides.

Thus the universe consists of a multitude of vortices,
which limit and circumscribe one another. The earth and
the planets are bodies carried round in the great vortex of
the solar system ; and by the pressure of the subtile mat-
ter, which circulates with great rapidity, and great centri-
fugal force, the denser bodies, which have less rapidity,
and less centrifugal force, are forced down towards the
sun, the centre of the vortex. In like manner, each planet
is itself the centre of a smaller vortex, by the subtile matter
of which the phenomena of gravity are produced, just as
with us at the surface of the earth.

The gradation of smaller vortices may be continued in the
same manner, te explain the cohesion of the grosser bodies,
and their other sensible qualities. But I forbear to enter
into the detail of a system, which is now entirely exploded,
and so inconsistent with the views of nature which have
become familiar to every one, that such details can hardly
be listened to with patience. Indeed, the theory of vorti-
ces did not explain a single phenomenon in a satisfactory
manner, nor is there a truth of any kind which has been
brought to light by means of it. None of the peculiar pro-

perties of the planetary orbits were taken into the account ; _

none of the laws of Kepler were considered ; nor was any

explanation given of those laws, more than of any other that

exe?. tv.) DISSERTATION SECOND. 149

might be imagined. The philosophy of Descartes could
explain all things equally well, and might have been ac-
commodated to the systems of Ptolemy or Tycho, just as
well as to that of Copernicus. It forms, therefore, no link
in the chain of physical discovery ; it served the cause of
truth only by exploding errours more pernicious than its
own; by exhausting a source of deception, which might
have misled other adventurers in science, and by leaving
a striking proof, how little advancement can be made in
philosophy, by pursuing any path but that of experiment
and induction. Descartes was, nevertheless, a man of
great genius, a deep thinker, of' enlarged views, and entire-
ly superiour to prejudice. Yet, in as far as the explanation
of astronomical phenomena is concerned (and it was his
main object,) he did good only by showing in what quarter
the attempt could not be made with success; he was the
forlorn hope of the new philosophy, and must be sacrificed
for the benefit of those who were to follow.

Gassendi, the contemporary and countryman of Descar-
tes, possessed great learning, with a very clear and sound
understanding. He was a good observer, and an enlight-
ened advocate of the Copernican system. He explained,
in a very satisfactory manner, the connexion between the
laws of motion and the motion of the earth, and made ex-
periments to show, that a body carried along by another
acquires a motion which remains after it has ceased to be so
carried. Gassendi first observed the transil of a planet over
the disk of the sun,—that of Mercury, in 1631. Kepler
had predicted this transit, but did not live to enjoy a
spectacle which afforded so- satisfactory a proof of the
truth of his system, and of the accuracy of his astronomical
tables.

The first transit of Venus, which was observed, hap-
pened a few years later, in 1639, when it was seen in En-

19

150 DISSERTATION SECOND. {pany y.

gland by Hortex Mina his friend Crabtree, and by them
only. Horrox, who was a young man of great genius, had
himself calculated the transit, and foretold the time very
accurately, though the astronomical tables of that day gave
different results, and those of Kepler, in which he confided
the most, were, in this instance, considerably in errour.
Horrox has also the merit of being among the first who
rightly appreciated the discoveries of the astronomer just
named. He had devoted much time to astronomical ob-
servation, and, though he died very young, he left be-
hind him some preparations for computing tables of the
moon, ona principle which was new, and which Newton
himself thought worthy of being adopted in his theory of
the inequalities of that planet.

The first complete system of astronomy, in which the
elliptick orbits were introduced, was the Astronomia Phi-
lolaica of Bullialdus (Bouillaud,) published in 1645. They
were introduced, however, with such hypothetical addi-
tions, as show that the idea of a centre of uniform motion
had not yet entirely disappeared. It is an idea, indeed,
which gives considerable relief to the imagination, and it
besides leads to methods of calculation more simple than
the true theory, and Bullialdus may have flattered himself
that they were sufficiently exact. He conceives the ellip-
tick orbit as a section of an oblique cone, the axis of which
passes through the superiour focus of the ellipse, while
the planet moves in its circumference in such a manner,
that a plane passing through it and through the axis, shall
be carried round with a uniform angular velocity. It is
plain that the cone and its axis are mere fictions, arbitrari-
ly assumed, and not even possessing the advantage of sim-
plicity. The author himself departs from this hypothesis,
and calculates the places of a planet, on the supposition
that it moves in the circumference of an epicycle, and the

> eee

szer. 1v.) DISSERTATION SECOND. 151

epicycle in the circumference of an eccentrick deferent,
both angular motions being unifotm, that of the planet in
the epicycle being retrograde, and double the other. The
figure thus described may be shown to be an ellipse, but
the line drawn from the planet to the focus does not cut off
areas proportional to the time. ’

An hypothesis advanced by Ward, Bishop of Salisbury,
was simpler and more accurate than that of the French
astronomer. According to it, the line drawn from a planet
to the superiour focus of its elliptick orbit, turns with a
uniform angular velocity round that point. In orbits’ of
small eccentricity, this is nearly true, and almost coincides
in such cases with Kepler’s principle of the uniform de-
scription of areas. Dr. Ward, however, did not consider
the matter in that light ; he assumed his hy pothesis as true,
guided, it would seem, by nothing but the opinion, that a
centre of uniform motion must somewhere exist, and pleased
with the simplicity thus introduced into astronomical calcu-
lation. It is, indeed, remarkable, as Montucla has cbserv-
ed, how little the most enlightened astronomers of that time
seem to have studied or understood the laws discovered by
Kepler. Riccioli, of whom we are just about to speak,
enumerates all the suppositions that had been laid down con-
cerning the velocities of the planets, but makes no mention
of their describing equal areas in equal times round the sun.
Even Cassini, great as he was in astronomy, cannot be
entirely exempted from this censure.

Riccioli, a good observer, and a learned and diligent
compiler, has ‘collected all that was known in astronomy
about the middle of the seventeenth century, in a volumi-
nous work, the New Almagest. Without much originality,
he was a very useful author, having had, as the historian of
astronomy remarks, the courage and the industry to read,
to know, and to abridge every thing. He was, neverthe-

152 DISSERTATION SECOND. [PART 1.

less, an enemy to the Copernican system, and has the dis-
_ credit of having measured the evidence for and against that
system, not by the weight, but by the number of the argu-
ments. ‘The pains which he took to prop the falling edi-
fice of deferents and epicycles, added to his misapprehend-
ing and depreciating the discoveries of Kepler, subject him
to the reproach of having neither the genius to discover
truth, nor the good sense to distinguish it when discovered.
He was, however, a priest and a jesuit; he had seen the
fate of Galileo ; and his errours may have arisen from want of
courage, more than from want of discernment.

Of the phenomena which the telescope in the hands of
Galileo had made known, the most paradoxical were those
exhibited by Saturn; sometimes attended by two globes,
one on each side, without any relative motion, but which
would, at stated times, disappear for a while, and leave the
planet single, like the other heavenly bodies. Nearly forty
years had elapsed, without any further insight into these
mysterious appearances, when Huygens began to examine
the heavens with telescopes of his own construction, better
and more powerful than any which had yet been employed.
The two globes that had appeared insulated, were now seen
connected by a circular and luminous belt, going quite
round the planet. At last, it was found that all these ap-
pearances resulted from a broad ring surrounding Saturn,
and seen obliquely from the earth. The gradual manner in
which this truth unfolded itself is very interesting, and has
been given with the detail that it deserves by Huygens in
his Systema Salurnium.

The attention which Huygens had paid to the ring of
Saturn, led him to the discovery of a satellite of the same
planet. His telescopes were not powerful enough to dis-
cover more of them than one; he believed, indeed, that
there were no more, and that the number of the planets now

a

a

———

seer. 1V.} DISSERTATION SECOND. 153

discovered was complete. ‘The reasoning by which he
convinced himself, is a proof how slowly men are cured of
their prejudices, even with the best talents and the best in-
formation. - The planets, primary and secondary, thus made
up twelve, the double of six, the first of the perfect num-
bers. In 1671, however, Cassini discovered another satel-
lite, and afterwards three more, making five in all, which the
more perfect telescopes of Dr. Herschell have lately aug-
mented to seven.

To the genius of Huygens astronomy is indebted for an
addition to its apparatus, hardly less essential than the
quadrant and the telescope. An accurate measure of time
is of use even in the ordinary business of life, but to the
astronomer is infinitely valuable. The dates of his obser-
vations, and an accurate estimate of the time elapsed be-
tween them, is necessary, in order to make them lead to any
useful consequences. Besides this, the only way of mea-
suring with accuracy those arches in the heavens, which
extend from east to west, or which are parallel to the equa-
tor, depends on the earth’s rotation, because such an arch
bears the same proportion to the entire circumference of a
circle, that the time of its passage under the meridian bears
to an entire day. The reckoning of time thus furnishes
the best measure of position, as determined by arches
parallel to the equator, whether on the earth or in the hea-
vens.

Though the pendulum afforded a measure of time, in
itself of the greatest exactness, the means of continuing its
motion, without disturbing the time of its vibrations, was
yet required to be found, and this, by means of the clock,
Huygens contrived most ingeniously to effect. ° Each vi-
bration of the pendulum, by means of an arm at right an-
gles to it, allows the tooth of a wheel to escape, the wheel
being put in motion by a weight. The wheel is so contriv-

154 DISSERTATION SECOND. [Parr 1.

ed, that the force with which it acts is just sufficient to
restore to the pendulum the motion which it had lost by the
resistance of the air, and the friction at the centre of mo-
tion. Thus the motion of the clock is continued without
. any diminution of its uniformity, for any length of time.

The telescope had not yet served astronomy in all the
capacities in which it could be useful. Huygens, of whose
inventive genius the history of science has so much to re-
cord, applied it to the measurement of small angles, form-
ing it into the instrument which has since been called a
micrometer. By introducing into the focus of the teles-
cope a round aperture of a given size, he contrived to
measure the angle which that aperture subtended to the
eye, by observing the time that a star placed near the
equator required to traverse it. When the angle subtend-
ed by any other object in the telescope was to be measur-
ed, he introduced into the focus a thin piece of metal, just
sufficient to cover the object inthe focus. The proportion
of the breadth of this plate, to the diameter of the aperture
formerly measured, gave the angle subtended by the image
in the focus of the telescope. This contrivance is describ-
ed in the Systema Saturnium, at the end.

The telescope has farther contributed materially to the
accuracy of astronomical observation, by its application
to instruments used for measuring, not merely small an-
gles, but angles of any magnitude whatever. The te-
lescope here comes in place of the plain sights with
which the index or allidad of an instrument used to be
directed to an object, and this substitution has been ac-
companied with two advantages. The disk of a star
is never so well defined to the naked eye as it is in the
telescope. Besides, in using plain sights, the eye adapts
itself to the farther off of the two, in order that its
aperture may be distinctly seen. Whenever this ad-

sect. tv.) DISSERTATION SECON 155

justment is made, the object seen through the aperture
necessarily appears indistinct to the eye, which is then
adapted toa near object. This circumstance produces an
uncertainty in all such observations, which, by the use of
the telescope, is entirely removed.

But the greatest advantage arises from the magnifying
power of the telescope, from which it follows, that what is
a mere point to the naked eye, is an extended line which
can be divided into a great number of parts when seen
through the former. The best eye, when not aided by
glasses, is not able to perceive an object which subtends an
angle less than half a minute, or thirty seconds. When
the index of a quadrant, therefore, is diyected by the naked
eye to any point in the heavens, we cannot be sure that
it is nearer than half a minute on either side of that point.
But when we direct the axis of a telescope, which magnifies
thirty times, to the same object, we are sure that it is within
the thirtieth part of half a minute, that is, within one second
of the point aimed at. ‘Thus the accuracy, caeteris pari-
bus, is proportional to the magnifying power.

The appiication of the telescope, however, to astronomi-
cal instruments, was not introduced without opposition. He-
velius of Dantzic, the greatest observer who had been since
Tycho Brahé, who had furnished his observatory with the
best and largest instruments, and who was familiar with the
use of the telescope, strenuously maintained the superiority
of the plain sights. His principal argument was founded
on this,—that, in plain sights, the line of collimation is de-
termined in its position by two fixed points at a considera-
ble distance from one another, viz. the centres of the two
apertures of the sights, so that it remains invariable with
respect to the index.

In the case of the telescope there was one fixed point,
the intersection of the wires in the focus of the eye glass ;

oe.

ard

156 DISSERTATION SECOND. . frarr i.

bat Hevelius did not think that the other point, viz. the
optical centre of.the object glass, was equally well defined.
This doubt, however, might have been removed by a di-
rect appeal to experiment, or te angles actually measured
on the ground, first by an instrument, and then by trigono-
metrical operations. From thence it would soon have been
discovered, that the centre of a lens is in fact a point de-
fined more accurately than can be done by any mechanical
construction.

This method of deciding the question was not resorted

to. Hevelius and Hooke had a very serious controversy
concerning it, in which the advantage remained with the lat-
ter. It should have been observed that the French astro-
nomer, Picard, was the first who employed instruments fur-
nished with telescopick sights, about the year 1665. It ap-
péars, however, that Gascoigne, an English gentleman, who
fell at the battle of Marston-moor, in 1644, had anticipated
the French astronomer in this invention, but that it had
remained entirely unknown. He had also anticipated the
invention of the micrometer. The vast additional accuracy
thus given to instruments formed a new era in the history
of astronomical observations.
_ Though Galileo had discovered the satellites of Jupiter,
their times of revolution, and even some of their inequali-
ties, it yet remained to define their motions with precision,
and to construct tables for calculating their places. This task
was performed by the elder Cassini, who was invited from
Italy, his native country, by Louis the Fourteenth, and
settied in France in 1669. His tables of the satellites had
been published at Bologna three years before, and he con-
tinued to improve them, by a series of observations made
in the observatory at Paris, with great diligence and accu-
racy.

smcr. 1¥.] DISSERTATION SECOND. : 157

The theory of the motions of these small bodies is a
research of great difficulty, and had been attempted by
many astronomers before Cassini, with very little success.
The planes of the orbits, their inclinations to the orbit of
Jupiter, and the lines in which they intersected that orbit,
were all to be determined, as well as the times of revolution,
and the distances of each from its primary. Add to this,
that it is only in a few points of their orbits that they can
be observed with advantage. 'The best are at the times of
immersion into the shadow of Jupiter, and emersion from
it. The same excellent astronomer discovered four satel-
lites of Saturn, in addition to that already observed by
Huygens. He also discovered the rotation of Jupiter and
of Mars upon their axes.

The. constant attention bestowed on the eclipses of the
satellites of Jupiter, made an inequality be remarked in the
periods of their return, which seemed to depend on the
position of the earth relatively to Jupiter and the sun, and
not, as the inequalities of that sort might have beenexpect-
ed to do, on the place of Jupiter in his orbit. From the
opposition of Jupiter to the sun, till the conjunction, it was
found, that the observed emersion of the satellites from the
shadow fell more and more behind the computed ; the diffe-
rences amounting, near the conjunction, to about fourteen
minutes. When, after the conjunction, the immersions
were observed, an acceleration was remarked jusi equal
to the former retardation, so that, at the opposition, the
eclipse happened fourteen minutes sooner than by the cal-
cujation.

The first person who offered an explanation of these facts
was Olaus Roemer, a Danish astronomer. He observed
that the increase of the retardation corresponded nearly
to the increase of the earth’s distance from Jupiter, and con-
versely, the acceleration to the diminution of that distance.

20

2

158 DISSERTATION SECOND. [rant 1.

Hence it occurred to him, that it was to the time which
light requires to traverse those distances that the whole
series of phenomena was to be ascribed. This explana-
tion was so simple and satisfactory, that it was readily
received.

Though Roemer was the first who communicated this
explanation to the world, yet it seems certain that it had
before occurred to Cassini, and that he was prevented
from making it known by a consideration whieh does him
great honour. ‘The explanation which the motion of light
afforded, seemed not to be consistent with two circumstan-
ces involved in the phenomenon. If, such was the cause
of the alternate acceleration and retardation above describ-
ed, why was it observed only in the eclipses of the first
satellite, and not in those of the other three? This difficul-
ty appeared so great to Cassini, that he suppressed the
explanation which he would otherwise have given.

The other difficulty occurred to Maraldi. Why did not
an equation or allowance of the same kind arise from the
position of Jupiter, with respect to his aphelion, for, all
other things being the same, his distance from the earth
must be greater, as he was nearer to that point of his orbit?
Both these difficulties have since been completely removed.
If the aforesaid inequality was not for some time observed
in any satellite but the first, it was only because the motions
of the first are the most regular, and were the soonest un-
derstood, but it now appears that the same equation belongs
to all the satellites. The solution of Maraldi’s difficulty
is similar ; for the quantity of what is called the equation
of the light, is now known to be affected by Jupiter’s
place in his orbit.

Thus, every thing conspires to prove the reality of the
motion of light, so singular on account of the immensity of
the velocity, and the smallness of the bodies to which it is
communicated.

sucr. 1v.) DISSERTATION SECOND. 159

5. EstastisHMent or Acapemigs, &c.

About the middie of the seventeenth century were form-
ed those associations of scientifick men, which, under the
appellation of Academies or Philosophical Societies, have
contributed so much to the advancement of knowledge in
Europe. The Academia del Cimento of Florence, found-
ed in 1651, carried in its name the impression of the new
philosophy. {t was in the country of Galileo where the
first institution for the prosecution of experimental know-
ledge might be expected to arise, and the monuments which -
it has left behind it will ever create regret for the shortness
of its duration. ii

England soon after showed the same example. It, has
been already remarked, that, during the civil wars, a num-
ber of learned and scientifick men sought, in the retirement of
Oxford, an asylum from the troubles to which the country
was thena prey. They had metas early as 1645; most
of them were attached to the royai cause; and after the
restoration of Charles the Second, they were incerporated
by a royal charter in 1662.

The first idea of this institution seems to have been sug-
gested by the writings of Baeon, who, in recommending the
use of experiment, had severely censured the schools, coi-
leges, and academies of his own time, as adverse to the
advancement of knowledge ;' and, in the Nova Atlantis, had

* «Tn moribus et institutis scholarum, academiarum, collegio-
rum, et similium conventuum quae doctorum hominum sedibus et
eruditionis culturae destinata sunt, omnia progressui scientiarum
adversa inveniuntur. Lectiones enim exercitia ita sunt disposi-
ta, ut aliud a consuetis haud facile cuiquam in mentem veniat
cogitare, aut contemplari. Si vero unus aut alter fortasse judicii
libertate uti sustinuerit, is sivi soli hanc operam imponere possit ;
ab aliorum antem consortio nihil capiet utilitatis. Sin et hoe

160 DISSERTATION SECOND. [PART 1.

given a most interesting sketch of the form of a society, di-
rected to scientifick improvement. In Germany, the Acade-
mia Naturae Curiosorum dates its commencement from
1652, and the historian of that institution ascribes the spirit
which produced it to the writings of the philosopher just nam-
ed. These examples, and a feeling that the union and co-
operation of numbers was necessary to the progress of expe-
rimental philosophy, operated still more extensively. The
Royal Academy of Sciences at Paris was founded in 1666,
in the reign of Louis the Fourteenth, and during the admin-
istration of Colbert. The Institute of Bologna in Italy be-
longs nearly to the same period; but almost all the other
philosophical associations, of which there are now so many,
had their beginning in the eighteenth century.

Frequent communication of ideas, and a regular method
of keeping up such communication, are evidently essential
to works in which great labour and industry are to be em-
ployed, and to which much time must necessarily be devot-
ed; when the philosopher must not always sit quietly in

toleraverit, tamen in capessenda fortuna industriam hane et mag-
napimitatem sibi non levi impedimento fore experietur. Studia
enim hominum in ejusmodi locis, in quorundam auctorum scripta,
veluti in carceres, conclusa sunt; a quibus si quis dissentiat, con-
tinvo ut homo turbidus et rerum novarum cupidus corripitur.
In artibus autem et scienttis tanquam in metalli-fodinis omntamovis
operibus et ultcrioribus progressibus circumsirepere debent.”—Nov.
Org. lib. i. cap. 90.

{t would be gratifying to be able to observe, that the universi-
ties of Europe had contributed to the renovation of science.
The fact is otherwise ;—they were often the fastnesses from
which prejudice and errour were latest ef being expelled. They
joined in persecuting the reformers of science. It has been seen,

_ that the masters of the university of Paris were augry with Ga-
lileo for the experiments on the descent of bodies. Even the
university of Oxford brought on itself the indelible disgrace of
persecuting, in Friar Bacon, the first man who appears to have
had a distinct view of the means by which the knowledge of the
laws of nature must be acquired.

axcr. 1W.] DISSERTATION SECOND. 161

his cabinet, but must examine nature with his own eyes,
and be present in the work shop of the mechanick, or the
laboratory of the chemist. These operations are facilitated
by the institutions now referred to, which, therefore, are of
more importance to the physical sciences than to the other
branches of knowledge. They who cultivate the former
are also fewer in number, and being, of course, farther se-
parated, are less apt to meet together in the common inter-
course of the world. The historian, the critick, the poet,
finds every where men who can enter in some degree at
least into his pursuits, who can appreciate his merit, and
deriye pleasure from his writings or his conversation.
The mathematician, the astronomer, the. mechanician, sees
few men who have much sympathy with his pursuits, or
who do not look with indifference on the objects which
he pursues. The world, to him, consists of a few indi-
viduals, by the censures or approbation of whom the pub-
lick opinion must be finally determined ; with them it is
material that he should have more frequent intercourse than
could be obtained by casual rencounter ; and he feels that
the society of men engaged in pursuits similar to his own,
is a necessary stimulus to his exertions. Add to this, that
such societies become centres in which information concern-
ing facts is collected from all quarters. For all these rea-
sons, the greatest benefit has resulted from the scientifick
institutions, which, since the middle of the seventeenth
century, have become so numerous in Europe.

The Royal Society of London is an association of
men, who, without salaries or appointment from Govern-
ment, defray by private contribution the expense of their
meetings, and of their publications. This last is another
important service, which a society so constituted renders

to science. .

162 DISSERTATION SECOND. [pang 5.

The demand of the publick for memoirs in mathematicks
and natural philosophy, many of them perhaps profound
and difficult, is not sufficiently great to defray the expense
of publication, if they come forward separately and uncon-
nected with one another. Ina collective state they are
much more likely to draw the attention of the publick ; the
form in which they appear is the most convenient both for
ihe reader and the author; and if, after all, the sale of the
work is unequal to the expense, the deficiency is made up
from the funds of the society. An institution of this kind,
therefore, is a patriotick and disinterested association of
the lovers of science, who engage not only to employ them-
selves in discovery, but, by private contribution, to de-
fray the expense of scientifick publications.

The Academy of Sciences in Paris was not exactly an
institution of the same kind. It consisted of three -classes
of members, one of which, the Pensionnaires, twenty in
number, had salaries paid by government, and were bound
in their turns to furnish the meetings with scientifick me-
moirs, and each of them also, at the beginning of every
year, was expected to give an account of the work in which
he was tobe employed. This institution has been of incre-
dible advantage to science. To detacha number of ingeni-
ous men from every thing but scientifick pursuits ; to deliver
them alike from the embarrassments of poverty or the
temptations of wealth; to give them a place and station in
society the most respectable and independent, is to remove
every impediment, and to add every stimulus to exertion,
To this institution, accordingly, operating upon a people
of great genius, and indefatigable activity of mind, we are
to ascribe that superiority in the mathematical sciences,
which, for the last seventy years, has been so conspi-
cuous.

axcr. 1¥.] DISSERTATION SECOND. 163

The establishment of astronomical observatories, as na-
tional or royal works, is connected in Europe with the in-
stitution of scientifick or philosophical societies. The
necessity of the former was, indeed, even more apparent
than that of the latter. A science, which has the heavenly
bodies for its objects, ought, as far as possible, to be ex-
empted from the vicissitudes of the earth. As it gains
strength but slowly, and requires ages to complete its dis-
coveries, the plan of observation must not be limited by the
life of the individual who pursues it, but must be followed
out in the same place, year after year, to an unlimited
extent. A perception of this truth, however indistinct,
seems, from the earliest times, to have suggested the utili-
ty of observatories to those sovereigns who patronised
astronomy, whether they looked to that science for real or
imaginary instruction. The circle of Osymandias is the
subject of one of the most ancient traditions in science, and
has preserved the name of a prince which otherwise would .
have been entirely unknown. A building, dedicated to
astronomy, made a conspicuous part of the magnificent es- ©
tablishment of the school of Alexandria. During the middle
ages, in the course of the migrations of science toward the
east, sumptuous buildings, furnished with astronomical in-
struments, rose successively in the plains of Mesopotamia,
and among the mountains of Tartary. An observatory in
the gardens of the Caliph of Bagdat contained a quadrant
of fifteen cubits ' in radius, anda sextant of forty.* Instru-
ments of a still larger size distinguished the observatory of
Samarcande, and the accounts would seem incredible, if
the ruins of Benares did not, at this moment, cat the
reality of similar constructions.

* Twenty-two feet three inches.
* Sixty feet five inches.

164 DISSERTATION SECOND. [part t

On the revival of letters in Europe, establishments of the
same kind were the first decisive indications of a taste for
science. We have seen the magnificent observatory on
which Tycho expended his private fortune, and employed
the munificence of his patron, become a sad memorial (after
the signal services which it had rendered to astronomy) of
the instability of whatever depends on individual greatness.
The observatories at Paris and London were secured from
a similar fate, by being made national establishments, where
a succession of astronomers were to devote themselves to
the study of the heavens. The observatory at Paris was
begun in 1667, and that at Greenwich in 1675. In the first
of these, La Hire and Cassini, in the second, Flamstead
and Halley, are at the head of a series of successors, who
have done honour to their respective nations. If there be
in Britain any establishment, in the success and conduct of
which the nation has reason to boast, it is that of the Royal
Observatory, which, in spite of a climate which so continu-
ally tries the patience, and so ofien disappoints the hopes
of the astronomer, has furnished a greater number of obser-
vations to be conipletely relied on, than all the rest of Bu-
rope put together, and afforded the data for those tables, in
which the French mathematicians have expressed, with
such accuracy, the past, the present, and the future condi-
tion of the heavens.

6. Figure anp Macenitupe oF THE Earru.

The progress made during the seventeenth century, in
ascertaining the magnitude and figure of the earth, is par-
ticularly connected with the establishments which we have
just been considering. Concerning the figure of the earth,
no accurate information was derived from antiquity, if we

sect. 1V.] DISSERTATION SECOND. 165

except that of the mathematical principle on which it was to
be determined. The measurement of an arch of the meri-
dian was aitempted by Eratosthenes of Alexandria, in per-
fect conformity with that principle, but by means very
inadequate to the importance and difficulty of the problem.
By measuring the sun’s distance from the zenith of Alexan- —
dria, on the solstitial day, and by knowing, as he thought
he did, that, on the same day, the sun was exactly in the
zenith of Syené, he found the distance in the heavens be-
tween the parallels of those places to be 7° 12’, or a 50th
part of the circumference of a great circle. Supposing,
then, that Alexandria and Syené were in the same meridian,
nothing more was required than to find the distance be-
tween them, which, when multiplied by 50, would give the
circumference of the globe. ‘The manner in which this
was attempted by Eratosthenes is quite characteristick of
the infant state of the arts of experiment and observation.
He took no trouble to ascertain whether Alexandria and
Syené were due north and south of one another: the truth
is, that the. latter is considerably east of the former, so
that, though their horizontal distance had been accurately
known, a considerable reduction would have been necessa-
ry, on account of the distance of the one from the meridian
of the other. It does not appear, however, that Eratosthe-
nes was at any more pains to ascertain the distance than
the bearing of the two places. He assumed the former
just as it was commonly estimated ; and, indeed, it appears
that the distance was not measured till long afterwards,
when it was done by the command of Nero.

It was in this way that the ancients made observations
and experiments; the mathematical principles might be
perfectly understood, but the method of obtaining accurate
data for the application of those principles was not a subject
of attention. The power of resolving the problem was the

2!

166 DISSERTATION SECOND. [PART 1.

main object; and the actual solution was a matter of very
inferiour importance. The slowness with which the art
of making accurate experiments and observations has been
matured, and the great distance it has kept behind theory,
is a remarkable fact in the history of the physical sciences.
It has been remarked, that mathematicians had found
out the area of the circle, and calculated its circumference
to more than a hundred places of decimals, before artists
had divided an arch into minutes of a degree; and that
many excellent treatises had been written on the properties
of curves, before a straight line had been drawn of any
considerable length, or measured with any tolerable wong
ness, on the surface of the globe."

The next measurement on record is that of the astrono-
mers of Almamon, in the plains of Mesopotamia, and the
manner of conducting the operation appears to have been
far more accurate than that of the Greek philosophers ;
but, from a want of knowledge of the measures employed,
it has conveyed no information to posterity.

The first arch of the meridian measured in modern times
with an accuracy any way corresponding to the difficulty
of the problem, was by Snellius,a Dutch mathematician,
who has given an account of it in a volume which he calls
Eratosthenes Batavus, published in 1617. The arch was
between Bergen-op-zoom and Alkmaar; its amplitude was
1° 11' 30", and the distance was determined by a series of
triangles, depending on a base line carefully measured.
The length of the degree that resulted was 55,021 toises,
which, as was afterwards found, is considerably too small.
Certain errours were discovered, and when they were cor-
rected, the degree came out 57,033 toises, which is not far
from the truth. The corrections were made by Snellius

' Edinburgh Review, Vol. V. p. 391.

oxcr. 1¥.] DISSERTATION SECOND. 167
himself, who measured his base over again, and also the
angles of the triangles. He died, however, before he could
publish the result. Muschenbroek, who calculated the
whole anew from his papers, came’ to the conclusion just
mentioned, which, of course, was not known till long after
the time when the measure was executed. No advantage,
accordingly, was derived to the world from this measure-
ment till its value was lost in that of other paoapersianis
still more accurately conducted.

A computation which, for the time, deserves considera-
ble praise, is that of Norwood, in 1635, who measured the
distance between London and York, taking the bearings
as he proceeded along the road, and reducing all to the
direction of the meridian, and to the horizontal plane. The
difference of latitude he found, by observation of the solsti-
ces, to be 2° 28', and from that and his measured distance,
he concluded the degree to be 367, 176 feet English, or
57,800 toises. This has been found to be a near approxima-
tion; yet his method was not capable of great accuracy,
nor did he always execute it in the best manner. ‘‘ Some-
times,” says he, “I measured, sometimes I paced, and I
believe I am within a scantling of the truth.”’

Fernel, a French physician, measured with a wheel from
Paris to Amiens, which are nearly in the same meridian,
and he determined the degree from thence to be 56,746
French toises ; a result which falls short of the truth, though
not very considerably.

These investigations, it is plain, could not but leave con-
siderable uncertainty with respect to the magnitude of the
earth. The Academy of Sciences became interested in
the question, and the measurement of an arch in the meridi-
an was undertaken under its auspices, and executed by the
Abbé Picard, already known for his skill in the operations
of practical geometry. He followed a method similar to

168 DISSERTATION SECOND. (PART 1.

that of Snellius, according to which, the distance between
Amiens and Malvoisine was found froma series of triangles,
and a base of 5663 1-6 toises. He determined the differ-
ence of latitude by means of a zenith sector of ten feet radi-
us, and found it to be 1° 22'55". The whole distance was
78,850 toises, whence the degree came out 57,060 toises.
This was the first measurement of a degree of the meridian,
on which perfect reliance could be placed.

Hitherto no doubt had been entertained of the spherical
figure of the earth, and, of consequence, of the equality of
all the degrees of the meridian, so that if one was known,
the whole circumference was determined. Men, with the
precipitation which they so often manifest, of assuming,
without sufficient evidence, the conclusion which appears
most simple, were no sooner satisfied that the earth was
round, than they supposed it to be truly spherical. An
observation soon occurred, which gave reason to suspect,
that much more must be done before its figure or its magni-
tude were completely ascertained.

With a view of observing the sun’s altitude in the vicinity
of the equator, where the distance from the zenith being
inconsiderable, the effects of refraction must be of small
account, it was agreed, by the same academy, to send an
astronomer, M. Richer, to make observations at the island
of Cayenne, in South America. ’

Richer observed the solstitial altitude of the sun at that
place in 1672, and found the distance of the tropicks to be
46° 57' 4"; and, therefore, the obliquity of the ecliptick
23° 28' 32", agreeing almost precisely with the determina-
tion of Cassini.

The most remarkable circumstance, however, which oc-
curred in the course of this voyage, was, that the clock,
though furnished with a pendulum of the same length which
vibrated seconds at Paris, was found, at Cayenne, to lose

szcr, 1v.] DISSERTATION SECOND. 169

two minutes and a half a-day nearly. This created great
astonishment in France, especially after the accuracy of it
was confirmed by the observations of Varin and Deshayes,
who, some years afterwards, visited different places on the
coast of Africa and America, near the line, and found the
necessity of shortening the pendulum, to make it vibrate
seconds in those latitudes. The first explanation of this
remarkable phenomenon was given by Newton, in the third
book of his Principia, published in 1687, where it is de-
duced as a necessary consequence of the earth’s rotation
on its axis, and of the centrifugal force thence arising.
That force changes both the direction and the intensity of
gravity, giving to the earth an oblate spheroidal figure, more
elevated at the equator than the poles, and making bodies
fall, and pendulums vibrate, more slowly in low than in
high latitudes.

This solution, however, did not, any more than the book
in which it was contained, make its way very readily into
France. The first explanation of the retardation of the
pendulum, which was received there, was given by Huy-
gens in 1690. Huygens deduced it also from a centrifugal
force, arising from the earth’s rotation, and the view which
he took was simpler, though much less accurate than that
of Newton. It had, indeed, the simplicity which often
arises from neglecting one of the essential conditions of a
problem; but it was nevertheless ingenious, and involved
a very accurate knowledge of the nature of centrifugal
force. Iam thus brought to touch on a subject which be-
longs properly to the second part of this Dissertation, for
which the fuller discussion of it must of course be re-
served.

2
170 DISSERTATION SECOND. [parr 3.

SECTION V.

OPTICKS.

1. Orricatn KnowLeEpGE oF THE ANCIENTS.

On account of the rectilineal propagation of light, the
phenomena of opticks are easily expressed in the form of
mathematical propositions, and seem, as it were, spontane-
ously to offer themselves to the study of geometers. Eu-
clid perceiving this affinity, began to apply the science
which he had already cultivated with so much success, to
explain the laws of vision, before a similar attempt had
been made with respect to any other branch of terrestrial
physicks, and at least fifty years before the researches of
Archimedes had placed mechanicks among the number of
the mathematical sciences.

In the treatise ascribed to Euclid, there are, however;
only two physical principles which have completely stood
the test of subsequent improvement. The first of these is
the proposition just referred to, that a point in any object
is seen in the direction ofa straight line drawn from the eye
to that point; and the second is, that when a point in an
object is seen by reflection from a polished surface, the
lines drawn from the eye and from the object to the point
whence the reflection is made, are equally inclined to the
reflecting surface. These propositions are assumed as
irue; they were, no doubt, known before the time of Eu-
clid, and it is supposed that the discovery of them was the
work of the Platonick school. The first of them is the
foundation of Opticks proper, or the theory of vision by

szer. v.] DISSERTATION SECOND. 171

_ direct light ; the second is the foundation of Catoptricks,
or the theory of vision by reflected light. Dioptricks, of
vision by refracted light, had not yet become an object of
attention.

Two other principles which Euclid adopted as postulates
in his demonstrations, have not met with the same entire
confirmation from experiment, and are, indeed, true only
in certain cases, and not universally, as he supposed. The
first of these is, that we judge of the magnitude of an ob-
ject altogether by the magnitude of the optical angle, or the
angle which it subtends at the eye. It is true that this angle
is an important element in that judgment, and Euclid, by dis-
covering this, came into the possession of a valuable truth ;
but by a species of sophistry, very congenial to the human
mind, he extended the principle too far, and supposed it
to be’ the only circumstance which determines our judg-
ment of visible magnitude. It is, indeed, the only measure
which we are furnished with directly by the eye itself;
but there are few cases in which we form our estimate
without first appealing to the commentary afforded by the
sensations of touch, or the corrections derived from our own
motion.

Another principle, laid down by the same geometer, is in
circumstances nearly similar to the preceding. According
to it, the place of any point of an object seen by reflection,
is always the intersection of the reflected ray with the per-
pendicular drawn from that point to the reflecting surface.
The proof offered is obscure and defective ; the proposi-
tion, however, is true of plain speculums always, and of
spherical as far as Euclid’s investigations extended, that is,
while the rays fall on the speculum with no great obliquity.
His assumption, therefore, did not affect the truth of his
conclusions, though it would have been a very unsafe guide
in more general investigations. ‘The book is in many other

172 DISSERTATION SECOND. [pant i.

respects imperfect, the reasoning often unsound, and the
whole hardly worthy of the great geometer whose name it
bears. There is, however, no doubt that Euclid wrote on
the subject of opticks, and many have supposed that this
treatise is a careless extract, or an unskilful abridgment of
the original work.

Antiquity furnished another mathematical treatise on
opticks, that of the astronomer Ptolemy. This treatise,
though known in the middle ages, and quoted by Roger
Bacon, had disappeared, and was supposed to be entirely
lost, till within these few years, when a manuscript on,
opticks, professing to be the work of Ptolemy, and to be
translated from the Arabick, was found in the King’s libra- |
ry at Paris. The most valuable part of this work is that
which relates to refraction, from whence it appears that
many experiments had been made on that subject, and the
angles of incidence and refraction, for different transparent
substances, observed with so much accuracy, that the
same ratio very nearly of the sines of these angles, from
air into water, or into glass, is obtained from Ptolemy’s
numbers, which the repeated experiments of later times
have shown to be true. ‘The work, however, in the state
in which it now appears, is very obscure, the reasoning
often deficient in accuracy, and the mathematical part
much Jess perfect than might have been expected. Mo-
_dern writers, presuming partly on the reputation of Ptole-
my, and partly guided by the authority of Roger Bacon,
had ascribed to this treatise more merit than it appears
to possess; and, of consequence, had allowed less to
the Arabian author Alhazen, who comes next in the
order of time, than of right belongs to him. Montucla,
on the authority of Bacon, says, that Ptolemy ascribed
the increase of the apparent magnitude of the heaven-
ly bodies near the horizon, to the greater distance at

secr. v.] DISSERTATION SECOND. 173

which they are supposed to be, onaccount of the number
of intervening objects across which they are seen. Ptole-
my’s explanation, however, as stated by Delambre,' from the
manuscript just mentioned, is quite different from this, and
amounts to no more than the vague and unsatisfactory re-
mark, that an observer looks at the bodies near the zenith
in a constrained posture, and in a situation to which the
eye is not accustomed. ‘The former explanation, therefore,
given by Alhazen, but supposed to have been borrowed
from Ptolemy, must now be returned to iis right owner.
{tis the best explanation yet known.

These are the only mathematical treatises on opticks of
any consideration which the ancients have transmitted to
us; but many metaphysical speculations on light and
vision are to be found in the writings of the philosophers.
Aristotle defined light much as he had defined motion ; the
act or energy of a transparent body, in as much as it is

* Connatssance des Tems, 1816, p. 245, &c. The glimpses
of fruth, notdestined to be fuily discovered till many ages after-
wards, which are found in the writings of the ancients, are always
interesting. Ptolemy distinguishes what has since been called
the virtual focus, which takes place in certain cases of reflection
from spherical specula. He remarks, that colours are confounded
by the rapidity of motion, and gives the instance of a wheel
painted with different colours, and turned quickly round.

* Another Greek treatise on opticks, that of Heliedorus of
Larissa, has been preserved, and was first published by Erasmus
Bartholinus at Paris, in 1657. It is a superficial work, which,
to a good deal of obscure and unsound metaphysicks, adds the
demonstration of a few very obvious truths. The author holds
the opinion, that vision is performed by the emission of some-
thing from the eyes; and the reason which he assigns is, that
the eyes are convex, and more adapted to emit than to receive.
His metaphysicks may be judged of from this specimen. He
has not been made mention of by any ancient author, and the
time when he wrote is unknown. As he quotes, however, the
wrilings of Ptolemy and Hero, he must have been later than the
first century.

22

174 » DISSERTATION SECOND. [parr 5,

transparent. The reason for calling light an act of a
transparent body is, that, though a body may be transpa-
rent in power or capacity, it does not become actually
transparent but by means of light. Light brings the trans-
parency into action; it is, therefore, the act of a transpa-
rent body. In such miserable puerilities did the genius of
this great man exhaust itself, owing to the unfortunate di-
rection in which his researches were carried on.

In his farther speculations concerning light, he denied
it to be a substance; and his argument contains a singular
mixture of the ingenious and the absurd. The time, he
says, in which light spreads from one place to another is
infinitely small, so that light has a velocity which is infinite-
ly great. Now, bodies move with a velocity inversely as
the quantities of matter which they contain ; light, there-
fore, cannot contain any matter, that is, it cannot be ma-
terial.' That the velocity of light was infinitely great,
seemed to him to follow from this, that its progress, esti-
mated either in the direction of north and south, or of east
and west, appeared to be instantaneous. In the opinion
of the Platonists, and of the greater part of the ancients,
vision was performed by means of certain rays which pro-
ceeded from the eye to the object, though they did not
become the instruments of conveying sensations to the
mind, but in consequence of the presence of light. In this
theory, we can now see nothing but a rude and hasty at-
tempt to assimilate the sense of sight to that of touch,
without inquiring sufficiently into the particular characters
of either.

Epicurus, and the philosophers of his school, as we learn
from Lucretius, entertained more correct notions of vision,

' The truth of the mathematical proposition, that se = 0,
was perceived by Aristotle. A strong intellect is always Visi-
ble in the midst of his greatest errours.

mer'v.) DISSERTATION SECOND. “175

though they were still far from the truth. They conceiv-
ed vision to be performed in consequence of certain simu-
lacra, or images continually thrown off from the surfaces
of bodies, and entering the eye. This was the substitute
in their philosophy for rays ef light, and had at least the
merit of representing that which is the medium of vision,
or which forms the communication between the eye and
external objects, as something proceeding from the latter.
The idea of simulacra, or spectra, flying off continually
from the surfaces of bodies, and entering the eye, was per-
haps as near an approach to the true theory of vision, as
could be made before the structure of the eye was under-
stood. Fr

In the arts connected with opticks, the ancients had
made some progress. They were sufficiently acquainted
with the laws of reflection to construct mirrors both plane
and spherical. ‘They made them also conical; and it ap-
pears from Plutarch, that the fire of Vesta, when extin-
guished, was not permitted to be rekindled but by the rays
of the sun, which were condensed by a conical speculum
of copper. The mirrors with which Archimedes set fire
to the Roman gallies have been subjects of much discus-
sion, and the fact was long disbelieved, on the ground of
being physically impossible. The experiments of Kircher
and Buffon showed that this impossibility was entirely ima-
ginary, and that the effect ascribed to the specula of the
Greek geometer might be produced without much difficul-
ty. There remains now no doubt of their reality. A pas-
sage from Aristophanes’ gives reason to believe that, in
his time, lenses of glass were used for burning, by collect-
ing the rays of the sun; but in a matter that concerns the
history of science, the authority of a comick poet and. 2

' In Nubibus, Act. 2, sc. 1, v. 20.

—
176 _ DISSERTATION SECOND. [ran t.

satirist would not deserve much attention, if it were not
confirmed by more sober testimony. Pliny, speaking of
rock crystal,’ says, that a globe or ball of that substance
was sometimes used by the physicians for collecting the
rays of the sun, in order to perform the operation of caute-
ry. In another passage, he mentions the power of a glass
globe filled with water, to produce a strong heat when ex-
posed to the rays of the sun, and expresses his surprise
that the water itself should all the while remain quite cold.

With respect to the power of glasses to magnify objects
seen through them, or to render such objects more distinct,
the ancients appear to have observed ill, and to have reason-
ed worse. ‘“ Literae quamvis minutae et obscurae per vi-
tream pilam aqua plenam majores clarioresque cernuntur.
Sidera ampliora per nubem adspicienti videntur : quia acies
nostra in humido labitur, nec apprehendere quod vult
fideliter potest.” This passage, and the speculations con-
cerning the rainbow in the same place, when they are con-
sidered as containing the opinions of some of the most able
and best informed men of antiquity, must be admitted to
mark, in a very striking manner, the infancy of the physical
sciences. /

2. From ALHAZEN To KEPLER.

An interval of nearly a thousand years divided Ptolemy
from Alhazen, who, in the history of optical discovery, ap-
pears as his immediate successor. This ingenious Ara-
biaa lived in the eleventh century, and his merit can be
more fairly, and will be more highly appreciated, now that
the work of his predecessor has become known. The mer-

* Hist. Nat. Lib. 37, cap. 10.
* Seneca, Nat. Quest. Lib. i. cap. 6.

sner. 'v.] DISSERTATION SECOND. 177

it of his book on Opticks was always admitted, but he was
supposed to have borrowed much from Ptolemy, without
acknowledging it; and the prejudices entertained in favour
of a Greek author, especially of one who had been for so
many years a legislator in science, gave a false impression,
both of the genius and the integrity of his modern rival.—
The work of Alhazen is, nevertheless, in many respects, su-
periour to that of Piolemy, and in nothing more than in the
geometry which it employs. The problem known by his
name, to find the point in a spherical speculum, at which a
ray coming from one given point shall be reflected to ano-
ther given point, is very well resolved in his book, though
a problem of so much difficulty, that Montucla hazards the
opinion, that no Arabian geometer was ever equal to the so-
lution of it." It is now certain, however, that the solution,
from whatever quarter it came, was not borrowed from Pto-
lemy, in whose work no mention is made of any such ques-
tion; and it may very well be doubted, whether, had this
problem been proposed to him, the Greek geometer would
have appeared to as much advantage as the Arabian. »
The account which the latter gives of the augmentation
of the diameters of the heavenly bodies near the horizon
has been already mentioned. He treated also of the re-
fraction of light by transparent bodies, and particularly of
the atmospherick refraction, but not with the precision o!
Ptolemy, whose optical treatise Delambre seems to think it

* Barrow, in his 9th lecture, says of this Problem, that it may
truly be called ducenyavoy, as hardly any one more difficult had
then been attempted by geometers. He adds, that, after trying
the analysis in many different ways, he had found nothing prefera-
ble to the solution of Alhazen, which he therefore gives only
freed from the prolixness and obseurity with which the original
is chargeable. Lectiones Opticae, Sect. 9. p.65. A very elegant
solution of the same problem is given by Simson, at the end of
his Conick Sections.

178 DISSERTATION SECOND. [rane c.

probable that he had never seen. The anatomical struc-
ture of the eye was known to him; concerning the uses of
the different parts he had only conjectures to offer ; but on
seeing single with two eyes, he made this very important
remark, that, when corresponding parts of the retina are af-
fected, we perceive but one image.

Prolixity and want of method are the faults of Alhazen.
Vitello,t a learned Pole, commented on his works, and has
very much improved their method and arrangement in a
treatise published in 1270. He has also treated more fully
of the subject of refraction, and reduced the results of his
experiments into the form of a table exhibiting the angles
of refraction corresponding to the angles of incidence, which
he had tried in water and glass. It was not, however, till
long after this period that the law which connects these an-
gles was discovered. The cause of refraction appeared to
him to be ihe resistance which the rays suffer in passing
into the denser-medium of water or glass, and one can see
in his reasoning an obscure idea of the resolution of forces.
He also treats of the rainbow, and remarks, that the alti-
tudes of the sun and bow together always amount to 42 de-
grees. He next considers the structure of the eye, of
which he has given a tolerably accurate description, and
proves, as Alhazen had before done,? that vision is not per-
formed by the emission of rays from the eye.

Roger Bacon, distinguished for pursuing the path of true
philosophy in the midst of an age of ignorance and errour,
belongs to the same period ; and applied to the study of op-
ticks with peculiar diligence. It does not appear, however,
that he added much to the discoveries of Alhazen and Pto-

* The name of this author is commonly written Vitellio. He
may be supposed to have known best the orthography of his own
mame. (

* Alhazen, Opt. lib. 1.

sxor. v.] ‘DISSERTATION SECOND. 179

lemy, with whose writings, particularly those of the former,
he seems to have been well acquainted. In some things he
was much behind the Arabian optician, as he supposed
with the ancients that vision is performed by rays emitted
from the eye. It must, however, be allowed, that the ar-
guments employed on both sides of this question are so
weak and inconclusive, as very much to diminish the merit of
being right, and the demerit of being wrong. What is most
to the credit of Bacon, is the near approach he appears to
have made to the knowledge of lenses, and their use in as-
sisting vision. Alhazen had remarked, that small objects,
letters, for instance, viewed through a segment of a glass
sphere, were seen magnitied, and that it is the larger segment
which magnifies the most. The spherical segment was sup-
posed to be laid with its base on the letters, or other mi-
nute objects which were to be viewed. Bacon recommends
the smaller segment, and observes, that the greater, though
it magnify more, places the object farther off than its natu-
ral position, while the other brings it nearer. This shows
sufficiently, that he knew how to trace the progress of the
rays of light through a spherical transparent body, and un-
derstood, what was the thing least obvious, how to determine
the place of the image. Smith, in his Opticks, endeavours
io show, that these conclusions were purely theoretical, and
that Roger Bacon had never made any experiments with
such glasses, notwithstanding that he speaks as if he had
done so.' This severe remark proceeds on some slight in-
accuracy in Bacon’s description, which, however, does not
seem sufficient to authorize so harsh a conclusion. The
probability appears rather to be, as Molineux supposed,
that Bacon had made experiments with such glasses, and
was both practically and theoretically acquainted with their

' Smith’s Opticks. vol. If. Remarks, 6 76.

180 DISSERTATION SECOND. [ean 1,

properties. At the same time, it must be acknowledged,
that his credulity on many points, and his fondness for the
marvellous, which, with every respect for his talents, it is
impossible to deny, take something away from the force of
his testimony, except when it is very expressly given.
However that may be in the present case, it is probable,
that the knowledge of the true properties of these glasses,
whether it was theoretical or practical, may have had a
share in introducing the use of lenses, and in the invention
of spectacles, which took place not long after.

It would be desirable to ascertain the exact period of an
invention of such singular utility as this last; one that dif-
fuses its advantages so widely, and that contributes so much
to the solace and comfort of old age, by protecting the most
intellectual of the senses against the general progress of
decay. In the obscurity of a dark age, careless about re-
cording discoveries of which it knew not the principle or
the value, a few faint traces and imperfect indications serve
only to point out certain limits within which the thing
sought for is contained. Seeking for the origin of a dis-
covery, is like seeking for the source of a river where
innumerable streams have claims to the honour, between
which it is impossible to decide, and where the only thing
that can be known with certainty is the boundary by which
they are all circumscribed. 'The reader will find the evi-
dence concerning the invention of spectacles very fully
discussed in Smith’s Opticks ; from which the most proba-
ble conclusion is, that the date goes back to the year 1313,
and cannot with any certainty be traced farther. *

The lapse of more than two hundred years brings us
down to Maurolycus, and to an age when men of science

' Smith’s Opticks, vol. Il. Remarks, § 75.

suer. v.] DISSERTATION SECOND. i$]

ceased to be so thinly scattered over the wastes of time.
Maurolycus, whose knowledge of the pure mathematicks
has been already mentioned, was distinguished for his skill
in opticks. He was acquainted with the crystalline lens,
and conceived that its office is to transmit to the optick
nerve the species of external objects; and in this process
he does not consider the retina as any way concerned.
This theory, though so imperfect, led him nevertheless to
form a right judgment of the defects of short-sighted and
long-sighted eyes. In one of his first works, Theoremata
de Lumine et Umbra, he also gives an accurate solution
of a question proposed by Aristotle, viz. why the light of
the sun, admitied through a small hole, and received ona
plane ata certain distance from it, always illuminates a
round space, whatever be the figure of the hole itself,
whereas, through a large aperture, the illuminated space
has the figure of the aperture. To conceive the reason
of this, suppose that the figure of the hoie is a triangle ; it
is plain that at each angle the illuminated space will be
terminated by a circular arch of which the centre corres-
ponds to the angular point, and the radius to the angle
subtended by the sun’s semidiameter. Thus the illumi-
nated space is rounded off at the angles; and when the
hole is so small that the size of those roundings bears a
large proportion to the distance of their centres, the figure
comes near toa circle, and may be to appearance quite
round. This is the true solution, and the same with that
of Maurolycus. The same author appears also to have
observed the caustick curve formed by reflection from a
concave speculum.

A considerable step in optical discovery was made at
this time by Baptista Porta, a Neapolitan, who invented
the Camera Obscura, about the year 1560, and described

23

182 DISSERTATION SECOND. [eanr 3.

it in a work, entitled Magia Naturalis. The light was
admitted through a small hole in the window-shutter of a
dark room, and gave an inverted picture of the objects
from which it proceeded, on the opposite wall. A lens
was not employed in the first construction of this appara-
tus, but was afterwards used; and Porta went so far as to
consider how the effect might be produced without inver-
sion. He appears to have been a man of great ingenuity ;
and though much of the Magia Naturalis is directed to
frivolous objects, it indicates a great familiarity with ex-
periment and observation. ‘It is remarkable, that we find
mention made in it of the reflection of cold by a speculum, *
an experiment which, of late, has drawn so much attention,
and has been supposed to be so entirely new. ‘The cold
was perceived by making the focus fall on the eye, which,
in the absence of the thermometer, was perhaps the best
measure of small variations of temperature. Porta’s book
was extremely popular ; and when we find it quickly trans-
lated into Italian, French, Spanish, and Arabick, we see
how much the love of science was now excited, and what
effects the art of printing was now beginning to produce.
Baptista Porta was a man of fortune, and jhis house was so
much the resort of the curious and learned at Naples, that
it awakened the jealousy with which the court of Rome
watched the progress of improvement. How grievous it
is to observe the head of the Christian church in that and
the succeeding age, like the Anarch old in Milton, reign-
ing in the midst of darkness, and complaining of the en-
croachments which the realm of light was continually
making on his ancient empire ! .

_The constitution of the eye, and the functions of the
different parts of which it consists, were not yet fully un-

' Magia Naturalis, Lib. 17, cap. 4, p. 553. Amsterdam edit.

.

excr. v.] DISSERTATION SECOND. 183

derstood. Maurolycus had nearly discovered the secret,
and it was but a thin, though, to him, an impenetrable veil,
which still concealed one important part of the truth.
This veil was drawn aside by the Neapolitan philosopher ;
but the complete discovery of the truth was left to Kepler,
who, to the glory of finding out the true laws of the plane-
tary system, added that of first analyzing the whole scheme
of nature in the structure of the eye. He perceived the
exact resemblance of this organ to the dark chamber, the
rays entering the pupil being collected by the crystalline
lens, and the other humours of the eye, into foci, which
paint on the retina the inverted images of external ob-
jects. By another step of the process, te which our ana-
lysis can never be expected to extend, the mind perceives
the images thus formed, and refers them at the same time
to things without.

It seemed a great difficulty, that, though the images be ©
inverted, the objects are seen erect ; but when it is con-
sidered that each point in the object is seen in the direc-
tion of the line, in which the light passes from it to the re-
tina, through the centre of the eye, it will appear, that the
upright position of the object is a necessary consequence
of this arrangement.

Kepler’s discovery is explained in his Paralipomena in
Vitellionem,*’ (Remarks on the Opticks of Vitello,) a work
of great genius, abounding with new and enlarged views,
though mixed occasionally with some unsound and visiona-
ry speculations. This book appeared in 1604. In the
next article we shall have occasion to return to the con-
sideration of other parts of Kepler’s optical discoveries.

-' Caput 5. de Modo Visionis.

184 DISSERTATION SECOND. [PART a.

3. From Kerrier tro THE COMMENCEMENT OF New-
Ton’s Orticat DiscoveRiEs.

The rainbow had, from the earliest times, been an ob-
ject of interest with those who bestowed attention on op-
fical appearances, but it is much too complicated a phe-
nomenon to be easily explained. In general, however, it
was understood to arise from light reflected by the drops
of rain falling from a cloud opposite to the sun. The diffi-
culty seemed to be how to account for the colour, which
is never produced in white light, such as that of the sun,
by mere reflection. Maurolycus advanced a considerable
step when he supposed that the light enters the drop, and
acquires colour by refraction ; but in tracing the course of
the ray he was quite bewildered. Others supposed the
refraction and the colour to be the effect of one drop, and
the reflection of another; so that two refractions and one
reflection were employed, but in such a manner as to be
still very remote from the truth.

Antonio de Dominis, Archbishop of Spalatro, had the
good fortune to fall upon the true explanation. Having
placed a bottle of water opposite to the sun, and a little
above his eye, he saw a beam of light issue from the under
side of the bottle, which acquired different colours, in the
same order, and with the same brilliancy as in the rain-
bow, when the bottle was a little raised or depressed.
From comparing all the circumstances, he perceived that
ihe rays had entered the bottle, and that, after two re-
fractions from the convex part, and a reflection from the
concave, they were returned to the eye tinged with differ-
ent colours, according to the angle at which the ray had
entered. The rays that gave the same colour made the
same angle with the surface, and hence all the drops that

ager. v.] DISSERTATION SECOND. 185

gave the same colour must be arranged in a circle, the
centre of which was the point in the cloud opposite to the
sun. This, though not a complete theqry of the rainbow,
and though it left a great deal to occupy the attention,
first of Descartes, and afterwards of Newton, was perfectly
just, and carried the explanation as far as the principles
then understood allowed it to go. The discovery itself
may be considered as an anomaly im science, as it is one of
a very refined and subtle nature, made by a man who has
given no other indication of much scientifick sagacity or
acuteness. In many things his writings show great igno-
rance of principles of opticks well known in his own time,
so that Boscovich, an excellent judge in such matters, has
said of him, “homo opticarum reruni, supra id quod patia-
tur ea aetas, imperitissimus.’”? The book containing this
discovery was published in 1611.

A discovery of the same period, but somewhat earlier,
willalways be considered as among the most remarkable
in the whole circle of human knowledge. It is the inven-
tion of the telescope, the work in which (by following un-
consciously the plan of nature in the formation of the eye)
man has come the nearest to the construction of a new or-
gan of sense. For this great invention, in its original form,
we are indebted to accident, or to the triais of men whe
had little knowledge of the principles of the science on
which they were conferring so great a favour. A series of
scientifick improvements, continued for more than two hun-
dred years, has continually added to the perfection of this
noble instrument, and has almost entitled science to con-
sider the telescope as its own production,

it will readily be believed, that the origin of such an
invention has been abundantly inquired into. The result,

* De Radiiz Lucis in Vitris perspectiyis et Iride.—Venetiis,
in 4to.

186 DISSERTATION SECOND. [Parr 1.

however, as is usual in such cases, has not been quite sa-
tisfactory ; and all that is known with certainty is, that the
honour belongs to the town of Middleburgh in Zealand,
and that the date is between the last ten years of the six-
teenth century, and the first ten of the seventeenth. Two
different workmen belonging to that town, Zachariah’ Jans,
and John Lapprey, have testimonies in their favour, be-
tween which it is difficult to decide; the former goes back
to 1590, the latter comes down to about 1610. It is not
of much consequence to settle the priority in a matter
which is purely accidental ; yet one would not wish to for-
get or mistake the names of men whom even chance had
rendered so great benefacturs to science. What we know
with certainty is, that the account of the effect produced
by this new combination of glasses being carried to Galileo
in 1610, led that great philosopher to the construction of
the telescope, and to the interesting discoveries already
enumerated. By what principle he was guided to the
combination, which consists of one convex and one con-
cave lens, he has not explained, and we cannot now exactly
ascertain. He had no doubt observed, that a convex lens,
such as was common in spectacles, formed images of ob-
jects, which were distinctly seen when thrown on a wall or
ona screen. He might observe also, that if the image,
instead of falling on the screen, were made to fall on the
eye, the vision was confused and indistinct. In the trials
to remedy this indistinctness, by means of another glass,
it would be found that a concave lens succeeded when
placed before the eye, the eye itself being also a little more
advanced than the screen had been.

This instrument, though very imperfect, compared with
those which have been since constructed, gave so much
satisfaction, that it remained long without any material
improvement. Descartes, whose treatise on Opticks was

suer. 7.) DISSERTATION SECOND. 187

writtten near thirty years after the invention of the tele-
scope, makes no mention of any but such as is composed of
a convex objectglass, and a concave eyeglass. The
theory of it, indeed, was given by Kepler in his Dioptricks
(1611,) when he also pointed out the astronomical tele-
scope, or that which is composed of two convex lenses,
and inverts the objects. He did not, however, construct
a telescope of that kind, which appears to have been first
done by Scheiner, who has given an account of it in the
Rosa Ursina (in 1650,) quoted by Montucla.?

After the invention of the telescope, that of the micro-
scope was easy ; and it is also to Galileo. that we are indebt
ed for this instrument, which discovers an immensity on the
one side of man, scarcely less wonderful than that which
the telescope discovers on the other. The extension and
divisibility of matter are thus rendered to the natural phi-
losopher almost as unlimited, as the extension and the di-
visibility of space are to the geometer.

The theory of the telescope, now become the main ob-
ject in optical science, required that the law of refraction
should, if possible, be accurately ascertained. This had
not-yet been affected, and Kepler, whose Dioptricks was
the most perfect treatise on refraction which had yet ap-
peared, had been unable to determine the general _princi-
ple which connects the angles of incidence and refraction.
In the case of glass, he had found by experiment, that
_those angles, when small, are nearly in the ratio of three to
two, and on this hypothesis he had found the focus of a
double convex lens, when the curvature of both sides is
equal, to be the centre of curvature of the side turned to-
ward the object,—a proposition which is known to coin-
cide with experiment. From the same approximation, he
derived other conclusions, which were found useful in

' Vol. II. p. 234, 2d edit.

188 DISSERTATION SECOND. [rant a.

practice, in the cases where the angles just mentioned
were very small.

The discovery of the true law of refraction was the work
of Snellius, the same mathematician whose labours concern-
ing the figure of the earth were before mentioned. In or-
der to express this law, he supposed a perpendicular to the
refracting surface, at the point where the refraction is made,
and also another line parallel to this perpendicular at any
given distance from it. The refracted ray, as it proceeds,
will meet this parallel, and the incident ray is supposed to
be produced, till it do so likewise. Now, the general
truth which Snellius found to hold, whatever was the posi-
tion of the incident ray, is, that the segments of the re-
fracted ray and of the incident ray, intercepted by these
parallels, had always the same ratio to one another. If
either of the media were changed, that through which the
‘incident ray, or that through which the refracted ray
passed, this ratio would be changed, but while the
media remained the same, the ratio continued unaltera-
ble. It is seldom that a general truth is seen at first
under the most simple aspect : this law admits of being
more simply expressed, for, in the triangle formed by
the two segments of the rays, and by the parallel which
they intersect, the said segments have the same ratio with
the sines of the opposite angles, that is, with the sines of
the angles of incidence and refraction. The law, there-
fore, comes to this, that, in the refraction of light, by the
same medium, the sine of the angle of incidence has to the
sine of the angle of refraction always the same ratio. This
last simplification did not occur to Snellius; it is the work
of Descartes, and was first given in his Dioptricks, in
1637, where no mention is made of Snellius, and the law of
refraction appears as the discovery of the author. This
naturally gave rise to heavy charges against the candour
and integrity of the French philosopher. The work of

szcr. y.] DISSERTATION SECOND. 189

Snellius had never been published, and the author himself
was dead ; but the proposition just referred to had been
communicated to his friends, and had been taught by his
countryman, Professor Hortensius, in his lectures. There
is no doubt, therefore, that the discovery was first made
by Snellius, but whether Descartes derived it from him, or
was himself the second discoverer, remains undecided.
The question is one of those, where a man’s conduct ina
particular situation can only be rightly interpreted from his
general character and behaviour. If Descartes had been
uniformly fair and candid in his intercourse with others,
one would have rejected with disdain a suspicion of the
kind just mentioned. But the truth -is, that he appears
throughout a jealous and suspicious man, always inclined
to depress and conceal the merit of others. In speaking
of the inventor of the telescope, he has told minutely all
that is due to accident, but has passed carefully over all
that proceeded from design, and has incurred the reproach
of relating the origin of that instrument, without mention-
ing the name of Galileo. In the same manner he omits to
speak of the discoveries of Kepler, so nearly connected
with his own; and in tmweating of the rainbow, he has made
no mention of Antonio de Dominis. It is impossible that
all this should not produce an unfavourable impression, and
hence it is, that even the warmest admirers of Descartes
do not pretend that this conduct toward Snellius can be
completely justified.

Descartes would have conceived his philosophy to be
disgraced if it had borrowed any general principle from
experience, and he therefore derived, or affected to derive,
the law of refraction from reasoning or from theory. In
this reasoning, there were so many arbitrary suppositions
concerning the nature of light, and the action of transparent

bodies, that no confidence can be placed in the conclusions
24

190 DISSERTATION SECOND. [par? 1.

deduced fromit. It is indeed quite evident, that, indepen-
dently of experiment, Descartes himself could have put no
trust in it, and it is impossible not to feel, how much more
it would have been for the credit of that philosopher, to
have fairly confessed that the knowledge of the law was
from experiment, and that the business of theory was te
deduce from thence some inferences, with respect to the
constitution of light and of transparent bodies. This f
conceive to be the true method of philosophizing, but it is
the reverse of that which Descartes pursued on all occa-
sions.

The weakness of his reasoning was perceived and at-
tacked by Fermat, who, at the same time, was not very
fortunate in the theory which he proposed to substitute for
that of his rival. The latter had Jaid it down as certain,
that light, of which he supposed the velocity infinite, or
the propagation instantaneous, meets with less obstruction
in dense than in rare bodies, for which reason, it is refract-
ed toward the perpendicular, in passing from the latter
into the former. ‘This seemed to Fermat a very improba-
ble supposition, and he conceived the contrary to be true,
viz. that light in rare bodies ha% less obstruction, and
moves with greater velocity than in dense bodies. On
this supposition, and appealing, not to physical, but to final
causes, Fermat imagined to himself that he could deduce
the true law of refraction. He conceived it to be a fact,
that light moves always between two points, so as to go
from the one to the other in the least time possible. Hence,
in order to pass from a given point in a rarer medium where
it moves faster, to a given point in a denser medium where
it moves slower, so that the time may be a minimum, it
must continue longer in the former medium than if it held
a rectilineal course, and the bending of its path, on entering
the latter, will therefore be toward the perpendicular. Qn

°

exer, V.] DISSERTATION SECOND. 191

instituting the calculus, according to his own doctrine of
maxima and minima, Fermat found, to his surprise, that
the path of the ray must be such, that the sines of the an-
gles of incidence and refraction have a constant ratio to
one another. Thus, did these philosophers, setting out
from suppositions entirely contrary, and following routes
which only agreed in being quite unphilosophical and arbi-
trary, arrive, by a very unexpected coincidence, at the
same conclusion. Fermat could no Jonger deny the law
of refraction, as laid down by Descartes, but he was less
than ever disposed to admit the justness of his reasoning.
Descartes proceeded from this to a problem, which,
though suggested by optical considerations, was purely
geometrical, and in which his researches were completely
successful. It was well known, that, in the ordinary cases
of refraction by spherical and other surfaces, the rays are
not collected into one point, but have their foci spread
over a certain surface, the sections of which are the curves
called caustick curves, and that the focus of opticians is
only a point in this surface, where the rays are more con-
densed, and, of course, the illumination more intense than in
other parts of it. It is plain, however, that if refraction is
to be employed, either for the purpose of producing light
or heat, it would be a great advantage to have all the rays,
which come from the same point ofan object, united ac-
curately, after refraction, in the same point of the image.
This gave rise to an inquiry into the figure which the
superficies, separating two transparent media of different
refracting powers, must have, in order that all the rays
diverging from a given point might, by refraction at the said
superficies, be made to converge to another given point.
The problem was resolved by Descartes in its full extent ;

! Cartesii Dioptrices, cap. 8vum ; Geometria, lib. 2dus,

192 DISSERTATION SECOND. » [PART i.

and he proved, that the curves, proper for generating such
superficies by their revolution, are all comprehended
under one general character, viz. that there are always
two given points, from which, if straight lines be drawn to
any point in the curve, the one of these, plus or minus,
that which has a given ratio to the other, is equal to a given
line.

It is evident, when the given ratio here mentioned is a
ratio of equality, that the curve is a conick section, and
the two given points its two foci. The curves, in general,
are of the fourth or the second order, and have been dis-
tinguished by the name of the ovals of Descartes.

From this very ingenious investigation no practical re-
sult of advantage in the construction of lenses has been
derived. The mechanical difficulties of working a super-
ficies into any figure but a spherical one are so great, thaf,
notwithstanding all the efforts of Descartes himself, and of
many of his followers, they have never been overcome, so
that the great improvements in optical instruments have
arisen in a quarter entirely different.

Descartes gave also a full explanation of the rainbow, ’
as far as colour was not concerned, a part of the probiem
which remained for Newton to resolve. The path of the
ray was traced, and the angles of the incident ray, with
that which emerges after two refractions and one reflection,
was accurately determined. Descartes paid little atten-
iion to those who had gone before him, and, as already re-
marked, never once mentioned the Archbishop of Spala-
tro. Like Aristotle, he seems to have formed the de-
sign of cutting off the memory of all his predecessors, but
the invention of printing had made this a far more hope-
less undertaking than it was ia the days of the Greek philo-
sopher.

* Meteorum, cap. Svum.

sect, v.] DISSERTATION SECOND. 193

After the publication of the Dioptricks of Descartes, ia
1637, a considerable interval took place, during which, op-
ticks, and indeed science in general, made but little pro-
gress, till the Optica Promota of James Gregory, in 1663,
seemed to put them again in motion. ‘The author of this
work, a profound and inventive geometer, had applied
diligently to the study of opticks and the improvement of
optical instruments. The Optica Promota embraced seve-
ral new inquiries concerning the illumination and distinct-
ness of the images formed in the foci of lenses, and con-
tained an account of the Reflecting Telescope still known
by the name of its author. The consideration which sug-
gested this instrument was the imperfection of the images
formed by spherical lenses, in consequence of which, they
are not in plane, but in curved surfaces. The desire of
removing this imperfection led Gregory to substitute re-
flection for refraction in the construction of telescopes ;
and by this means, while he was seeking to remedy a
small evil, he provided the means of avoiding a much great-
er one, with which he was not yet acquainted, viz. that
which arises from the unequal refrangibility of light. The
attention of Newton was about the same time drawn to the
same object, but with a perfect knowledge of the defect
which he wanted to remove. Gregory thought it necessa-
ry that the specula should be of a parabolick figure ; and

the execution proved so difficult, that the instrument, dur-
ing his own life, was never brought to any perfection.
~The specula were afterwards constructed of ithe ordinary
spherical form, and the Gregorian telescope, till the time
of Dr. Herschell, was more in use than the Newtonian.

Gregory was professor of mathematicks at St. Aadrews,
and afterwards for a short time, at Edinburgh. His writings
strongly mark the imperfect intercourse which subsisted
at ‘that time between this country and the Continent.--

194 DISSERTATION SECOND. [rant &

Though the Opticks of Descartes had been published
twenty-five years, Gregory had not heard of the discovery
of the jaw of refraction, and had found it out only by his
own efforts ;—happy in being able, by the fertility of his
genius, to supply the defects of an insulated and remote
situation. |
A course of lectures on opticks, delivered at Cambridge
in 1668, by Dr. Barrow, and published in the year follow-
ing, treated of all the more difficult questions which had
occurred in that state of the science, with the acuteness
and depth which are found in all the writings of that geo-
meter. This work contains some new views in opticks,
and a great deal of profound mathematical discussion.
About this time Grimaldi, a learned jesuit, the compa-
nion of Riccioli in his astronomical labours, made known
some optical phenomena which had hitherto escaped ob-
servation. ‘They respected the action of bodies on light,
and when compared with reflection and refraction, might
be called, in the language of Bacon’s philosophy, crepus-
cular instances, indicating an action of the same kind,.but
much weaker and less perceptible. Having stretched a
hair across a sunbeam, admitted through a hole in the

window-shutter ef a dark chamber, he was surprised to find

the shadow much larger than the natural divergence of the

rays could have led him to expect. Other facts of the —
same kind made known the general law of the diffraction |
or inflexion of light, and showed that the rays are acted on
by bodies, and turned out of their rectilineal course, even a

when not in contact, but at a measurable distance from the
surfaces or edges of such bodies. Grimaldi gave an ac-
count of those facts in a treatise printed at Bologna in
1665. '

' Physico-Mathesis de Lumine, Coloribus, &c. in 4to.

secr. v.] | DISSERTATION SECOND. 195

Opticks, as indeed all the branches of natural philoso-
phy, have great obligations to Huygens. The former was
among the first scientifick objects which occupied his
mind; and his Dioptricks, though a posthumous work, is
most of it the composition of his early youth. It is writ-
ten with great perspicuity and precision, and is said to
have been a favourite book with Newton himsélf. Though
beginning from the first elements, it contains a full deve-
lopment of the matters of greatest difficulty in the con-
struction of telescopes, particularly in what concerns the
indistinctness arising from the imperfect foci into which
rays are united by spherical lenses ; and rules are deduc-
ed for constructing telescopes, which, though of different
sizes, shall have the same degree of distinctness, illumina-
tion, &c. Huygens was besides a practical optician; he
polished lenses, and constructed telescopes with his own
hands, and some of his objeet-glasses were of the enormous
focal distance of 130 feet. To his Dioptricks is added a
valuable treatise De Formandis Vitris.

In the history of opticks, particular attention is due to
his theory of light, which was first communicated to the
Academy of Sciences of Paris, in 1678, and afterwards
published, with enlargements, in 1690. '

Light, according to this ingenious system, consists in
certain undulations communicated by luminous bodies to
the etherial fluid which fills all space. This fluid is com-
posed of the most subtle matter, is highly elastick, and the
undulations are propagated through it with great velocity
in spherical superficies proceeding from a centre. Light,
in this view of it, differs from that of the Cartesian sys-
tem, which is supposed to be without elasticity, and to
convey impressions instantaneously, as a staff does from
the object it touches to the hand which holds it.

+ Traité de la Lumiére. Leyd. 1690,

196 DISSERTATION SECOND. (rans r,

It is not, however, in this general view, that the inge-
nuity of the theory appears, but in its application to ex-
plain the equality of ihe angles of incidence and reflection; |
and, most of all, the constant ratio which subsists between
the sines of the angles of incidence and of refraction. Few
things are to be met with more simple and beautiful than
this last application of the theory; but that which is most
remarkable of all is, the use made of it to explain the
double refraction of Iceland crystal. This crystal, which
is no other than the calcareous spar of mineralogists, has
not only the property of refracting light in the usual man-
ner of glass, water, and other transparent bodies, but it has
also another power of refraction, by which even the rays
falling perpendicularly on the surface of the crystal are
turned out of their course, so that a double image is form-
ed of all objects seen through these crystals. This pro-
perty belongs not only to calcareous spar, but, in a greater
or less degree, to all substances which are both crystallized
and transparent.

The common refraction is explained by Huygens, on-
the supposition, that the undulations in the luminous fluid
are propagated in the form of spherical waves. The double
refraction is explained on the supposition, that the undula-
tions of light, in passing through the calcareous spar, as-
sume a spheroidal form; and this hypothesis, though it
does not apply with the same simplicity as the former, yet
admits of such precision, that a proportion of the axes
the spheroids may be assigned, which will account for ee
precise quantity of the extraordinary refraction, and for all
the phenomena dependent on it, which Huygens had stu-
died with great care, and had reduced to the smallest num-
ber of general facts. 'That these spheroidal undulations
actually exist, he would, after all, be a bold theorist who
should affirm; but that the supposition of their existence is

*¥

sxcr. v.] DISSERTATION SECOND. 197

an accurate expression of the phenomena of double refrac-
tion, cannot be doubted. When one enunciates the hypo-
thesis of the spheroidal undulations, he, in fact, expresses
in a single sentence all the phenomena of double refraction.
The hypothesis is therefore the means of representing
these phenomena, and the laws which they obey, to the
imagination or the understanding, and there is, perhaps,
no theory in opticks, and but very few in natural philoso-
phy, of which more can be said. Theory, therefore, in
this instance, is merely to be regarded as the expression
of a general law, and in that light, I think, it is considered
by La Place.

To carry the theory of Huygens farther, and to render
it quite satisfactory, a reason ought to be assigned, why the
undulations of the luminous fluid are spheroidal in the case
of crystals, and spherical in all other cases. This would
be to render the generalization more complete ; and till
that is done, and a connexion clearly established between
the structure of crystallized bodies, and the property of
double refraction, the theory will remain imperfect. The
attention which at present is given to this most singular
and interesting branch of opticks, and the great number of
new phenomena observed and classed under the head of
the Polarisation of Light, make it almost certain that this
object will be either speedily accomplished, or that science
has here reached one of the immoveable barriers by which
the circle of human knowledge is to be for ever circum.

scribed.

END OF PART SECOND.

25

ADVERTISEMENT.

Tr was stated in a former Advertisement,* that the Second
Part of Mr. Srewart’s Dissertation on the History of Meta-
physical, Ethical, and Political Philosophy, would accom-
pany the present Half-Volume. Circumstances having
rendered it necessary to postpone the appearance of that
Part, till a later period in the progress of this Work, it
was agreed, that the remaining Part of Prorgssor Puay-
rain’s Dissertation on the History of the Mathematical
and Physical Sciences should, in the mean time, take its
place. It is to this arrangement, that the World is in-
debted for a Piece, which, though it only forms part of
a greater design, cannot but be regarded as a most valu-
able contribution to the History of Science ; whilst it de-
rives a melancholy interest, from its being the last literary
object that employed the talents, and engaged the solici-
tude, of its eminent author.

* Advertisement to Volume Third, of the Supplement to the
Encyclopedia Britannica.

2

Mr. Puayratir’s Dissertation was intended to furnish an
historical sketch of the principal discoveries and improve-
ments in Science, from the revival of Letters to the be-
ginning of the present century; and, in that portion of it
which is prefixed to the Second Volume of this work, the
history is brought down to the period marked by the
commencement of Newton’s discoveries. The remaining
half was to have completed the design, in three parts or
’ subdivisions ; the First, comprehending the period of New-
ton and Leisnitz; the Second, that of Euner and D’-
AtemBerT; and the Third, that of Lacranese and La-

PLACE.

Mr. Puayrarr was proceeding, with his accustomed di-
ligence and ardour, in the execution of this interesting
and congenial task, when he was seized with the illness
of which he died. The first subdivision of his plan, which
embraces a view of the advances made in the most re-
markable period of the history of Science, was happily
completed, and the printing finished, while he was yet
able to correct the Press. It is now given to the Pub-
lic, under the painful impression that it must too proba-
bly be considered as a Fragment ; for the editor fears,
that the materials collected for the completion of the
Dissertation, though containing the results of much ela-
borate inquiry, and profound reflection, cannot be put into
a shape that would justify their publication as a work of

Proressor PLAYFAIR.

Edinburgh, December 1819.

DISSERTATION SECOND:

EXHIBITING A GENERAL VIEW OF THE

jProgress of fiathematical and JOhpstcal
Sctence,

SINCE THE REVIVAL OF LETTERS IN EUROPE.

PART II.

BY JOHN PLAYFAIR,

Late Professor of Natural Philosophy in the University of Edinburgh, Fellow of the Royal
Society of London, and Secretary of the Royal Society of Edinburgh.

=(9 bs ae
Nh RlOetaa
‘hong?

eh | vines a ihn 404
Wik aes, nco bli 9 mere

y
Ay hist.
lA
ay A! *y bce Ab on

DISSERTATION SECOND.

SKETCH OF THE PROGRESS OF NATURAL PHILOSOPHY FROM
THE REVIVAL OF LEARNING TO THE PRESENT TIME.

peewee news

PART SECOND.

FROM THE COMMENCEMENT OF NEWTON’S DISCOVERIES
TO THE YEAR 1818.

Ty the former part of this sketch, the history of each division
of the sciences was continued without interruption, from the
beginning to the end. During the period, however, on
which I am now to enter, the advancement of knowledge has
been so rapid, and marked by such distinct steps, that seve-
ral pauses or resting-places occur of which it may be advisa-
ble to take advantage. Were the history of any particular
science to be continued for the whole of the busy interval
which this second part embraces, it would leave the other
sciences too far behind; and would make it difficult to per-
ceive the mutual action by which they have so much assisted
the progress of one another. Considering some sort of sub-
division, therefore, as necessary, and observing, in the inter-
val which extends from the first of Newton’s discoveries to
the year 1818, three different conditions of the Physico-

4 DISSERTATION SECOND. {pant xt.

Mathematical sciences, well marked and distinguished by
great improvements, I have divided the above interval into
three corresponding parts. The first of these, reaching
from the commencement of Newton’s discoveries in 1663,
to a little beyond his death, or to 1730, may be denominat-
ed, from the men who impressed on it its peculiar charac-
ter, the period of Newton and Leibnitz. 'The second, which,
for a similar reason, I call that of Euler and D’Alembert,
may be regarded as extending from 1730 to 1780; and the
third, that of Lagrange and Laplace, from 1780 to 1818.

PERIOD FIRST.

SECTION If.
THE NEW GEOMETRY.

Tue seventeenth century, which had advanced with such
spirit and success in combating prejudice, detecting error,
and establishing truth, was destined to conclude with the
most splendid series of philosophical discoveries yet recorded
in the history of letters. It was about to witness, in suc-
cession, the invention of Flaxions, the discovery of the
Composition of Light, and of the Principle of Universal
Grayitation,—all three within a period of little more than
twenty years, and all three the work of the same individual.
It is to the first of these that our attention at present is
to be particularly directed.

sucr. 1.] DISSERTATION SECOND. 5

The notion of Infinite Quantity had, as we have already
seen, been for some time introduced into Geometry, and
having become a subject of reasoning and calculation, had,
in many instances, after facilitating the process of both, led
to conclusions from which, as if by magic, the idea of infinity
had entirely disappeared, and left the geometer or the alge-
braist in possession of valuable propositions, in which were
involved no magnitudes but such as could be readily exhibit-
ed. The discovery of such results had increased both the
interest and extent of mathematical investigation.

It was in this state of the sciences, that Newton began his
mathematical studies, and, after a very short interval, his
mathematical discoveries.’ ‘The book, next to the elements,
which was put into his hands, was Wallis’s Arithmetic of
Infinites, a work well fitted for suggesting new views in
geometry, and calling into activity the powers of mathe-
matical invention. Wallis had effected the quadrature of all
those curves in which the value of one of the co-ordinates
can be expressed in terms of the other, without involving
either fractional or negative exponents. Beyond this point
neither his researches, nor those of any other geometer,
had yet reached, and from this point the discoveries of
Newton began. The Savilian Professor had himself been
extremely desirous to advance into the new region, where,
among other great objects, the quadrature of the circle
must necessarily be contained, and he made a very noble
effort to pass the barrier by which the undiscovered coun-
try appeared to be defended. He saw plainly, that if the
equations of the curves which he had squared were ranged
in a regular series, from the simpler to the more com-
plex, their areas would constitute another corresponding

' He entered at Trinity College, Cambridge, in June 1660.
The date of his first discoveries is about 1663.

6 DISSERTATION SECOND. [paRnT 11.

series, the terms of which were all known. He farther
remarked, that, in the first of these series, the equation to
the circle itself might be imtroduced, and would occupy
the middle place between the first and second terms of
the series, or between an equation to a straight line and
an equation to the common parabola. He concluded,
therefore, that if, in the second series, he could interpo-
late a term in the middle, between its first and second
terms, this term must necessarily be no other than the area
of the circle. But when he proceeded to pursue this very
refined and philosophical idea, he was not so fortunate ; and
his attempt toward the requisite interpolation, though it did
not entirely fail, and made known a curious property of the
area of the circle, did not lead to an indefinite quadrature of
that curve.". Newton was much more judicious and success-
ful in his attempt. Proceeding on the same general princi-
ple with Wallis, as he himself tells us, the simple view
which he took of the areas already computed, and of the
terms of which each consisted, enabled him to discover
the law which was common to them all, and under which
. the expression for the area of the eircle, as well as of in
numerable other curves, must needs be comprehended.
In the case of the circle, as in all those where a frac-
tional exponent appeared, the area was exhibited in the
form of an infinite series.

The problem of the quadrature of the circle, and of so
many other curves, being thus resolved, Newton immedi-
ately remarked, that the law of these series was, with a

! The interpolation of Wallis failed, because he did not em-
ploy literal or general exponents. His theorem, expressing the
area of the entire circle by a fraction, of which the numerator
and denominator are each the continued product of a certain
series of numbers, is a remarkable anticipation of some of Eu-
ler’s discoveries, Calc. Int. Tom. I. cap. 8.

h
;

7

sKcr. 1.] DISSERTATION SECOND. 7

small alteration, the law for the series of terms which
expresses the root of any binomial quantity whatsoever.
Thus he was put in possession of another valuable dis-
covery, the Binomial Theorem, and at the same time
perceived that this last was in reality, in the order of
things, placed before the other, and afforded a much easier
access to such quadratures than the method of interpola-
tion, which, though the first road, appeared now, neither
to be the easiest nor the most direct.

It is but rarely that we can lay hold with certainty of the
thread by which genius has been guided in its first dis-
coveries. Here we are proceeding on the authority of
the author himself, for in a letter’ to Oldenburgh,' Se-
cretary of the Royal Society of London, he has entered
into considerable detail on this subject, adding (so ready
are the steps of invention to be forgotten), that the facts
would have entirely escaped his memory, if he had not
been reminded of them by some notes which he had
made at the time, and which he had accidentally fallen on.
The whole of the letter just referred to, is one of the most
valuable documents to be found in the history of inven-
tion.

In all this, however, nothing occurs from which it can be
inferred that the method of fluxions had yet occurred to the
inventor. His discovery consisted in the method of reduc-
ing the value of y, the ordinate of a curve, into an infinite
series of the integer powers of x the abscissa, by division,
or the extraction of roots, that is, by the Binomial Theorem ;
after which, the part of the area belonging to each term
could be assigned by the arithmetic of infinites, or other
methods aiready known. He has assured us himself, how-
ever, that the great principle of the new geometry was

“ Commercium Epistolicum, Art. 55.

S DISSERTATION SECOND. [PART 11.

known to him, and applied to investigation as early as 1665
or 1666.' Independently of that authority, we also know,
on the testimony of Barrow, that soon after the period just
mentioned, there was put into his hands by Newton a manu-
script treatise,? the same which was afterwards published
under the title of dnalysis per quationes Numero Termino-
rum Infinitas, in which, though the instrument of investiga-
tion is nothing else than infinite series, the principle of flux-
ions, if not fully explained, is at least distinctly pointed out.
Barrow strongly exhorted his young friend to publish this
treasure to the world; but the modesty of the author, of
which the excess, if not culpable, was certainly in the pre-
sent instance very unfortunate, prevented his compliance.
All this was previous to the year 1669; the treatise itself
was not published till 1711, more than forty years after it
was written.

For a long time, therefore, the discoveries of Newton
were known only to his friends, and the first work in which
he communicated any thing to the world on the subject of
fluxions, was in the first edition of the Principia, in 1687, in the
second Lemma of the second book, to which, in the disputes
that have since arisen about the invention of the new analy-
sis, reference has been so often made. The principle of the
fluxionary calculus was there pointed out, but nothing ap-
peared that indicated the peculiar algorithm, or the new
notation, which is so essential to that calculus. About this
Newton had yet given no information; and it was only from
the second volume of Wallis’s Works, in 1693, that it be-
came known to the world.’ It was no less than ten years

? Quadrature of Curves, Introduction.

2 Com. Epist, No. I. 11. II. &c.

* Wallis says, that he had inserted in the English edition of
his book, published in 1685, several extracts from Newton’s
Letters, ** Omissis multis alits inibi notatu dignis, co quod sperave-

seer, 1.] DISSERTATION SECOND. 9

after this, in 1704, that Newton himself first published a work
on the new calculus, his Quadrature of Curves, more than
twenty-eight years after it was written.

These discoveries, however, even before the press was
employed as their vehicle, could not remain altogether un-
_ known in a country where the mathematical sciences were
cultivated with zeal and diligence. Barrow, to whom they
were first made known by the author himself, communicated
them to Oldenburgh, the Secretary of the Royal Society,
who had a very extensive correspondence all over Europe.
By him the series for the quadrature of the circle were made
known to James Gregory,' in Scotland, who had occupied
himself very much with the same subject. They were also
communicated to Leibnitz in Germany, who had become
acquainted with Oldenburgh in a visit which he made to
England in 1673. At the time of that visit, Leibnitz was
but little conversant with the mathematics ; but having after-
wards devoted his great talents to the study of that science,
he was soon in a condition to make new discoveries. He
invented a method of squaring the circle, by transforming it
into another curve of an equal area, but having the ordinate
expressed by a rational fraction of the absciss, so that its
area could be found by the methods already known. In this
way he discovered the series, so remarkable for its simplici-
ty, which gives the value of a circular arch in terms of the

rim clarissimuim virum voluisse tum illa, tum alia que apud ipsum
premit edidisse. Cum vero illud nondum fecerit abet eorum non-
nulla hic atlingere ne pereant.”. Among these last is an account
of the fluxionary notation, according to which the fluxions of
flowing quantities are distinguished by points, and also of certain
applications of this new algorithm, extracted from two letters of
Newton, written in 1792.—Opera, Tom. II. p. 390, &c.—There
is no evidence of his notation having existed earlier than that
date, though it be highly probable that it did.

1 Note A, at the end.
2

’

10 DISSERTATION SECOND. {ran i.

tangent. This series he communicated to Oldenburgh in
in 1674, and received from him in return an account of the
progress made by Newton and Gregory in the invention of
series. In 1676, Newton described his method of quadra-
tures at the request of Oldenburgh, in order that it might be
transmitted to Leibnitz in the two letters already mentioned,
as of such value by recording the views which guided that
great geometer in his earliest, and some of his most impor-
‘tant discoveries. 'The method of fluxions is not communi-
cated in these letters; nor are the principles of it in any
way suggested ; though there are, in the last letter, two sen-
tences in transposed characters, which ascertain that New-
ton was then in possession of that method, and employed in
speaking of it the same language in which it was afterwards
made known. In the following year, Leibnitz, in a letter to
Oldenburgh, introduces differentials, and the methods of his
calculus for the first time. This letter,' which is very im-
portant, clearly proves that the author was then in full pos-
session of the principles of his calculus ; and had even in-
vented the algorithm and notation.

From these facts, and they are all that bear directly on
the question concerning the invention of the infinitesimal
analysis, if they be fairly and dispassionately examined, |
think that no doubt cai remain, that Newton was the first
inventor of that analysis, which he called by the name of
Fluxions ; but that, in the communications made by him, or
his friends, to Leibnitz, there was nothing that could convey
any idea of the principle on which that analysis was founded,
or of the algorithm which it involved. The things stated
were merely results; and though some of those relating to
the tangents of curves might shew the author to be in pos-
session of a method of investigation different from infinite

" Commercium Epistolicum, No. 66.

azcr. 1.] DISSERTATION SECOND. ii

series, yet they afforded no indication of the nature of that
method, or the principles on which it proceeded.

In what manner Newton’s communications in the two let-
ters already referred to, may have acted, in stimulating the
curiosity and extending or even directing the views of such
a man as Leibnitz, I shall not presume to decide (nor even,
if such effect be admitted, will it take from the originality of
his discoveries); but that in the authenticated communications
which took place between these philosophers, there was
nothing which could make known the nature of the fluxiona-
ry calculus, I consider as a fact most fully established.

Of the new or infinitesimal analysis, we are, therefore, to
consider Newton as the first inventor, Leibnitz as the se-
cond; his discovery, though posterior in time, having been
made independently of the other, and having no less claim
to originality. It had the advantage also of being first made
known to the world; an account of it, and of its peculiar al-
gorithm, having been inserted in the first volume of the
Acta Eruditorum, in 1684. ‘Thus, while Newton’s dis-
covery remained a secret, communicated only to a few
friends, the geometry of Leibnitz was spreading with great
rapidity over the Continent. Two most able coadjutors, the
brothers James and John Bernoulli, joined their talents to
those of the original inventor, and illustrated the new me-
thods by the solution of a great variety of difficult and inte-
resting problems. The reserve of Newton still kept his
countrymen ignorant of his geometrical discoveries, and the
first book that appeared in England on the new geome-
iry, was that of Craig, who professedly derived his knowledge
from the writings of Leibnitz and his friends. Nothing,
however, like rivalship or hostility between these inventors
had yet appeared ; each seemed willing to admit the origin-
ality of the other’s discoveries ; and Newton, in the passage

12 DISSERTATION SECOND. {pany ar.

of the Principia just referred to, gave a highly favourable
opinion on the subject of the discoveries of Leibnitz.

The quiet, however, that now prevailed between the En-
glish and German philosophers, was clearly of a nature to be
easily disturbed. With the English was conviction, and, as
we have seen, a well grounded conviction, that the first dis-
covery of the Infinitesimal Analysis was the property of New-
ton ; but the analysis thus discovered was yet unknown to
the public, and was in the hands of the inventor and his
friends. With the Germans, there was the conviction, also
well founded, that the invention of their countryman was
perfectly originial ; and they had the satisfaction to see his
calculus everywhere adopted, and himself considered all
over the continent as the sole inventor. The friends of
Newton could not but resist this latter claim, and the friends
of Leibnitz, seeing that their master had become the great
teacher of the new calculus, could not easily bring them-
selves to acknowledge that he was not the first discoverer.
The tranquillity that existed under such circumstances, if
once disturbed, was not likely to be speedily restored.

Accordingly, a remark of Fatio de Duillier, a mathemati-
cian, not otherwise very remarkable, was sufficient to light
up a flame which a whole century has been hardly sufficient
to extinguish. Ina paper on the line of swiftest descent,
which he presented to the Royal Society in 1699, was this sen-
tence: “ J hold Newton to have been the first inventor of this
calculus, and the earliest, by several years, induced by the
evidence of facts; and whether Leibnitz, the second inven-
tor, has borrowed any thing from the other, I leave to the
judgment of those who have seen the letters and manuscripts
of Newton.” Leibnitz replied to this charge in the Leipsic
Journal, without any asperity, simply stating himself to have
been, as well as Newton, the inventor; neither contesting

— a «

*

seer. 1.] DISSERTATION SECOND. 13

nor acknowledging Newton’s claim to priority, but asserting
his own to the first publication of the calculus.

Not long after this, the publication of Newton’s Quadra-
jure of Curves, and his Enumeration of the lines of the third
order (1705), afforded the same journalists an opportunity of
showing their determination to retort the insinuations of Du-
illier, and to carry the war into the country of the enemy.
After giving a very imperfect synopsis of the first of these
books, they add: “ Pro differentiis igitur Leibnitianis D.
Newtonus adhibet, semperque adhibuit fluxiones ; que sunt
proxime ut fluentium augmenta, equalibus temporis particulis
quam minimis genita 3 visque tum in suis Prineipiis Nature
Mathematicis, tum in aliis post editis,.eleganter est usus ;
quamadmodum Honoratus Fabrius in sua Synopst Geometricé
motuum progressus Cavalieriane methodo substitut.”'

In spite of the politeness and ambiguity’ of this passage,
the most obvious meaning appeared to be, that Newton had
been led to the notion of fluxions by the differentials of
Leibnitz, just as Honoratus Fabri had been led to substitute
the idea of progressive motion for the indivisibles of Cavalieri.
A charge so entirely unfounded, so inconsistent with acknow-
ledged facts, and so little consonant to declarations that had
formerly come from the same quarter, could not but call
forth the indignation of Newton and his friends, especially as
it was known, that these journalists spoke the language of
Leibnitz and Bernoulli. In that indignation they were per-
fectly justified ; but when the minds of contending parties
have become irritated in a certain degree, it often happens
that the injustice of one side is retaliated by an equal injus-
tice from the opposite. Accordingly, Keill, who, with more
zeal than judgment, undertook the defence of Newton’s

* Com. Epist. No. 97. Newtoni Opera, Tom. IV. p. 577,
* Note B, at the end.

14 DISSERTATION SECOND. [pant 1.

claims, instead of endeavouring Jo establish the priority of
his discoveries, by an appeal to facts and to dates that could
be accurately ascertained (in which he would have been
completely successful), undertook to prove, that the commu-
nications of Newton to Leibnitz, were sufficient to put the
latter in possession of the principles of the new analysis, after
which he had only to substitute the notion of differentials for
that of fuxions. In support of a charge which it would have
required the clearest and most irresistible evidence to justify,
he had, however, nothing to offer but equivocal facts and
overstrained arguments, such as could only convince those
who were already disposed to believe. They were, accord-
ingly, received as sound reasoning in England, rejected as
absurd in Germany, and read with no effect by the mathe-
maticians of France and Italy.

Leibnitz complained of Keill’s proceeding to the Royal
Society of London, which declined giving judgment, but ap-
pointed a commission of its members to draw up a full and
detailed report of all the communications which had passed
between Newton and Leibnitz, or their friends, on subjects
connected with the new analysis, from the time of Collins
and Oldenburgh to the date of Keill’s letter to Sir Hans
Sloane in 1711, the same that was now complained of. This
report forms what is called the Commercium Epistolicum ;
it was published by order of the Royal Society the year fol-
lowing, and contains an account of the facts, which, though
in the main fair and just, does not give that impression of the
impartiality of the reporters which the circumstances so im-
periously demanded. Leibnitz complained of this publica-
tion; and alleged, that though nothing might be inserted that
was not contained in the original letters, yet certain passages
were suppressed which were favourable to his pretensions.
He threatened an answer, which, however, never appeared.
Some notes were added to the Commercium, which contain

s

seer. 1.] DISSERTATION SECOND. 15

a good deal of asperity and unsupported insinuation ; the
Recensio, or review of it, inserted i in the Philosophical ina
actions for 1715, though written with ability, is still more
hable to the same censure.

In the year (1713) which followed the publication of the
Commercium Epistolicum, a paragraph was circulated among
the mathematicians of Europe, purporting to be the judgment
of a mathematician on the invention of the new analysis.
The author was not named, but was generally understood to
. be John Bernoulli, of which, indeed, the terms in which
Leibnitz speaks of the judgment leave no room to doubt.
Bernoulli was without question well acquainted with the sub-
ject in dispute; he was a perfect master of the calculus ; ha
had been one of the great instruments of its advancement,
and, except impartiality, possessed every requisite for a
judge. Without offence it might be said, that he could
scarcely be accounted impartial. He had been a party
in all that had happened ;—warmly attached as he was
to the one side, and greatly exasperated against the other.
his temper had been more frequently ruffled, and his pas-
sions or prejudices more violently excited, than those of any
other individual. With all his abilities, therefore, he was
not likely to prove the fairest and most candid judges in a
cause that might almost be considered as his own. His sen-
tence, however, is pronounced in calm and temperate lan-
guage, and amounts to this, That there is no reason to believe,
that the fluxionary calculus was invented before the diffe-
rential. Ishall refer to a note’ the discussion of the evi-
dence which he points out as the ground of this decision,
though the facts already stated might be considered as sufli-
cient to enable the reader to form an opinion on the subject.
The friends of Leibnitz hurt their own cause, by attempting

* Note C, at the end.

16 DISSERTATION SECOND. [eanr 1.

to fix on Newton a charge of plagiarism, which was refuted
by such a chain of evidence, by so many dates distinctly as-
eertained, and so many concessions of their own. A candid
review of the evidence led to the conviction, that both New-
ton and Leibnitz were original inventors. When the English
mathematicians accused Leibnitz of borrowing from Newton,
they were, therefore, going much farther than the evidence
authorized them, and were mistaking~their own partialities
for proofs. They maintained what was not true, but what,
nevertheless, was not physically impossible, the discovery of
Newton being certainly prior to that of Leibnitz. The Ger-
man mathematicians, on the other hand, when they charged
Newton with borrowing from Leibnitz, were maintaining what
was not only false, but what involved an impossibility. This
is the only part of the dispute, in which any thing that could
be construed into mala fides can be said to have appeared.
Iam far, however, from giving it that construction; men of
auch high character, both for integrity and talents, as Leib-
nitz and Bernoulli, ought not to be lightly subjected to so
cruel an imputation. Partiality, prejudice, and passion, are
sufficient to account for much injustice, without a decided
intention to do wrong.

In the state of hostility to which matters were now brought,
ihe new analysis itself was had recourse to, as affording to
either side abundant means of annoying its adversaries, by an
inexhaustible supply of problems, accessible to those alone
who were initiated in the doctrines, and who could command
the resources of that analysis. ‘The power of resolving such
problems, therefore, seemed a test whether this analysis was
understood or not. Already some questions of this kind had
been proposed in the Leipsic Journal, not as defiances, but
as exercises in the new geometry. Such was the problem of
the Catenaria, or the curve, which a chain of uniform weight
makes when suspended from two points. This had been

sacr. 1.] DISSERTATION SECOND. 17

proposed hy Bernoulli in 1690; and had been resolved by
Huygens, Leibnitz, and himself.

A question had been proposed, also, concerning the line
of swiftest descent in 1697, or the line along which a
body must descend, in order to go from one point to an-
other not perpendicularly under it, in the least time pos-
sible. Though a straight line be the shortest distance
between two points, it does not necessarily follow, that
the descent in that line will be most speedily performed,
for, by falling in a curve that has at first a very rapid de-
clivity, the body may acquire in the beginning of its motion
so great a velocity, as shall carry it over a long line in
jess time than it would describe a short one, with a ve-
locity more slowly acquired. This, however, is a problem
that- belongs to a class of questions of peculiar difficulty ;
and accordingly it was resolved only by a few of the most
distinguished mathematicians. The solutions which ap-
peared within the time prescribed were from Leibnitz,
Newton, the two Bernoullis, and M. de l’Hopital. New-
ton’s appeared in the Philosophical Transactions without a
name; but the author was easily recognised. John Ber-
noulli, on seeing it, is said to have exclaimed, Ex unguce
leonem !

The curve that has the property required is the cycloid ;
Newton has given the construction, but has not accompa-
nied it with the analysis. He added afterwards "the de-
moustration of a very curious theorem for determining the
time of the actual descent. Leibnitz resolved the problem
the same day that he received the programme in which it
was proposed.

The problem of orthogonal trajectories, as it is called,
had been long ago proposed in the Acta Eruditorum, with
an invitation to all who were skilled in the new analysis
to attempt the solution. The problem had not, at first.

3

18 DISSERTATION SECOND. [parT 17.

met with the attention it was supposed to deserve, but
John Bernoulli having resumed. the consideration of it,
found out what appeared a very perfect and very gene-
ral solution; and the question was then (1716) proposed
anew by Leibnitz, for the avowed purpose of trying the
skill of the English mathematicians. The question is, a
system of curves described according to a known law be-
ing given (all the hyperbolas, for instance, that are de-
scribed between the same assymptotes; or all the parabo-
las that have the same directrix, and that pass through
the same point, &c.), to describe a curve which shall cut
them all at right angles. This may be considered as the
first defiance professedly aimed at the English mathemati-
cians. ‘The problem was delivered to Newton on his re-
turn from the Mint, when he was much fatigued with the
business of the day; he resolved it, however, the same
evening, and his solution, though without a name, is given
in the Philosophical Transactions for 1716.

This solution, however, only gave rise to new quarrels,
for hardly any thing so excellent could come from the
one side, that it could meet with the entire approbation
of the other. -Newton’s, indeed, was rather the plan or
projet of an investigation, than an actual solution; and, in
the general view which it took of the question, could
hardly provide against all the difficulties that might occur
in the application to particular cases. This was what Ber-
noulli objected to, and affected to treat the solution as of
no value. Brook Taylor, secretary of the Royal Society,
and well known as one of .the ablest geometers of the
time, undertook the defence of it, but concluded with
using language very reprehensible, and highly improper to
be directed by one man of science against another. Hav-

* Vol. XXIX. p. 399.

skcr. 1.] DISSERTATION SECOND. i9

ing sufficiently, as he supposed, replied to Bernoulli and
his friends, he adds, “if they are not satisfied with the
solution, it must be ascribed to their own ignorance.” It
strongly marks the temper by which both sides were now
animated, when a man like Taylor, eminent for profound
science, and, in general, very much disposed to do justice
to the merits of others, should so forget himself as to re-
proach with ignorance of the calculus, one of the men
who understood it the best, and who had contributed the
most to its improvement. The irritability and prejudices
of Bernoulli admitted of no defence, and he might very
well have been accused of viewing the solution of New-
ton through a medium disturbed by their action; but to
suppose that he was unable to understand it, was an im-
pertinence that could only react on the person who was
guilty of it. Bernoulli was not exemplary for his pa-
tience, and it will be readily believed, that the incivility
of Taylor was sufficiently revenged. It is painful to see
men of science engaged in such degrading altercation, and
I should be inclined to turn from so disagreeable an ob-
ject, if the bad effects of the spirit thus excited were not
such as must again obtrude themselves on the notice of the
reader.

Taylor not long after came forward with an open defi-
ance to the whole Continent, and proposed a problem,
Omnibus geometris non Anglis,—a problem, of course,
which he supposed that the English mathematicians alone
were sufliciently enlightened to resolve. He selected one,
accordingly, of very considerable difliculty,—the integration
of a fluxion of a complicated form; which, nevertheless,
admitted of being done in a very elegant manner, known,
[ believe, at that time to very few of the English mathemati-

* Eorum imperitie tribuendum est.

20 DISSERTATION SECOND. [Parr i,

cians, to Cotes, to himself, and, perhaps, one or two more.
The selection, nevertheless, was abundantly injudicious ;
for Bernoulli, as long ago as» 1702, had explained the
method of integrating this, and such like formulas, both in
the Paris Memoirs and in the Leipsic Acts. ‘The ques-
tion, accordingly, was no sooner proposed than it was an-
swered in a-manner the most clear and satisfactory ; se
the defiance of Taylor only served to display the address
and augment the triumph of his adversary.

The last and most unsuccessful of these challenges was
that of Keill, of whose former appearance in this contro-
versy we have already had so much more reason to com-
mend the zeal than the discretion. Among the problems
in the mixt mathematics which had excited most attention,
and which seemed best calculated to exercise the resour-
ces of the new analysis, was the determination of the path
of a projectile in a medium which resists proportionally
to the square of the velocity, that being nearly the law
of the resistance which the air opposes to bodies moving
with great velocity. The resistance of fluids had been
treated of by Newton in the second book of the Princi-
pia, and he had investigated a great number of curious
and important propositions relative to its effects. He had
considered some of the simpler laws of resistance, but of
the case just mentioned he had given no solution, and,
after approaching as near as possible to it on all sides,
had withdrawn without making an attack. A problem so
formidable was not likely to meet with many who, even
in the more improved state at which the calculus had
now arrived, could hope to overcome its difliculties.—
Whether Keill had flattered himself that he could resolve
the problem, or had forgotten, that when a man proposes
a question of defiance to another, he ought to be sure
that he can answer it himself, may be doubted ; but this

seer. 1.) DISSERTATION SECOND, 21

is certain, that, without the necessary preparation, he
boldly challenged Bernoulli to produce a solution.

Bernoulli resolved the question in a very short time,
not only for a resistance proportional to the square, but
to any power whatsoever of the velocity, and by the con-
ditions which he aflixed to the publication of his solution,
took care to expose the weakness of his antagonist. He
repeatedly offered to send his solution to a confidential
person in London, providing Keill would do the same.
Keill never made any reply to a proposal so fair, that
there could only be one reason for declining ity Bernoulli,
of course, exulted over him cruelly, breaking out in a
torrent of vulgar abuse, and losing sight of every maxim
of candour and good taste.

Such, then, were the circumstances under which the
infinitesimal analysis,—the greatest discovery ever made in
the mathematical sciences;—was ushered into the world.
Every where, as it became known, it enlarged the views,
roused the activity, and increased the power of the ge-
ometer, while it directed the warmest sentiments of his
gratitude and admiration toward the great inventors. In
one respect, only, its effects were different from those
which one would have wished to see produced. It ex-
cited jealousy between two great men who ought to have
been the friends of one another, and disturbed in both
that philosophic tranquillity of mind, for the loss of which
even glory itself is scarcely an adequate recompense.

In order to form a correct estimate of the magnitude
and value of this discovery, it may be useful to look back
at the steps by which the mathematical sciences had been
prepared for it. When we attempt to trace those steps
to their origin, we find the principle of the infinitesimak
analysis making its first appearance in the method of Ex-
haustions, as exemplified in the writings of Euclid and Ar-

22 DISSERTATION SECOND. [rant it.

chimedes. These geometers observed, and, for what we
know, were the first to observe, that the approach which
a rectilineal figure may make to one that is curvilineal,
by the increase of the number of its sides, the diminution
of their magnitude, and a certain enlargement of the an-
gles they contain, may be such that the properties of the
former shall coincide so nearly with those of the latter,
that no real difference can be supposed between them
without involving a contradiction ; and it was i ascertaining
the conditions of this approach, and in showing the contra-
diction to be unavoidable, that the method of Exhaustions
consisted. ‘The demonstrations were strictly geometrical,
but they were. often complicated, always indirect, and of
course synthetical, so that they did not explain the means
by which they had been discovered.

At the distance of more than two thousand years, Cava-
lieri advanced a step farther, and, by the sacrifice of some
apparent, though of no real accuracy, explained, in the me-
thod of indivisibles, a principle which could easily be made
to assume the more rigid form of Exhaustions. This was a
very important discovery ;—though the process was not
analytical, the demonstrations were direct, and, when applied
to the same subjects, led to the same conclusions which the
ancient geometers had deduced ; by an indirect proof also,
such as those geometers had adopted, it could always be
shown that an absurdity followed from supposing the results
deduced from the method of indivisibles to be other than
rigorously true.

The method of Cavalieri was improved and extended by
a number of geometers of great genius who followed him;
Torricelli, Roberval, Fermat, Huygens, Barrow, who all ob-
served the great advantage that arose from applying the
general theorems concerning variable quantity, to the cases
where the quantities approached to one another infinitely

7 —

srer. 1,] DISSERTATION SECOND. 23

near, that is, nearer than within any assigned difference.’
There was, however, as yet, no calculus adapted to these
researches, that is, no general method of reasoning by help
of arbitrary symbols.

But we must go back a step, in point of time, if we would
trace accurately the history of this last improvement. Des-
cartes, as has been shown in the former part of this outline,
made a great revolution in the mathematical sciences, by
applying algebra to the geometry of curves; or, more gene-
rally, by applying it to express the relations of variable quan-
tity. This added infinitely to the value of the algebraic
analysis, and to the extent of its investigations. The same
great mathematician had observed the advantage that would
be gained in the geometry of curves, by considering the
variable quantities in one state of an equation as differing in-
finitely little from the corresponding quantities in another
state of the same equation. By means grounded on this he
had attempted to draw tangents to curves, and to determine
their curvature ; but itis seldom the destination of Nature
that a new discovery should be begun and perfected by the
game individual; and, in these attempts, though Descartes
did not entirely fail, he cannot be considered as having been
successful.?

At last came the two discoverers, Newton and Leibnitz,
who completely lifted up the veil which their predecessors
had been endeavouring to draw aside. They plainly saw,
as Descartes indeed had done in part, that the infinitely
small variations of the ordinate and absciss are closely con-
nected with many properties of the curve, which have but
avery remote dependence on the ordinates and abscissz
themselves. Hence they inferred, that, to obtain an equa-
tion expressing the relations of these variations to one an-

1 Note D, at the end. * Dissert. Second, Part I. p. 18.

24 DISSERTATION SECOND. [PART If.

other, was to possess the most direct access to the knowledge
of those properties. They observed also, that when an
equation of this kind was deduced from the general equation,
it admitted of being brought to great simplicity, and of being
resolved much more readily than the other. In effect, it
assumed the form of a simple equation ; but, in order to make
this deduction in the readiest and most distinct way, the in-
troduction of new symbols, or of a new algorithm, was
necessary, the invention of which could cost but little to the
creative genius of the men of whom I now speak. They
appear, as has been already shown, to have made their dis-
coveries separately ;—Newton first,—Leibnitz afterwards, at
a considerable interval, yet the earliest, by several years, in
communicating his discoveries to the world.

Thus, though there had been for ages a gradual approach
to the new analysis, there were in that progress some great
and sudden advances, which elevated those who made them
to a much higher level than their predecessors. A great
number of individuals co-operated in the work; but those
who seem essential, and in the direct line of advancement,
are Euclid, Cavalieri, Descartes, Newton and Leibnitz. If
any of the others had been wanting, the world would have
been deprived of many valuable theorems, and many colla-
teral improvements, but not of any general method essential
to the completion’ of the infinitesimal analysis.

The views, however, of this analysis, taken by the two
inventors, were not precisely the same. Leibnitz, consider-
ing the differences of the variable quantities as infinitely
small, conceived that he might reject the higher powers of
those differences without any sensible error; so that none
of those powers but the first remained in the differential
equation finally obtained. ‘The rejection, however, of the
higher powers of the differentials, was liable to objection, for
it had the appearance of being only an approximation, and

ee

:
a

SS a ee a ee ee

seer. 1.] DISSERTATION SECOND. 25

did not come up to the perfect measure of geometrical pre-
cision. The analysis, thus constituted, necessarily divided
itself into two problems ;—the first is,—having given an
equation involving two or more variable quantities, to find
the equation expressing the relation of the differentials, or
infinitely small variations of those quantities; the second is
the converse of this ;—having given an equation involving
two or more variable quantities, and their differentials, to
exterminate the differentials, and so to exhibit the variable
quantities in a finite state. This last process is called inte-
gration in the language of the differential analysis, and the
finite equation obtained is called = integral of the given
differential equation.

Newton proceeded in some respects differently, and so as
to preserve his calculus from the imputation of neglecting or
throwing away any thing merely because it was small. In-
stead of the actual increments of the flowing or variable
quantities, he introduced what he called the fluxions of those
quantities,—meaning, by fluxions, quantities which had to
one another the same ratio which the increments had in their
ultimate or evanescent state. He did not reject quantities,
therefore, merely because they were so small that he might
do so without committing any sensible error, but because he
must reject tliem, in order to commit no error whatsoever.
Fluxions were, with him, nothing else than measures of the
velocities with which variable or flowing quantities were
supposed to be generated, and they might be of any magni-
tude, providing they were in the ratio of those velocities, or,
which is the same, in the ratio of the nascent or evanescent
increments.' The fluxions, therefore, and the flowing quan-

+ « J consider mathematical quantities in this place not as con-
sisting of small parts, but as described by a continued motion.
Lines are described and thereby generated, not by the apposition
of paris, but by the continued motion of points, superfices by the
motion of lines,” &c.—Quadrature of Curves, Introduction.

4

26 DISSERTATION SECOND. [pany 4.

tities or fluents of Newton correspond to the differentials and
the sums or integrals of Leibnitz; and though the symbols
which denote fluxions are different from those used to ex-
press differentials, they answer precisely the same purpose.
The fluxionary and differential calculus may therefore be
considered as two modifications of one general method,
aptly distinguished by the name of the infinitesimal analysis.

By the introduction of this analysis, the domain of the
mathematical sciences was incredibly enlarged in every di-
rection. The great improvement which Descartes had
made by the application of algebraic equations to define
the nature of curve lines was now rendered much more efli-
cient, and carried far beyond its original boundaries. From
the equation of the curve the new analysis could deduce the
properties of the tangents, and, what was much more diffi-
cult, could go back from the properties of the tangents to the
equation of the curve. From the same equation it was able
to determine the curvature at every point; it could measure
the length of any portion of the curve or the area corres-
ponding to it. Nor was it only toalgebraic curves that those
applications of the calculus extended, but to curves transcen-
dental and mechanical, as in the instances of the catenaria,
the cycloid, the elastic curve, and many others. ‘The same
sort of research could be applied to curve sur..ces described
according to any given law, and also to the solids contained
by them.

The problems which relate to the maxima and minima,
or the greatest and least values of variable quantities, are
among the most interesting in the mathematics; they are
connected with the highest attainments of wisdom and the
greatest exertions of power; and seem like so many im-
moveable columns erected in the infinity of space, to mark
the eternal boundary which separates the regions of possibi-
lity and impossibility from one another. For the solution of

Se te ee ee ee

DISSERTATION SECOND. 27

sect. 1.]

these problems, a particular provision seemed to be made in
the new geometry.

When any function becomes either the greatest or the
least, it does so by the velocity of its increase or of its de-
crease ceasing entirely, or, in the language of algebra, be-
coming equal to nothing. But when the velocity with which
the function varies becomes nothing, the flaxion which is
proportional to that velocity must become nothing also.
Therefore, it is only necessary to take the fluxion of the
given function, and by supposing it equal to nothing, an equa-
tion will be obtained in finite terms (for the fluxion will en-
tirely disappear), expressing the relation of the quantities
when the function assigned is the greatest or the least pos-
sible.

Another kind of maximum or minimum, abounding also in
interesting problems, is more difficult by far than the preced-
ing, and, when taken generally, seems to be only accessible
to the new analysis. Such cases occur when the function of
the variable quantities which is to be the greatest or the least
is not given, but is itself the thing to be found ; as when it is
proposed to determine the line by which a heavy body can
descend in the least time from one point to another. Here
the equation between the co-ordinates of the curve to be
found is, of course, unknown, and the function of those
co-ordinates which denotes the time of descent cannot
therefore be algebraically expressed, so that its fluxion
cannot be taken in the ordinary way, and thus put equal
to nothing. The former rule, then, is not applicable in
such cases, and it is by no means obvious in what man-
ner this difficulty is to be overcome. The general pro-
blem exercised the ingenuity of both the Bernoullis, as it
has since done of many other mathematicians of the greatest
name. As there are in such problems always two condi-
tions, according to the first of which, a certain property is to

28 DISSERTATION SECOND. PraRT 11.

remain constant, or to belong to all the individuals of the
species, and, according to the second, another property is to
be the greatest or the least possible ; and as, in some of the
simplest of such questions,’ the constant quantity is the cir-
cumference or perimeter of a certain curve, so problems of
this kind -have had the name of Jsoperimetrical given them,
a term which has thus come to denote one of the most curi-
ous and difficult subjects of mathematical investigation.

The new analysis, especially according to the view taken
of it by Newton, is peculiarly adapted to physical researches,
as the hypothesis of quantities being generated by continu-
ed motion, comes there to coincide exactly with the fact.
The momentary increments or the fluxions represent so pre-
cisely the forces by which the changes in nature are produc-
ed, that this doctrine seemed created for the express purpose
of penetrating into the interior of things, and taking direct
cognizance of those animating powers which, by their sub-
tility, not only elude the observation of sense, but the ordi-
nary methods of geometrical investigation. The infinitesi-
mal analysis alone affords the means of measuring forces,
when each acts separately, and instantaneously under condi-
tions that can be accurately ascertained. In comparing the
effects of continued action, the variety of time and circum-
stance, and the continuance of effects after their causes have
ceased, introduce so much uncertainty, that nothing but
vague and unsatisfactory conclusions can be deduced. The
analysis of infinites goes directly to the point; it measures
the intensity or instantaneous effort of the force, and, of

' The most simple problem of the kind is strictly and literally
Tsoperimetrical, viz. of all curves having the same perimeter to
find that which has the greatest area. Elementary geometry
had pronounced this curve to be the circle long before there
was any idea of an entire class of problems characterized by
similar conditions, Vid. Pappi Alexandrini Collect. Math. Lib.
V. Prop. 2. &c.

oe

|
|

sxor. 1.| DISSERTATION SECOND. 29

course, removes all those causes of uncertainty which pre-
vailed when the results of continued action could alone be
estimated. It is not even by the effects produced in a
short time, but by effects taken in their nascent or eva-
nescent state, that the true proportion of causes must be
ascertained.

Thus, though the astronomers had proved that the planets
describe ellipses round the sun as the common focus, and
that the line from the sun to each planet sweeps over areas
proportional to the time; had not the geometer resolved the
elliptic motion into its primary elements, aud compared them
in their state of evanescence, it would never have been dis-
covered that these bodies gravitate to the sun with forces
which are inversely as the square of their distances from the
centre of that luminary. Thus, fortunately, the first dis-
covery of Newton was the instrument which was to conduct
him safely through all the intricacies of his future investiga-
tions.

_ The calculus, as already remarked, necessarily. divides
itself into two branches ; one which, from the variable quan-
tities, finds the relation of their fluxions or differentials ;
another which, from the relation of these last, investigates
the relation of the variable quantities themselves. The first
of these problems is always possible, and, in general, easy to
be resolved ; the second is not always possible, and when
possible, is often very difficult, but in various degrees, ac-
cording to the manner in which the differentials and the va-
riable quantities are combined with one another.

If the function, into which the differential stands multi-
plied, consist of a single term, or an aggregate of terms, in
each of which the variable quantity is raised to a power ex-
pounded by a number positive, negative, or fractional, the
integration can be effected with ease, either in algebraic or
logarithmic terms ; and the calculus had not been long known
hefore this problem was completely resolved.

30 DISSERTATION SECOND. [rant ut.

The second case of this first division is, when the given
function is a fraction having a binomial or multinomial de-
nominator, the terms of which contain any powers whatever
of the variable magnitude, but without involving the radical
sign. Ifthe denominator contain only the simple power of
the variable quantity, the integral is easily found by loga-
rithms ; if it be complex, it must be resolved either into sim-
ple or quadrafic divisors, which, granting the solution of
equations, is always possible, at least by approximation, and
the given fraction is then found equal to an aggregate of sim-
ple fractions, having these divisors for their denominators,
and of which the fluents can always be exhibited in algebraic
terms, or in terms of logarithms and circular arches. This
very general and important problem was resolved by J. Ber-
noulli as early as the year 1702.

The denominator is in this last case supposed rational ; but
if it be irrational, the integration requires other means to be
employed. Here Leibnitz and Bernoulli both taught, how,
by substitutions, as in Diophantine problems, the irrationality
might be removed, and the integration of course reduced to
the former case. Newton employed a different method, and,
in his Quadrature of Curves, found the fluents, by comparing
the given fluxion with the formulas immediately derived from
the expression of circular or hyperbolic areas. The integra-
tions of these irrational formule, whichever of the methods
be employed, often admit of being effected with singular
elegance and simplicity ; but a general integration of all the
formule of this kind, except by approximation, is not yet
within the power of analysis.

The second general division of the problem of integration,
viz. when the two variable quantities and their differentials
are mixed together on each side of the equation, is a more
difficult subject of inquiry than the preceding. It may in-
deed happen, that an equation, which at first presents itself

a

sxcr. 1.) DISSERTATION SECOND. 31

under this aspect, can, by the common rules of algebra, have
the quantities so separated, that on each side of the sign of
equality there shall be but one variable quantity with its flux-
ion; and when this is done, the integration is reduced to one
of the cases already enumerated.

When such a separation cannot be made, the problem is
among the most difficult which the infinitesimal analysis pre-
sents, at the same time that it is the key to a vast number of
interesting questions both in the pure and mixed mathema-
tics. The two Bernoullis applied themselves strenuously to
the elucidation of it; and to them we owe all the best and
most accurate methods of resolving such questions which ap-
peared in the early history of the calculus, and which laid
the foundation of so many subsequent discoveries. ‘This is
a fact which cannot be contested; and it must be acknow-
ledged also, that, on the same subject, the writings of the
English mathematicians were then, as they continue to be at
this day, extremely defective. Newton, though he had
treated of this branch of the infinitesimal analysis with his
usual ingenuity and depth, had done so only in his work on
Fluxions, which did not see the light till several years after
his death, when, in 1736, it appeared in Colson’s translation.
But that work, even had it come into the hands of the public
in the author’s lifetime, would not have remedied the defect
of which I now speak. When the fluxionary equation could
not be integrated by the simplest and most elementary rules,
Newton had always recourse to approximations by infinite
series, in the contrivance of which he indeed displayed great
ingenuity and address. But an approximation, let it be ever
so good, and converge ever so rapidly, is always inferior to an
accurate and complete solution, if this last possess any tolerable
degree of simplicity. ‘The series which affords the approxi-
mation cannot converge always, or in all states of the varia-
ble quantity ; and its utility, on that account, is so much limit-

32 DISSERTATION SECOND. [PART 12.

ed, that it can hardly lead to any general result. Besides, it
does not appear that these series can always be made to
involve the arbitrary or indeterminate quantity, without
which no fluent can be considered as complete. For these
reasons, such approximations should never be resorted to till
every expedient has been used to find an accurate solution.
To this rule, however, Newton’s method does not conform,
but employs approximation in cases where the complete inte-
eral can be obtained. ‘The tendency of that method, there-
fore, however great its merit in other respects, was to give
a direction to research which was not always the best, and
which, in many instances, made it fail entirely short of the
object it ought to have attained. It is true, that many flux-
jonary equations cannot be integrated in any other way ; but
by having recourse to it indiscriminately, we overlook the
eases in which the integral can be exactly assigned. Accor-
dingly, Bernoulli, by following a different process, remarked
entire classes of fluxionary or differential equations, that ad-
mitted of accurate integration. Thus he found, that diffe-
rential equations, if homogeneous,’ however complicated,
may always have the variable quantities separated, so as to
come under one of the simpler forms already enumerated.
By the introduction, also, of exponential equations, which
had been considered in England as of little use, he materially
improved this branch of the calculus.

To all these branches of analysis we have still another to
add of indefinite extent, arising out of the consideration of the
fluxions or differentials of the higher orders, each of these
orders being deduced from the preceding, just as first flux-
ions are from the variable quantities to which they belong.
To understand this, conceive the successive values of the

' Homogeneous equations in the differential calculus, are
those in which the sum of the exponents of the variable quanti-
ties is the same in all the terms.

srer. 1.] DISSERTATION SECOND. oo

first fluxions of any variable quantity, to constitute a new se-
ries of variable quantities flowing with velocities, the measures
of which form the fluxions of the second order, from which,
in the same manner, are deduced fluxions of the third and of
still higher orders. ‘The general principles are the same as
in the fluxions of the first order, but the difficulties of the
calculus are greater, particularly in the integrations; for to
rise from second fluxions to the variable quantities them-
selves two integrations are necessary; from third fluxions
three, and so on.

The tract which first made known the new analysis was
that of Leibnitz, published as already remarked, in the first
volume of the Acta Eruditorum for 1684, where it occupies
no more than six pages,' and is the work of an author not
yet become very familiar with the nature of his own inven-
tion. It was suflicient, however, to explain that invention to
mathematicians ; but, nevertheless, some years elapsed be-
fore it drew much attention. ‘The Bernoullis were the first
who perceived its value, and made themselves masters of the
principles and methods contained, or rather suggested, in it.
Leibnitz published many other papers in the 4cta Eruditorum
and the journals of the times, full of original views and im-
portant hints, thrown out very briefly, and requiring the elu-
cidations which his friends just mentioned were always so
willing and so able to supply. The number of literary and
scientific objects which divided the attention of the author
himself was so great, that he had not time to bestow on the
illustration and developement of the most important of his
own discoveries, and the new analysis, for all that he has
taught, would have been very little known, and very imper-

* Nova Methodus pro Maximis et Minimis, &e. Leibnitii Opera,
Tom. Il. p. 167.

5

34 DISSERTATION SECOND. [PART 18.

fectly unfolded, if the two excellent geometers just named.
had not come to his assistance. Their tracts were also, like
his, scattered in the different periodic works of that time,
and several years elapsed before any elementary treatise
explained the general methods, and illustrated them by ex-
amples. The first book in which this was done, so far at
least as concerned the differential or direct calculus, was the
Analyse des Infiniment Petits of the Marquis de TP Hopital,
published in 1696, a work of great merit, which did
“much to diffuse the knowledge of the new analysis. It

was well received at that time, and has maintained its

character to the present day. The author, a man of genius,
indefatigable and ardent in the pursuits of science, had
enjoyed the viva voce instructions of John Bernoulli, on
the subject of the new geometry, and therefore came for-
. ward with every possible advantage.

It was long after this before the works of the Bernoul-
lis were collected together, those of James in two quarto
volumes, and of John in four.’ In the third of these last
volumes is a tract of considerable length, with the title of
Lectiones de Methodo Integralium, written in 1691 and
1692, for the use of M. de l’Hopital, to whose book on
the differential calculus it seems to have been intended as
a sequel. It is a work of great merit; and affords a dis-
tinct view of many of the most general methods of inte-
gration, with their application to the most interesting pro-
blems; so that, though the earliest treatise on that sub-
ject, it remains at this day one of the best compends of
the new analysis of which the mathematical world is in
possession. Indeed, the whole of the volumes just refer-
red to are highly interesting, as containing the original

* Those of James were published at Geneva in 1744 ; of John
_ at Lausanne and Geneva in 1742,

OE a a

scr. 1.] DISSERTATION SECOND. 35

germs of the new analysis, and as being the work of men
always inspired by genius, sometimes warmed by opposi-
tion, and generally animated by the success which accom-
panied their researches.

But we must now look at the original works of the
earliest inventor. Newton, besides his letters published in
the Commercium Epistolicum, is the author of three tracts
on the new analysis that have all been occasionally men-
tioned. None of them, however, appeared nearly so soon
as a great number of the pieces which have just been
enumerated. The Quadrature of Curves, written as early
as 1665 or 1666, did not appear till 1704; and though it
be a treatise of great value, and containing very impor-
tant and very general theorems concerning the quadrature
of curves, it must be allowed, that it is not well adapted
to make known the spirit and the views of the infinitesi-
mal analysis. After a short introduction, which is indeed
analytical, and which explains the idea of a fluxion with
great brevity and clearness, the treatise sets out with pro-
posing to find any number of curves that can be squared ;
and here the demonstrations become all synthetical, with-
out any thing that may be properly called analytical in-
vestigation. By synthetical demonstrations | do not mean
‘yeasonings where the algebraic language is not used, but
reasonings, whatever language be employed, where the
solution of the proposed question is first laid down, and
afterwards demonstrated to be true. Such is the method
pursued throughout this work, and it is wonderful how
many valuable conclusions concerning the areas of curves,
and their reduction to the areas of the circle and hyper-
bola, are in that manner deduced. But though truths can
be very well conveyed in the syuthetical way, the methods
of investigating truth are not communicated by it, nor the
powers of invention directed to their proper objects. As

36 DISSERTATION SECOND. [pant u.

an elementary treatise on the new analysis, the Quadra-
ture of Curves is therefore imperfect, and not calculated,
without great study, to give to others any portion of the
power which the author himself has exerted. The pro-
blem of finding fluents, though it be that on which the
whole quadrature of curves depends, is entirely kept out
of view, and never once proposed in the course of a
work, which, at the same time, is full of the most elabo-
rate and profound reasonings.

Newton had a great fondness for the synthetical method,
which is apparent even in the most analytical of his works.

In his Fluaxions, when he is treating of the quadrature
of curves, he says, “After the area of a curve has been
found and constructed, we should -consider about the de-
monstration of the construction, that, laying aside all alge-
braical calculation, as much as may be, the theorem may
be adorned and made elegant, so as to become fit for
public view.”?! This -is followed by two or three exam-
ples, in which the rule here given is very happily illustrated.
When the analysis of a problem requires, like the quad-
rature of curves, the use of the inverse method of flux-
ions, the reversion of that analysis, or the synthetical de-
monstration, must proceed by the direct method, and there-
fore may admit of more simplicity than the others, so as,
in the language of the above passage, to be easily adorn-
ed and made elegant.

The book of Fluxions is, however, an excellent work, en-
tering very deeply into the nature and spirit of the calcu-
lus,—illustrating its application by well chosen examples,
—and only failing, as already said, by having recourse,
for finding the fluents of fluxionary equations, too exclu-
sively to the method of series, without treating of the ca-
ses in which exact solutions can be obtained.

* Newton’s Fluxiens, Colson’s Translation, p, 116, § 107.

ee

SECT. 1] DISSERTATION SECOND. Si

Of the works that appeared in the early stages of the
calculus, none is more entitled to notice than the Harmo-
nia Mensurarum of Cotes. The idea of reducing the areas
of curves to those of the circle and hyperbola, in those cases
which did not admit of an accurate comparison with rectilineal
spaces, had early occurred to Newton, and was very fully
exemplified in his Quadrature of Curves. Cotes extended
this method :—his work appeared in 1722, and gave the
rules for finding the fluents of fractional expressions, whe-
ther rational or irrational, greatly generalized and highly
improved by means of a property of the circle discovered
by himself, and justly reckoned among the most remarka-
ble propositions in geometry. It is singular that a work
so profound, and so useful as the Harmonia Mensurarum,
should never have acquired, even among the mathemati-
cians of England, the popularity which it deserves ; and
that, on the Continent, it should be very little known,
even after the excellent commentary and additions of
Bishop Walmsley. The reasons, perhaps, are, that, in ma-
ny parts, the work is obscure; that it does not explain
the analysis which must have led to the formule contain-
ed in the tables ; and that it employs an unusual language
and notation, which, though calculated to keep in: view the
analogy between circular and hyperbolic areas, or between
the measures of angles and of ratios, do not so readily ac-
commodate themselves to the business of calculation as those
which are commonly in use. Demoivre, a very skilful and
able mathematician, improved the method of Cotes ; and ex-
~plamed many things in a manner much more clear and ana-
lytical than had hitherto been done.!

’ Demoivre, Miscellanea Analytica. See also the work of
an anonymous author, Epistola ad Amicum de Cotesii Inventis.

38 DISSERTATION SECOND. rane ti.

Another very original and profound writer of this period
was Brook Taylor, who has already been often mentioned,
and who, in his Method of Increments, published in 1715,
added a new branch to the analysis of variable quantity.
According to this method, quantities are supposed to change,
not by infinitely small, but by finite increments, or such as
may be of any magnitude whatever. There are here, there-
fore, as in the case of fluxions or differentials, two general
questions: A function of a variable quantity being given,
to find the expression for the finite increment of that function,
the increment of the variable quantity itself being a finite
magnitude. ‘This corresponds to the direct method of flux-
ions; the other question corresponds to the inverse, viz. A
function being given containing variable quantities, and their
increments any how combined, to find the function from
which it is derived. The author has considered both these
problems, and in the solution of the second, particularly, has
displayed much address. He has also made many ingenious
applications of this calculus both to geometrical and physical
questions, and, above all, to the summation of series, a pro-
blem for the solution of which it is peculiarly adapted.

Taylor, however, was more remarkable for the ingenuity
and depth, than for the perspicuity of his writings; even a
treatise on Perspective, of which he is the author, though in
other respects excellent, has always been complained of as
_ obscure ; and it is no wonder if, on a new subject, and one
belonging to the higher geometry, his writings should be still
more exposed to that reproach. ‘This fault was removed,
and the whole theory explained with great clearness, by M.
Nicol, of the Academy of Sciences of Paris, in a series of —
Memoires from the year 1717 to 1727.

A single analytical formula in the Method of Increments
has conferred a celebrity on its author, which the most
voluminous works have not often been able to bestow. It

sncr. 1.) DISSERTATION SECOND. 39

is known by the name of Taylor’s Theorem, and expresses
the value of any function of a variable quantity in terms of
the successive orders of increments, whether finite or in-
finitely small. If any one proposition can be said to com-
prehend in it a whole science it is this: for from it almost
every truth and every method of the new analysis may be
deduced. It is difficult to say, whether the theorem does
most credit to the genius of the author, or the power of the
language which is capable of concentrating such a vast body
of knowledge in a single expression. Without an acquaint-
ance with algebra, it is impossible, I believe, to conceive the
manner in which this effect is produced.

By means of its own intrinsic merit, and the advantageous
display of it made in the works now enumerated, the new
analysis, long before the expiration of the period of which I
am here treating, was firmly established all over Europe.
It did not, however, exist everywhere in the same condition,
nor under the same form; with the British and Continental
mathematicians, it was referred to different origins; it was
in different states of advancement; the notation and some of
the fundamental ideas were also different. ‘The authors
communicated little with one another, except in the way of
defiance or reproach ; and, from the angry or polemical tone
which their speculations often assumed, one could hardly
suppose, that they were pursuing science in one of its most
abstract and incorporeal forms. y

Though the algorithm employed, and the books consulted
on the new analysis, were different, the mathematicians of
Britain and of the Continent had kept pace very nearly with
one another during the period now treated of, except in one
branch, the integration of differential or of fluxional equa-
tions. In this, our countrymen had fallen considerably be-
hind, as has been already explained; and the distance be-
tween them and their brethren on the Continent continued

40 DISSERTATION SECOND. [part ut.

to increase, just in proportion to the number and importance
of the questions, physical and mathematical, which were
found to depend on these integrations. The habit of study-
ing only our own authors on these subjects, produced at first
by our admiration of Newton and our dislike to his rivals,
and increased by a circumstance very insignificant in itself,
the diversity of notation prevented us from partaking in the
pursuits of our neighbours ; and cut us off in a great measure
from the vast field in which the genius of France, of Ger-
many, and Italy, was exercised with so much activity and
success. Other causes may have united in the production
of an effect, which the mathematicians of this country have
had much reason to regret; but the evil had its origin in
the spirit of jealousy and opposition, which arose from the
controversies that have just passed under our review. The
habits so produced continued long after the spirit itself had
subsided.

It must not be supposed, that so great a revolution in
science, as that which was made by the introduction of the
new analysis, could be brought about entirely without oppo-
sition, as in every society there are some who think them-
selves interested to maintain things in the condition wherein
they have found them. ‘The considerations are indeed suffi-
ciently obvious which, in the moral and political world,
tend to produce this effect, and to give a stability to human
institutions, often so little proportionate to their real value
or to their general utility. Even in matters purely intel-
lectual, and in which the abstract truths of arithmetic and
geometry seem alone concerned, the prejudices, the selfish-
ness, or vanity of those who pursue them, not unfrequently
combine to resist improvement, and often engage no in-
considerable degree of talent in drawing back instead of
pushing forward the machine of science. ‘The introduction
ef methods entirely new must often change the relative place

sxcr. 1.] DISSERTATION SECOND. 4]

of the men engaged in scientific pursuits ; and must oblige
many, after descending from the stations they formerly occu-
pied, to take a lower position in the scale of intellectual ad-
vancement. The enmity of such men, if they be not ani-
mated by a spirit of real candour and the love of truth, is
likely to be directed against methods, by which their vanity
is mortified, and their importance lessened. ‘Though such’
changes as this must have everywhere accompanied the as-
cendency acquired by the calculus, for the credit of mathe-
maticians it must be observed, that,no one of any considera-
ble eminence has had the misfortune to enrol his name
among the adversaries of the new science ; and that Huy-
gens, the most distinguished and most profound of the older
mathematicians then living, was one of the most forward to
acknowledge the excellence of that science, and to make
himself master of its rules, and of their application.

Nevertheless, certain adversaries arose successively in
Germany, France, and England, the countries in which the
new methods first became known.

Nieuentit, an author commendable as a naturalist, and as a
writer on morals, but a very superficial geometer, aimed the
first blow at the Differential Calculus. He objected to the
explanation of Leibnitz, and to the notion of quantities infi-
nitely smail.'| It seemed as if he were unwilling to believe
in the reality of objects smaller than those discovered by his
own microscope, and were jealous of any one who should
come nearer to the limit ‘of extension than he himself had
done. Leibnitz thought his objections not undeserving of a
reply; but the reply was not altogether satisfactory. A se-
cond was given with better success ; and afterwards Herman

* He published Analysis Infinitorum at Amsterdam, in 1695 ;
and another tract, Considerationes circa Calculi Differentialis
Principia, in the year following. This last was answered by
Herman.

6

42 DISSERTATION SECOND. [PaRT it.

and Bernoulli each severally defeated an adversary, who was
but very ill able to contend with either of them.

Soon after this, the calculus had to sustain an attack from
two French academicians, which drew more attention than .
that of the Dutch naturalist. One of these, Rolle, was a
mathematician of no inconsiderable acquirement, but whose
chief gratification consisted in finding out faults in the works
of others. He founded his objections to the differential calcu-
lus, not on the score of principles or of general methods, but
on certain cases which he had sought out with great-industry,
in which those methods seemed to him to lead to false and
contradictory conclusions. On examination, however, it
turned out, that in every one of those instances the error was
entirely his own; that he had misapplied the rules, and that
his eagerness to discover faults had led him to commit them.
His errors were detected and pointed out with demonstra-
tive evidence by Varignon, Saurin, and some others, who
were among the first to perceive the excellence and to de-
fend the solidity of the new geometry. These disputes were
of consequence enough to occupy the attention of the Acade-
my of Sciences during a great part of the year 1701.

The Abbé Gallois joined with Rolle in his hostility to the
calculus, and though he added very little to the force of the
attack, he kept the field after the other had retired from the
combat. “Fontenelle, in his El/oge on the Abbé, has given an
elegant turn to the apology he makes for him.—‘ His taste
for antiquity’*made him suspicious of the geometry of infi-.
nites. He was, in general, no friend to any thing that was
new, and was always prepared with a kind of Ostracism to
put down whatever appeared too conspicuous for a free state
like that of letters. The geometry of infinites had both
these faults, and particularly the latter.”

After all these dispute were quieted in France, and the
new analysis appeared completely victorious, it had an at-

sacr. 1.J DISSERTATION SECOND. 43

tack to sustain in England from a more formidable quarter.
Berkeley Bishop of Cloyne, was a man of first-rate talents,
distinguished as a metaphysician, a philosopher, anda divine.
His geometrical knowledge, however, which, for an attack
on the method of fluxions, was more essential than all his
other accomplishments, seems to have been little more than
elementary. The motive which induced him to enter on
discussions so remotely connected with his usual pursuits has
been variously represented; but, whatever it was, it gave
rise to the Analyst, m which the author professes to demon-
strate, that the new analysis is inaccurate in its principles,
and that, if it ever lead to true conclusions, it is from an acci-
dental compensation of errors that cannot be supposed al-
ways to take place. The argument is ingeniously and plau-
sibly conducted, and the author sometimes attempts ridicule
with better success than could be expected from the subject;
thus, when he calls ultimate ratios the ghosts of departed
quantities, it is not easy to conceive a witty saying more hap-
pily fastened on a mere mathematical abstraction.

The Analyst was answered by Jurin, under the signature
of Philalethes ; and to this Berkeley replied in a tract en-
titled 4 Defence of Freethinking in Mathematics. Replies
were again made to this, so that the argument assumed the
form of a regular controversy ; in which, though the defen-
ders of the calculus had the advantage, it must be acknow-
ledged that they did not always argue the matter quite fairly,
nor exactly meet the reasoning of their adversary. The true
answer to Berkeley was, that what he conceived to be an
accidental compensation of errors was not at all accidental,
but that the two sets of quantities that seemed to him neg-
lected in the reasoning were in all cases necessarily equal,
and an exact balance for one another. ‘The Newtonian idea,
of a fluxion contained in it this truth, and so it was argued by
Jurin and others, but not in a manner so logical and satisfac-

44 DISSERTATION SECOND. [part 1.

tory as might have been expected. Perhaps it is not too
much to assert, that this was not completely done till La
Grange’s Theory of Functions appeared. Thus, if the author
of the Analyst has had misfortune to enrol his name on
the side of error, he has also had the credit of proposing diffi-
culties of which the complete solution is only to be derived
from the highest improvements of the calculus.

This controversy made some noise in England, but I do
not think that it ever drew much attention on the Continent.
The Analyst, | imagine, notwithstanding its acuteness, never
crossed the Channel. Montucla evidently knows it only by
report, and seems as little acquainted with the work as with its
author, of whom he speaks very slightly, and supposes he has
sufficiently described him by saying, that he has written a book
against the existence of matter, and another in praise of tar-
water. But it is less from the opinions which men support
than from the manner in which they support them, that their
talents are to be estimated, If we judge by this criterion,
. we shall pronounce Berkeley to be a man of genius, whether
he be employed in attacking the infinitesimal analysis, in dis-
proving the existence of the external world, or in celebrating
the virtues of tar-water.’

* Though Berkeley reasons very plausibly, and with conside-
rable address, he hurts his cause by the comparison so often in-
troduced between the mysteries of religion and what he accounts
the mysteries of the new geometry. From this it is natural to

‘infer, that the author is avenging the cause of religion on the in-
fidel mathematician to whom his treatise is addressed, and an ar-
gument that is suspected to have any other object than that at
which it is directly aimed, must always lose somewhat of its
weight.

The dispute here mentioned did not take place till about the
year 1734; so that I have here treated of it by anticipation, be-
ing unwilling to resume the subject of controversies which,
though perhaps useful at first for the purpose of securing the
foundations of science, are long since set to rest, and never like-
jy to be revived,

sgcr. 11.] DISSERTATION SECOND. 45
SECTION Il.

MECHANICS, GENERAL PHYSICS, &c.

Tue discoveries of Galileo, Descartes, and other mathe-
maticians of the seventeenth century, had made known some
of the most general and important laws which regulate the
phenomena of moving bodies. The inertia, or the ten-
dency of body, when left to itself, to preserve unchanged its
condition either of motion or of rest; the effect of an impulse
communicated to a body, or of two simultaneous impulses,
had been carefully examined, and had led to the discovery of
the composition of motion. The law of equilibrium, not in
the lever alone, but in all the mechanical powers, had been
determined, and the equality of action to reaction, or of the
motion lost to the motion acquired, had not only been esta-
blished by reasoning, but confirmed by experiment. The
fuller elucidation and farther extension of these principles
were reserved for the period now treated of.

The developement of truth is often so gradual, that it is
impossible to assign the time when certain principles have
been first introduced into science. ‘Thus, the principle of
Virtual Velocities, as it is termed, which is now recognized as
regulating the equilibrium of all machines whatsoever, was
perceived to hold in particular cases long before its full ex-
tent, or its perfect universality, was understood. Galileo
made a great step toward the establishment of this principle
when he generalized the property of the lever, and showed,
that an equilibrium takes place whenever the sums of the op-
posite momenta are equal, meaning by momentum the product
of the force into the velocity of the point at which it is appli-
ed. This was carried farther by Wallis, who appears to have

46 DISSERTATION SECOND. [parr 1,

been the first writer who, in his Mechanica, published in
1669, founded an entire system of statics on the principle
of Galileo, or the equality of the opposite momenta. The
proposition, however, was first enunciated in its full generali-
ty, and with perfect precision,’ by John Bernoulli, in a letter
to Varignon, so late as the year 1717. Varignon inserted
this letter at the end of the second edition of his Projet d’une
Nouvelle Mecanique, which was not published till 1725. The
first edition of the same book appeared in 1687, and had the
merit of deriving the whole theory of the equilibrium of the
mechanical powers, from the single principle of the compo-
sition of forces. At first sight, there appear in mechanics
two independent principles of equilibrium, that of the lever,
or of equal and opposite momenta, and that of the composi-
tion of forces. ‘To show that these coincide, and that the
one may be deduced from the other, is, therefore, doing a
service to science, and this the ingenious author just nam-
ed accomplished by help of the property of the parallelo-
gram, which he seems to have been the first who demon-
strated.

The Principia Mathematica of Newton, published also in
1687, marks a great era in the history of human knowledge,
and had the merit of effecting an almost entire revolution in
mechanics, by giving new powers and a new direction to its

‘ The principle of Virtual Velocities may be thus enunciated :
if a system of bodies be in a state of equilibrium, in consequence
of the action of any forces whatever, on certain points in the
system ; then were the equilibrium to be for a moment destroy-
ed, the small space moved over by each of these points will ex-
press the virtual velocity of the power applied to it, and if each
force be multiplied into its virtual velocity, the sum of all the
products where the velocities are in the same direction, will be
equal to the sum of all those in which they are in the opposite.

The distinction between actual and virtual velocities was first
made by Bernoulli, and is very essential to thinking as well
as to speaking with accuracy on the nature of equilibriums.

skcT. 1.] DISSERTATION SECOND. 47

researches. In that work the composition of forces was treat-
ed independently of the composition of motion, and the equi-
librium of the Fever was deduced from the former, as well as
in the treatise already mentioned. From the equality of ac-
tion and re-action it was also inferred, that the state of the
centre of gravity of any system of bodies, is not chang-
ed by the action of those bodies on one another. This
is a great proposition in the mechanics of the universe, and is
one of the steps by which that science ascends from the earth
to the heavens ; for it proves that the quantity of motion ex-
isting in nature, when estimated in any one given direction,
continues always of the same amount.

But the new applications of mechanical reasoning,—the
reduction of questions concerning force and motion to ques-
tions of pure geometry,—and the mensuration of mechanical!
action by its nascent effects,—are what constitute the great
glory of the Principia, considered as a treatise on the theory
of motion. A transition was there made from the considera-
tion of forces acting at stated intervals, to that of forces act-
ing continually,—and from forces constant in quantity and
direction to those that converge to a point, and vary as any
function of the distance from that point; the proportionality
of the areas described about the centre of force, to the times
of their description; the equality of the velocities generated
in descending through the same distance by whatever route ;”
the relation between the squares of the velocities produced
or extinguished, and the sum of the accelerating or retarding
forces, computed with a reference, not to the time during
which, but to the distance over which they have acted.
These are a few of the mechanical and dynamical discoveries
contained in the same immortal work; a fuller account of
which belongs to the history of physical astronomy.

The end of the seventeenth and the beginning of the
eighteenth centuries were rendered illustrious, as we have

.

48 DISSERTATION SECOND. [PAR? 11.

already seen, by the mathematical discoveries of two of
the greatest men who have ever enlightened the world. A
slight sketch of the improvements which the theory of
mechanics owes to Newton has been just given; those
which it owes to Leibnitz, though not equally important,
nor equally numerous, are far too conspicuous to be pas-
sed over in silence. So far as concerns general princi-
ples they are reduced to three,—the argument of the suffi-
cient reason,—the law of continuity,—and the measurement
of the force of moving bodies by the square of their veloci-
ties; which last, being a proposition that is true or false ac-
cording to the light in which it is viewed, I have supposed it
placed in that which is most favourable.

With regard to the first of these,—the principle of the
sufficient reason,—according to which, nothing exists in any
state without a reason determining it to be in that state
rather than in any other,—though it be true that this pro-
position was first distinctly and generally announced by the
philosopher just named, yet is it certain that, long before his
time, it had been employed by others in laying the founda-
tions of science. Archimedes and Galileo had both made
use of it, and perhaps there never was any attempt to place
the elementary truths of science on a solid foundation in
which this principle had not been employed. We have an
example of its application in the proof usually given, that a
body in metion cannot change the direction of its motion,
abstraction being made from all other bodies, and from alk
external action; for it is evident, that no reason exists to
determine the change of motion to be in one direction more
than another, and we therefore conclude that no such change
ean possibly take place. Many other instances might be
produced where the same principle appears as an axiom of
the clearest and most undeniable evidence. Wherever, in-
deed, we can pronounce it with certainty that the conditions

sxe. 11.) DISSERTATION SECOND. 49

which determine two different things, whether magnitudes or
events, are in two cases precisely the same, it cannot be
doubted that these events or magnitudes are in all respects
identical.

However sound this principle may be in itself, the use
which Leibnitz sometimes made of it has tended to bring it
into discredit. He argued, for example, that of the particles
of matter no two can possess exactly the same properties, or
can perfectly resemble one another, otherwise the Supreme
Being could have no reason for employing one of them ina
particular position more than another, so that both must ne-
cessarily be rejected. To argue thus, is to suppose that we
completely understand the manner in which motives act on
the mind of the Divinity,’ a postulate that seems but ill suit-
ed to the limited sphere of the human understanding. But,
if Leibnitz has misapplied his own principle and extended
its authority too far, this affords no ground for rejecting it
when we are studying the ordinary course of nature, and ar-
guing about the subjects of experiment and observation. In
fact, therefore, the sciences which aspire to place their
foundation on the solid basis of necessary truth, are much
indebted to Leibnitz for the introduction of this principle into
philosophy.

Another principle of great use in investigating the laws
of motion, and of change in general, was brought into
view by the same author,—the law of Continuity,—accord-
ing to which, nothing passes from one state to another

1 The argument of Leibnitz seems evidently inconclusive.
For, though there were two similar and equal atoms, yet as they
could not co-exist in the same space, they would not, so far as
position is concerned, bear the same relation to the particles
that surrounded them; there might exist, therefore, consider-
ing them as part of the materials to be employed in the con-
struction of the universe, very good reasons for assigning diffe-
rent situations to each.

‘

50 DISSERTATION SECOND. [panT si.

without passing through all the intermediate states. Leib-
nitz considers himself as the first who made known this
law ; but it is fair to remark, that, in as much as motion
is concerned, it was distinctly laid down by Galileo,’ and
ascribed by him to Plato. But, though Leibnitz was not
the first to discover the law of continuity, he was the first
who regarded it as a principle in philosophy, and used it for
trying the consistency of theories, or of supposed laws of na-
ture, and the agreement of their parts with one another. It
was in this way that he detected the error of Descartes’s
conclusions concerning the collision of bodies, showing, that
though one case of collision must necessarily graduate into
another, the conelusions of that philosopher did by no means
pass from one to another by such gradual transition. In-
deed, for the purpose of such detections, the knowledge of
this law is extremely useful; and I believe few have been
much occupied in the investigations either of the pure or
mixed mathematics, who have not often been glad to try
their own conclusions by the test which it furnishes.

Leibnitz considered this principle as known d priori, be-
cause if any saltus were to take place, that is, if any change
were to happen without the intervention of time, the thing
changed must be in two different conditions at the same indi-
vidual instant, which is obviously impossible. Whether this
reasoning be quite satisfactory or not, the conformity of the
law to the facts generally observed, cannot but entitle it to
great authority in judging of the explanations and theories of
natural phenomena.

It was the usual error, however, of Leibnitz and his follow-
ers, to push the metaphysical principles of science into ex-
treme cases, where they lead to conclusions to which it was

* Opere di Galileo, Tom. Ill. p. 150, and Tom. II. p. 32.
Edit. Padova, 1744.

SuCT. 11. DISSERTATION SECOND. 51

hardly possible to assent. The Academy of Sciences at Pa-
ris having proposed as a prize question, the Investigation of
the Laws of the Communication of Motion,’ John Bernoulli
presented an Essay on the subject, very ingenious and pro-
found, in which, however, he denied the existence of hard
bodies, because, in the collision of such bodies, a finite
change of motion must take place in an instant, an event
which, on the principle just explained, he maintained to be
impossible. Though the essay was admired, this conclusion
was objected to, and D’Alembert, in his E/oge on the author,
remarks, that, even in the collision of elastic bodies, it is diffi-
cult to conceive how, among the parts which first come into
contact, a sudden change, or a change per saltwm, can be
avoided. Indeed, it can only be avoided by supposing
that there is no real contact, and that bodies begin to act
upon one another when their surfaces, or what seems to
be their surfaces, are yet at a distance.

Maclaurin and some others are disposed, on account of
the argument of Bernoulli, to reject the law of continuity
altogether. This, however, I cannot help thinking, is to
deprive ourselves of an auxiliary that, under certain re-
strictions, may be very useful in our researches, and is
often so, even to those who profess to reject its assist-
ance. It is admitted that the law of continuity generally
leads right, and if it sometimes lead wrong, the true bu-
siness of philosophy is to define when it may be trusted
to as a safe guide, and what, on the other hand, are the
circumstances which render its indications uncertain.

The discourse of Bernoulli, just referred to, brought
another new conclusion into the field, and began a con-
troversy among the mathematicians of Europe, which last-
ed for many years. It was a new thing to see geome-

‘* Jn 1724.

52 DISSERTATION SECOND. [ramp 1s

ters contending about the truths of their own science, and
opposing one demonstration to another. ‘The spectacle
must have given pain to the true philosopher, but may
have afforded consolation to many who had looked with
envy on the certainty and quiet prevailing in a region
from which they found themselves excluded.

Descartes had estimated the force of a moving body by
the quantity of its motion, or by the product of iis velo-
city into its mass. The mathematicians and philosophers
who followed him did the same, and the product of these
quantities was the measure of force universally adopted.
No one, indeed, had ever thought of questioning the con-
formity of this measure to the phenomena of nature, when,
in 1686, Leibnitz announced in the Letpsic Journal, the
demonstration of a great error committed by Descartes and
others, in estimating the force of moving bodies. In this
paper, the author endeavoured to show, that the force of
a moving body is not proportional to its velocity simply,
but to the square of its velocity, and he supported this
new doctrine by very plausible reasoning. A body, he
says, projected upward against gravity, with a double ve-
locity, ascends to four times the height; with the triple
velocity, to nine times the height, and so on; the height
ascended to being always as the square of the velocity.
But the height ascended to is the effect, and is the na-
tural measure of the force, therefore the force of a mov-
ing body is as the square of its velocity. Such was the
first reasoning of Leibnitz on this subject,—simple, and
apparently conclusive; nor should it be forgotten that,
daring the long period to which the dispute was length-
ened out, and notwithstanding the various shapes which it
assumed, the reasonings on his side were nothing more
than this original argument, changed in its form, or ren-
dered more complex by the combination of new circum-

exer. 11.] DISSERTATION SECOND. 53

stances, so as to be more bewildering to the imagination,
and more difficult either to apprehend or to refute.’

John Bernoulli was at first of a different opinion from
his friend and master, but came at length to adopt the
same, which, however, appears to have gone no farther till
the discourse was submitted to the Academy of Sciences,
as has been already mentioned. ‘The mathematical world
could not look with indifference on a question which seem-
ed to affect the vitals of mechanical science, and soon
separated into two parties, in the arrangement of which,
however, the effects of national predilection might easily
be discovered. Germany, Holland, and Italy, declared for
the vis viva; England stood firm for .the old doctrine ;
and France was divided between the two opinions. No
controversy, perhaps, was ever carried on by more illus-
trious disputants; Maclaurin, Stirling, Desaguliers, Jurin,
Clarke, Mairan, were all engaged on the one side, and
on the opposite were Bernoulli, Herman, Poleni, S’Graves-
ende, Muschenbroek; and it was not till long after the
period to which this part of the Dissertation is confined, that
the debate could be said to be brought to a conclusion.
That | may not, however, be obliged to break off a sub-
ject of which the parts are closely connected together, I
shall take the liberty of transgressing the limits which the
consideration of time would prescribe, and of now stating,
as far as.my plan admits of it, all that respects this cele-
brated controversy.

A singular circumstance may be remarked in the whole
of the dispute. ‘The two parties who adopted such diffe-

* To mere pressure, Leibnitz gave the name of vis moriua,
and to the force of moving bodies the name of vis viva. The
former he admitted to be proportional to the simple power of the
virtual velocity, and the second he held to be proportional to the
square of the actual velocity.

J

54 DISSERTATION SECOND. [PART 11,

rent measures of force, when any mechanical problem was
proposed concerning the action of bodies, whether at rest
or in motion, resolved it in the same manner, and arrived
exactly at the same conclusions. It was therefore evident,
that, however much their language and words were opposed,
their ideas or opinions exactly agreed. In reality, the two
parties were not at issue on the question; their positions,
though seemingly opposite, were not contrary to one an-
other ; and after debating for nearly thirty years, they found
out this to be the truth. That the first men in the scientific
world should have disputed so long with one another, with-
out discovering that their opposition was only in words, and
that this should have happened, not in any of the obscure
and tortuous tracts through which the human mind must
srope its way in anxiety and doubt, but in one of the clearest
and straightest roads, where it used to be guided by the light
of demonstration, is one of the most singular facts in the his-
tory of human knowledge.

The degree of acrimony and illiberality which were some-
times mixed in this controversy was not very creditable to
the disputants, and proved how much more men take an in-
terest in opinions as being their own, than as being simply
in themselves either true or false. The dispute, as con-
ducted by S’Gravesende and Clarke, took this turn, es-
pecially on the part of the latter, who, in the schools of
theology having sharpened both his temper and his wit, ac-
companied his reasonings with an insolence and irritability
peculiarly ill suited to a discussion about matter and motion.
His paper on this subject, in the Philosophical Transactions,’
contains many just and acute remarks, accompanied with the
most unfair representation of the argument of his antagonists,

* Vol. XXXV. (1728), p. 381. Hutton’s Abridgment, Vol.
Vil. p. 279:

veer. u.] DISSERTATION SECOND. 55

as if the doctrine of the vis viva were a matter of as palpa-
ble absurdity as the denial of one of the axioms of geometry.’
Now, the truth is, that the argument in favour of living for-
ces is not at all liable to this reproach. One of the effects
produced by a moving body is proportional to the square of
the velocity, while another is proportional to the velocity
simply; and, according to which of these ways the force it-
selfis to be measured, may involve the propriety or impro-
priety of mathematical language, but cannot be charged with
absurdity or contradiction. Absurdity, indeed, was a re-
proach that neither side had any right to cast on the other.
A dissertation of Mairan, on the force of moving bodies, in
the Memoirs of the Academy of Sciences for 1728, is one
of those in which the common measure of force is most ably
supported. Nevertheless, for a long time after this, the opi-
nious on that subject in France continued still to be divided.
In the list of the disputants we should hardly expect to find
a lady included, if we did not know that the name of Madame
du Chastellet, along with those of Hypatia and Agnesi, was
honourably enrolled in the annals of mathematical learning.
Her writings on this subject are full of ingenuity, though, from

’ In all the-arguments for the vis viva, this learned metaphysi-
cian saw nothing but a conspiracy formed against the Newtonian
philosophy. ‘An extraordinary instance,” says he, ‘‘ of the
maintenance of the most palpable absurdity we have had in late
years of very eminent mathematicians, Leibnitz, Bernoulli, Her-
man, Gravesende, who, in order to raise a dust of opposition
against the Newtonian philosophy, some years back insisted with
great eagerness-on a principle which subverts all science, and
which easily may be made appear, evento an ordinary capacity,
to be contrary to the necessary and essential nature of things.”’
This passage may serve as a proof of the spirit which pre-
vailed among the philosophers of that time, making them ascribe
such illiberal views to one another, and distorting so entirely
both their own reasoning and those of their adversaries. The
spirit awakened by the discovery of fluxions had not yet sub-
sided.

56 DISSERTATION SECOND. [PART 1.

the fluctuation’ of her opinions, it seems as if she had not yet

entirely exchanged the caprice of fashion for the austerity of
science. About the same time Voltaire engaged in the argu-
ment, and ina Memoire,’ presented to the Academy of Sci-
ences in 1741, contended that the dispute was entirely about
words. His reasoning is on the whole sound, and the suf-
frage of one who united ‘the character of a wit, a poet, and a
philosopher, must be of great importance in a country where
the despotism of fashion extends even to philosophical
opinion.

The controversy was now drawing to a conclusion,® and in
effect may be said to have been terminated by the publica-
tion of D’Alembert’s Dynamique in 1743. Tam not certain,
however, that all the disputants acquiesced in this decision,
at least till some years later. Dr. Reid, in an essay On
Quantity, in the Philosophical Transactions for 1748, has

' Mad. du Chastellet,in a Dissertation on Fire, published in
1740, took the side of Mairan, and bestowed great praise on his
discourse on the force of moving bodies. Having, however,
afterwards become a convert to the philosophy of Leibnitz, she
espoused the cause of the Vis Piva, and wrote against Mairan.
At this time too she drew up a compend of the Leibnitian philo-
sophy for the use of her son, which displays ingenuity and acute-
wess, and is certainly such a present as very few mothers have
ever been ina condition to make to their children. Soon after-
wards the same lady, having become a Newtonian, returned to
her former opinion about the force of moving bodies, and in the
end, gave to her countrymen an excellent translation of the
Principia of Newton, with a commentary on a part of it, far
superior to any other that has yet appeared.

2 Doutes sur la Mesure des Forces Motrices ; GQluvres de Vol-
taire, Tom. XXXIX. p. 91. 8vo. edit. 1785.

* Two very valuable papers that appeared at this late period
of the dispute are found in the Philosophical Transactions ; one
by Desaguliers, in 1733, full of excellent remarks and valuable
experiments ; another by Jurin, in 1745, containing a very full
state of the whole controversy.

|

sxer. 11] DISSERTATION SECOND. 57

treated of this controversy, and remarked, that it had been
dropt rather than concluded. In this I confess | differ from
the learned author. The controversy seemed fairly ended,
the arguments exhausted, and the conclusion established, that
the propositions maintained by both sides were true, and
were not opposed to one another. ‘Though the mathemati-
cal sciences cannot boast of never having had any debates,
they can say that those that have arisen have always been
brought to a satisfactory termination.

The observations with which I am to conclude the pre-
sent sketch, are not precisely the same with those of the
French philosopher, though they rest nearly on the same
foundation.

As the effects of moving bodies, or the changes they pro-
duce, may vary considerably with accidental circumstances,
we must, in order to measure their force, have recourse to
effects which are uniform, and not under the influence of
variable causes. First, we may measure the force of one
moving body by its effect upon another moving body; and
here there is no room for dispute, nor any doubt that the
forces of such bodies are as the quantities of matter multipli-
ed into the simple power of the velocities, because the for-
ces of bodies in which these products are equal, are well
known, if opposed, to destroy One another. Thus one ef-
fect of moving bodies affords a measure of their force, which
. does not vary as the square ; but as the sumple power of the
velocity.

There is also another condition of moving bodies which
may be expected to afford a simple and general measure of
their force. When a moving body is opposed by pressure,
by a vis mortua, or a resistance like that of gravity, the
quantity of such resistance required to extinguish the motion,
and reduce the body to rest, must serve to measure the force
of that body. It is a force which, by repeated impulses, has

: 8

58 DISSERTATION SECOND. [parT 11.

annihilated another, and these impulses, when properly
collected into one sum, must evidently be equal to the
force which they have extinguished. It happens, however,
that there are two ways of computing the amount of these
retarding forces, which lead to different results, both of
them just, and neither of them to be assumed to the exclu-
sion of the other.

Suppose the body, the force of which is to be measur-
ed, to be projected perpendicularly upward with any velo-
city, then, if we would compute the quantity of the force
of gravity which is employed in reducing it to rest, we may
either inquire into the retardation which that force produ-
ces during a given time, or while the body is moving
over a given space. In other words, we may either in-
quire how long the motion will continue, or how far it will
carry the body before it be entirely exhausted. If the
length of the time that the uniform resistance must act
before it reduce the body to rest be taken for the efiect,
and consequently for the measure of the force of the body,
that force must be proportional to the velocity, for to this
the time is confessedly proportional. If, on the other
hand, the length of the line which the moving body de-
scribes, while subjected to this uniform resistance, be taken
for the effect and the measure of the force, the force must
be as the square of the velocity, because to that quantity the
line in question is known to be proportional. Here, there-
fore, are two results, or two values of the same thing, the
force of a moving body, which are quite different from one
another; an inconsistency which evidently arises from this,
that the thing denoted by the term force, is too vague and
indefinite to be capable of measurement, unless some farther
condition be annexed. ‘This condition is no other than a
specification of the work to be performed, or of the effect to
be produced by the action of the moving body. ‘Thus, when

|
;
.

scr, 11.) DISSERTATION SECOND. 59

to the question concerning the force of the moving body, you
add that it is to be employed in putting in motion another
body, which is itself free to move, no doubt remains that the
force is as the velocity multiplied into the quantity of matter.
So also, if the force of the moving body is to be opposed by
a resistance like that of gravity, the length of time that the
motion may continue is one of its measurable effects, and
that effect is, like the former, proportional to the velocity.
There is a third effect to be considered, and one which al-
Ways occurs in such an experiment as the last,—the height
to which the moving body will ascend. This limitation
gives to the force a definite character, and it is now measur-
ed by the square of the velocity. In fact, therefore, it is not
a precise question to ask, What is the measure of the force —
of a moving body ? You must, in addition, say, How is the
moving body to be employed, or in which of its different ca-
pacities is it that you would measure its effect? In this state
of the question there is no ambiguity, nor any answer to be
given but one. Hence it was that the mathematicians and
philosophers who differed so much about the general ques-
tion of the force of moving bodies, never differed about the
particular applications of that force. It was because the con-
dition necessary for limiting the vagueness and ambiguity of
the data, in all such cases, was fully supplied.

In the argument, therefore, so strenuously maintained on
the force of moving bodies, both sides were partly in the
right and both partly in the wrong. Kach produced a mea-
sure of force which was just in certain circumstances, and
thus far had truth on his side: but each argued that his was
the only true measure, so that all others ought to be reject-
ed; and here each of them was in error. Hence, also, it is
not an accurate account of the controversy to say that it was
about words merely; the disputants did indeed misunder-
stand one another, but their error lay in ascribing generality

60 DISSERTATION SECOND. [Parr 1.

to propositions that were true only in particular cases, to
which indeed the ambiguity and vagueness of the word force
materially contributed. It does not appear, however; that
any good would now accrue from changing the language of
dynamics. If, as has been already said, to the question, How
are we to measure the force of a moving body? be added
the nature of the effect which is to be produced, all ambigui-
ty will be avoided.

It is, I think, only farther necessary to observe, that, when
the resistance opposed to the moving body is not uniform but
variable, according to any law, it is not simply either the
time or the space which is proportional to the velocity or to
the square of the velocity, but functions of those quautities.
These functions are obtained from the integration of certain
fluxionary expressions, in which the-meastires above describ-
ed are applied, the resistance being regarded as uniform for
an infinitely small portion of the time or of the space.

Many years after the period | am now treating of, the con-
troversy about the vis viva seemed to revive in England, on
the occasion of an Essay on Mechanical Force, by the late
Mr. Smeaton, an able engineer, who, to great practical skill,
and much experience, added no inconsiderable knowledge of
the mathematics.’

The reality of the vis viva, then, under certain conditions,
is to be considered as a matter completely established.
Another inquiry concerning the nature of this force, which
also gave rise to considerable debate, was, whether, in the
communication of motion, and in the various changes through
which moving bodies pass, the quantity of the vis viva re-
mains always the same? It had been observed, in the colli-
sion of elastic bodies, that the vis viva, or the sum made up
by multiplying each body into the square of its velocity, and

' Note E, at the end.

szcr. 11] DISSERTATION SECOND, 61

adding the products together, was the same after collision
that it was before it, and it was concluded with some precipita-
tion, by those who espoused the Leibnitian theory, that a simi-
lar result always took place in the real phenomena of nature.
Other instances were cited ; and it was observed, that a par-
ticular view of this principle which presented itself to Huy-
gens, had enabled him to find the centre of oscillation of a
compound pendulum, at a time when the state of mechanical
science was scarcely prepared for so difficult an investigation.
The proposition, however,"is true only when all the changes
are gradual, and rigorously subjected to the law of continui-
ty. Thus, in the collision of bodies imperfectly elastic (a
case which continually occurs in nature), the force which,
during the recoil, accelerates the separation of the bodies,
does not restore to them the whole velocity they had lost,
and the vis viva, after the collision, is always less than it was
before it. The cases in which the whole amount of the vis
viva is rigorously preserved, may always be brought under
the thirty-ninth proposition of the first book of the Principia,
where the principle of this theory is placed on its true
foundation.

So far as General Principles are concerned, the preceding
are the chief mechanical improvements which belong to the
period so honourably distinguished by the names of Newton
and Leibnitz. The application of these principles to the
solution of particular problems would afford materials for
more ample discussion than suits the nature of a historical
outline. Such problems as that of finding the centre of
oscillation,—the nature of the catenarian curve,—the deter-
mination of the line of swiftest descent,—the retardation pro-
duced to motion in a medium that resists according to the
square of the velocity, or indeed according to any function
of it,—the determination of the elastic curve, or that into
which an elastic spring forms itself when a force is applied

62 DISSERTATION SECOND. [pan 11.

to bend it,—all these were problems of the greatest interest,
and were now resolved for the first time; the science of me-
chanics being sufficient, by means of the composition of for-
ees, to find out the fluxionary or differential equations which
expressed the nature of the gradual changes which in all these
cases were produced, and the calculus being now sufliciently
powerful to infer the properties of the finite from those of
the infinitesimal quantities.

The doctrine of Hydrostatics was cultivated in England
by Cotes. The properties of thé atmosphere, or of elastic
fluids, were also experimentally investigated ; and the ba-
rometer, after the ingenuity of Pascal had proved that the
mercury stood lower the higher up into the atmosphere the
instrument was carried, was at length brought to be a measure
of the height of mountains. Mariotte appears to have been
the first who proposed this use of it, and who discovered
that, while the height from the ground increases in arith-
metical, the density of the atmosphere, and the column of
mercury in the barometer, decrease in geometrical progres-
sion. Halley, who seems also to have come of himself to
the same conclusion, proved its truth by strict geometrical
reasoning, and showed, that logarithms are easily applicable
on this principle to the problem of finding the height of
mountains. This wasin the year 1685. Newton two years
afterwards gave a demonstration of the same, extended to
the case when gravity is not constant, but varies as any
power of the distance from a given centre.

To the assiduous observations and the indefatigable ac-
tivity of Halley, the natural history of the atmosphere, of
the ocean, and of magnetism, are all under the greatest
obligations. For the purpose of inquiring into these ob-
jects, this ardent and philosophical observer relinquished the
quiet of academical retirement, and, having gone to St.
Helena, by a residence of a year in that island, not only

excr, 11.] DISSERTATION SECOND. 63

made an addition to the catalogue of the stars, of 360 from
the southern hemisphere, but returned with great acquisi-
tions both of nautical and meteorological knowledge. His
observations on evaporation were the foundation of two
valuable papers on the origin of fountains ; in which, for the
first time, the sufficiency of the vapour taken up into the at-
mosphere, to maintain the perennial flow of springs and ri-
vers, was established by undeniable evidence. The diffi-
culty which men found in conceiving how a precarious and
accidental supply like that of the rains, can sufficiently pro-
vide for a great and regular expenditure like that of the
rivers, had given rise to those various opinions concerning
the origin of fountains, which had hitherto divided the scien-
tific world. A long residence on the summit of an insulat-
ed rock, in the midst of a vast ocean, visited twice every
year by the vertical sun, would have afforded to an observer,
less quick-sighted than Halley, an opportunity of seeing the
work of evaporation carried on with such rapidity and co-
piousness as to be a subject of exact measurement. From
this extreme case, he could infer the medium quantity, at
least by approximation; and he proved that, in the Medi-
terranean, the humidity daily raised up by evaporation is
three times as great as that which is discharged by all the
rivers that flow into it. The origin of fountains was no
longer questioned, and of the multitude of opinions on that
subject, which had hitherto perplexed philosophers, all but
one entirely disappeared.'

Beside the voyage to St. Helena, Halley made two
others; the British government having been enlightened,
and liberal enough to despise professional etiquette, where
the interests of science were at stake, and to entrust to a
Doctor of Laws the command of a ship of war, in which

* Philosophical Transactions, 1687, Vol. XVI. p. 366.

64 DISSERTATION SECOND. [PART 11,

he traversed the Atlantic and Pacific Oceans in various di-
rections, as far as the 53d degree of south latitude, and re-
turned with a collection of facts and observations for the
improvement of geography, meteorology, and navigation, far
beyond that which any individual traveller or voyager had
hitherto brought together.

The variation of the compass was long before this time
known to exist, but its laws had never yet been ascertained.
These Halley now determined from his own observations,
combined with those of former navigators, in so far as to
trace, on a nautical chart, the lines of the same variation
over a great part both of the Atlantic and Pacific Oceans,
affording to the navigator the ready means of correcting the
errors which the deviation of the needle from the true
meridian was calculated to produce. In his different tra-
verses he had four times intersected the line of no varia-
tion, which seemed to divide the earth into two parts, the
variations on the east side being towards the west, and on
the west side towards the east. These lines being found
to change their position in the course of time, the place as-
signed to the magnetical poles could not be permanent.
Any theory, therefore, which could afford an explanation of
their changes must necessarily be complex and difficult to
be established. The attempt of Halley to give such an ex-
planation, though extremely ingenious, was liable to great
objections ; and while it has shared the fate of most of the
theories which have been laid down before the phenomena
had been sufficiently explored, the general facts which he
established have led to most of the improvements and dis-
coveries which have since been made respecting the po-
larity of the needle.

Besides the conclusion just mentioned, Dr. Halley de-
rived, from his observations, a very complete history of
the winds which blow in the tropical regions, viz. the

enor. 11,] _ DISSERTATION SECOND. 65

trade-wind, and the monsoons, together with many inte-
resting facts concerning the phenomena of the tides. The
chart which contained an epitome of all these facts was pub-
lished m°1701.

The above are only a part of the obligations which the
sciences are under to the observations and reasonings of this
ingenious and indefatigable inquirer. Halley was indeed
one of the ablest and most accomplished men of his age.
A scholar well versed in the learned languages, and a ge-
ometer profoundly skilled in the ancient analysis, he res-
tored to their original elegance some of the precious frag-
ments of that analysis, which time happily had not eftire-
ly defaced. He was well acquainted also with the alge-
.braical and fluxionary calculus, and was both in theory
and practice a profound and laborious astronomer. Fi-
nally, he was the friend of Newton, and often stimulated,
with good effect, the tardy purposes of that great philo-
sopher. Few men, therefore, of any period, have more
claims than Halley on the gratitude of succeeding ages.

The invention of the thermometer has been already
noticed, and the improvements made on that instrument
about this period, laid the foundation of many future dis-
coveries. ‘The discovery of two fixed températures, each
marked by the same expansion of the mercury in the
thermometer, and the same condition of the fluid in which
it is immersed, was made about this time. The differen-
ces of temperature were thus subjected to exact measure-
ment; the phenomena of heat became, of course, known
with more certainty and precision ; and that substance or
virtue, to which nothing is impenetrable, and which finds
its way through the rarest and the densest bodies, appa-
rently with the same facility,—which determines so many
of our sensations, and of which the distribution so mate-
rially influences all the phenomena of animal and vegeta-

¥

66 | DISSERTATION SECOND. [PART i3.

ble life, came now to be known, not indeed in its essence,
but as to all the characters in which we are practically or
experimentally concerned. The treatise on Fire, in Boer-
haave’s Chemistry, is a great advance beyond any thing on
that subject hitherto known, and touches, notwithstanding
many errors and imperfections, on most of the great truths,
which time,, experience, and ingenuity, have since brought
into view.

It was in this period also, that electricity may be said
first to have taken a scientific form. The power of amber
to attract small bodies, after it has been rubbed, is said to
have been known to Thales, and is certainly made men-
tion of by Theophrastus. The observations of Gilbert, a
physician of Colchester, in the end of the sixteenth cen-
tury, though at the distance of two thousand years, made
the first addition to the transient and superficial remarks
of the Greek naturalist, and afford a pretty full enumera-
tion of the bodies which can be rendered electrical by
friction. The Academia del Cimento, Boyle, and Otto
Guericke, followed in the same course; and the latter is
the first who mentions the crackling noise and faint light
which electricity sometimes produced. These, however,
were hardly perceived, and it was by Dr. Wall, as de-
scribed in the Philosophical Transactions, that they were
first distinctly observed.’ By a singularly fortunate anti-
cipation, he remarks of the light and crackling, that they
seemed in some degree to represent thunder and light-
ning.

After the experiments of Hauksbee in 1709, by which
the knowledge of this mysterious substance was considera-
bly advanced, Wheeler and Gray, who had discovered that

* Wall’s paper is in the Transactions for 1708, Vol. XXVI.

No. 314, p. 69.—Hauksbee on Electrical Light, in the same vol-
ume. See Abridgment, Vol. V. p. 408, 411.

\

eect. 11.] DISSERTATION SECOND. 67

one body could communicate electricity to another with-
out rubbing, being willing to try to what distance the elec-
trical virtue might be thus conveyed, employed, for the
purpose of forming the communication, a hempen rope,
which they extended to a considerable length, supporting
it from the sides, by threads which, in order to prevent
the dissipation of the electricity, they thought it proper to
make as slender as possible. They employed silk threads
with that view, and found the experiment to succeed.—
Thinking that it would succeed still better, if the supports
were made still more slender, they tried very fine metal-
lic wire, and were surprised to find, that the hempen rope,
thus supported, conveyed no electricity at all. It was,
therefore, as being silk, and not as being small, that the
threads had served to retain the electricity. This acci-
dent led to the great distinction of substances conducting,
and not conducting electricity. An extensive field of in-
quiry was thus opened, a fortunate accident having sup-
plied an instantia crucis, and enabled these experimenters
to distinguish between what was essential and what was
easual in the operation they had performed. ‘The history
of electricity, especially in its early stages, abounds with
facts of this kind; and no man, who would study the na-
ture of inductive science, and the rules for the interpre-
tation of nature, can employ himself better than in tracing
the progress of these discoveries. He will find abundant
reason to admire the ingenuity as well as the industry of
the inquirers, but he will often find accident come in very
opportunely to the assistance of both. The experiments
of Wheeler and Gray are described in the Zransactions,
for 1729.

68 DISSERTATION SECOND. [eanT 1.
‘ SECTION III, .

OPTICS.

Tue mvention of the telescope and the microscope, the
discoveries made concerning the properties of light and the
laws of vision, added to the facility of applying mathematical
reasoning as an instrument of investigation, had long given
a peculiar interest to optical researches. The experiments
and inquiries of Newton on that subject began in 1666, and
soon made a vast addition both to the extent and importance
ofthe science. He was at that time little more than twenty-
three years old; he had already made some of the greatest and
most original discoveries in the pure mathematics; and the
same young man, whom we have been admiring as the most
profound and inventive of geometers, is to appear, almost at
the same moment, as the most patient, faithful, and sagacious
interpreter of nature. ‘These characters, though certainly
not opposed to one another, are not often combined; but to
be combined in so high a degree, and in such early life, sn
hitherto without example.

In hopes of improving the telescope, by giving to the glas-
ses a figure different from the spherical, he had begun to
make experiments, and had procured a glass prism, in order,
as he tells us, to try with it the celebrated phenomena of
colours.' These trials led to the discovery of the different

* Phil. Trans. Vol. VI. (1672), p. 3075. Also Hutton's
Abridgment, Vol. I. p. 678. The account of the experiments is
in a letter to Oldenburgh, date February 1672; it is the first of
Newton’s works that was published. It is plain from what is
said above, that the phenomena of the ptismatic spectrum were
not unknown at that time, however little they were understood,
and however imperfectly observed,

sect. 111. DISSERTATION SECOND. 69

refrangibility of the rays of light, and are now too well known
to stand in need of a particular description.

Having admitted a beam of light into a dark chamber,
through a hole in the window-shutter, and made it fall on a
glass prism, so placed as to cast it on the opposite wall, he
was delighted to observe the brilliant colouring of the sun’s
image, and not less surprised to observe its figure, which, in-
stead of being circular, as he expected, was oblong in the
direction perpendicular to the edges of the prism, so as to
have the shape of a parallelogram, rounded at the two ends,
and nearly five times as long as it was broad.

When he reflected on these appearances, he saw nothing
that could expain the elongation of the image but the suppo-
sition that some of the rays of light, in passing through the
prism, were more refracted than others, so that rays which
were parallel when they fell on the prism, diverged from one
another after refraction, the rays that differed in refrangibi-
lity differing also in colour. ‘The spectrum, or solar image,

would thus consist of a series of circular images partly cover-
"ing one another, and partly projecting one beyond another,
from the red or least refrangible ray 8, in succession, to the
orange, yellow, green, blue, indigo, and violet, the most re-
frangible of all.

It was not, however, till he tried every other hypothesis
which suggested itself to his mind bythe test of experiment,
and proved its fallacy, that he adopted this as a true inter-
pretation of the phenomena. Even after these rejections,
his explanation had still to abide the sentence ae an emper i-
mentunt Crucis.

Having admitted the light and applied a prism as before,
he received the coloured spectrum ona board at the dis-
‘tance of about twelve feet from the first, and also pierced
with.a small hole. The coloured light which passed through
this second hole was made to fall on a prism, and afterwards

70 DISSERTATION SECOND. [part ii.

received on the opposite wall. It was then found that the
rays which had been most refracted, or most bent from their
course by the first prism, were most refracted also by the
second, though no mew colours were produced. “ So,” says
he, “the true cause of the length of the image was detected
to be no other than that light consists of rays differently re-
frangible, which, without any respect to a difference in their
incidence, were, according to their degrees of refrangibility,
transmitted towards divers parts of the wall.’”

It was also observed, that when the rays which fell on the
second prism were all of the same colour, the image formed
by refraction was truly*circular, and of the same colour with
the incident light. This is one of the most conclusive and
satisfactory of all the experiments.

When the sun’s light is thus admitted first through one
aperture, and then through another at some distance from the
first, and is afterwards made to fall ona prism, as the rays
come only from a part of the sun’s disk, the spectrum has
nearly the same length as before, but the breadth is greatly |
diminished ; in consequence of which, the light at each point
is purer, it is free from penumbra, and the confines of the
different colours can be more accurately traced. It was in
this way that Newton measured the extent of each colour,
and taking the mean of a great number of measures, he as-_
signed the following proportions, dividing the whole length of
the spectrum, exclusive of its rounded terminations, into 360
equal parts ; of these the

Red occupied - 45
Orange - - ey
Yellow - ~ 48
Green - - 60

1 Phil. Trans, Vol. VI. (1672), No. 80. p. 3075.

bace. 111.], DISSERTATION SECOND. 71

Blue - - 60
Indigo - 40
Violet - - 80

Between the divisions of the spectrum, thus made by the dif-
ferent colours, and the divisions of the monochord by the
notes of music, Newton conceived that there was an analogy,
and indeed an identity of ratios; but experience has since
shown that this analogy was accidental, as the spaces occupi-
ed by the different colours do not divide the spectrum: in the
same ratio, when prisms of different kinds of glass are em-
ployed. |

Such were the experiments by which Newton first “ un-
twisted all the shining robe of day,” and made known the tex-
ture of the magic garment which nature has so kindly spread
over the surface of the visible world. From them it followed,
that colours are not qualities which light derives from refrac-
tion or reflection, but are original and connate properties con-
nected with the different degrees of refrangibility that belong
to the different rays. The same colour is always joined
to the same degree of refrangibility, and conversely, the
same degree of refrangibility to the same colour.

Though the seven already enumerated are primary and
simple colours, any of them may also be produced by a mix-
ture of others. A mixture of yellow and blue, for instance,
makes green; of red and yellow, orange ; and, in general, if
two colours, which are not very far asunder in the natural
series, be mixed together, they compound the colour that
is in the middie between them.

But the most surprising composition of all, Newton ob-
serves, is that of whiteness ; which is not produced by one
sort of rays,. but by the mixture of all the colours in a
certain proportion, namely, in that proportion which they
have in the solar spectrum. ‘This fact may be said te

yp DISSERTATION SECOND. [paRT 1.

be made out both by analysis and composition. ‘The white
light of the sun can be separated, as we have just seen,
into the seven simple colours ; and if these colours be united
again they form white. Should any of them have been
wanting, or not in its due proportion, the white produced is
_ defective. y

It appeared, too, that natural bodies, of whatever colour,
if viewed by simple and homogeneous light, are seen of the
colour ofthat light and of no other. Newton tried this very
satisfactory experiment on bodies of all colours, and found it
to hold uniformly ; the light was never changed by aaa colour
of the body that reflected it.

Newton, thus furnished writh so many new and accurate
notions concerning the nature and production of colour, pro-
ceeded to apply them to the explanation of phenomena.
The subject which naturally offered itself the first to this analy-
sis was the rainbow, which, by the grandeur and simplicity
of its figure, added to the brilliancy of its colours, in every
age has equally attracted the attention of the peasant and of
the philosopher. That two refractions and one reflection
were at least a part of the machinery which nature employed
in the construction of this splendid arch, had been known
from the time of Antonio de Deminis ;} and the manner in
which the arched figure is produced had been shown by
Descartes ; so that it only remained to explain the nature of
the colour and its distribution. As the colours were the same
with those exhibited by the prism, and succeeded in the same
order, it could hardly be doubted that the cause was the
same. Newton showed the truth of his principles by calcu-
lating the extent of the arch, the breadth of the colour-
ed bow, the position of the secondary bow, its distance
from the primary, and by explaining the inversion of the

' Note F, at the end.

srer. u1.] DISSERTATION SECOND. 73

colours." There is not, perhaps, in science any happier ap-
plication of theory, or any in which the mind rests with ful-
ler confidence.

Other meteoric appearances seemed to be capable of simi-
lar explanations, but the phenomena being no where so
regular or so readily subjected to measurement as those of
the rainbow, the theory cannot be brought to so severe a
test, nor the evidence rendered so satisfactory.

But a more difficult task remained,—to explain the perma-
nent colour of natural bodies. Here, however, as it cannot
be doubted that all colour comes from the rays of light, so
we must conclude that one body is red and another violet,
because the one is disposed to reflect the red or least refran-
gible rays, and the other to reflect the violet or the most re-
frangible. Every body manifests its disposition to reflect the
light of its own peculiar colour, by this, that if you cast on it
pure light, first of its own colour, and then of any other, it |
will reflect the first much more copiously than the second.
If cinnabar, for example, and ultra-marine blue, be both ex-
posed to the same red homogeneous light, they will both ap-
pear red; but the cinnabar strongly luminous and resplen-
dent, and the ultra-marine of a faint obscure red. If the
homogeneal light thrown on them be blue, the converse of
the above will take place.

Transparent bodies, particularly fluids, often transmit light
of one colour and reflect light of another. Halley told New-
ton, that, being deep under the surface of the sea ina div-
ing-bell, in a clear sunshine day, the upper side of his hand,
on which the sun shone darkly through the water, and
through a small glass window in the diving-bell, appeared of
a red colour, like a damask rose, while the water below, and
the under part of his hand, looked green,?

* Optics, Book I. prop. 9. 2 Optics, p. 11. Horseley’s edit.
10

Fe DISSERTATION SECOND. [pant ti.

But, in explaining the permanent colour of bodies, this
difficulty always presents itself{—Suppose that a body re-
flects red or green light, what is it that decomposes the
light, and separates the red or the green from the
rest? Refraction is the only means of decomposing light,
and separating the rays of one degree of refrangibility
and of one colour, from those of another. This appears
to have been what led Newton to study the colours pro-
duced by light passing through thin plates of any transpa-
rent substance. The appearances are very remarkable, and
had already attracted the attention, both of Boyle and of
Hook, but the facts observed by them remained insulated
in their bands, and unconnected with other optical phe-
nomena.

It probably had been often remarked, that when two trans-
parent bodies, such as glass, of which the surfaces were con-
vex in a certain degree, were pressed together, a black spot
was formed at the contact of the two, which was surround-
ed with coloured rings, more or less regular, according to the
form of the surfaces. In order to analyse a phenomenon that
seemed in itself not a little curious, Newton proposed to
make the experiment with surfaces of a regular curvature,
such as was capable of being measured. He took two object
glasses, one a plano-convex for a fourteen feet telescope,
the other a double convex for one of about fifty feet, and
upon this last he laid the other with its plane side downwards, |
pressing them gently together. At their contact in the cen-
tre was a pellucid spot, through which the light passed with-
out suffering any reflection. Round this spot was a coloured
circle or ring, exhibiting blue, white, yellow, and red.. This
was succeeded by a pellucid or dark ring, then a coloured
ring of violet, blue, green, yellow, and red, all copious and
vivid except the green. The third coloured ring consisted
of purple, blue, green, yellow, and red. ‘The fourth consist-

sncr. 111,] DISSERTATION SECOND. 79

ed of green and red ; those that succeeded became gradually
more dilute and ended in whiteness. It was possible to count
as far as seven. :

The colours of these rings were so marked by peculian-
ties in shade and vivacity, that Newton considered them as
belonging to different orders; so that an eye accustomed
to examine them, on any particular colour of a natural ob-
ject being pointed out, would be able to determine to what
order in this series it belonged.

Thus we have a system of rings or zones surrounding a
dark central spot, and themselves alternately dark and
coloured, that is, alternately transmitting the hght and re-
flecting it. It is evident that the thickness of the plates of
air interposed between the glasses, at each of those rings,
must be a very material element in the arrangement of this
system. Newton, therefore, undertook to compute their
thickness. Having carefully measured the diameters of the
first six coloured rings, at the most lucid part of each, he
found their squares to be as the progression of odd numbers
1, 3,5, 7, &c. The squares of the distances from the centre
of the dark spot to each of these circumferences, were,
therefore, in the same ratio, and consequently the thickness
of the plates of air, or the intervals between the glasses,
were as the numbers 1, 3, 5, 7, &c.

When the diameters of the dark or pellucid rings which
separated the coloured rings were measured, their squares
were found to be as the even numbers 0, 2, 4, 6, and, there-
fore, the thickness of the plates through which the light was
wholly transmitted were as the same numbers. A great
many repeated measurements assured the accuracy of these
determinations.

As the curvature of the convex glass on which the flat
surface of the plano-convex rested, was known, and as the
diameters of the rings were measured in inches, it was easy

76 DISSERTATION SECOND. [Parr 17.

to compute the thickness of the plates of air, which corres-
ponded to the different rings.

An inch being divided into 178000 parts, the distance of
the lenses for the first series, or for the luminous rings, was
PR ye ee
178000 178000 178000’

aM di

f Pe eaten aa tebe) 7/3
For the second series 178000 178000”

When the rings were examined by looking through the
lenses in the opposite direction, the central spot appeared
white, and, in other rings, red was opposite to blue, yellow
to violet, and green to a compound of red and violet; the
colours formed by the transmitted and the reflected light
being, what is now called, complementary, or nearly so, of
one another; that is, such as when mixed produce white.

When the fluid between the glasses was different from air,
as when it was water, the succession of rings was the same ;
the only difference was, that the rings themselves were nar-
rower.

When experiments on thin plates were made in such a
way that the plate was of a denser body than the surround-
ing medium, as in the case of soap-bubbles, the same pheno-
mena were observed to take place. These phenomena
Newton also examined with his accustomed accuracy, and
even bestowed particular care on having the soap-bubbles as
perfect and durable as their frail structure would admit. In
the eye of philosophy no toy is despicable, and no occupa-
tion frivolous, that can assist in the discovery of truth.

To the different degrees of tenuity, then, in transparent
substances, there seemed to be attached the powers of sepa-
rating particular colours from the mass of light, and of render-
ing them visible sometimes by reflection, and, in other cases,
by transmission. As there is reason to think, then, that the

sxcr. 111.] DISSERTATION SECOND. . 77
minute parts, the mere particles of ‘all bodies, even the most
. opaque, are transparent, they may very well be conceived to
act on light after the manner of the thin plates, and to pro-
. duce each, according to its thickness and density, its appro-
priate colour, which, therefore, becomes the colour.of the
surface. ‘Thus the colours in which the bodies round us ap-
pear everywhere arrayed, are reducible to the action of the
parts, which constitute their surfaces, on the refined and active
fluid which pervades, adorns, and enlightens the world.

But the same experiments led to some new and unexpect-
ed conclusions, that seemed to reach the very essence of the
fluid of which we now speak. It was impossible to observe,
without wonder, the rings alternately luminous and dark that
were formed between the two plates of glass in the preced-
ing experiments, and determined to be what they were by
the different thickness of the air between the plates, and
having to that thickness the relations formerly expressed.
A plate of which the thickness was equal to a certain quan-
tity multiplied by an odd number, gave always a circle of the
one kind; but if the thickness of the plate was equal to the
same quantity multiplied by an even number, the circle was
of another kind, the light, in the first case, being reflected,
in the second transmitted. Light penetrating a thin transpa-
rent plate, of which the thickness was m, 3m, 5m, &c. was
decomposed and reflected; the same hght penetrating the
same plate, but of the thickness 0, 2m, 4m, was transmitted,
though, in a certain degree, also decomposed. The same
light, therefore, was transmitted or reflected, according as
the second surface of the plate of air through which it passed
was distant from the first, by the intervals 0, 2, 4m, or m, 3m,
5m; so that it becomes necessary to suppose the same ray
to be successively disposed to be transmitted and to be re-
flected at points of space separated from one another by the
same interval m. ‘This constitutes what Newton called Fits

,

78 DISSERTATION SECOND. [Pant 18.

of easy transmission and easy reflection, and forms one of the
most singular parts of his optical discoveries. It is so unlike
any thing which analogy teaches us to expect, that it has
often been viewed with a degree of incredulity, and regard-
ed as at best but a conjecture introduced to account for cer-
tain optical phenomena. This, however, is by no means a
just conclusion, for it is, in reality, a necessary inference
from appearances accurately observed, and is no less en-
titled to be considered as a fact than those appearances
themselves. The difficulty of assigning a cause for such
extraordinary alternations cannot be denied, but does not
entitle us to doubt the truth of a conclusion fairly deduced
from experiment. The principle has been confirmed by
phenomena that were unknown to Newton himself, and pos-
sesses this great and unequivocal character of philosophic
truth, that it has served to explain appearances which were
not observed till long after the time when it first became
known.

We cannot follow the researches of Newton into what re-
gards the colours of thick plates, and of bodies in general.
We must not, however, pass over his explanation of refrac-
tion, which is among the happiest to be met with in any part
of science, and has the merit of connecting the principles of
Optics with those of Dynamics.

The theory from which the explanation we speak of is
deduced, is, that ght is an emanation of particles, moving in
straight lines with incredible velocity, and attracted by the
particles of transparent bodies. . When, therefore, light falls
obliquely on the surface of sucha body, its motion may be
resolved into two, one parallel to that surface, and the
other perpendicular to it. Of these, the first is not affected
by the attraction of the body, which is perpendicular to its
own surface ; and, therefore, it remains the same in the re-
fracted that it was in the incident ray. But the velocity

———

ser. 111.] ' DISSERTATION SECOND, 79

perpendicular to the surface is increased by the attraction
of the body, and, according to the principles of dynamics
(the 39th, Book I. Princip.), whatever be the quantity of
this velocity, its square, on entering the same transparent
body, will always be augmented by the same quantity. But
it is easy to demonstrate that, if there be two right-angled
triangles, with a side in the one equal to a side in the other,
the hypothenuse of the first being given, and the squares of
their remaining sides differing by a given space, the sines of
the angles opposite to the equal sides must have a given
ratio to one another.’ This amounts to the same with say-
ing, that, in the case before us, the sine of the angle of inci-
dence is to the sine of the angle of refraction in a given
ratio. ‘The explanation of the law of refraction thus given
is so highly satisfactory, that it affords a strong argument in
favour of the system which considers light as an emanation
of particles from luminous bodies, rather than the vibrations
of an elastic fluid. It is true that Huygens deduced from
this last hypothesis an explanation of the law of refraction,
on which considerable praise was bestowed in the former
part ofthis Dissertation. It is undoubtedly very ingenious,
but does not rest on the same solid and undoubted princi-
ples of dynamics with the preceding, nor does it leave the
mind so completely satisfied. Newton, in his Principia, has
deduced another demonstration of the same optical proposi-
tion from the theory of central forces.?

The different refrangibility of the rays of light forms no
exception to the reasoning above. ‘The rays of each. par-
ticular colour have their own particular ratio subsisting be-
tween the sines of incidence and refraction, or in each, the
square that is added to the square of the perpendicular

Optics, Book Il. Part iii. prop. 10.
* Prin. Math, Lib. 1, prop. 94. Also Optics, Book I, prop. 6.

‘
-

80 DISSERTATION SECOND. [ran 11.

velocity has its own value, which continues the same while
‘the transparent medium is the same.

Light, in consequence of these views, became, in the
hands of Newton, the means of making important discoveries
eoncerning the internal and chemical constitution of bodies.
The square that is added to that of the perpendicular veloci-
ty of light in consequence of the attractive force of the trans-
parent substance, is properly the measure of the quantity of
that attraction, and is the same with the difference of the
squares of the velocities of the incident and the refracted
light. ‘This is readily deduced, therefore, from the ratio of
the angle of incidence to that of refraction ; and when this is
done for different substances, it is found, that the above
measure of the refracting power of transparent bodies is
nearly proportional to their density, with the exception of
those which contain much inflammable matter in their com-
position, or sulphur as it was then called, which is always
accompanied with an increase of refracting power.'

Thus the refracting power, ascertained as above, when
divided by the density, gives quotients not very different
from one ‘another, till we come to the inflammable ‘bodies,
where a great increase immediately takes place. In air,
for instance, the quotient is 5208, in rock-crystal 5450,
and the same nearly incommon glass. But in spirit of wine,
oil, amber, the same quotients are 10121, 12607, 13654.
Newton found in the diamond, that this quotient is still
greater than any of the preceding, being 14556.? Hence
he conjectured, what has since been so fully verified by
experiment, that the diamond, at least in part, is an inflam-
mable body. Observing, also, that the refracting power of
water is great for its density, the quotient, expounding it, as
above, being 7845, he concluded, that gn inflammable sub-

' Newton’s Optics, Ibid. ? Ibid.

seer. 01] BISSERTATION SECOND. Si

stance enters into the composition of that fluidj—a conclu-
sion which has been confirmed by one of the most certain but
most unexpected results of cheinical analysis. The views
thus suggested by Newton have been successfully pursued by
future inguirers, and the action of bodies on light is now
regarded as one of the means of examining into their in-
terval constitution.

I should have before remarked, that the alternate dispo-
sition to be easily reflected and easily transmitted, serves to
explain the fact, that all transparent substances reflect a por-
tion of the incident light. The reflection of light from the
surfaces of opaque bodies, and from the anterior surfaces of
transparent bodies, appears to be produced by a repulsive
force exerted by those surfaces at a determinate but very
small distance, in consequence of which there is stretched
out over them an elastic web, through which the particles
of light, notwithstanding their incredible velocity, are not
always able to penetrate.'” In the case of a transparent
body, the light which, when it arrives at this outwork, as
it may be called, is in a fit of easy reflection, obeys of
course the repulsive force, and is reflected back again.
The particles, on the other hand, which are in the state
which disposes them to be transmitted, overcome the repul-
sive force, and, entering into the interior of the transpa-
rent body, are subjected to the action of its attractive
force, and obey the law of refraction already explained.
If these rays, however, reach the second surface of the trans-
parent body (that body being suppposed denser than the
medium surrounding it), in a direction having a certain obli-
quity to that surface, the attraction will not suffer the rays

‘A velocity that enables light to pass from the sun to the
earth in 8’ 13”, as is deduced from the eclipses of Jupiter’s
satellites.

11

82 DISSERTATION SECOND. [pana 41.

to emerge into the rarer medium, but will force them to
return back into the transparent body. Thus the reflec-
tion of hght at ‘the second surface of a transparent body
is produced, not by the repulsion of the medium in which
it was about to enter, but by the attraction of that which
it was preparing to leave.

The first account of the experiments from which all
these conclusions were deduced, was given in the Philoso-
phical Transactions for 1672, and the admiration excited
by their brilhancy and their novelty may easily be imagin-
ed. Among the men of science, the most enlightened
were the most enthusiastic in their praise. Huygens,
writing to one of his friends, says of them, and of the
truths they were the means of making known, “ Quorum
respeciu omnia huc usque edita jeyunia sunt et prorsus
puerilia.”? Such were the sentiments of the person who,
of all men living, was the best able to judge, and had the
best right to be fastidious in what related to optical ex-
periments and discoveries. But all were not equally can-
did with the Dutch philosopher; and though the discove-
ry now communicated had every thing to recommend it
which can arise from what is great, new and singular ;
though it was not a theory or a system of opmions, but
the generalization of facts made known by experiments ;
and though it was brought forward in the most simpte
and unpretending form, a host of enemies appeared, each
eager to obtain the unfortunate pre-eminence, of being the
first to attack conclusions, which the unanimous voice of
posterity was to confirm. In this contention, the envy
and activity of Hook did not fail to give him the advan-
tage, and he communicated his objections to Newton’s con-
\ clusions concerning the refrangibility of light in less than a
month afier they had been read in the Royal Society.
Te admitted the accuracy of the experiments themselves,

eRer. 111.) DISSERTATION SECOND. 83

but denied that the cause of the colour is im any quality
residing permanently in the rays of light, any more than
that the sounds emitted from the pipes of an organ exist
originally in the air. An imaginary analogy between sottnd
and light seems to have been the basis of all his optical theo-
ries. He conceived that colour is nothing but the disturbance
of light by pulses propagated through it; that blackness
proceeds from the scarcity, whiteness from the plenty, of
undisturbed light; and that the prism ects by exciting
different pulses in this fluid, which pulses give rise to the
sensations of colour. This obscure and unintelligible theo-
ry (if we may honour what is unintelligible with the
name of a theory) he accompanied with a multitude of
captious objections to the reasonings of Newton, whom
he was not ashamed to charge with borrowing from him
without acknowledgment. To all this Newton replied,
with the solidity, calmness, and modesty, which became the
understanding and the temper of a true philosopher.

The new theory of colours was quickly assailed by
several other writers, who seem all to have had a better
apology than Hooke for the errors into which they fell.
Among them one of the first was Father Pardies, who
wrote against the experiments, and what he was pleased
to call the hypothesis, of Newton. A satisfactory and
‘calm reply convinced him of his mistake, which he had
the candour very readily to acknowledge. A countryman
of his, Mariotte, was more difficult to be reconciled, and,
though very conversant with experiment, appears never
to have succeeded in repeating the experiments of New-
ton. Desaguliers, at the request of the latter, repeated
the experiments doubted of before the Royal Society,
where Monmort, a countryman anda friend of Mariotte,
was present.'

; Montucla, Tom, Il.

84 DISSERTATION SECOND. [PART 11.

MM. Linus and Lucas, both of Leige, objected te
Newton’s experiments as inaccurate; the first, because,
on attempting to repeat them, he had not obtained the
game results; and the second, because he had not been
able to perceive that a red object and a blue required
the focal distance to be different when they were viewed
through a telescope. Newton replied with great patience
and good temper to both.

The series was closed, in 1727, by the work of an
Italian author, Rizetti, who, in like manner, called in
question the accuracy of experiments which he himself
had not been able to repeat. Newton was now no more,
but Desaguliers, in consequence of Rizetti’s doubts, insti-
tuted a series of experiments which seemed to set the
matter entirely at rest. ‘These experiments are described
in the Philosophical Transactions for 1728.

An inference which Newton had immediately drawn
from the discoveries above described was, that the great
source of imperfection in the refracting telescope was the
different refrangibility of the rays of light, and that there
were stronger reasons than either Mersenne or Gregory
had suspected, for looking to reflection for the improve-
ment of optical instruments. It was evident, from the
different refrangibility of light, that the rays coming from
the same point of an object, when decomposed by the
refraction of a lens, must converge to different foci; the
red rays, for example, to a point more distant from the
lens, and the violet to one nearer by about a fifty-fourth
part of the focal distance. Hence it was not merely
from the aberration of the rays caused by the spherical
figure of the lens, that the imperfection of the images
formed by refraction arose, but from the very nature of
refraction itself. It was evident, at the same time, that
in a combination of lenses with opposite figures, one con-

seieeihsi1,] DISSERTATION SECOND. 85

vex, for instance, and another concave, there was a ten-
dency of the two contrary dispersions to correct one
another. But it appeared to Newton, on examining dif-
erent refracting ‘substances, .that the dispersion of the
coloured rays never could be corrected except when the
refraction itself was entirely destroyed, for he thought
he had discovered that the quantity of the refraction and
of the dispersion in different substances bore always the
same proportion to one another. This is one of the few
instances in which his conclusions ‘have not been confirm-
ed by subsequent experiment; and it will, accordingly, fall
under discussion in another part of this discourse.

Having taken the resolution of constructing a reflecting
telescope, he set about doing so with his own hands.
There was, indeed, at that time, no other means by
which such a work could be accomplished; the art of the
ordinary glass-grinder not being sufficient to give to me-
tallic specula the polish which was required. It was on
this account that Gregory had entirely failed in realizing
his very ingenious optical invention.

Newton, however, himself possessed excellent hands for
mechanical operations, and could use them to better pur-
pose than is common with men so much immersed in
deep and abstract speculation. It appears, indeed, that
mechanical invention was one of the powers of his mind
which began to unfold itself at a very early period. In
some letters subjoined to a memoir drawn up after his
death by his nephew Conduit, it is said, that, when a boy,
Newton used to amuse himself with constructing machines,
mills, &c. on a small! scale, in which he displayed great
ingenuity ; and it is probable that he then acquired that
use of his hands which is so difficult to be learned at a
later period. To this, probably, we owe the neatness
and ingenuity with which the optical experiments above

&6 DISSERTATION SECOND. [PART 1.

referred to were contrived and executed,—experiments of
so difficult a nature, that any. error in the manipulation
would easily defeat the effect, and appears actually to
have done so with many of those who objected to his
experiments.’ .

He succeeded perfectly in the construction of his teles-
scope, and his first communication with Oldenburg, and
the first reference to his optical experiments, is+connect-
ed with the construction of this instrument, and mention-
ed in a letter dated the 11th January 1672. He had
then been proposed as a member of the Royal Society

*The Memoir of Conduit was sent to Fontenelle when he
was preparing the Eloge on Newton, but he seems to have
paid little attention to it, and has passed over the early part
of his life with the remark, that ene may apply to him what
Lucan says of the Nile, that it has not been ‘ permitted to
mortals to see that river in a feeble state.” If the letters
above referred to had formed a part of this communication,
I think the Secretary of the Academy would have sacrificed
a fine comparison to an instructive fact. In other respects
Conduit’s Memoir did not convey much information that could
be of use. His instructions to Fontenelle are curious enough ;
he- vids him be sure to state, that Leibnitz had borrowed the
Differential Calculus from the Method of Fluxions. He con-
jared him in another place not to omit to mention, that Queen
Caroline used to delight much in the conversation of New-
ton, and nothing could do more honour to Newton than the
commendation of a Queen, the Minerva of her age. Fonte-
nelle was too much a philosopher, and a man of the world
(and bad himself approached too near to the persons of
princes), to be of Mr. Conduit’s opinion, or to think that the
approbation of the most illustrious princess could add dignity
to the man, who had made the three greatest discoveries yet
known, and in whose hands the sciences of Geometry, Optics,
and Astronomy, had all taken new forms. If he had been
called to write the Eloge of the Queen of England, he would,
no doubt, have remarked her relish for the conversation of
Newton.

On the whole, the Eloge on Newton has great merit, and,
to be the work of one who was at bottom a Cartesian, is a
singular example of candour and impartiality.

srer, 11.) DISSERTATION SECOND. 37

by the Bishop of Sarum, and he says, “If the honour of
being a member of the Society shall be conferred on me,
I shall endeavour to testify my gratitude by communi-
cating what my poor and solitary endeavours can effect
toward the promoting its philosophical designs.”? Such
was the modesty of the man who was to effect a greater
revolution in the state of our knowledge of nature than
‘any individual had yet done, and greater, perhaps, than
any individual is ever destined to bring about. Success,
however, never altered the temper in which he began
his researches.

Newton, after considering the reflection and refraction
of light, proceeded, in the third and last Book of his
Optics, to treat of its inflexion, a subject which, as has
been remarked in the former. part of this discourse, was
first treated of by Grimaldi. Newton having admitted a
ray of light through a hole in a window-shutter into a
dark chamber, made it pass by the edge of a knife, or,
in some experiments, between the edges of two knives,
fixed parallel, and very near to one another; and, by re-
ceiving the light on a sheet of paper at different distan-
ces behind the knives, he observed the coloured fringes
which had been described by the Italian optician, and.
on examination, found, that the rays had been acted on
in passing the knife edges both by repulsive and attrac-
tive forces, and had begun to be so acted on in a sensi-
ble degree when they were yet distant by g55 of an inch
of the edges of the knives. His experiments, however,
on this subject were interrupted, as he informs us, and
do not appear to have been afterwards resumed. They
enabled him, however, to draw this,conclusion, that the
path of the ray in passing by the knife edge was bent
in opposite directions, so as to form a serpentine line.

' Birch’s History of the Reyal Soctety, Vol. 1. p. 3.

88 DISSERTATION SECOND. [part 11,

convex and concave toward the knife, according to the
repulsive or attractive forces which acted at different dis-
tances ; that it was also reasonable to conclude, that the
phenomena of the refraction, reflection, and inflexion of
light were all produced by the same force variously modifi-
ed, and that they did not arise from the actual contact or
collision of the particles of light with the particles of bodies.

The Third Book of the Optics concludes with those

celebrated Queries which carry the mind so far beyond the

~ bounds of ordinary speculation, though still with the sup-
port and under the direction either of direct experiment or
‘ elose analogy. They are a collection of propositions rela-
tive chiefly to the nature of the mutual action of light
and of bodies on one another, such as appeared to the
author highly probable, yet wanting such complete evi-
ence as might entitle them to be admitted as princi-
ples established. Such enlarged and comprehensive views,
so many new and bold conceptions, were never before
combined with the sobriety and caution of philosophical
induction. The anticipation of future discoveries, the as-
semblage of so many facts from the most distant regions of
human research, all brought to bear on the same points, and
to elucidate the same questions, are never to be sufliciently
admired. At the moment when they appeared, they must
have produced a wonderful sensation in the philosophic
world, unless, indeed, they advanced too far before the age,
and contained too much which the comment of time was yet
required to elucidate.

It is in the Queries that we meet with the ideas of this
philosopher concerning the Elastic Ether, which he con-
ceived to be the means of conveying the action of bodies
from one part of the universe to another, and to which the
phenomena of light, of heat, of gravitation, are to be ascrib-
ed. Here we have his conclusions concerning that polarity

scor, ut] DISSERTATION SECOND. 8S

or peculiar virtue residing in the opposite sides of the
rays of light, which he deduced from the enigmatical phe-
nomena of doubly refracting crystals. Here, also, the
first step is made toward the doctrine of elective attrac-
tions or of chemical affinity, and to the notion, that the
phenomena of chemistry, as well as of cohesion, depend
on the alternate attractions and repulsions existing be-
tween the particles of bodies at different distances. ‘The
comparison of the gradual transition from repulsion to
attraction at those distances, with the positive and nega-
tive quantities in algebra, was first suggested here, and is
the same idea which the ingenuity of Boscovich after-
wards expanded into such/a beautiful‘and complete sys-
tem. Others who have attempted such flights, had ended
in mere fiction and romance; it is only for such men as
Bacon or Newton to soar beyond the region of poetical
fiction, still keeping sight of probability, and alighting
again safe on the terra firma of philosophic truth.'

' The optical works of Newton are not often to be found al}
brought together into one body. The first part of them con-
sists of the papers in the Philosophical Transactions, which
gave the earliest account of his discoveries, and which have
been already referred to. ‘They are in the form of Letters to
Oldenburg, the Secretary of the Society, as are also the an-
. swers to Hooke, and the others who objected to these discove-
ries ; the whole forming a most interesting and valuable series,
which Dr. Horsely has published in the fourth volume of his
edition of Newton’s works, under the title of Letters relating to
the Theory of Lights and Colours. The next work, in point of
time, consists of the Lectiones Optice, or the optical lectures
which the author delivered at Cambridge. The Optics, in
three books, is the last and most complete, containing all the
reasoning concerning optical phenomena above referred to.
The first edition was in 1704, the second, with additions, in
1717. Nemwtont Opera, Tom. IV. Horsely’s edition. _

12

90 DISSERTATION SECOND: ie eet
SECTION IV.

ASTRONOMY.

Tre time was now come when the world was to be
enlightened by a new science, arising out of the com-
parison of the phenomena of motion as observed in the
heavens, with the laws of motion as known on the earth.
Physical astronomy was the result of this comparison, a
science embracing greater objects, and destined for a
higher flight than any other branch of natural knowledge.
Ti is unnecessary to observe, that it was by Newton that
the comparison just referred to was instituted, and the
riches of the new science unfolded to mankind.

This young philosopher, already signalized ‘by great
discoveries, had scarcely reached the age of twenty-four,
when a great public calamity forced him into the situa-
tion where the first step in the new science is said to
have been suggested; and that, by some of those com-
mon appearances, in which an ordinary man sees nothing
to draw his attention, nor even the man of genius, ex-
cept at those moments of inspiration when the mind sees
farthest into the intellectual world. In 1666, the plague
forced him to retire from Cambridge ito the country ;
and, as he sat one day alone, in a garden, musing on the
nature of the mysterious force by which the phenomena
at the earth’s surface are so much regulated, he observ-
ed the apples falling spontaneously from the trees, and
the thought occurred to him, since gravity is a tendency
not confined to bodies on the very surface of the earth,
but since it reaches to the tops of trees, to the tops of
the highest buildings, nay, to the summits of the most

ezer. 1V.] DISSERTATION SECOND. 91

lofty mountains, without its intensity or direction suffer-
ing any sensible change, Why may it not ‘reach to a
much greater distance, and even to the moon itself?
And, if so, may not the moon be retained in her orbit
by gravity, and forced to describe a curve like a pro-
jectile at the surface of the earth”?

Here another consideration very naturally occurred.
Though gravity be not sensibly weakened at the small
distances from the surface to which our experiments ex-
tend, it may be weakened at greater distances, and at
the moon may be greatly diminished. To estimate the
quantity. of this diminution, Newton appears to have rea-
soned thus: If the moon be retained in her orbit by her
gravitation to the earth, it is probable that the planets
are, in like manner, carried round the sun by a power
of the same kind with gravity, directed to the centre of
that luminary. He proceeded, therefore, to inquire, by
what law the tendency, or gravitation of the planets to
the sun, must diminish, in order that, describing, as they
do, orbits nearly circular round the sun, their times of
revolution and their distances may have the relation to
one another which they are known to have from obser-
vation, or from the third law of Kepler.

This was an investigation which, to most even of the
philosophers and mathematicians of that age, would have
proved an insurmountable obstacle to their farther pro-
gress; but Newton was too familiar with the geometry of
evanescent or infinitely small quantities, not to discover
very soon, that the law now referred to, would require
the force of gravity to diminish exactly as the square of
the distance increased. The moon, therefore, being dis-
tant from the earth about sixty semidiameters of the

* Pemberton’s View of Newton’s Philosophy, Pref.

92 DISSERTATION SECOND. [par if.

earth, the force of gravity at that distance must be re-
duced to the 3600th part of what it is at the earth’s
surface. Was the deflection of the moon then from the
tangent of her orbit, in a second of time, just the 3600th
part of the distance which a heavy body fails in a second
at the surface of the earth? This was a question that
could be precisely, answered, supposing the moon’s dis-
tance known not merely in semidiameters of the earth
but in feet, and her angular velocity, or the time of her
revolution in her orbit, to be also known.

~In this calculation, however, being at a distance from
books, he took the common estimation of the earth’s
circumference that was in use before the measurement of
Norwood, or of the French Academicians, according to
which, a degree is held equal to 60 English miles. This
being in reality a very erroneous supposition, the result
of the calculation did not represent the force as adequate
to the supposed effect; whence Newton concluded that
some other cause than gravity must act on the moon,
and on that account he laid aside, for the time, all farther
speculation on the subject. It was in the true spint of
philosophy that he so readily gave up an hypothesis, in
which he could not but feel some interest, the moment
he found it at variance with observation. He was sensi-
ble that nothing but the exact cemcidence of the things
compared could establish the conclusion he meant to de-
duce, or authorize him to proceed with the superstruc-
ture, for which it was to serve as the foundation.

It appears, that it was not till some years after this,
that his attention was called to the same subject, by a
letter from Dr. Hooke, proposing, as a question, To de-
termine the line in which a body let fall from a_height
descends to the ground, taking into consideration the
motion of the earth on its.axis. This induced him to

sner. 1v.] DISSERTATION SECOND. 93

resume the subject of the moon’s motion; and the mea-
sure of a degree by Norwood having now furnished more
exact data, he found that his calculation gave the pre-
cise quantity for the moon’s momentary deflection from
the tangent of her orbit, which was deduced from astro-
nomical observation. ‘The moon, therefore, has a ten-
dency to descend toward the earth from the same cause
that a stone at its surface has; and if the descent of the
stone in a second be diminished in the ratio of 1 to
3600, it will give the quantity by which the moon de-
scends in a second, below the tangent to her orbit, and
thus is obtained an experimental proof of the fact, that
gravity decreases as the square of the distance increa-
ses. He had already found that the times of the plane-
tary revolutions, supposing their orbits to be circular,
led to the same conclusion; and he now proceeded,
with a view to the solution of Hooke’s problem, to in-
quire what their orbits must be, supposing the centripetal
force to be inversely as the square of the distance, and
the initial or projectile force to be any whatsoever. On
this subject Pemberton says, he composed (as he calls it)
a dozen propositions, which probably were the same with
those in the beginning of the Principia,—such as the de-
scription of equal areas in equal times, about the centre
of force, and the elipticity of the orbits described under
the influence of a centripetal force that varied inversely
as the square of the distances.

What seems very difficult to be explained is, that,
after having made trial of his strength, and of the power
of the instruments of investigation which he was now in
possession of, and had entered by means of them on the
noblest and most magnificent field of investigation that
was ever yet opened to any of the human race, he again
desisted from the pursuit, so that it was not till severa!

94 DISSERTATION SECOND. [rant it.

years afterwards, that the conversation of Dr. Halley, whe
made him a visit at Cambridge, mduced him to resume
and extend his researches.

He then found, that the three great facts in astronomy,
which form the laws of Kepler, gave the most complete
evidence to the system of gravitation. The /irst of them,
the proportionality of the areas described by the radius vec-
tor to the times in which they are described, is the peculiar
character of the motions produced by an original impulse
impressed on a body, combined with a centripetal force
‘continually urging it toa given centre. The second law, |
that the planets deseribe ellipses, having the sun in one of
the foci, common to them all, coincides with this proposi-
tion, that a body under the influence of a centripetal force,
varying as the square of the distance inversely, and having any
projectile force whatever originally impressed on it, must
describe a conic section having one focus in the centre of
force, which section, if the projectile force does not exceed
-a certain limit, will become an ellipse. The third law,
that the squares of the periodic times are as the cubes of
the distances, is a property which belongs to the bodies de-
scribing elliptic orbits under the conditions just stated.
Thus the three great truths to which the astronomy of the
planets had been reduced by Kepler, were all explained in
the most satisfactory manner, by the supposition, that the
planets gravitate to the sun with a force, which varies in
the inverse ratio of the square of the distances. It added
much to this evidence,: that the observations of Cassini had
proved the same laws to prevail among the satellites of
Jupiter. :

But did the principle which appeared thus to unite the
great bodies of the universe act only on those bodies?
Did it reside merely in their centres, or was it a force
common to all the particles of matter? Was ita fact that

sner. 1V.] DISSERTATION SECOND. 95

every particle of matter had a tendency to unite with every
other? Or was that tendency directed only to particular
centres? It could hardly be doubted that the tendency was
common to all the particles of matter. The centres of the
great bodies had no properties as mathematical points, they
had none but what they derived from the material particles
distributed around them. But the question admitted of
being brought to a better test than that of such general rea-
soning as the preceding. The bodies between which this
tendency had been observed to take place, were all round
bodies, and either spherical or nearly so, but whether great
or small, they seemed to gravitate toward one another ac-
cording to the same law. ‘The planets gravitated to the
sun, the moon to the earth, the satellites of Jupiter toward
Jupiter; and gravity, in all these instances, varied inversely
as the squares of the distances. Were the bodies ever so
small—were they mere particles—provided only they were
round, it was therefore safe to infer, that they would tend
to unite with forces inversely as the squares of the distances.
It was probable, then, that gravity was the mutual tendency
of all the particles of matter toward one another; but this
could not be concluded with certainty, till it was found,
whewier great spherical bodies composed of particles gravi-
tating according to this law, would themselves gravitate ac-
cording to the same. Perhaps no man of that age but
Newton himself was fit to undertake the solution of this
problem. His analysis, either in the form of fluxions or in
that of prime and ultimate ratios, was able to reduce it to the
quadrature of curves, and he then found, no doubt infinitely
to his satisfaction, that the law was the same for the sphere
as for the particles which compose it; that the gravitation
was directed to the centre of the sphere, and was as the
quantity of matter contained in it, divided by the square
of the distance from its centre. Thus a complete expres-

96 DISSERTATION SECOND. [FART 11,

sion was obtained for the law of gravity, involving both the
conditions on which it must depend, the quantity of matter
in the gravitating bodies, and the distance at which the
bodies were placed. There could be no doubt that this
tendency was always mutual, as there appeared nowhere
any exception to the rule that action and reaction are
"equal; so that if a stone gravitated to the earth, the earth
gravitated equally to the stone; that is to say, that the two
bodies tended to approach one another with velocities which
were inversely as their quantities of matter." There ap-
peared to be no limit to the distance to which this action
reached; it was a force that united all the parts of matter
to one another, and if it appeared to be particularly directed
to certain points, such as the centres of the sun or of the
planets, it was only on account of the quantity of matter
collected and distributed uniformly round those points,
through which, therefore, the force resulting from the com-
position of all those elements must pass either accurately or
nearly.

A remarkable inference was deduced from this view of
- the planetary motions, giving a deep insight into the con-
stitution of our system in a matter that seems the most re-
condite, and the furthest beyond the sphere which necessari-
ly cireumscribes human knowledge. The quantity of mat-
ter, and even the density of the planets, was determined.
We have seen how Newton compared the intensity of gravi-
tation at the surface of the earth, with its intensity at the
moon, and by a computation somewhat similar, he compar-

1df M and M’ are the masses of two spheres, and zthe dis-

2 1 M- iM’. Z f :
iance of their centres, Bi is the accelerating force with
x
: ¢? Haas i ot ae
which they tend to unite ; but the velocity of the approach of

MM’
M will be —, and of M’, 5.

«

sxer. 1V.] DISSERTATION SECOND. rhe iif

ed the intensity of the earth’s gravitation to the sun,
with the moon’s gravitation to the earth, each being
measured by the contemporaneous and momentary deflex-
ion from a tangent to the small arch of its orbit. A
more detailed investigation showed, that the intensity of
the central force in different orbits, is as the mean dis-
tance divided by the square of the periodic time; and
the same intensity being also as the quantities of matter
divided by the squares of the distances, it follows, that
these two quotients are equal to one another, and that,
therefore, the quantities of matter are as the mean dis-
tances divided by the squares of the periodic times.
Supposing, therefore, in the instance, just mentioned, that
the ratio of the mean distance of the sun from the earth
to the mean distance of the moon from the earth is
given (which it is from astronomical observation) ; as the
ratio of their periodic lines is also known, the ratio of
the quantity of matter in the sun to the quantity of mat-
ter in the earth, of consequence is found, and the same
holds good for all the planets which have satellites moving
round them. Nothing certainly can be more unexpecied,
than that the quantities of matter in bodies so remote,
should admit of being compared with one another, and
with the earth. Hence also their mean densities, or
mean specific gravities, became known. For from their
distances and the angles they subtended, beth known
from observation, their magnitudes or cubical contents
were easily inferred, and the densities of all bodies are,
as their quantities of matter, divided by their magnitude.
The Principia Philosophie Naturalis, which contained all
these discoveries, and established the principle of univer-
sal gravitation, was given to the world in 4687, an era,
on that account, for ever memorable in the history of
human knowledge.

98 DISSERTATION SECOND. [pan? 11.

The principle of gravity which was thus fully established,
and its greatest and most extensive consequences deduced,
was not now mentioned for the first time, though for the first
time its existence as a fact was ascertained, and the law it
observes was discovered. Besides some curious references to
weight and gravity, contained in the writings of the ancients,
we fmd something more precise concerning it in the writings
of Copernicus, Kepler, and Hooke.

Anaxagoras is said to have held that “ the heavens are
kept in their place by the rapidity of their revolution, and
would fall down if that rapidity were to cease.””!

Plutarch, in ike manner, says, the moon is kept from
falling by the rapidity of her motion, just as a stone
whirled round in a sling is prevented from falling to the
ground.”

Lucretius, reasoning probably after Democritus, holds,
that the atoms would all, from their gravity, have long since
united in the centre of the universe, if the universe were not
infinite, so as to have no centre.°

An observation of Pythagoras, supposed to refer to the
doctrine of gravity, though in reality extremely vague, has
been abundantly commented on by Gregory and Maclaurin.
A musical string, said that philosopher, gives the same sound
with another of twice the length, if the latter be straitened by
four times the weight that straitens the former; and the
gravity of a planet is four times that of another which is at
twice the distance. These are the most precise notices, as
far as I know, that exist in the writings of the ancients, con-
cerning gravity as a force acting on terrestrial bodies, or as
extending even to those that are more distant. They are

' Colum omne vehementi circuitu constare, alias remissione
lapsurum. (Diog. Laert. an Anax. Lib. II. Sect. 12.
} 5

? De facte in Orbe Lune. * Lib. I. vy. 983

exer, 1v.] DISSERTATION SECOND. OY

the reveries of ingenious men who had no steady principles
deduced from experience and observation to direct their in-
quiries; and who, even when in their conjectures they hit on
the truth, could hardly distinguish it from error.

Copernicus, as might be expected, is considerably more
precise. “I do not think,” says he, “that gravity is any-
thing but a natural appetency of the parts (of the earth) given
by the providence of the Supreme Being, that, by uniting to-
gether, they may assume the form of a globe. It is proba-
ble, that this same affection belongs to the sun, the moon,
and the fixed stars, which all are of a round form.’”

The power which Copernicus here speaks of, has nothing
to do, in his opinion, with the revolutions of the earth or the
planets in their different orbits. It is merely intended as an
explanation of their globular forms, and the consideration
that does the author most credit is, that of supposing the
force to belong, not to the centre, but to all the parts of the
earth.

Kepler, in his immortal work on the Motions of Mars,
treats of gravity as a force acting naturally from planet to
planet, and particularly from the earth to the moon. “ If
the moon and the earth were not retained by some animal or
other equivalent force each in its orbit, the earth would as-
cend to the moon by a 54th part of the interval between
them, while the moon moved over the remaining 53 parts,
that is, supposing them both of the same density.’” This
passage is curious, as displaying a singular mixture of know-
ledge and error on the subject of the planetary motions.
The tendency of the earth and moon being mutual, and pro-
ducing equal quantities of motion in those bodies, bespeaks

* Revolutionum, Lib. 1. cap. 9. p. 17.

? On that supposition their quantities of matter would be as
their bulks, or as 1 to 53.

100 DISSERTATION SECOND. [PART 11.

an accurate knowledge of the nature of that tendency, and of
the equality, at least in this instance, between action and re-
action. Then, again, the idea of an animal force or some
other equally unintelligible power being necessary to carry
on the circular motion, and to prevent the bodies from mov-
ing directly toward each other, is very strange ; considering
that Kepler knew the inertia of matter, and ought, therefore,
to have understood the nature of centrifugal force, and its
power to counteract the mutual gravitations of the two bo-
dies. In this respect, the great astronomer who was laying
the foundation of all that is known of the heavens, was not
so far advanced as Anaxagoras and Plutarch ;—so slow and
unequal are the steps by which science advances to perfec-
tion. The mutual gravity of the earth and moon is not sup-
posed by Kepler to have any concern in the production of
their circular motions; yet he holds the tides to be pro-
duced by the gravitation of the waters of the sea toward the
moon.’

The length to which Galileo advanced in this direction,
and the point at which he stopped, are no less curious to be
remarked. ‘Though so well acquainted with the nature of
gravity on the earth’s surface,—the object of so many of his
researches and discoveries, and though he conceived it to
exist in all the planets, nay, in all the celestial bodies, and to

be the cause of their round figure, he did not believe it to be
m power that extended from one of those bodies to another.
He seems to have thought that gravity was a principle
which regulated the domestic economy of each particular
body, but had nothing to do with their external relations ; so
that he censured Kepler for supposing, that the phenomena
of the tides are produced by the gravitation of the waters of
the ocean to the moon.’

? Astronomia Stelle Martis. Introd. Parag. 8.
® Dial. 4to. Tom. IV. p. 325, Edit. de Padova.

SECT, 1Y.} DISSERTATION SECOND. 101

Hooke did not stop short in the same unaccountable man-
ner, but made a nearer approach to the truth than any one
had yet done. In his attempt to prove the motion of the
earth, published in 1674, he lays it down as the principle on
which the celestial motions are to be explained, that the
heavenly bodies have an attraction or gravitation toward
their own centres, which extends to other bodies within the
sphere of their activity ; and that all bodies would move in
straight lines, if some force like this did not act on them
continually, and compel them to describe circles, ellipses, or
other curve lines. The force of gravity, also, he considered
as greatest nearest the body, though the law of its variation
he could not determine. These are great advances ;—
though, from his mention of the sphere of aetivity, from his
considering the force as residing in the centre, and from his
ignorance of the law which it observed, it is evident, that
beside great vagueness, there was much error in his notions
about gravity. Hooke, however, whose candour and up-
rightness bore no proportion to the strength of his under-
standing, was disingenuous enough, when Newton had deter-
mined that law, to lay claim himself to the discovery.

This is the farthest advance that the knowledge of the
cause of the celestial motions had made before the investiga-
tions of Newton; it is the precise point at which this know-
ledge had stopped; having met with a resistance which re-
quired a mathematician armed with all the powers of the
new analysis to overcome. ‘The doctrine of gravity was yet
no more than a conjecture, of the truth or falsehood of which .
the measurements and reasonings of geometry could alone
determine.

Thus, then, we are enabled accurately to perceive in
what Newton’s discovery consisted. It was in giving the
evidence of demonstration to a principle which a few saga-
cious men had been sufficiently sharp-sighted to see obscure-

102 DISSERTATION SECOND. fine at:

ly or inaccurately, and to propose as a mere conjecture. Jn
the history of human knowledge, there is hardly any discove-
_ry to which some gradual approaches had not been made
before it was completely brought to light. To have found
out the means of giving certainty to the thing asserted, or of
disproving it entirely ; and, when the reality of the principle
was found out, to measure its quantity, to ascertain its laws,
and to trace their consequences with mathematical precision,
-—in this consists the great difficulty and the great merit of
such a discovery as that which is now before us. In this
Newton had no competitor : envy was forced to acknowledge
that he had no rival, and consoled itself with supposing that
he had no judge.

Of all the physical principles that have yet been made
known, there is none so fruitful in consequences as that of
gravitation; but the same skill that had directed Newton to
the discovery, was necessary to enable him to trace its conse-
quences.

The mutual gravitation of all bodies being admitted, it was
evident, that while the planets were describing their orbits
round the greatest and most powerful body in the system,
they must mutually attract one another, and thence, in their
revolutions, some irregularities, some deviations from the
description of equal areas in equal times, and from the laws
of the elliptic motion, might be expecied. Such irregulari-
ties, however, had not been observed at that time in the mo-
tion of any of the planets, except the moon, where some of
them were so conspicuous as to have been known to Hippar-
chus and Ptolemy. Newton, therefore, was very natural-
ly led to inquire what the different forces were, which, ac-
cording to the laws just established, could produce irregulaxt-
ties in the case of the moon’s motion. Beside the force of
the earth, or rather of the mutual gravitation of the moon
and earth, the moon must be acted on by the sun; and the

srer. 1v.] DISSERTATION SECOND. 103

same force which was sufficient to bend the orbit of the earth
into an ellipse, could not but have a sensible effect on the
orbit of the moon. Here Newton immediately observed,
that it is not the whole of the force which the sun exerts on
the moon, that disturbs her motion round the earth, but only
the difference between the force just mentioned, and that
which the sun exerts on the earth,—for it is anly that diffe-
rence that affects the relative positions of the two bodies. To
have exact measures of the disturbing forces, he supposed
the entire force of the sun on the moon to be resolved into
two, of which one always passed through the centre of the
earth, and the other was always parallel to the line joining
the sun and earth,—consequently, to the direction of the
force of the sun on the earth. ‘The former of these forces
being directed to the centre of the earth, did not prevent the
moon from describing equal areas in equal times round the
earth. The effect of it on the whole, however, he showed
to be, to diminish the gravity of the moon to the earth
by about one 358th part, and to increase her mean distance
in the same proportion, and her angular motion by about a
179th.

From the moon thus gravitating to the centre of the earth,
not by a force that is altogether inversely as the square of
the distance, but by such a force diminished by a small part
that varies simply as the distance, it was found, from a very
subtle investigation, that the dimensions of the elliptic orbit
would not be sensibly changed, but that the orbit itself
would be rendered moveable, its longer axis having an angu-
lar and progressive motion, by which it advanced over a
certain atc during each revolution of the moon. This af-
forded an explanation of the motion of the apsides of the
lunar orbit, which had been observed to go forward at the
rate of 3° 4’, nearly, during the time of the moon’s revolu-
tion, in respect of the fixed stars.

104 DISSERTATION SECOND. [part i.

This was a new proof of the reality of the principle of
gravitation, which, however, was rendered less conclusive by
the consideration, that the exact quantity of the motion of the
apsides observed, did not come out from the diminution of
the moon’s gravity as above assigned. (There was a sort of
cloud, therefore, which hung over this point of the lunar the-
ory, to dissipate which, required higher improvements in the
calculus than it was given to the inventor himself to accom-
plish. It was not so with respect to another motion to which
the plane of the lunar orbit is subject, a phenomenon which
had been long known in consequence of its influence on the
eclipses of the sun and moon. This was the. retrogradation
of the line of nodes, amounting to 3’ 10” every day. New-
ton showed that the second of the forces into which the solar
action is moved being exerted, not in the plane of the moon’s
orbit, but in that of the ecliptic, inclined to the former at an
angle somewhat greater than five degrees, its effect must be
to draw down the moon to the plane of the ecliptic sooner
than it would otherwise arrive at it; in consequence of
which, the intersection of the two planes would approach, as
it were, toward the moon, or move in a direction opposite to
that of the moon’s motion, or become retrograde. From
the quantity of the solar force, and the inclination of the
moon’s orbit, Newton determined the mean quantity of
this retrogradation, as well as the irregularities to which
it is subject, and found both to agree very accurately with
observation.

Another of the lunar inequalities,—that discovered by
Tycho, and called by him the Variation, which consists in
the alternate acceleration and retardation of the moon in
each quarter of her revolution, was accurately determined
from theory, such as it is found by observation; and the
same is true as to the annual equation, which had been long
confounded with the equation of time. With regard to the

wer. 1v.] DISSERTATION SECOND. 105

other inequalities, it does not appear that Newton attempted
an exact determination of them, but satisfied himself with
this general truth, that the principle of the sun’s disturbing
force led to the supposition of inequalities of the same
kind with those actually observed, though whether of the
same exact quantity it must be difficult to determine. It
was reserved, indeed, for a more perfect state of the cal-
culus to explain the whole of those irregularities, and to
deduce their precise value from the theory of gravity.—
Theory has led to the knowledge of many inequalities,
which observation alone would have been unable to dis-
cover.

While Newton was thus so successfully oecupied in
iracing the action of gravity among those distant bodies,
he did not, it may be supposed, neglect the consideration
of its effects on the objects which are nearer us, and
particularly on the Figure of the Earth. We have seen
that, even with the limited views and imperfect informa-
tion which Copernicus possessed on this subject, he as-
cribed the round figure of the earth and of the planets to
the force of gravity residing in the particles of these bo-
dies. Newton, on the other hand, perceived that, in the
earth, another force was combined with gravity, and that
the figure resulting from that combination could not be
exactly spherical. The diurnal revolution of the earth,
he knew, must produce a centrifugal force, which would
act most powerfully on the parts most distant from the
axis. The amount of this centrifugal force is greatest at
the equator, and being measured by the momentary recess
of any point from the tangent, which was known from the
earth’s rotation, it could be compared with the force of
gravity at the same place, measured in like manner by
the descent of a heavy hody in the first moment of its
fall. When Newton made this comparison, he found that

14

106 DISSERTATION SECOND. {parr xt

the centrifugal force at the equator is the 289th part of
gravity, diminishing continually as the cosine of the lati-
tude, on going from thence toward the poles, where it
ceases altogether. From the combination of this force,
though small, with the force of gravity, it follows, that
the line in which bodies actually gravitate, or the plumb-
line, cannot tend exactly to the earth’s centre, and that
a true horizontal line, such as is drawn by levelling, if
continued from either pole, in the plane of a meridian
all round the earth, would not be a circle but an ellipse,
having its greatest axis in the plane of the equator, and
its least in the direction of the axis of the earth’s rota-
tion. Now, the surface of the ocean itself actually tra-
ces this level as it extends from the equator to either
pole. The terraqueous mass which we call the globe must
therefore be what geometers call an oblate spheroid, or
a solid generated by the revolution of the elliptic meridi-
an about its shorter axis.

In order to determine the proportion of the axes of
this spheriod, a problem, it will readily be believed, of
no ordinary difficulty, Newton conceived, that if the wa-
ters at the pole and at the equator were to communicate
by a canal through the interior of the earth, one branch
reaching from the pole to the centre and the other at
right angles to it, from the centre to the circumference
of the equator, the water in this canal must be in eguili-
brio, or the weight of fluid mm the one branch just equal
“to that in the other. Including, then, the consideration
of the centrifugal force which acted on one of the bran-
ches but not on the other, and considering, too, that the
figure of the mass being no longer a sphere, the attrac-
tion must not ‘be supposed to be directed to the centre,
but must be considered as the result of the action of all
the particles of the spheriod on the fluid in the canals;

secr. 1v.] DISSERTATION SECOND. 107

by a very subtle process of reasoning, Newton found that
the longer of the two canals must be to the shorter as
230 to 229. This, therefore, is the ratio of the radius
of the equator to the polar semi-axis, their difference
amounting, according to the dimensions then assigned to
the earth, to about 17, English miles. In this investiga-
tion, the earth is understood to be homogeneous, or eve-
rywhere of the same density.

It is very remarkable, that though the ingenious and
profound reasoning on which this conclusion rests is not
entirely above objection, and assumes some things without
sufficient proof, yet, when these defects were corrected
in the new investigations of Maclaurin and Clairaut, the
conclusion, supposing the earth homogeneous, remained
exactly the same. The sagacity of Newton, like the
Genius of Socrates, seemed sometimes to inspire him with
wisdom from an invisible source. By a profound study
of nature, her laws, her analogies, and her resources, he
seems to have acquired the same sort of tact or feeling
in matters of science, that experienced engineers and other
artists sometimes acquire in matters of practice, by which
they are often directed right, when they can scarcely de-
scribe in words the principle on which they proceed.

From the figure of the earth thus determined, he showed
that the intensity of gravity at any point of the surface,
is inversely as the distance of that point from the centre ;
and its increase, therefore, on going from the equator to
the poles, is as the square of the sine of the latitude, the
same ratio in which the degrees of the meridian increase.?
As the intensity of gravity diminished on going from the
poles to the equator, or from the higher to the lower
latitudes, it followed, that a pendulum of a given length

» Princip. Lib. Ul. prop. 20.

108 DISSERTATION SECOND. [panT au.

would vibrate slower when carried from Europe into the
torrid zone. The observations of the two French astro-
nomers, Varin and De Hayes, made at Cayenne and Mar-
tinique, had already confirmed this conclusion.

The problem which Newton had thus resolved enabled him
to resolve one of still greater difficulty. The precession,
that is, the retrogradation of the equinoctial points, had
been long known to astronomers ; its rate had been mea-
sured by a comparison of ancient and modern observa-
tions, and found to amount nearly to 50” annually, so as
to complete an entire revolution of the heavens in 25,920
years. Nothing seemed more difficult to explain than this
phenomenon, and no idea of assigning a physical or me-
chanical cause for it had yet occurred, I believe, to the
boldest and most theoretical astronomer. The honour of
assigning the true cause was reserved for the most cau-
tious of philosophers. He was directed to this by a cer-
tain analogy observed between the precession of the equi-
noxes and the retrogradation of the moon’s nodes, a phe-
nomenon to which his calculus had been already success-
fully applied. The spheroidal shell or ring of matter
which surrounds the earth, as we have just seen, in the
direction of the equator, being one half above the plane
of the ecliptic and the other half below, is subjected to
the action of the solar force, the tendency of which is to
make this ring turn on the line of its intersection with
the ecliptic, so as ultimately to coincide with the plane
of that circle. This, accordingly, would have happened
long since, if the earth had not revolved on its axis. The
effect of the rotation of the spheroidal ring from west
to east, at the same time that it is drawn down toward
the plane of the ecliptic, is to preserve the inclination of
these two planes unchanged, but to make their intersec-
tion move in a direction opposite to that of the diurnal

sre. 1v.] DISSERTATION SECOND. 109

rotation, that is, from east to west, or contrary to the or-
der of the signs.

The calculus in its result justified this general conclu-
sion; 10” appeared the part of the effect due to the
moon’s attraction, 40” to the attraction of the sun ; and lL
know not if there be any thing respecting the constitution
of our system, in which this great philosopher gave a
stronger proof of his sagacity and penetration, than in the
explanation of this phenomenon. The truth, however, is,
that his data for resolving the problem were in some de-
gree imperfect, all the circumstances were not included, and
some were erroneously applied, yet the great principle
and scope of the solution were right, and the approxima-
tion very near to the truth. “ Il a été bien servi par son
genie,’ says the eloquent and judicious historian of as-
tronomy ; “ Vinspiration de cette faculté divine lui a fait
appercevoir des determinations, qui n’etoient pas encore
accessibles; soit qu’il efit des preuves qu’il a supprimées,
soit quil ett dans Vesprit un sorte d’estime, une espéce de
balance pour approuver certaines vérités, en pesant les ve-
vités prochaines, et jugeant les unes par les autres.”

It was reserved for a more advanced condition of the
new analysis, to give to the solution of this problem all the
accuracy of which it is susceptible. It is a part, and a
distinguishing part, of the glory of this system, that it was
susceptible of more perfection than it received from the
hands of the author; and that the century and a half
which has nearly elapsed since the first discovery of it,
has been continually adding to its perfection. This cha-
racter belongs to a system which has truth and nature for
its basis, and had not been exhibited in any of the physi-
cal theories that had yet appeared in the world. The

* Bailly, Hist. de PAstron, Mod. Tom. Il. livre xii. § 28.

110 DISSERTATION SECOND. [part ut,

philosophy of Plato and Aristotle were never more per-
fect than when they came from the hands of their respec-
tive authors, and a legion of commentators, with all their
efforts, did nothing but run round perpetually in the same
circle. Even Descartes, though he. had recourse to phy-
sical principles, and tried to fix his system on a firmer
basis than the mere abstractions of the mind, left behind
him a work which not only could not be improved, but
was such, that every addition attempted to be made de-
stroyed the equilibrium of the mass, and pulled away
the part to which it was mtended that it should be at-
tached. The philosophy of Newton has proved suscep-
tible of continual improvement ; its theories have explain-
ed facts quite unknown to the author of it; and the ex-
ertions of La Grange and La Place, at the distance of
an hundred years, have perfected a work which it was
not for any of the human race to begin and to complete.

Newton next turned his attention to the phenomena of
the Tides, the dependence of which on the moon, and in
part also on the sun, was sufliciently obvious even from
common observation. That the moon is the prime ruler
of the tide, is evident from the fact, that the high water,
at any given place, occurs always nearly at the moment
when the moon is on the same meridian, and that the
retardation of the tide from day to day, is the same with
the retardation of the moon in her diurmal revolution.—
That the sun is also concerned in the production of the
tides is evident from this, that the highest tides happen
when the sun, the moon, and the earth, are all three in
the same straight line; and that the lowest, or neap tides,
happen when the lines drawn from the sun-and moon to
the earth make right angtes with one another. The eye
of Newton, accustomed to generalize and to penetrate
beyond the surface of things, saw that the waters of the

axer, 1v.] DISSERTATION SECOND. i)

sea revolving with the earth, are nearly in the condition
of a satellite revolving about its primary; and are liable
to the same kind of disturbance from the attraction of a
third body. The fact in the history of the tides which
seems most difficult to be explained, received, on this
supposition, a very easy solution. It is known, that high
water always takes place in the hemisphere where the
moon is, and in the opposite hemisphere where the moon
is not, nearly at the same time. This seems, at first sight,
very unlike an effect of the moon’s attraction; for, though
the water in the hemisphere where the moon is, and
which, therefore, is nearest the moon, may be drawn up
toward that body, the same ought not to happen in the
opposite hemisphere, where the earth’s surface is most ,
distant from the moon. But if the action of the moon
disturb the equilibrium of the ocean, just as the action of
one planet disturbs the motion of a satellite moving round
another, it is exactly what might be expected. It had
been shown, that the moon, in conjunction with the sun,
has her gravitation to the earth diminished, and when in
opposition to the sun has it diminished very nearly by the
same quantity. ‘The reason is, that at the conjunction, or
the new moon, the moon is drawn to the sun more than
the earth is; and that, at the opposition, or full moon,
the earth is drawn toward the sun more than the moon
nearly by the same quantity; the relative motion of the
_two bodies is therefore affected the same way in both ca-
ses, and the gravity of the moon to the earth, or her
tendency to descend toward it, is in both cases lessened.
It is plain, that the action of the moon on the waters
of the ocean must be regulated by the same principle.
in the hemisphere where the moon is, the water is more
drawn toward the moon than the mass of the earth is,
and its gravity being lessened, the columns toward the

112 DISSERTATION SECOND. [part i

middle of the hemisphere lengthen, in consequence of the
pressure of the columns which are at a distance from the
middle point, of which the weight is less diminished, and
towards the horizon must even be increased. In the op-
posite hemisphere, again, the mass of the earth is more
drawn to the moon than. the waters of that hemisphere,
and their relative tendencies are changed in the same
direction, and nearly by the same quantity. If the action
of the moon on all the parts of the earth, both sea and
land, were the same, no tide whatever would be pro-
duced. .

Thus, the same analysis of the force of gravity which
explained the inequalities of the moon, was shown by
Newton to explain those inequalities in the elevation of
the waters of the ocean to which we give the name of
tides. On the principle also explained in this analysis, it
is, that the attraction of the sun and moon conspire to
elevate the waters of the ocean whether these luminaries
be in opposition or conjunction. In both cases the solar
and lunar tides are added together, and the tide actually
observed is their sum. At the quadratures, or the first
and third quarters, these two sides are opposed to one
another, the high water of the lunar tide coinciding with
the low water of the solar, and conversely, so that the
tide actually observed is the difference of the two.

The other phenomena of the tides were explained ina
manner no less satisfactory, and it only remained to in-
quire, Whether the quantity of the solar and lunar forces
were adequate to the effect thus ascribed to them? The
lunar force there were yet no data for measuring, but a
measure of the solar force, as it acts on the moon, had
been obtained, and it had been shown that in its mean
quantity it amounted to ;4; of the force which retains
the moon in her orbit. This last is z45 of the force of

eusr. 1.) DISSERTATION SECOND. 1135

gravity at the earth’s surface, and, therefore, the force
with which the sun disturbs the moon’s motion is 745 Xx¢55
of gravity at the earth’s surface. This is the solar disturbing
force on the moon when distant sixty semidiameters from the
earth’s centre, but on a body only one semidiameter distant
from that centre, that is, on the water of the ocean, the
disturbing force would be sixty times less, and thus is
found to be no more than zsz¢ys00 Of gravity at the earth’s
surface.

Now, this being the mean force of the sun, is that by which
he acts on the waters, 90 degrees distant from the point
to which he is vertical, where it is added to the force of gra-
vity, and tends to increase the weight and lower the level
of the waters. At the point where the sun is vertical, the
force to raise the water is about double of this, and, there-
fore, the whole force tending to raise the level of the high,
abeve that of the low water, is three times the preceding,
or about the y3;7see0 Of gravity. Small as this force is,
when it is applied to every particle of the ocean, it is capa-
ble of producing a sensible effect. The manner in which
Newton estimates this effect can only be considered as af-
fording an approximation to the truth. In treating of the
figure of the earth, he had shown that the centrifugal force,
amounting to 54, of gravity, was able to raise the level of
the ocean more than seventeen miles, or, more exactly,
85,472 French feet. Hence, making the effect proportional
to the forces, the elevation of the waters produced by the
solar force will come out 1.92 feet.

But, from the comparison of the neap and spring tides,
that is, of the difference and the sum of the lunar and solar
forces, it appears, that the force of the moon is to that of the
sun as 4.48 to 1. As the solar force raises the tide 1.92 feet,
the lunar will raise it 8.63 fect, so that the two together will

15

-414 DISSERTATION SECOND. {rant a1

produce a tide of 10.4’ French feet, which agrees not ill
with what is observed in the open sea, at a distance from
land. | ,
The calculus of Newton stopped not here. From the
force that the moon exerts on the waters of the ocean, he
found the quantity of matter in the moon to that in the
earth as 1 to 39.78, or, in round numbers, as 1 to 40. He
also found the density of the moon to the density of the
earth as 11 to 9.

Subsequent investigations, as we shall have occasion to re-
mark, have shown that much was yet wanting to a complete
theory of the tides; and that even after Maclaurin, Ber-
noulli, and Euler’ had added their efforts to those of Newton,
there remained enough to give full employment to the calcu-
lus of Laplace. As an original deduction, and as a first ap-
proximation, that of which I have now given an account, will
be for ever memorable.

The motion of Comets yet remained to be discussed.
They had only lately been acknowledged to belong to the
heavens, and to be placed beyond the region of the earth’s
atmosphere; but with regard to their motion, astronomers
were not agreed. Kepler believed them to move in straight
lines ; Cassini thought they moved in the planes of great cir-
cles, but with little curvature. Hevelius had come much
nearer the truth; he had shown the curvature of their paths
to be different in different parts, and to be greatest when
they were nearest the sun; and a parabola having its vertex
in that point seemed to him to be the line in which the comet
moved. Newton, convinced of the universality of the prin-
ciple of gravitation, had no doubt that the orbit of the comet

! Newtoni, Prin. Lib. IL. Prop. 36 ad 37.

? See the solutions of these three mathematicians in the Com-
mentary of Le Seur and Jacquier on the Third Book of the
Principia.

SRC. 1V.] DISSERTATION SECOND. 115

must be a conic section, having the sun in one of its
foci, and might either be an ellipse, a parabola, or even
an hyperbola, according to the relation between the force
of projection and the force tending to the centre. As
the eccentricity of the orbit on every supposition must be
great, the portion of it that fell within our view could not
differ much from a parabola, a circumstance which rendered
the calculation of the comet’s place, when the position of the
orbit was once ascertained, more easy than in the case ‘of the
planets. Thus far theory proceeded, and observation must
then determine with what degree of accuracy this theory
represented the phenomena. From three observations of
the comet, the position of the orbit could be determined,
though the geometric problem was one of great difficulty.
Newton gave a solution of it; and it was by this that his the-
ory was to be brought to the test of experiment. If the
orbit thus determined was not the true one, the places of the
comet calculated on the supposition that it was, and that it
described equal areas in equal times about the sun, could not
agree with the places actually observed. Newton showed,
by the example of the remarkable comet then visible (1680),
that this agreement was as great as could reasonably be ex-
pected; thus adding another proof to the number of those
already brought to support the principle of universal gravita-
tion. The comets descend into our system from all different
quarters in the heavens, and, therefore, the proofs that they
afforded went to show, that the action of gravity was confined
to no particular region of the heavens.

Thus far Newton proceeded in ascertaining the existence,
and in tracing the effects, of the principle of gravitation, and
had done so with a success of which there had been no in-
stance in the history of human knowledge. At the’same time
that it was the most successful, it was the most difficult re-
search that had yet been undertaken. The reasonings up-

116 DISSERTATION SECOND. [pant 1.

ward from the facts to the general principle, and again down
from that principle to its effects, both required the applica-
tion of a mathematical analysis which was but newly invent-
ed; and Newton had not only the difficulties of the investi-

gation to encounter, but the instrument to invent, without -

which the investigation could not have been conducted.
Every one who considers all this, will readily join in the senti-
ment with which Bailly closes a eulogy as just as it is elo-
quent. Si, comme Platon a pensé, il existoit dans la na-
ture une echelle Metres et de substances intelligentes jus-
qwa PEtre Supremé, VPespéce humaine, defendant ses droits,
aurou une foule de grands hommes a presenter ; mais New-

ton, suivi de ses vérités pures, montreroit le plus haut de-

gré de force de Vesprit humain, et suffiroit seul pour lu
assigner sa vrat place.

Though the creative power of genius was never more
clearly evinced than in the discoveries of this great philoso-
pher, yet the influence of circumstances, always extensive
and irresistible in human affairs, can readily be traced. The
condition of knowledge at the time when Newton appeared,
was favourable to great exertions; it was a moment when
things might be said to be prepared fora revolution in the ma-
thematical and physical sciences. The genius of Copernicus
had unfolded the true system of the world; and Galileo had
shown its excellence, and established it by arguments, the
force of which were generally acknowledged. Kepler had
done still more, having, by an admirable effort of generali-
sation, reduced the facts concerning the planetary motions
to three general laws. Cassini’s observations had also ex-
tended the third of these laws to the satellites of Jupiter,
showing that the squares of their periodic times were as the
cubes of their distances from the centre of the body round

1 Hist. de Astron. Mod. Tom. I. .

“
a

sueT. 1v.] DISSERTATION SECOND. 117

>

which they revolved. The imaginary apparatus of cycles
and epicycles,—the immobility of the earth,—the supposed
essential distinction between celestial and terrestrial sub-
stances, those insuperable obstacles to real knowledge,
which the prejudice of the ancients had established as
physical truths, were entirely removed; and Bacon had
taught the true laws of philosophising, and pointed out the
genuine method of extracting knowledge from experiment
and observation. The leading principles of mechanics
were established ; and it was no unimportant circumstance,
that the Vortices of Descartes had exhausted one of the
sources of error, most seducing on account of its simpli-
city.

All this had been done when the genius of Newton
arose upon the earth. Never till now had there been set
before any of the human race so brilliant a career to run,
or so noble a prize to be obtained. In the progress of
knowledge, a moment had arrived more favourable to the
developement of talent than any other, either later or ear-
lier, and in which it might produce the greatest possible
effect. But, let it not be supposed, while | thus admit the in-
fluence of external circumstances on the exertions of intel-
lectual power, that Iam lessening the merit of this last, or tak-
ing any thing from the admiration that is due to it. I am, in
truth, only distinguishing between what it is possible, and what
it is impossible, for the human mind to effect. With all the
aid that circumstances could give, it required the highest
degree of intellectual power to accomplish what Newton
performed. We have here a memorable, perhaps a sin-
gular instance, of the highest degree of intellectual power,
united to the most favourable condition of things for its
exertion. Though Newton’s situation was more favourable
than that of the men of science who had gone before him, it
was not more so than that of those men who pursued the

118 DISSERTATION SECOND. [Pant a.

‘game objects at the same time with himself, placed in a
situation equally favourable.

When one considers the splendour of Newton’s dis-
coveries, the beauty, the simplicity, and grandeur of the
system they unfolded, and the demonstrative evidence by
which that system was supported, one could hardly doubt,
that, to be received, it required only to be made known,
and that the establishment of the Newtonian philosophy
all over Europe would very quickly have followed the
publication of it. In drawing this conclusion, however, we
should make much too small: an allowance. for the influ-
ence of received opinion, and the resistance that mere
habit is able, for a time, to oppose to the strongest evi-
dence. The Cartesian system of vortices had many fol-
lowers in all the countries of Europe, and particularly in
France. In the universities of England, though the Aris-
totelian physics had made an obstinate resistance, they had
been supplanted by the Cartesian, which became firmly
established about the time when their foundation began to
be sapped by the general progress of science, and particu-
larly by the discoveries of Newton. For more than thirty .
years after the publication of those discoveries, the system
of vortices kept its ground, and.a translation from the
French into Latin of the Physics of Rohault, a work entire-
ly Cartesian, continued at Cambridge to be the text for
philosophical instruction. About the year 1718, a new
and more elegant translation of the same book was pub-
lished by Dr. Samuel Clarke, with the addition of notes,
in which that profound and ingenious writer explained the
views of Newton on the principal objects of discussion, so
that the notes contained virtually a refutation of the text;
they did so, however, only virtually, all appearance of ar-
gzument and controversy being carefully avoided. Whether
this escaped the notice of the learned Doctors or not is

exer. 1v.) DISSERTATION SECOND. 119

uncertain, but the new translation, from its better Latinity,
and the name of the editor, was readily admitted to all
the academical honeurs which the old one had enjoyed.
Thus, the stratagem of Dr. Clarke completely succeeded ;
the tutor might prelect from the text, but the pupil would
sometimes look into the notes, and error is never so sure
of being exposed as when the truth is placed close to it,
side by side, without any thing to alarm prejudice, or
awaken from its lethargy the dread of innovation. Thus,
therefore, the Newtonian philosophy first entered the uni-
versity of Cambridge under the protection of the Carte-
sian.!

If such were the obstacles to its progress that the new
philosophy experienced ‘in a country that was proud of
having given birth to its author, we must expect it to ad-

* The universities of St. Andrews and Edinburgh were, I be-
lieve, the first in Britain where the Newtonian philosophy was
made the subject of the academical prelections. For this dis-
tinction they are indebted to James and David Gregory, the first
in some respects the rival, but both the friends, of Newton.
Whiston bewails in the anguish of his heart the difference in this
respect between those universities and his own. David Grego-
ry taught in Edinburgh for several years prior to 1690, when he
removed to Oxford ; and Whiston says, ‘‘ He had already caused
several of his scholars to keep acts, as we cail them, upon seve-
ral branches of the Newtonian philosophy, while we at Cam-
bridge (poor wretches) were ignominiously studying the ficti-
tious hypotheses of the Cartesian.”” (Whiston’s Memoirs of his
own Life.) 1 do not, however, mean to say, that from this date
the Cartesian philosophy was-expelled from those universities ;
the Physics of Rohault were still in use as-a text, at least occa-
sionally, to a much later period than this, and a great deal, no
doubt, depended on the character of the individual professors.
Keil introdueed the Newtonian philosophy in his lectures at Ox-
ford in 1697 ; but the instructions of the tutors, which constitute
the real and efficient system of the university, were not cast in
that mould till long afterwards. The publication of S’Graves-
ande’s Elements proves that the Newfonian philosophy was
taught in the Dutch universities before the date of 1720.

120 DISSERTATION SECOND. [parr it.

vance very slowly indeed among foreign nations. In
France, we find the first astronomers and mathematicians,
such men as Cassini and Maraldi, quite unacquainted with
it, and employed in calculating the paths of the comets they
were observing, on hypotheses the most unfounded and
imaginary; long after Halley, following the principles of
Newton, had computed tables from which the motions of all
the comets that ever had appeared, or ever could appear,
might be easily deduced. Fontenelle with great talents
and enlarged views, and, as one may say, officially in-
formed of the progress of science all over Europe, con-
tinued a Cartesian to the end of his days. Mairan in his
youth was a zealous defender of the vortices, though he
_ became afterwards one of the most strenuous supporters
of the doctrine of gravitation.

A Memoir of the Chevalier Louville, among those of
the Academy of Sciences for 1720, is the first in that col-
lection, and, I believe, the first published in France, where
the elliptic motion of the planets is supposed to be pro-
duced by the combination of two forces, one projectile and
the other centripetal. Maupertuis soon after went much
farther; in his elegant and philosophic treatise, Figure des
Astres, published about 1730, he not only admitted the ex-
istence of attraction as a fact, but even defended it, when
considered as an universal property of body, against the
reproach of being a metaphysical absurdity. These were
considerable advances, but they were made slowly; and it
was true, as Voltaire afterwards remarked, that though the
author of the Principia survived the publication of that
great work nearly forty years, he had not, at the time of
his death, twenty followers out of England.

We should do wrong, however, to attribute this slow
conversation of the philosophic world entirely to prejudice,
inertness, or apathy. ‘The evidence of the Newtonian

sxer. tv.) DISSERTATION SECOND. 42]

philosophy was of a nature to require time in order to
‘make an impression. It implied an application of mathe-
matical reasoning which was often difficult; the doctrine
of prime and ultimate ratios was new to most readers, and
could be familiar only to those who had studied the infi-
nitesimal analysis. |

The principle of gravitation itself was considered as dif-
ficult to be admitted. When presented indeed as a mere
fact, like the weight of bodies at the earth’s surface, or
their tendency to fall to the ground, it was free from ob-
jection; and it was in this light only that Newton wished
it to be considered.'’ But though this appears to be the
sound and philosophical view of the subject, there has al-
ways appeared a strong desire in those who speculated
concerning gravitation, to go farther, and to inquire into
the cause of what, as a mere fact, they were sufficiently
disposed to admit. If you said that you had no explana-
tion to give, and was only desirous of having the fact ad-
mitted; they alleged, that this was an unsatisfactory pro-
ceeding,—that it was admitting the doctrine of occult cau-
ses,—that it amounted to the assertion, that bodies acted in
places where they were not,—a proposition that, metaphy-
sically considered, was undoubtedly absurd. ‘The desire
to explain gravitation is indeed so natural, that Newton
himself felt its force, and has thrown out, at the end of
his Optics, some curious conjectures concerning this general
affection of body, and the nature of that elastic ether to

1 « Vocem attractionis hic generaliter usurpo pre corporum
conatu quocungue accedendi ad invicem ; sive conatus iste fiat ab
aclione corporum se mutuo petentium, vel per spiritus emissos se
mutuo agitaniium ; sive is ab actione etheris, aut aeris medit
cujuscunque, corporet vel incorporei, orttur, corpora innataniia in
st invicem utcunque impellentes.” Principia Math. Lib. 1. Schol.
ad finem. prop. 69.

16

122 DISSERTATION SECOND. [PART 13.

which he thought that it was perhaps to be ascribed. “Is
not this medium (the ether) much rarer within the dense
_ bodies of the sun, stars, and planets, than in the empty
celestial spaces between them? And, in passiag from them
to great distances, does it not grow denser and denser
perpetually, and thereby cause the gravity of those great
bodies to one another, every body endeavouring to go from
the denser parts of the medium to the rarer?!

Notwithstanding the highest respect for the author of
these conjectures, | cannot find any thing like a satisfac-
tory explanation of gravity in the existence of this elastic
ether. It is very true that an elastic fluid, of which the
density followed the inverse ratio of the distance from a
given point, would urge the bodies immersed in it, and
impervious to it, toward that point, with forces inversely as
the squares of the distances from it; but what could main-
tain an elastic fluid in this condition, or with its density
varying according to this law, is a thing as mexplicable
as the gravity which it was meant to explain. The nature
of an elastic fluid must be, in the absence of all inequality
of pressure, to become everywhere of the same density.
If the causes that produce so marked and so general a
deviation from this rule be not assigned, we can only be
said to have substituted one difficulty for another.

A different view of the matter was taken by some of
the disciples and friends of Newton, but which certainly
did not lead to any thing more satisfactory. That philo-
sopher himself had always expressed his decided opinion*

E Optics, Query 21, at the end of the Third Book.

* The passages quoted sufficiently prove that Newton did
‘not consider gravity as a property inherent in matter. The fol-
lowing passage in one of his Letters to Dr. Bentley is still more
explicit: ‘It is inconceivable that inanimate brute matter
should, without the mediation of something else, which is not

ever. 1v.] DISSERTATION SECOND. 123

that gravity could not be considered as a property of mat-
ter; but Mr. Cotes, in the preface to the second edition
of the Principia, maintains, that gravity is a property which
we have the same right to ascribe to matter, that we have
to ascribe to it extension, impenetrability, or any other
property. This is said to have been inserted without the
knowledge of Newton,—a freedom which it is diflicult to
conceive that any man could use with the author of the
Principia. However that be, it is certain that these dif_i-
culties have been always felt, and had their share in re-
tarding the progress of the philosophy to which they seem-
ed to be inseparably attached.

There were other arguments of a less abstruse nature,
and more immediately connected with experiment, which,
for a time, resisted the progress of the Newtonian philo-
sophy, though they contributed, in the end, very materi-

material, operate upon and affect other matter without mutual
contact ; as it must do, if gravitation, in the sense of Epicurus,
be essential or inherent in it. That gravity should be innate,
inherent, and essential to matter, so that one body may act on.
another, at a distance, through a vacuum, without the mediation
of any thing else, by and through which their action and force
may be conveyed from one to another, is, to me, so great an ab-
surdity, that | believe no man who, in philosophical matters, has
a competent faculty of thinking, can ever fall into it.”” (Nemtont
Opera, Tom. 1V. Horseley’s edit. p. 438.) On this passage I
cannot help remarking, that it is not quite clear in what manner
the interposition of a material substance can convey the action of
distant bodies to one another. In the case of percussion or pres-
sure, this is indeed very intelligible, but it is by no means so in
the case of attraction. if two particles of matter, at opposite
extremities of the diameter of the earth, attract one another, this
effect is just as little intelligible, and the modus agendi is just as
mysterious, on the supposition that the whole globe of the earth
is interposed, as on that of nothing whatever being interposed, or
of acomplete vacuum existing between them. Itis not enough that
each particle attracts that in contact with it ; it must attract the
particles that are distant, and the intervention of particles be»
tween them, does not render this at all more intelligible.

424 DISSERTATION SECOND. [rant mn,

ally to its advancement. Nothing, indeed, is so hostile to
the interests of truth, as facts inaccurately observed; of
which we have a remarkable example in the measurement
of an arch of the meridian across France, from Amiens to
Perpignan, though so large as to comprehend about seven
degrees, and though executed by Cassini, one of the first
astronomers in Europe. According to that measurement,
the degrees seemed to diminish on going from south to
north, each being less by about an 800th part than that
which immediately preceded it toward the south. From
this result, which is entirely erroneous, the conclusion first
deduced, was correct, the error in the reasoning, by a very
singular coincidence, having corrected the error in the
data from which it was deduced. Fontenelle argued that,
as the degrees diminished in length on going toward the
poles, the meridian must be less than the circumference
of the equator, and the earth, of course, swelled out in the
plane of that circle, agreeably to the facts that had been
observed concerning the retardation of the pendulum when
carried to the south. This, however, was the direct con-
trary of the conclusion which ought to have been drawn,
as was soon perceived by Cassini and by Fontenelle him-
self. The degrees growing less as they approached the pole,
was an indication of the curvature growing greater, or of
the longer axis of the meridian being the line that passed
through the poles, and that coincided with the axis of the
earth. The figure of the earth must, therefore, be that
of an oblong spheroid, or one formed by the revolution
of an ellipsis about its longer axis. This conclusion seem-
ed to be strengthened by the prolongation of the meridian
from Amiens northward to Dunkirk in 1713, as the same
diminution. was observed; the medium length of the de-
gree between Paris and Dunkirk being 56970 toises, no
less than 137 less than the mean of the degrees toward

sacr. tv.} DISSERTATION SECOND. 125

the south.’ All this seemed quite inconsistent with the
observations on the pendulum, as well as with the conclu-
sions which Newton had deduced from the theory of gra-
vity. The Academy of Sciences was thus greatly per-
plexed, and uncertain to what side to incline. In these
circumstances, J. Cassini, whose errors were the cause of
all the difficulty, had the merit of suggesting the only
means by which the question concerning the figure of the
earth was likely to receive a satisfactory solution,—the
measurement of two degrees, the one under the equator,
and the other as near the pole as the nature of the thing
would admit. But it was not till considerably beyond the
limits of the period of which I am_ now treating, that
these measures were executed; and that the increase of
the degrees toward the poles, or the oblateness of the
earth’s figure, was completely ascertained. Cassini, on
resuming his own operations, discovered, and candidly ac-
knowledged, the errors in his first measurement; and thus
the objections which had arisen in this quarter against the
theory of gravity, became irresistible arguments in its fa-
vour. This subject will occupy much of our attention in
the history of the second period, till which, the establish-
ment of the Newtonian philosophy on the Continent, can-
not be said to have been accomplished.

In addition to these discoveries in physical astronomy,
this period affords several on the descriptive parts of the
science, of which, however, | can only mention one, as far
too important to be passed over in the most general out-
line. It regards the apparent motion in the fixed stars,
known by the name of the Aberration, and is the discove-
ry of Dr. Bradley, one of the most distinguished asirono-
mers of whom England has to boast. Bradley and_ his

“

* Memoires de Acad. des Sciences, 1718, p. 245.

126 DISSERTATION SECOND. [panr.ts.

friend Molyneux, in the end of the year 1725,’ were occu-
pied in searching for the parallax of the fixed stars by
means of a zenith sector, constructed by Graham, the most
skilful instrument maker of that period. The sector was
erected at Kew; it was of great radius, and furnished with
a telescope twenty-four feet in length, with which they pro-
posed to observe the transits of stars near the zenith, ac-
cording to a method that was first suggested by Hooke,
and pursued by him so far as to induce him to think that
he had actually discovered the parallax of y Draconis, the
bright star in the head of the dragon, on which he made
his observations. ‘They began their observations of the
transits of the same star on the 3d of December, when the
distance from the zenith at which it passed was carefully
marked. By the observations of the subsequent days the
star seemed to be moving to the south; and about the
beginning of March, in the following year, it had got 20”
to the south, and was then nearly stationary. In the
beginning of June it had come back to the same situa-
tion where it was first observed, and from thence it continu-
ed its motion northward till September, when it was about
20” north of the point where it was first seen, its whole
change of declination having amounted to 40”.

This motion occasioned a good deal of surprise to the
two observers, as it lay the contrary way to what it would
have done if it had proceeded from the parallax of the
star. The repetition of the obserfations, however, con-
firmed their accuracy; and they were afterwards pursued
by Dr. Bradley, with another sector constructed also by
Graham, of a less radius, but still of one sufficiently great
to measure a star’s zenith distance to half a second. It
embraced a larger arch, and admitted of the observations

>
> Phil. Trans. Vol. XXXV. p. 697.

scr. 1v.] DISSERTATION SECOND. 127

being extended to stars that passed at a more considerable
distance from the zenith.

Even with this addition the observations did not put
Bradley in possession of the complete fact, as they only
gave the motion of each star in declination, without giving
information about what change might be produced in its
right ascension.

Had the whole fact, that is, the motion in right ascen-
sion as well as in declination, been given from observation,
it could not have been long before the cause was dis-
covered. With such information, however, as Dr. Brad-
ley had, that discovery is certainly to be regarded as a
great effort of sagacity. He has not told us the steps by
which he was led to it; only we see that, by the method
of exclusion, he had been careful to narrow the field of
hypothesis, and had assured himself that the phenomenon
was not produced by any nutation of the earth’s axis; by
any change in the direction of the plumb-line, or by re-
fraction of any kind. All these causes being rejected, it
occurred to him that the appearances might arise from the
progressive motion of light combined with the motion of
the earth in its orbit. He reasoned somewhat in this man-
ner. If the earth were at rest, it is plain that a telescope,
to admit a ray of light coming from a star to pass along
its axis, must be directed to the star itself. But, if the
earth, and, of course, the telescope be in motion, it must be
inclined forward, so as to be in the diagonal of a parallelo-
gram, the sides of which represent the motion of the earth,
and the motion of light, or in the direction of those mo-
tions, and in the ratio of their velocities. It is with the
telescope just as with the vane at the mast-head of a
ship; when the ship is at anchor, the vane takes exactly
the direction of the wind; when the ship is under weigh,
it places itself in the diagonal of a parallelogram, of which

4

128 DISSERTATION SECOND. [Pant it.

one side represents the velocity of the ship, and the other
the velocity of the wind. -If, instead of the vane, we con-
ceive a hollow tube, moveable in the same manner, the
case will become more exactly parallel to that of the teles-
cope. The tube will take such a position that the wind
may blow through it without striking against the sides,
and its axis will then be the diagonal of the parallelogram
just referred to.

The telescope, therefore, through which a star is viewed,
and by the axis of which its position star is determined,
must make an angle with the straight line drawn to the
star, except when the earth moves directly upon the star,
or directly from it. Hence it follows, that if the star be
in the pole of the ecliptic, the telescope must be pointed
forward, in the direction of the earth’s motion, always by
the same angle, so that the star would be seen out of its
true place by that angle, and would appear to describe a
circle round the pole of the ecliptic, the radius of which,
subtended at the earth, an angle, of which the sine is to
unity, as the velocity of the earth to the velocity of light.
If the star be any where between the plane of the eclip-
tic and the pole, its apparent path will be an ellipse, the
longer axis of which is the same with the diameter of the
former circle, and the shorter equal to the same quantity,
multiplied by the sine of the star’s latitude. If the star
be in the plane of the ecliptic, this shorter axis vanishes,
and the apparent path of the star is a straight line, equal
to the axis just mentioned.

Bradley saw that Romer’s observation concerning the
time that light takes to go from the sun to the earth, gave
a ready expression for the velocity of light compared with
that of the earth. The proportion, however, which he
assumed as best suited to his observations was somewhat
different ; it was that of 10313 to 1, which made the ra-
dius of the circle of aberration 20’, and the transverse

| sagen. 1v,] DISSERTATION SECOND. 129

axis of the ellipse in every case, or the whole ehange of posi-
tion, 40”. It was the shorter axis which Bradley had ac-
tually actually observed in the case of y Draconis, that
star being very near the solstitial colure, so that its chan-
ges of declination and of latitude are almost the same. In
order to show the truth of his theory, he computed the
aberration of different stars, and, on comparing the results
with his observations, the coincidence appeared almost per-
fect, so that no doubt remained concerning the truth of
the principle on which he had founded his calculations.
He did not explain the rules themselves: Clairaut publish-
ed the first investigation of these in the Memoirs of the
Academy of Sciences for 1737. Simpson also gave a de-
monstration of them in his Essays, published in 1740.

It has been remarked, that the velocity of light, as as-
sumed by Bradley, did not exactly agree with that which
Romer had assigned; supposing the total amount of the
aberration 404”, it gave the time that hght takes to come
from the sun to the earth 8’ 13”; but it is proper to add,
that since the time of this astronomer, the velocity of light
deduced from the eclipses of Jupiter’s satellites has been
found exactly the same.

It is remarkable that the phenomenon thus discovered
by Bradley and Molyneux, when in search of the parallax
of the fixed stars, is in reality as convincing a proof of
ihe earth’s motion in its orbit, as the discovery of that
parallax would have been. It seems, indeed, as satisfac-
tory as any evidence that can be desired. One only re-
egrets, in reflecting on this discovery, that the phenomenon
of the aberration was not foreseen, and that, after being
predicted from theory, it had been ascertained from obser-
vation. As the matter stands, however, the discovery both
of the fact and the theory is highly creditable to its author.

17

130 DISSERTATION SECOND. [rant 1.

In the imperfect outline which I have now sketched of
one of the most interesting periods in the history of hu-
man knowledge, much has been omitted, and many great
characters passed over, lost, as it were, in the splendour
of the two great Inminaries which marked this epocha.
Newton and Leibnitz are so distinguished from the rest
even of the scientific world, that we can only compare
them with one another, though, in fact, no two intellec-
tual characters, who both reached the highest degree of ex-
cellence, were ever more dissimilar.

For the variety of his genius, and the extent of his re-
search, Leibnitz is perhaps altogether unrivalled. A law-
yer, a historian, an antiquary, a poet, and a philologist,—
a mathematician, a metaphysician, a theologian, and I will
add a geologer, he has in all these characters produced
works of great merit, and in some of them of the highest
excellence. It is rare that original genius has so little
of a peculiar direction, or is disposed to scatter its efforts
over so wide a field. Though a man of great inventive
powers, he occupied much of his time in works of mere
labour and erudition, where there was nothing to invent,
and not much of importance to discover. Of his in-
ventive powers as a mathematician we have already
spoken; as a metaphysician, his acuteness and depth are
universally admitted; but metaphysics is a science in
which there are few discoveries to be made, and the
man who searches in it for novelty, is more likely to
find what is imaginary than what is real. ‘The notion
of the Monads, those unextended units, or simple essen-
ces, of which, according to this philosopher, all things
corporeal and spiritual, material or intellectual, are form-
ed, will be readily allowed to have more in it of novel-

sKcr. 1v.] DISSERTATION SECOND. 13i

’ ty than truth. The pre-established harmony between the
body and the mind, by which two substances incapable
of acting on one another, are so nicely adjusted from the
beginning, that their movements for ever correspond, is a
system of which no argument can do more than prove
the possibility. And, amid all the talent and acuteness
with which these doctrines are supported, it seems to ar-
gue some unsoundness of understanding, to have thought
that they could ever find a place among the established
principles of human knowledge.

Newton did not aim at so wide a range. Fortunately
for himself and for the world, his genius was more de-
termined to a particular point, and its efforts were more
concentrated. ‘Their direction was to the accurate scien-
ces, and they soon proved equally inventive in the pure
and in the mixed mathematics. Newton knew how to
transfer the truths of abstract science to the study of things
actually existing, and, by returning in the opposite direc-
tion, to enrich the former by ideas derived from the lat-
ter. In experimental and inductive investigation, he was
as great as in the pure mathematics, and his discoveries
as distinguished in the one as in the other. In this double
claim to renown, Newton stands yet unrivalled; and though,
in the pure mathematics, equals may perhaps be found,
no one, I believe, will come forward as his rival both in
‘that science and in the philosophy of nature. His cau-
tion in adopting general principles ; his dislike to what was
vague or obscure ; his rejection of all theories from which
precise conclusions cannot be deduced; and his readiness
to relinquish those that depart in any degree from the
truth, are, throughout, the characters of his philosophy,
ahd distinguish it very essentially from the philosophy of
Leibnitz. The characters now enumerated are most of
them negative, but without the principles on which they

132 DISSERTATION SECOND. (part i,

are founded, invention can hardly be kept in the right
course. The German philosopher was not furnished with
them in the same degree as the English, and hence his
great talents have run very frequently to waste.

It may be doubted also, whether Leibnitz’s great me-
taphysical acuteness did not sometimes mislead him in the
study of nature, by inclining him to those reasonings
which proceed, or affect to proceed, continually from the
cause to the effect. The attributes of the Deity were
the axioms of his philosophy ; and he did not reflect that
this foundation, excellent in itself, lies much too deep for
a structure that is to be raised by so feeble an architect
as man; or, that an argument, which sets out with the
most profound respect to the Supreme Being, usually ter-
minates in the most unwarrantable presumption. His rea-
sonings from first causes are always ingenious; but nothing
can prevent the substitution of such causes for those that
are physical and efficient, from being one of the worst and
most fatal errors in philosophy.

As an interpreter of nature, therefore, Leibnitz stands
in no comparison with Newton. His general views in
physics were vague and unsatisfactory; he had no great
value for inductive reasoning; it was not the way of ar-
riving at truth which he was accustomed to take ; and
hence, to the greatest physical discovery of that age, and
that which was established by the most ample induction,
the existence of gravity as a fact in which all bodies
agree, he was always incredulous, because no proof of it,
a priori, could be given.

As to who benefited human knowledge the most, no
question, therefore, can arise; and if genius is to be weigh-
ed in this balance, it is evident which scale must prepon-
derate. Except in the pure mathematics, Leibnitz, with
all his talents, made no material or permanent addition

sKer. 1Y.] DISSERTATION SECOND. 133

to the sciences. Newton, to equal inventions in mathe-
matics, added the greatest discoveries in the philosophy
of nature; and, in passing through his hands, Mechanics,
Optics, and Astronomy, were not merely improved, but
renovated. No one ever left knowledge in a state so dif-
ferent from that in which he found it. Men were in-
structed not only in new truths, but in new methods of
discovering truth; they were made acquainted with the
great principle which connects together the most distant
regions of space, as well as the most remote periods of
duration ; and which was to lead to future discoveries,
‘far beyond what the wisest or most sanguine could an-
ticipate.

neha dien eit Der

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